Analysis and Design of Marine Structures (2015) [PDF]

Zitiervorschau

Editors Guedes Soares Shenoi

The MARSTRUCT series of conferences started in Glasgow, UK in 2007, the second event of the series took place in Lisbon, Portugal (2009), while the third was in Hamburg, Germany (2011), and the fourth in Espoo, Finland (2013). This conference series deals with Ship and Offshore Structures, addressing the following topics: – Methods and Tools for Loads and Load Effects – Methods and Tools for Strength Assessment – Experimental Analysis of Structures – Materials and Fabrication of Structures – Methods and Tools for Structural Design and Optimisation – Structural Reliability, Safety and Environmental Protection

Analysis and Design of Marine Structures

Analysis and Design of Marine Structures contains the papers presented at MARSTRUCT 2015, the 5th International Conference on Marine Structures (Southampton, UK, 25-27 March 2015).

Analysis and Design of Marine Structures

This book is essential reading for academics, engineers and all professionals involved in the area of design of marine and offshore structures.

Editors: C. Guedes Soares R.A. Shenoi

an informa business

ANALYSIS AND DESIGN OF MARINE STRUCTURES

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PROCEEDINGS OF THE 5TH INTERNATIONAL CONFERENCE ON MARINE STRUCTURES (MARSTRUCT 2015), SOUTHAMPTON, UK, 25–27 MARCH 2015

Analysis and Design of Marine Structures

Editors

C. Guedes Soares Instituto Superior Técnico, University of Lisbon, Portugal

R.A. Shenoi University of Southampton, UK

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CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2015 Taylor & Francis Group, London, UK Typeset by V Publishing Solutions Pvt Ltd., Chennai, India Printed and bound in Great Britain by CPI Group (UK) Ltd, Croydon, CR0 4YY All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: CRC Press/Balkema P.O. Box 11320, 2301 EH Leiden, The Netherlands e-mail: [email protected] www.crcpress.com – www.taylorandfrancis.com ISBN: 978-1-138-02789-3 (Hbk + CD-ROM) ISBN: 978-1-315-68505-2 (eBook PDF)

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Table of contents

Preface

xi

Organisation

xiii

Wave loads and responses Experimental investigation of the influence of hull damage on ship responses in waves S.S. Bennett & A.B. Phillips

3

Unstructured MEL scheme for 3D body nonlinear ship hydrodynamics A.C. Chapchap & P. Temarel

11

An initial estimation of DP requirements for a non-moored operational FPSO G.E. Hearn & G.C. Bratu

19

CFD based computation of bow impact loads for buckling assessment J. Oberhagemann, M. Radon, H. von Selle & D.K. Lee

27

Comparison of two practical methods for seakeeping assessment of damaged ships J. Parunov, M. Ćorak & I. Gledić

37

Smoothed Particle Hydrodynamics (SPH) method for modelling 2-dimensional free surface hydrodynamics M.Z. Ramli, P. Temarel & M. Tan

45

Wave-induced responses of a bulk carrier in heading and following seas X.L. Wang, R.M. Liu & J.J. Hu

53

Numerical simulation of the dynamics of a large moored tanker S. Zhang, Q. Jin, J. Xin, T. Li, P. Temarel & W. Geraint Price

61

Hydroelasticity Slamming impact loads on high-speed craft sections using two-dimensional modelling J. Camilleri, D.J. Taunton & P. Temarel

73

Non-linear hydroelastic and fatigue analyses for a very large bulk carrier B. Cristea, C.I. Mocanu & L. Domnisoru

83

Hydroelastic analysis of a flexible barge in regular waves using coupled CFD-FEM modelling P. Lakshmynarayanana, P. Temarel & Z. Chen

95

Vibrations Simplified method for natural frequency analysis of stiffened panel E. Avi, A. Laakso, J. Romanoff & I. Lillemäe Investigation of vibrational power flow patterns in damaged plate structures for damage localisation P. Boonpratpai & Y.P. Xiong

107

115

v

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Influence of the sea action on the measured vibration levels in the comfort assessment of mega yachts E. Brocco, L. Moro, P.N. Mendoza Vassallo, M. Biot, D. Boote, T. Pais & E. Camporese

125

A computational framework for underwater shock response of marine structures C. Diyaroglu, D. De Meo & E. Oterkus

131

Liquid sloshing analysis for independent tank with elastic supports W. Liu, H.X. Xue & W.Y. Tang

139

Numerical simulation of the dynamic behaviour of resilient mounts for marine diesel engines L. Moro, E. Brocco, P.N. Mendoza Vassallo, M. Biot & H. Le Sourne Characteristic of low and middle frequency underwater noise of a catamaran F.Z. Pang, Y.Z. Xue, D. Tang, S. Li & Y.P. Xiong Simulation of the vibration characteristics of a propulsion system excited by hull deformations Z. Tian, X.P. Yan, C. Zhang & Y.P. Xiong

149 159

167

A new approach to analyze the underwater vibration of double layer ribbed cylinder F. Wang, Y. Xiong, Z. Weng & L. He

175

Impact resistance assessment of shipboard equipment considering spectrum dip effect Y.J. Yang, M. Yu, X.B. Li & L.H. Feng

181

Natural frequencies of eccentric cylindrical shells filled with pressurized fluid G.J. Zhang, T.Y. Li, X. Zhu, L. Xiong & Y.P. Xiong

189

Magnetorheological elastomer materials and structures with vibration energy control for marine application G. Zhu, Y.P. Xiong, S. Daley & R.A. Shenoi

197

Fatigue Corrosion fatigue crack growth in offshore wind monopile steel HAZ material O. Adedipe, F. Brennan & A. Kolios

207

Improvements in fatigue strength assessment of marine propellers C. Bertoglio, S. Gaggero, C.M. Rizzo, M. Viviani, C. Vaccaro & F. Conti

213

Fatigue crack growth performance of laser hybrid and arc welds of AH36 naval steels E.V. Chatzidouros, T. Tsiourva, D.I. Pantelis & A. Lopez

225

Overload and dwell time effects on crack growth property of high strength titanium alloy TC4 ELI used in submersibles F. Wang, W.C. Cui, Y.Y. Wang & Y.S. Shen

233

Consideration of stress gradient effects for complex structures in local fatigue approaches C. Fischer & W. Fricke

241

Fatigue assessment of joints at bulb profiles by local approaches C. Fischer, W. Fricke & C.M. Rizzo

251

Fatigue life improvement of laser-welded web-core steel sandwich panels using filling materials D. Frank, J. Romanoff & H. Remes

261

A full-scale fatigue test on longitudinal-through-bulkhead detail by using equivalent angle bar Q. Yi, J. Yue, Y. Liu, W. Tang & Z. He

269

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Springing effect on the fatigue life of an 8000TEU container ship P.-K. Liao, Y.-J. Lee, H.-J. Lin, S.-C. Tsai, H.-L. Chien, B.C. Chang & G.M. Luo A fracture mechanics based approach for the analysis of crack growth at weld joints of ship structures B.Q. Lou, S. Zhang, J. Tong, S. Wong, F. Cheng & S. Hirdaris

277

285

A study on the effect of hull girder vibration on the fatigue strength M. Oka, T. Niwa & K. Takagi

293

Fatigue strength of welded extra high-strength and thin steel plates H. Remes, M. Peltonen, T. Seppänen, A. Kukkonen, S. Liinalampi, I. Lillemäe, P. Lehto, H. Hänninen, J. Romanoff & S. Nummela

301

Investigation of weld root fatigue of single-sided welded T-joints W. Sundermeyer, W. Fricke & H. Paetzold

309

Fatigue data of High-Frequency Mechanical Impact (HFMI) improved welded joints subjected to overloads H.C. Yildirim & G. Marquis Application of wave models to fatigue assessment of offshore floating structures T. Zou, M.L. Kaminski & X.L. Jiang

317 323

Structural analysis A simplified FE model for the non-linear analysis of container stacks subject to inertial loads due to ship motions E. Brocco, L. Moro & M. Biot

331

Structural Health Monitoring of marine structures by using inverse Finite Element Method A. Kefal & E. Oterkus

341

Method for estimating soft clay type seabed embedment by pipeline movement D.K. Kim, M.S. Liew, S.Y. Yu, K.S. Park & H.S. Choi

351

Assumptions and reality: Stress states in uniaxial tension test M. Kõrgesaar

359

Dynamic analysis of ring stiffened conical-cylindrical shell combinations with general coupling and boundary conditions X.L. Ma, G.Y. Jin, Y.P. Xiong & Z.G. Liu

365

The determination of ice-induced loads on the ship hull from shear strain measurements M. Suominen, J. Romanoff, H. Remes & P. Kujala

375

A study of cross deck effects on warping stresses in large container ships R. Villavicencio, S. Zhang & J. Tong

385

Analysis on influence of spherical bulkhead reinforcement on stability S. Yuan, X.H. Miao & B. Luo

395

Hull ultimate strength Structural capacity of an aging box girder accounting for the presence of a dent S. Saad-Eldeen, Y. Garbatov & C. Guedes Soares

403

Strength analysis of ship shaped structures subjected to asymmetrical bending moment M. Tekgoz, Y. Garbatov & C. Guedes Soares

415

A study on the effect of lateral loads on the hull girder ultimate strength of bulk carriers K. Toh & T. Yoshikawa

425

vii

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Model test on the ultimate longitudinal strength of a damaged box girder Y. Yamada & T. Takami

435

Ultimate strength Influence of shear-induced secondary bending on buckling of web-core sandwich panels J. Jelovica & J. Romanoff Influence of lateral pressure on load-shortening behavior of stiffened panels under combined loads L. Jiang & S. Zhang

445

453

Elastic shear buckling capacity of the longitudinally stiffened flat panels S. Kitarovic, J. Andric & K. Piric

463

Strength of aluminium alloy ship plating under combined shear and compression/tension M.S. Syrigou, S.D. Benson & R.S. Dow

473

Effects of initial imperfection shapes on plate ultimate strength under combined loads S. Zhang & L. Jiang

483

Damaged structures Statistics of still water bending moment of damaged ships B. Bužančić Primorac, M. Ćorak & J. Parunov

491

Multiphysics modelling of Stress Corrosion Cracking by using peridynamics D. De Meo, C. Diyaroglu, N. Zhu, E. Oterkus & M. Amir Siddiq

499

Progressive collapse of intact and damaged stiffened panels A. Leelachai, S.D. Benson & R.S. Dow

505

Comparative analysis of HCSR based on ultimate strength of intact ships and residual strength in damaged condition W. Lei, W.Z. Quan & H.J. Hao

513

Residual strength of a severely damaged box-girder with non-uniform and inter-crystalline corrosion S. Saad-Eldeen, Y. Garbatov & C. Guedes Soares

521

Structural design Investigation into the implications of increasing minimum manhole access requirements in marine structures P. Eames, H. Hickson & J.M. Underwood Structural design and optimisation of an aluminium trimaran D. Fuentes, M. Salas, G. Tampier & C. Troncoso Efficient optimization framework for robust and reliable structural design considering interval uncertainty Y. Liu & M.D. Collette Multi-objective optimization of lightweight modular sandwich panels A. Milat, T. Tomac, D. Frank, Đ. Dundara, O. Kuzmanović & V. Radolović

535 545

555 565

Fabrication technology Numerical investigation of welding induced distortion of a stiffened plate structure Z. Chen, Z.C. Chen & R.A. Shenoi Reduction in weld induced distortions of butt welded plates subjected to preventive measures M. Hashemzadeh, Y. Garbatov & C. Guedes Soares

573

581

viii

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Buckling mitigation of stiffened panels through thermomechanical tensioning N.R. Mandal, S. Kumar, D. Podder, A. Gadagi & T. Mathew Numerical analysis of residual stress in butt-welded High Tensile Strength steel structures Y.X. Wan, Y.P. Xiong & L.B. Li

589

597

Composite structures Comparison of effective breadth definitions large composite top-hat stiffened grillages E. Arnaud, A.J. Sobey & J.I.R. Blake

609

Static and fatigue tests of hybrid composite-to-steel butt joints E.A. Kotsidis, P. Yarza, N.G. Tsouvalis, R. de la Mano & E. Rodriguez-Senín

617

Mechanical characterization of yachts and pleasure crafts fillers G. Nebbia, C.M. Rizzo, M. Gaiotti & A. Caleo

627

Composite riser design and development—A review D.-C. Pham, S. Narayanaswamy, X. Qian, W. Zhang, A.J. Sobey, M. Achintha & R.A. Shenoi

637

Steel or composite car deck structure—a comparison analysis of weight, strength and cost J.W. Ringsberg

647

Crashworthiness Numerical crashworthiness analysis of an offshore wind turbine monopile impacted by a ship A. Bela, L. Buldgen, Ph. Rigo & H. Le Sourne

661

Development of a failure strain surface in average stress triaxiality and average lode angle domains of a low temperature high strength steel J. Choung, S.J. Park & G.T. Tayyar

671

Structural response of ship bottom floor plating during shoal grounding Z.G. Gao & Z.Q. Hu Non-linear finite element analysis of crashworthy shields of offshore wind turbine supporting structures B. Liu, Y. Garbatov & C. Guedes Soares Modelling of structural damage and environmental consequences of tanker grounding K. Tabri, R. Aps, A. Mazaheri, M. Heinvee, A. Jönsson & M. Fetissov

685

693 703

Renewable energies devices A study on the short and long-term analysis of nonlinear hydrodynamic force on an Energy Saving Device D.B. Lee, H.J. Kim & B.-S. Jang Numerical simulation of multi-body wave point-absorber L. Li, M. Tan & J.I.R. Blake

713 723

Optimized topology design for substructure of 10MW grade floating type Wave-Wind Hybrid Power Generation System C.Y. Song, H.C. Song, C.S. Shim, J.H. Park & Y.Y. Park

731

Fatigue reliability of an offshore wind turbine supporting structure accounting for inspection and repair B. Yeter, Y. Garbatov & C. Guedes Soares

737

ix

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Uncertainty and reliability System reliability analysis of a ship deck structure for buckling collapse and corrosion limit states B. Gaspar & C. Guedes Soares

751

Effect of the aspect ratio on the ultimate compressive strength of plate elements with non-uniform corrosion B. Gaspar, A.P. Teixeira & C. Guedes Soares

765

Uncertainty analysis of the energy absorbed in beam and plate elements under impulsive loading B. Liu & C. Guedes Soares

775

Effects of input uncertainty on composite patch disbond under impact loading S.C. TerMaath & D.C. Hart

785

Experimental study on the rear bearing load characteristics considering uncertainty external loading S.D. Zhang, Z.L. Liu, X.P. Yan, Y. Jin & B. Qin

793

Author index

799

x

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Preface

This book contains the papers presented at the 5th International Conference on Marine Structures, MARSTRUCT 2015, held in Southampton, UK between 25 and 27 March. This is the fifth in the MARSTRUCT Conference series and follows on from previous events held in Glasgow—Scotland, Lisbon—Portugal, Hamburg—Germany and Espoo—Finland in 2007, 2009, 2011 and 2013 respectively. The main objective of the MARSTRUCT Conferences is to provide a specialised forum for academics, researchers and industrial participants to discuss progress in their research directly related with structural analysis and design of marine structures. It was the intention that the MARSTRUCT Conferences be specifically dedicated to marine structures, which complements other conferences on general aspects of ships and offshore structures already available. This series of Conferences is one of the main activities of the MARSTRUCT Virtual Institute, an association of research groups interested in cooperating in the field of marine structures which was created in 2010 after the end of the Network of Excellence on Marine Structures (MARSTRUCT), which was funded by the European Union. The MARSTRUCT Virtual Institute was founded with the same members as the EU project but with the aim to extend that membership to other interested groups in the future. The Conference reflects the work conducted in the analysis and design of marine structures, including the full range of methods, modelling procedures and experimental results. The aim is to promote knowledge that enables marine structures to be more efficient, environmentally friendly, reliable and safe using the latest methods and procedures design and optimisation. This book also deals with the fabrication and new materials of marine structures. The 87 papers are categorized in the following themes and areas of research: • Methods and tools for establishing loads and load effects—Wave loads, Vibrations, Response to accident loads, • Methods and tools for strength assessment—Structures on ice, Impact and collision, Fatigue strength, Ultimate strength, • Experimental analysis of structures—Experimental analysis, • Materials and fabrication of structures—Composite structures, Weld simulations, • Methods and tools for structural design and optimisation—Structural analysis, Structural design, • Structural reliability, safety and environmental protection—Structural reliability models, • Renewable Energy. The articles in this book were accepted after a review process, based on the full text of the papers. Thanks are due to the Technical Programme Committee and to the Advisory Committee who had most of the responsibility for reviewing the papers. We are also grateful to the additional anonymous reviewers who helped the authors deliver better papers by providing them with constructive comments. We hope that this process contributed to a consistently good level of the papers included in the book. C. Guedes Soares & R.A. Shenoi

xi

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Organisation

CONFERENCE CHAIRMAN Prof. R.A. Shenoi, University of Southampton, UK TECHNICAL PROGRAMME COMMITTEE Prof. C. Guedes Soares, IST, University of Lisbon, Portugal (Chair) Prof. R.A. Shenoi, University of Southampton, UK Prof. M. Biot, University of Trieste, Italy Prof. L. Domnisoru, University “Dunarea de Jos” at Galati, Romania Prof. R.S. Dow, University of Newcastle-upon-Tyne, UK Prof. W. Fricke, Technical University Hamburg-Harburg, Germany Prof. Y. Garbatov, IST, University of Lisbon, Portugal Prof. J.M. Gordo, IST, University of Lisbon, Portugal Prof. J. Parunov, University of Zagreb, Croatia Prof. P. Rigo, University of Liège, Belgium Prof. J. Ringsberg, Chalmers University of Technology, Sweden Prof. E. Rizzuto, University of Genova, Italy Prof. Jani Romanoff, Aalto University, Finland Prof. M. Samuelidis, NTUA, Greece Prof. M. Taczala, West Pomeranian University of Technology, Poland Prof. P. Temarel, University of Southampton, UK Dr. H. von Selle, DNV GL, Germany Dr. A. Vredeveldt, TNO, The Netherlands ADVISORY COMMITTEE Prof. F. Brennan, Cranfield University, UK Prof. X. Chen, CSSRMT, Wuxi, China Dr. F. Cheng, Lloyd’s Rregister, UK Prof. S.R. Cho, University of Ulsan, Korea Prof. Y.S. Choo, National University of Singapore, Singapore Prof. M. Collette, Univesity of Michigan, USA Prof. W.C. Cui, CSSRC, China Prof. M. Fujikubo, Osaka University, Japan Prof. T. Fukasawa, Osaka Prefecture University, Japan Prof. C.-F. Hung, National Taiwan University, Taiwan ROC Prof. H.W. Leheta, Alexandria University, Egypt Prof. N.R. Mandal, Indian Institute of Technology, Kharagpur, India Dr. Y. Ogawa, National Maritime Research Institute, Japan Dr. N.G. Pegg, DND, Canada Prof. R. Sundaravadivelu, Indian Institute of Technology, Madras, India Dr. H. Thorkildsen, Det Norske Veritas, Norway Dr. G. Wang, American Bureau of Shipping, USA Prof. L. Zhu, Wuhan University of Technology, China Dr. N. Yamamoto, NKK, Japan

xiii

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LOCAL ORGANIZING COMMITTEE Prof. R.A. Shenoi, University of Southampton, UK Prof. P. Temarel, University of Southampton, UK Dr. M. Tan, University of Southampton, UK Prof. F. Cheng, Lloyd’s Register, Southampton, UK Mrs. A. Subaiah-Varma, University of Southampton, UK TECHNICAL PROGRAMME SECRETARIAT Maria de Fátima Pina, IST, University of Lisbon, Portugal Sandra Ponce, IST, University of Lisbon, Portugal

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Wave loads and responses

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Experimental investigation of the influence of hull damage on ship responses in waves S.S. Bennett & A.B. Phillips Fluid-Structure Interactions Group, Southampton Boldrewood Innovation Campus, University of Southampton, UK

ABSTRACT: A number of high profile ship damage events in recent years have highlighted ship safety concerns following such an incident. With damage incidents occurring with relatively high frequency, the survivability of damaged ships needs to be better understood and modelled. Damage may result in: water influx, free surface effects, and abnormal load distributions, these will influence ship stability, motions and global loading. This paper experimentally investigates the influence of abnormal loading on the motions and vertical bending moments experienced by a 1:44 scale Leander Class frigate operating in regular waves for intact, one and two compartment damage. Results show that the inclusion of an abnormal mass distribution due to damage has a significant effect on the magnitude of the motion and response RAOs in head and beam seas; the severity of this effect increases with forward speed. 1

INTRODUCTION

and the additional mass of flood water within the ship creating an abnormal load distribution will have a significant effect on ship responses.

A number of high profile ship damage incidents in recent years, due to both collision and grounding, have drawn attention to ship safety. Table 1 gives a brief summary of damage incidents in the last decade; this affirms the idea that these incidents are occurring with a greater regularity than might be expected with the modern technologies available for vessel routing and tracking. Furthermore, based on Table 1 it can be said that a collision incident may be more likely to lead to casualties, generally from the smaller of the vessels involved in the collision. Ships have been known to split in half due to damage incidents. This is more common when a grounding or structural failure incident is being considered (e.g. the cargo ship MV SMART off South Africa (gCaptain, 2013a), the cargo ship MV Rena off New Zealand in 2012 (BBC, 2012) and the MV MOL Comfort in 2013 (RINA, 2014)). Such a scenario is less likely to occur in a collision incident, when a ship simply being holed is a more likely scenario. With such damage incidents occurring with a relatively high frequency, the survivability of ships subject to damage needs to be better understood and modelled. However, the fact that a ship can break up due to a damage incident emphasizes the need to predict not only the motions, but also the global loads (specifically the vertical bending moment) that a ship hull is subject to when damage occurs. Effects due to damage such as water influx, movement of flood water (e.g. free surface effect),

1.1

Experimental investigations of damaged ship responses

A limited number of experimental investigations have been conducted into damaged ship responses. Perhaps one of the most extensive is that into the

Table 1. Recent incidents of ship groundings and collisions in the last decade (Vanem and Ellis, 2010; MIT, 2013; MAIB, 2014a; gCaptain, 2013a; USA Today, 2013; Maritime Executive, 2013; gCaptain, 2013b; gCaptain, 2014). Year Ship

Incident

2007 MV Sea Grounding Diamond 2012 Costa Concordia Grounding cruise ship 2013 MV Danio cargo Grounding vessel 2013 MV Smart coal Grounding cargo ship 2013 Eifuku Maru Collision No. 18/Jia Hui 2013 Sima Sapphire/ Collision Fishing vessel 2013 Maria Security Collision Vessel/Texal 68 Trawler 2014 MV Colombo Collision Express/MV Maersk Tanjong

Result

Casualties

Sunk

2 missing

Sunk

32 deaths

Refloated None & repaired Structure None compromised Capsize/afloat 6 missing Afloat/sunk

8 missing

Sunk/afloat

3 missing

Afloat

None

3

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six degree of freedom motions and global loads of a Ro-Ro ferry by Korkut et al. (2004) and Korkut et al. (2005). This investigated symmetric two compartment damage for a flexible model hull, and found that motions, vertical and horizontal bending were less than for an intact Ro-Ro in beam seas. Palazzi and de Kat (2004) look at the motions of a damaged frigate and the likelihood of capsize following damage, assuming a rigid body. A passenger ship hull is used by Lee et al. (2012) to assess vessel motions and the free surface of floodwater inside a damaged compartment. Results are used as validation for a Computational Fluid Dynamics (CFD) model based on the Navier Stokes equations for simulating a damaged ship scenario. To date none of the CFD investigations include structural response such as global loads. Tabri et al. (2009) investigate the specific case of a ship-ship collision, and the effect on sloshing in tanks on collision dynamics. Experiments showed that sloshing has a significant effect on a ship collision and should be included in any model of ship-ship collision. What none of these investigations do is model the entire damage incident from prior to the event including any manoeuvring during the damage incident through to the resulting attitude of the vessel; however incidents such as the Costa Concordia in 2012 illustrate that this is important and should be addressed. 1.2

Table 2. Principal particulars of representative ship.

2.1

Model

Ship

Length overall (m) Length between perpendiculars (m) Breadth (m) Draught (m) Displacement (kg, tonnes) Block coefficient LCG aft amidships (m) VCG above keel (m) Pitch gyradius (%LOA) Roll gyradius (%LOA) 2-node natural frequency (rad/s)

2.60 2.52 0.29 0.096 29.40 0.406 0.091 0.098 24.88 12.04 94.30

113.4 109.7 12.36 4.19 2921 0.485 3.96 4.28 25.26 14.28

paddle wavemaker. Wave reflections from the absorption beach (measured using the technique of Isaacson (1991)) were less than 10%. Measurements were taken of the encountered wave profile, heave, pitch, roll and vertical bending moment. Video recordings of tests were taken using a GoPro Hero 2 camera. 2.2 Representative flexible model hull Tests were conducted using a representative hull of a Leander class frigate with the principal particulars in Table 2 (Denchfield, 2011). The model hull was a segmented, flexible backbone model constructed of four rigid segments attached to a uniform aluminium backbone beam. This model hull was constructed for research into ship responses in rogue waves (Bennett et al., 2013; Bennett et al., 2014). The backbone beam was designed such that the model was capable of reproducing the 2-noded bending of the full-scale ship. Figure 1 presentes a schematic of the model hull, and the experimental set-up. A body plan of the model hull can be found in Bennett et al. (2013). Photos of the experimental set-up for head and beam seas are in Figures 2 and 3.

Project scope

The long-term aim of this project is to develop a numerical model based on hydroelastic principles that can predict the survivability of a damaged ship by modelling the entire damage event, whilst allowing for the inherent flexible nature of a ship hull and any influence this may have on ship responses. This paper addresses this problem through experimentally modelling the influence of abnormal loading on the motions and vertical bending moments experienced by a 1:44 scale Leander class frigate operating in regular waves. Experiments are conducted for intact, one and two compartment damage cases; results are presented for wave height to length ratios, H/LOA, of 0.013 and 0.027, for head and beam seas at zero forward speed, and for head seas at a Froude number of 0.154.

2

Parameter

2.3

Test programme

Tests were conducted in the intact condition, and with the addition of sufficient equivalent mass to simulate one and two compartment damage on the vessel, neglecting the effect of free surface and sloshing. For the damage condition tests it was assumed that segment 2 (where the damage was located) was divisible into three watertight compartments of equal length. The equivalent mass added to each compartment to simulate the mass of flood water was calculated based on the displacement and waterline at which the ship would float assuming firstly one compartment

EXPERIMENTAL INVESTIGATION Test facility and set-up

Experiments were conducted in a towing tank 60 m long, 3.7 m wide and 1.86 m deep with a maximum carriage speed of 4.5 ms−1. Unidirectional regular waves were generated using a single, motor-driven

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and secondly two compartment flooding. This equivalent mass was found to be 554.95 tonnes per compartment (6.68 kg at model scale) resulting in a draft with two compartment damage of 0.1204 m—a 25% increase from the intact condition. The three mass distributions of the model are given in Figure 4. Table 3 gives the key parameters affecting ship stability in the three tested conditions, at full scale. The equivalent mass technique used for the results in this paper eliminated freely moving flood water within the vessel that would normally be seen during a damage incident and contribute to the severity of motions and loads experienced by the vessel due to cross-flooding, sloshing and free surface effects—instead using a “lumped mass” approach to represent the presence of flood water. The reasoning behind this was firstly to gain an understanding of the influence of the increase in mass due to flood water, and associated changes in vessel condition (i.e. VCG, GM etc.) on the vessel responses. Secondly, it allows direct comparison with available numerical models that do not currently have the capacity to deal with a moving free surface or mass of water within the vessel in order to assess their ability to model the motions and global loads due to damage, and assess the importance of including three-dimensional effects. Once this is ascertained, the capacity to include flood water following damage can be included.

Figure 1. (a) Schematic of segmented flexible backbone model and (b) Schematic of experimental set-up showing measurement locations for data (dimensions in metres).

Figure 2.

Figure 4. Mass per segment at model scale including additional mass required for one and two compartment damage testing. (non-dimensional).

Experimental set-up in head seas.

Table 3. Variation in ship parameters with damage condition.

Figure 3.

Experimental set-up in beam seas.

Parameter

Intact

One-comp

Two-comp

Draught (m) Freeboard (m) VCG (m) GMT (m) GML (m) Trim (deg)

4.19 7.61 4.28 2.11 207.88 0.259

4.74 7.07 4.88 1.42 181.65 0.407

5.25 6.55 5.38 0.85 162.13 0.542

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Table 4.

Regular wave parameters.

Model scale

Full scale

λ/LOA λ (m) ω (rad/s) T (s) λ (m)

ω (rad/s) T (s)

0.75 1.00 1.25 1.50 2.00

0.85 0.74 0.66 0.60 0.52

2.4

1.95 2.60 3.25 3.90 5.20

5.62 4.87 4.35 3.98 3.44

1.12 1.29 1.44 1.58 1.82

85.06 113.41 141.77 170.12 226.82

7.40 8.52 9.51 10.44 12.02

Test sea conditions

Experiments were conducted in regular waves with the parameters given in Table 4, chosen to give a complete Response Amplitude Operator (RAO). All waves were tested at heights of 35 mm and 70 mm model scale, corresponding to H/LOA values of 0.013 and 0.027, and full scale wave heights of 1.53 m and 3.05 m. Experiments were conducted in head and beam seas at zero speed. The former were in addition conducted at a forward speed of 10 knots full scale (corresponding to 0.78 m/s at model scale) although results at H/LOA = 0.027 were not obtained for this case due to the limited freeboard in the two-compartment damage case. 3

Figure 5. Heave RAOs at zero speed in head seas for intact ship and one and two compartment damage (full scale) at (a) H/LOA = 0.013 and (b) H/LOA = 0.027.

RESULTS

3.1 Effect of damage in head seas— zero speed case Figures 5–7 present the heave, pitch and Vertical Bending Moment (VBM) RAOs measured experimentally in head seas at zero speed for the intact and one and two compartment damage conditions. Results are in each case for (a) H/LOA = 0.013 and (b) H/LOA = 0.027. Error bars indicating experimental uncertainty (calculated using the technique of Coleman and Steele, 1999) are shown, assuming a 95% confidence in results. The uncertainty levels are considered adequate. Results show that, whilst the one-compartment damage appears to have little effect on both the rigid body motions and the global loads, when the two-compartment damage case is considered there is a significant change in the ship responses. In the case of the rigid body motions this is visible as a resonance peak in the RAO at approximately 0.65 rad/s at the higher wave height and 0.5 rad/s at the lower wave height; for both wave heights and the vertical bending moment the effect is seen as a significant increase in the magnitude of the peak of the RAO curve between 0.75 and 1.0 rad/s. Furthermore, the influence of two-compartment damage is more visible at H/LOA = 0.027 than H/ LOA = 0.013. This is likely to be in part due to the

Figure 6. Pitch RAOs at zero speed in head seas for intact ship and one and two compartment damage (full scale) at (a) H/LOA = 0.013 and (b) H/LOA = 0.027.

fact that the wave height was larger, but also because at H/LOA = 0.027 the responses of the ship, even in the intact case, are approaching the nonlinear regime compared to those at H/LOA = 0.013.

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Figure 8. Heave RAOs at 10 knots in head seas for intact ship and one and two compartment damage (full scale) at H/LOA = 0.013.

Figure 7. VBM RAOs at zero speed in head seas for intact ship and one and two compartment damage (full scale) at (a) H/LOA = 0.013 and (b) H/LOA = 0.027.

3.2

Effect of damage in head seas— influence of forward speed

Figures 8–10 show the heave, pitch and vertical bending moment RAOs in head seas, for an example forward speed of 10 knots. Results are shown at full scale for intact, one compartment and two compartment damage. Error bars show an acceptable level of uncertainty in experimental data, assuming a 95% confidence in experimental results. As with at zero speed in head seas, there are some interesting trends visible in the results at 10 knots. In heave the resonant peak that is visible at an encounter frequency of approximately 0.8 rad/s does not occur in the one and two compartment damage conditions; the heave responses at the lower encounter frequencies are however larger than in the intact condition. In pitch the resonant peak in all conditions occurs at approximately 0.8 rad/s encounter frequency; however it is of larger amplitude in the damaged cases than in the intact condition. Furthermore the responses at the lower end of the frequency range are of the order of 50% larger in the damaged conditions compared to the intact case. The vertical bending moment data shows a double resonance, firstly at approximately 0.75 rad/s and secondly at approximately 1.1 rad/s. This is different to the trends in the intact condition where there is a single resonance at approximately

Figure 9. Pitch RAOs at 10 knots in head seas for intact ship and one and two compartment damage (full scale) at H/LOA = 0.013.

Figure 10. VBM RAOs at 10 knots in head seas for intact ship and one and two compartment damage (full scale) at H/LOA = 0.013.

0.8 rad/s. At both the lower and higher ends of the frequency range, the one and two compartment vertical bending moment RAOs are up to double those recorded for the intact case. Whilst one-compartment damage data was not available for the full frequency range due to equipment

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this. The trends seen here in terms of the influence of the damage levels on the response magnitudes are comparable to what was seen in the pitch results at zero speed in head seas; it is therefore likely that if speed were introduced to the beam seas case, the one-compartment damage would begin to have a more significant effect on the ship roll motions.

failure, it is likely based on the presented data that it would have followed the trend of the two compartment condition as seen in the heave and pitch results. 3.3

Effect of damage in beam seas— zero speed case

Figures 11 and 12 present the heave and roll RAOs for the ship in beam seas at zero speed, for the intact, one and two compartment damage cases. As previously, error bars are presented assuming a 95% confidence in experimental results. Figure 11 demonstrates that there is little variation in the heave response with the inclusion of damage, in beam seas at zero speed. However in beam seas the magnitude of roll is of more interest. In this case whilst there is little variation between the intact and one-compartment damage cases at zero speed, some significant differences are seen with two compartment damage. A resonant peak is seen at approximately 0.6 rad/s, and potentially also at 0.85 rad/s although further data needs to be collected at higher frequencies in order to confirm

3.4

Visual analysis of transient flooding effect

Analysis has been carried out using the GoPro videos taken during testing to ascertain the process for transient flooding of two compartments. Preliminary results are presented here. The regular wave in which the model was tested was the worst case scenario for the vessel. i.e. the peak of the Roll RAO obtained in previous experiments. Transient flooding was obtained by covering the hull opening with a latex seal and piercing this when required (when the regular wave motions had reached a steady state). In this way motions could be observed prior to the damage incident, during the damage incident and once the transient flooding had reached a steady state condition. The damage was chosen to occur on the side of the vessel encountering the waves and the damage length (across the two compartments) was 10% of the length between perpendiculars, distributed evenly between compartment 1 and compartment 2. The breadth of the damage was 10% of the ship breadth. These dimensions and location were chosen based on information available in previous literature (e.g. Zhu et al., 2002). The locations of the damaged compartments are shown in Figure 13.

Figure 11. Heave RAOs at zero speed in beam seas for intact ship and one and two compartment damage (full scale) at H/LOA = 0.013.

Figure 13. Damage location and flooding compartments for transient flooding tests in beam seas seen from (a) above and (b) profile view.

Figure 12. Roll RAOs at zero speed in beam seas for intact ship and one and two compartment damage (full scale) at H/LOA = 0.013.

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From Figure 14 it can be seen that onecompartment flooding is initiated 4 seconds after steady-state motion is achieved and twocompartment flooding another 5 seconds after that, at which point the first compartment has almost flooded to its maximum capacity. It then takes a further 10 seconds for both compartments to be flooded to their maximum capacity. The video showed that during the transient phase the important effect to take into account is the movement of the flood water and how this movement affects the ship responses; once the flooding reaches a steady state (both compartments flooded to maximum capacity) sloshing of the floodwater in the compartments is considered to be of greater relevance. The video analysis in this section shows some interesting preliminary results. More accurate determination of the flood times and flood levels will allow more precise characterisation of transient flooding in waves. Further aspects to consider will be how the flooding times scale from model to full scale, and what influence factors such as the permeability of a compartment will have on transient flooding—something which has been found to be of considerable importance (Khaddaj-Mallat et al., 2011). In addition, time to capsize should be considered, as this is an important concept and to date has not been included in any investigation of damage and flooding that includes hydroelastic effects. 4

CONCLUSIONS

This paper has presented an experimental investigation of the influence of one and two compartment damage on the motions and global loads of a Leander class frigate. An uncertainty analysis of the experimental data, assuming a 95% confidence interval, shows adequate levels of accuracy in the results. Experimental results show that two-compartment damage has a significantly larger effect on vessel responses, both motions and global loads, at zero speed in head and beam seas. One-compartment damage appears to have little effect compared to the intact case. As the ship speed increases in head seas, the difference between the magnitude of motions and global load responses in the one and two-compartment cases reduces. However both cases show considerably worse motions and global loads than are seen in the intact case. Furthermore, the location of the resonant responses changes as forward speed and damage are introduced compared to the intact condition. In particular, the 50–100% increases in the vertical bending moment response seen with the two compartment damage case at higher wave

Figure 14. Preliminary visual analysis of transient flooding of the Leander class frigate (two compartment damage).

Figure 14 presents stills taken from the GoPro camera positioned looking down at the model from the towing carriage. Stills are at stated time intervals. The depth of flood water in each compartment is marked by a red line on each image. The time-step at which one-compartment and twocompartment damage occurs is indicated.

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frequencies are of particular concern when considering the structural integrity of a vessel. Preliminary analysis of digital imagery of transient flooding in beam seas shows that such imagery can be used to characterize the transient flooding event in terms of when one-compartment, two-compartment initiate and how long before the steady-state (fully flooded compartment) condition is achieved. Challenges now associated with this are how this data can be used in a useful manner, and how it can be scaled to a full scale vessel. 4.1

between experimental results and classification society rules, Journal of Fluids and Structures, 49, 498–515. Coleman, H.W., and Steele, W.G. 1999. Experimentation and Uncertainty Analysis for Engineers, Wiley Interscience, New York, second edition. Denchfield, S.S. 2011. An Investigation of the Influence of Rogue Waves on a Travelling Ship, PhD Thesis, University of Southampton, UK. gCaptain, 2013a. http://gcaptain.com/bulk-carrier-mvsmart-aground-richards-bay/, accessed 21/10/14. gCaptain, 2013b. http://gcaptain.com/three-missingafter-dutch-security-vessel-sinks-after-collision-withfishing-trawler/, accessed 21/10/14. gCaptain, 2014b. http://gCaptain.com/containershipscollide-suez-canal/, accessed 19/11/2014. Isaacson, M. 1991. Measurement of regular wave reflection. Journal of Waterway, Port and Coastal Engineering, 117, 533–569. Khaddaj-Mallat, C., Rousset, J-M., and Ferrant, P. 2011. The transient and progressive flooding stages of damaged Ro-Ro vessels: a systematic review of entailed factors. Journal of Offshore Mechanics and Arctic Engineering, 133 (3). Korkut. E., Atlar, M., and Incecik, A. 2004. An experimental study of motion behaviour with an intact and damaged Ro-Ro ship model, Ocean Engineering, 31 (3–4), 483–512. Korkut, E., Atlar, M., and Incecik, A., 2005. An experimental study of global loads acting on an intact and damaged Ro-Ro ship model, Ocean Engineering, 32 (11–12), 1370–1403. Lee, S., You, J-M., and Lee, H-H. 2012. Experimental study on the six degree-of-freedom motion of a damaged ship for CFD validation, Proc. 29th Symposium on Naval Hydrodynamics, Gothenburg, Sweden, 26–31 August 2012. MAIB, 2014a. Report on the investigation of the grounding of Danio off Longstone, Farne Islands, England, Marine Accident Investigation Branch Report No 8/2014. Maritime Executive, 2013. http://www.maritime-executive. com /article/Ship-Collision-Leaves-1-Dead-7-MissingOff-Vietnam-2013–09–16/, accessed 11/10/13. MIT, 2013. Ministry of Infrastructures and Transports (Marine Casualties Investigative Body) Costa Concordia: Report on safety technical investigation. Palazzi, L., and de Kat, J. 2004. Model experiments and simulations of a damaged ship with air flow taken into account, Marine Technology, 41, 38–44. RINA, 2014. http://www.rina.org.uk/mol_comfort_accident.html, accessed 22/10/2014. Tabri, K., Matusiak, J., and Varsta, P., 2009. Sloshing interaction in ship collisions—an experimental and numerical study, Ocean Engineering, 36, 1366–1376. USA Today, 2013. http://www.usatoday.com/story/ news/world/2013/09/27/cargo-ships-japan-collision/ 2881593/, accessed 11/10/13. Vanem, E., and Ellis, J., 2010. Evaluating the cost-effectiveness of a monitoring system for improved evacuation from passenger ships, Safety Science, 48, 788–802. Zhu, L., James, P., and Zhang, S. 2002. Statistics and damage assessment of ship grounding, Marine Structures, 15, 515–530.

Future work

Future work will: • Carry out further testing in head seas at a range of forward speeds in order to obtain an accurate picture of the influence of damage on motions and global loads with forward speed, • Carry out further analysis of the influence of damage on motions and global loads in beam seas, • Extend the transient flooding analysis to obtain numerical data as well as digital imagery in order to characterise the one and two compartment transient flooding, • Investigate the time-to-capsize concept including hydroelastic effects, • Investigate the use of hydroelasticity modelling for predicting the global loads associated with abnormal load distributions due to flooding, • Investigate the use of a 3D partly nonlinear seakeeping model for predicting 6 degree of freedom motions associated with abnormal load distributions due to flooding.

ACKNOWLEDGEMENTS This project is supported by funds from the Lloyd’s Register Foundation, through the Lloyd’s Register University Technology Centre at the University of Southampton. The authors would also like to acknowledge the work of University of Southampton MSc students Ioannis Anastasopoulos and Mark Towells who carried out the experimental work during the course of their MSc dissertations. REFERENCES BBC, 2012. http://www.bbc.co.uk/news/world-asia-16458574, accessed 22/10/14. Bennett, S.S., Hudson, D.A. and Temarel, P. 2013. The influence of forward speed on ship motions in abnormal waves: experimental measurements and numerical predictions, Journal of Fluids and Structures, 39, 154–172. Bennett, S.S., Hudson, D.A., and Temarel, P. 2014. Global wave-induced loads in abnormal waves: comparison

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Unstructured MEL scheme for 3D body nonlinear ship hydrodynamics A.C. Chapchap & P. Temarel Fluid Structure Interactions Group, University of Southampton, Southampton, UK

ABSTRACT: The radiation forces for a body oscillating on the free surface are investigated through a solution in the time domain accounting for the changes in the submerged geometry of the body. The solution is obtained in the context of a Mixed Eulerian-Lagrangian (MEL) scheme using direct Boundary Element Method (BEM). An important issue when the body submerged geometry changes concerns the accurate evaluation of the velocity potential time derivative. In this work, two approaches are proposed to tackle this problem. Results are presented for a sphere and a Wigley hull undergoing forced oscillations in the heave mode with large amplitudes of motion. The predictions of the heave related hydrodynamic forces and coefficients are compared against available experimental measurements and other numerical methods. Sensitivity studies investigating the influences of mesh density, domain size, damping beach size and amplitude of forced motion are also discussed. 1

boundary value problem for the potential time derivative on the floating body surface (Neumann boundary) and for the normal acceleration on the free surface (Dirichlet boundary). These numerical approaches are detailed in section 3. Results for a sphere undergoing forced oscillations in the heave mode, for a range of amplitudes, obtained using both FDBNL and φt Exact BNL methods, are compared to other numerical predictions by Hongmei (2010) and Lin & Yue (1991). Results for the Wigley hull, undergoing forced oscillations in heave, are obtained using the φt Exact BNL method and compared against experimental data (by Journee (1992)).

INTRODUCTION

The extension of the methodology of linear time domain potential flow simulations by Chapchap et al. (2012) is extended to the case of body nonlinear oscillations in this paper. Under the assumptions of potential flow theory, in the time domain, there are basically three sources of nonlinearities that can change the hydrodynamic coefficients of a floating structure. The first source of nonlinearity is caused by the changes in the submerged geometry of the floating body. The second source is caused by the time evolution of water line, i.e. as the body enters and exits the water not only does its submerged geometry change, but also, the instantaneous water line evolves with time. This consideration means that in addition to hydrostatic effects, hydrodynamics influences neglected by linear theory are now included. The third source of nonlinearity is the nonlinear free surface boundary conditions, so that on the free surface Bernoulli’s equation cannot be linearized. Thus, a quadratic term, i.e. the velocity of the fluid, needs to be computed as well. In the present work, efforts are focused on the first source of nonlinearity. One of the most difficult numerical challenges is the changing of the domain with time, requiring a new mesh and a new influence matrix to be computed at each time step. In order to tackle this problem, two numerical approaches are proposed: (a) Finite Difference Body NonLinear approach (FDBNL), approximating the time derivative of the potential by means of a finite difference scheme and (b) φt Exact BNL uses the formulation proposed by Battistin & Iafrati (2003) and Wu & Eatock Taylor (1996), solving a second

2

PROBLEM FORMULATION

In order to describe the flow field, a fixed three dimensional orthonormal, right handed, Cartesian system Oxyz is used in the fluid domain Ω. This way, under the assumptions of inviscid and irrotational flow, the equation governing the fluid flow, namely Navier-Stokes equations, are reduced in Ω to the continuity or Laplace’s equation for the potential φ. In this setting, making use of Green’s second identity and taking the limit to the boundary of the domain ∂Ω, it can be shown (Liu, 2009) that Laplace’s equation can be written in terms of an integral equation defined only on ∂Ω, as:   c( x )φ (x ( )

    G(( x, y )∇φ ( y ) nd Γ G     − ∫ φ ((yy ) G G(( x, y ) nd ;

∫

∂Ω

∂Ω

(1)

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 where c( x) represents the interior solid angle and dΓ is the area of element ∂Ω. Equation (1) is still exact, but can be discretized/approximated in a numerical fashion, which can be written as:   c( x )φ ((x )

N

∑∫ j =1 N

Ej

Ej

Equation (6) can therefore be solved for the time-derivative of the velocity potential directly since the linearized dynamic free surface boundary condition (upon ignoring the damping term introduced in equation (5)) is generally written as:

(2)

∂φ = − gz. ∂t

∂φ = − gz − vφ . ∂t

(7)

On the floating body the wetted surface boundary condition to be imposed is more involved. Battistin & Iafrati (2003) showed that for the case of heave the boundary condition on the body is given by:

(3)

   In equation 3, n and V ( x,t ) denote the normal and body velocity, respectively. Substitution of equation (3) into (2) yields, for elements on the floating body surface, a Neumann problem for the potential on the floating body. On the other hand, on the free surface the potential is known from the time evolution of the kinematic and dynamic boundary conditions. In the present body nonlinear analysis, these are: ∂z ∂φ = − vz ∂t ∂z and

Ej

(6)

where Ej denotes the jth element and dEj its corresponding element area. Since equation (2) holds for every element from i = 1 to N, it can be recast as a linear system, solved subject to boundary conditions. On the floating body the impermeability boundary condition states that the normal velocity of the body should be equal to the normal velocity of the flow,      φ ( , ) ⋅ = ( x,, ) ⋅ n.

    ∂ ( ∇φ ( y j , t ) ⋅ n ) G ( xi , y j ) dE j dE Ej ∂t j =1 N     − ∑ ∫ φt ( y j G ( xi , y j ) ⋅ ndE j . N

∑∫

j =1

    G(( x, y j )∇φ ( y j ) ⋅ ndE j

    − ∑ ∫ φ ((yy j ) G( G ( x, y j ) ndE j . j =1

  c( xi )φt ( xi , t )

    ∂ 2φ = n ⋅ a − n ⋅ (V ⋅ ∇ )∇φ , ∂ ∂n

(8)

 where a represents acceleration. In the present work, only the problem of forced oscillations is addressed in the heave mode. Hence, the body displacement is Z = A sin(ω t), where A is the amplitude of motion. The pressure on the floating body is then calculated as: ⎛ ∂φ ⎞ p = −ρ ⎜ + ∇φ 2 ⎟ . ⎝ ∂t ⎠

(4)

The force is then given by:   F =∫ pndS

(5)

∂B ( t )

In equations 4 and 5, ν is a damping parameter used in the same fashion as applied by Chapchap et al. (2012). Its main role is to avoid wave reflection from the outer boundaries, satisfying the radiation condition. Integrating equations (4) and (5), the potential at a given time step is found. In the present approach a simple Euler method was used for integration. This potential is then fed into equation (3) for points on the free surface. Hence, for the free surface a Dirichlet problem is solved for the normal velocity. Because equation (2) has both Neumann and Dirichlet boundary conditions, it is referred as a mixed boundary value problem. In addition the solution of equation (2) is called the Eulerian phase, while the integration of equations (4) and (5) in time is called the Lagrangian phase. Equation (2) can also be differentiated in time. Assuming interchangeability of the derivative and integration operators leads to:

(9)

(10)

There are two avenues to calculate the pressure on equation (9). The first one is to solve equation (2) at each time step and estimate the potential derivative by a finite difference scheme, denoted as FDBNL. The second approach requires solving equation (2) and using the resulting potential to estimate its spatial derivatives. These quantities are then substituted into equation (8) and finally equation (6) is solved to obtain the time-derivative of the velocity potential. This approach will be referred to as φt Exact Body Non-Linear, and designated the φt Exact BNL. Both methods are summarized in section 3. 3

NUMERICAL METHODS

The mathematical formulation of Section 2 presents the implementation challenge of determining the spatial velocity potential derivatives requi-

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red in the wetted surface boundary condition and pressure evaluation of equations (8) and (9), respectively. One approach is to use higher order boundary elements of an isoparametric nature so that the associated shape-function permits the evaluation of the required derivatives locally, within each boundary element. These local derivatives can then be assigned within the global reference system. Additional challenges associated with the so called double node boundary conditions introduced to remove the water line singularity should be addressed using the intersection of Dirichlet and Neumann surfaces in accordance with (Xu, 1992). A novel approach in hydrodynamics is the authors’ transference of methods designed to address interpolation of scattered data. Its application is considered appropriate where grids are unstructured. Transference to the current problem has certain requirements given the computational grid is reconstructed at each time step, namely:

m    hw x ) = ∑ wi f x xi + w0 ,

(

i =1

)

(11)

where || . || refers to the Euclidean distance and  f is the RBF centered on xi . Equation (11), together with the restriction that the sum of the weights should be zero, lends itself to a linear system, whose influence matrix is given by A(i, j) = f (||x j xi ||). This linear system can be solved to obtain the weights. In the present case, in order to make the interpolator more robust a regularization parameter, λ, is introduced in the RBF interpolator. The regularization is imposed on the l2 norm (i.e. the sum of squares of the vector components). Hence, the problem of finding the weights in equation (11) can be stated as: find w* such that: w

ag i w

− The interpolation process needs to “generalize well”; that is, if the test error is small then the generalization error needs to be small; − The spatial derivatives sought require the interpolator to be differentiable up to at least second order. − Once a global approximation is being pursued, it is important to have some assurance about capturing the variability of spatially complex functions.

 1 m λ ( hw ( xi ) − yi )2 + w 2 . ∑ 2 i =1 2

(12)

It is not hard to show that the weights that solve equation (12) are given by: ( AT A λ I ) 1 AT y.

w

(13)

There are several possibilities for the function f, amongst them the most commonly used are the Gaussian, multiquadratic and inverse multiquadradic kernels. Motivated by the results of Chinchapatnam (2006) a multiquadratic kernel was chosen. Finally, after the weights are known, equation (11) can be differentiated analytically, yielding equations (14) and (15). At this point, the numerical model is complete.

The concern addressed in the first point is the need for centres of newly generated triangular elements to have centres close to the original triangular element so that interpolation errors cannot become too high. Since a Radial Basis Function (RBF) is suitable on scatter data sets the same approach is adopted here. The RBFs will be used to capture the behavior of the velocity potential over a floating body, and on the free surface, when its elevation is needed. Moreover, RBFs are differentiable and can be used to calculate the spatial derivatives. Furthermore, the universal approximation theorem for RBF networks by (Park & Sandberg 1991) provides the basis in terms of the errors that can be made on the approximation as a function of the number of points needed to fit the interpolator. The problem of finding a function approximator can be recast as a supervised learning problem, as follows: given a set of input vec  tors x1, x2 , …, xm in Rd (d = 2 or 3 for the present purposes) and a set of scalar labels y1, y2, …, ym one searches for a map from Rd to R (say,  x to h( x )) that is “close” to y for any given x and y. Since the RBF representations were chosen the   approximator h ( x ) is parameterized by weights w ,  i.e., hw ( x ) , and is given by:

 ∂hw ( x ) m = ∑ wi ∂x i =1  ∂hw ( x ) m = ∑ wi ∂y i =1  ∂hw ( x ) m = ∑ wi ∂z i =1  ∂f − xi = ∂x  ∂ f y − yi = ∂y  ∂f − zi = ∂z

(

)

(

)

(

)

∂f

(

∂f

(

 − xi ∂x  − xi

∂y  ∂ff x − xi

(

∂z

x − xi ; σ 2 + r2 y − yi ; σ 2 + r2 z − zi . σ 2 + r2

), ),

(14)

),

(15)

The main stages in the FDBNL approach, at a given time step, can be summarized as follows: 1. Create a mesh and solve equation (2) on that mesh.

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analysis. The norm of the first harmonic is given by | f1 | A323 + B332 , and the norm of the second harmonic | f2|, can be calculated accordingly, using A33(2) and B33(2). In all the figures and tables, the harmonics are non-dimensionalised according to equation (17).

2. Calculate the time derivative of the potential u sing a finite difference scheme in time. 3. Use equation (14) to calculate the spatial gradient of the velocity potential. This potential is tangential to the boundary, so it has to be combined with the normal velocity component (which is solved for in equation (2)). 4. Calculate the pressure according to equation (9). 5. Integrate equations (4) and (5) in order to march the boundary conditions and the free surface elevation in time.

1 t +T / 2 fˆ0 = ∫ Fh (t )dt; T t −T / 2 2 t +T / 2 A33 = ∫ Fh (t )sin(wt )dt; T t −T / 2 2 t +T / 2 B33 = ∫ Fh (t )cos(wt )dt; T t −T / 2 t + T / 2 2 A33(2) = ∫ Fh (t )sin(2wt )dt; T t −T / 2 2 t +T / 2 B33(2) = ∫ Fh (t )cos(2wt )dt; T t −T / 2 fˆ0 ; f0 = ρ gπ RA2 fˆ1 ; f1 = ρ gπ R 2 A fˆ2 . f2 = ρ gπ RA2

The mesh is created using the Gmsh library, developed by Geuzaine & Remacle (2009), and the geometry definition of the floating body allows for its underwater geometry to change according to a forced oscillation displacement. The φt Exact BNL approach can be summarized as: 1. Create a mesh and solve equation (2) on that mesh. 2. Use equation (14), and its derivative, to estimate the body boundary condition of equation (8). Impose the boundary condition on the free surface according to equation (7). 3. Solve equation (6), which provides the exact time derivative of the potential on the floating body. 4. Use equation (14) to calculate the spatial gradient of the velocity potential. This potential is tangential to the boundary, so it has be combined with the normal velocity component (which is solved for in equation (2)). 5. Calculate the pressure according to equation (9). 6. Integrate equations (4) and (5) in order to march the boundary conditions and the free surface elevation in time. 4

(16)

(17)

The topology of a typical mesh used in the BNL simulations is shown in Figure 1. On average a total of 8926 triangles, of which 300 are on the floating body, are used. An analysis of the effect of different terms in pressure evaluation (see equation (9)), on the harmonics was carried out, using the FDBNL approach and shown in Figure 2. The triangles line is the pressure field resulting only from the inertial term. The rectangles denote evaluation of velocity gradient using only the tangential velocity components to the floating body. Finally the circles denote evaluation of the velocity gradient by combining both tangential and normal velocities components. Interestingly, the effect of the quadratic term is more pronounced on the evaluation of f0 and | f2|. The results suggest that the magnitude of the normal gradient component is of great importance by comparison to the tangential potential component, since it changes the values of both f0 and | f2| when it is accounted for. In fact, compared to the results obtained by Lin & Yue (2001), there is some indication that the value of f0 is being overestimated (larger negative values) by the present simulation, since the sphere is undergoing forced oscillations, the normal velocity is given by the impervious boundary condition, which is exact. However, the values of f0 are consistently below the ones obtained by Lin & Yue (1991). That said, there is also evidence that the tangential gradient has not a considerable contribution to the pressure field, since its incorporation does not change significantly the values of f0 and | f2| across the amplitude range.

RESULTS FOR A HEAVING SPHERE

In this section, the results of both algorithms, FDBNL and by φt Exact BNL, for a heaving sphere, of radius R, are presented and compared with other available numerical predictions. A range of heave amplitudes are used. More specifically, in the body nonlinear analysis, new terms arise in the decomposition of the time series of the hydrodynamic force, Fh(t), in the frequency domain. This implies, within the context of Fourier decomposition, that first and higher order harmonic terms are relevant, as well as the mean term (Lin & Yue, 1991). In the context of forced oscillatory motions in the heave mode, Z = A sin(ωt), the hydrodynamic force can be decomposed, by means of a Fourier decomposition, according to equation 16. In addition to the hydrodynamic coefficients themselves, added mass A33 and damping B33, their norm is also of great interest for hydrodynamic

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Mesh used for the BNL simulation of the

Figure 3. Harmonics of the force time series using the φt exact BNL approach for the heaving sphere.

Figure 2. Harmonics of the force time series using the FDBNL approach for the heaving sphere.

Figure 4. Comparison of heave added mass & damping, using φt Exact BNL, FDBNL with other numerical predictions for sphere.

Furthermore, when comparing the harmonics calculated by algorithm FDBNL with the body nonlinear results from Lin & Yue (1991), it can be seen that the trends are qualitatively in line, but FDBNL has a tendency to over/underestimate the harmonics. The effect of the regularization term also influences the hydrodynamic coefficients, hence the harmonics. Figure 3, compares the harmonics obtained by the φt Exact BNL, with and without regularization. The results suggest improvements on the estimation of | f2|, but there is no consistent trend and as the amplitude increases/decreases a deterioration of the accuracy is observed, compared to Lin & Yue (1991). However, the agreement for | f0| is worse in the presence of the regularization term. The causes of this issue require further investigations. The agreement for the first harmonic, | f1|, is reasonable, but as the amplitude increases

a deterioration of the accuracy is observed, compared to the predictions by Lin & Yue (1991). The components of | f1| in phase with velocity and acceleration, namely damping and added mass terms, are shown in Figure 4, using both numerical techniques as well as other numerical predictions. One can see from Figure 4 that the heave damping values of the φt Exact BNL case, even for small amplitudes of oscillation, are very different from the linear prediction; so are the values by Lin & Yue (1991). The added mass and damping values obtained using FDBNL, for small oscillation amplitudes are very close to the linear values. FDBNL provides good predictions for heave damping, comparing well with the QBEM results by Hongmei (2010). However, its added mass predictions are overestimates. On the other hand φt Exact BNL provides good predictions for the

Figure 1. sphere.

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added mass, even at larger oscillation amplitudes. However, the damping can be underestimated at some large amplitudes of oscillation. The reason for this particular behaviour requires further investigation as it could be linked to the accuracy of the approximation of the time derivative of the velocity potential by the finite difference scheme. 5

Table 1. Main dimensions of Wigley hull, model IV. L L B B T Cb

1.0 5.0 3.2 0.46

RESULTS FOR A HEAVING WIGLEY HULL

The results obtained for the heaving sphere showed that the φt Exact BNL provides better overall agreement with other nonlinear numerical predictions. Therefore, this method is used for a Wigley hull undergoing forced oscillations in the heave mode (Journee, 1992). The dimensions of the Wigley hull are given in the Table 1, where the length is taken as 1 unit. The initial domain and typical mesh topology, mesh 1 shown in Figure 5, for the Wigley hull simulations comprises a box like domain, extended on the free surface, [−5 −5 −1] × [5 5 0] units, in order to mitigate issues related to wave reflection. On average the mesh has a total of 4300 triangles of which 300 are on the body surface. The numerical simulation comprises a set of two amplitudes A = 0.053 T and A = 0.27 T, T denoting mean draught, and three non-dimensional frequencies ω = ω/(g/L)0.5 = 2.49, 3.45 and 4.41 for each amplitude. For the lower amplitude the same mesh, namely mesh 1 and shown in Figure 5, was used for all frequencies. This was done to check that the BNL solution is close to the linear solution. The higher amplitude case, i.e. A = 0.27T, turned out to be more difficult from a numerical perspective. It required different meshes for different frequencies. The finer mesh (mesh 4), comprising on average a total of 11484 triangles of which 3216 are on the body surface, is shown in Figure 6. For the lower frequency, ω = 2.49 meshes 1 and 2 were used. Mesh 2 was obtained by halving the edge size of the triangles on the surface of the Wigley hull for mesh 1. The edge size of the elements on the walls remained the same while the edge size of the elements between the Wigley hull and the walls were linearly interpolated. Thus, the refinement is local to the hull wetted surface and to the free surface in its vicinity. The corresponding total hydrodynamic force and the quadratic term contribution are shown in Figure 7. As can be seen the effect of using different meshes is negligible. Furthermore the quadratic term contribution is small. For ω = 3.45, three meshes were used, namely mesh 1, mesh 2 and mesh 3. This was carried out to ensure better accuracy for the total hydrodynamic force and the quadratic term in pressure. At this frequency there are differences between mesh 1 and meshes 2 and 3, especially for the quadratic term, as

Figure 5. Mesh topology 1 (Mesh 1) used for the Wigley hull forced oscillations in heave.

Figure 6. Mesh topology 4 (Mesh 4) used for the Wigley hull forced oscillations in heave when ω = 4.41.

can be seen by comparing Figures 7 and 8. Mesh 3 is created by adjusting the edges of the elements of mesh 2 by the corresponding wave length of the higher frequency, so that the ratio of wavelength/ element edge is kept constant in the vicinity of the Wigley hull. The damping zone and free surface dimensions are also adjusted accordingly, 1.8 and

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Figure 7. Forces of the Wigley hull undergoing forced heave oscillations, ω = 2.49, A = 0.27T and λ = 0.01.

Figure 9. Comparison of heave hydrodynamic coefficients using linear analysis (Chapchap et al. 2012), φt exact BNL and experimental data (Journee, 1992).

The free surface damped/undamped zones were kept at the same size, with respect to the generated wave length, i.e. 1.8 and 3.25 wave lengths respectively. The hydrodynamic coefficients in heave obtained by the present simulations, as well as the results of linear analysis (Chapchap et al. 2012), are compared against the experimental data by Journee (1992) in Figure 9. For the lower amplitude A = 0.053T, the results from the Linear analysis and φt Exact BNL, compare relatively well. The agreement of the added mass is good. On the other hand for damping, φt Exact BNL shows a tendency to overestimate even for small amplitudes, despite the good qualitative agreement observed. For the higher amplitude (A = 0.27 T) the agreement between φt Exact BNL and experimental data is good in the lower frequencies, but deteriorates as the wave steepness kA increases, where k is the wave number. The deterioration is more pronounced in the damping coefficient as it increases with the frequency of forced oscillation. For the added mass the effect of increasing the amplitude of oscillation is less pronounced. This effect reduces the magnitude of the added mass coefficients as a function of the amplitude of oscillation. The deterioration on the predictions of the damping coefficient, compared to linear analysis and experimental data, as kA increases is a point that requires further investigation. It can be linked to inaccuracies in the approximation of the velocity potential spatial derivatives on the Wigley hull surface, which alters the pressure field. In this respect, the use of RBFs in the so called finite difference

Figure 8. Forces of the Wigley hull undergoing forced heave oscillations, ω = 3.45, A = 0.27T and λ = 0.01. Table 2. Wigley hull hydrodynamic coefficients calculated for ω = 3.45, compared against data from (Journee, 1992). Mesh

Aˆ33

Bˆ33

Mesh 1 Mesh 2 Experiment

1.10 1.17 1.23

3.55 3.34 3.21

3.25 wavelengths, respectively. The added mass and damping coefficients obtained by the three meshes are presented in Table 2 and compared to the experimental data, the latter for Froude number Fn = 0.2. There is more improvement from mesh 1 to mesh 2, rather than from mesh 2 to mesh 3. This suggests convergence has been achieved. In order to investigate the behaviour of the solution at the higher frequency, i.e. ω = 4.41, mesh 4 was created using the same procedure for mesh 3. Since the frequency is higher, the wave lengths are shorter. Thus, rescaling according to the corresponding wavelength, the box domain is reduced to [−1.8 −1.8 0] × [1.8 1.8 −1] units and the edge length of elements near the hull were rescaled (see Figure 6).

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insight into the behaviour of the φt Exact BNL algorithm. For low amplitudes, the agreement is in line with the predictions made by the linear analysis. On the other hand when using the higher amplitude A = 0.27 T, the results showed considerable disagreement as the frequency of oscillation increased, compared to both linear and experimental results. The general behaviour observed in the body nonlinear predictions, i.e. lower added masses and higher damping coefficients as the amplitude of oscillation increases, is not in line with the experimental measurements. This may raise an interesting point regarding the applicability of the body nonlinear analysis, namely that it can improve results with reference to linear theory, only in a region where the wave steepness, kA, is kept small.

mode can be an alternative approach (Chinchapatnam, 2006), since it seeks to approximate the derivatives of the function locally instead of seeking a global pproximation. The experimental measurements indicate that as the amplitude increases a slightly higher added mass in the lower and higher frequency ranges with no appreciable change in the medium frequency range is observed. The current predictions, in the low to medium frequency range, show a small increase of added mass with increasing amplitude. The variation of the damping coefficient indicated by the experimental measurements is similar to that of the added mass behaviour in the lower and higher frequencies, but in the medium frequency range the damping coefficient decreases with increasing amplitude of oscillation. The current predictions indicate an increase in the damping coefficient with increasing oscillation amplitude, especially toward the high end of the medium frequency range. 6

REFERENCES Battistin, D. & Iafrati, A. 2003. Hydrodynamic loads during water entry of two dimensional and axisymmetric bodies. Journal of Fluids and Structures 17:643–664. Chapchap, A.C., Miao, S.H., Temarel & Hirdaris, S.E. 2012. Time domain hydroelasticity analysis: The three dimensional linear problem. 6th nternational Conference on Hydroelasticity in Marine Technology, Tokyo, Japan. Chinchapatnam, P.P. 2006. Radial Basis Function based meshless methods for fluid flow problems. PhD thesis. University of Southampton. Geuzaine, C. & Remacle, J.F. 2009. Gmsh: a threedimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79(11):1309–1331. Hongmei, Y. 2010. Computation of fully nonlinear three-dimensional wave-body interactions. PhD thesis. MIT. Hulme, A. 1982. The wave forces acting on a floating hemisphere undergoing forced periodic oscillations. Journal of Fluid Mechanics 8:22–44. Journee, M.J. 1992. Experiments and calculations on four wigley hull forms. Report 909, Delft. Lin, W.M. & Yue, D. 1991. Numerical solutions for large amplitude ship motions in time domain. 18th Symposium on Naval Hydrodynamics. Liu, Y. 2009. Fast Multipole Boundary Element Method. Cambridge University Press. Park, J. & Sandberg, M. 1991. Universal approximation using radial basis-function networks. Neural Computation 3:246–257. Wu, G.X. & Eatock Taylor, R. 1996. Transient motion of a floating body in steep water waves. 11th Internationl Workshop on Water Waves and Floating Bodies. Xu, H. 1992. Numerical study of fully nonlinear water waves in three dimensions. PhD thesis. MIT.

CONCLUSIONS

In this work, the body nonlinear problem was addressed for a sphere and for a Wigley hull undergoing forced oscillations in the heave mode. The results were compared to other numerical methods and available experimental data. The paper focused on assessing the influence of how the potential time derivative is evaluated for the heaving sphere, by adopting a finite difference based interpolation scheme (FDBNL) and a direct solution method (φt Exact BNL). The latter is used for the heaving Wigley hull, where the sensitivity of the predictions to different meshes is examined over a range of oscillation frequencies. For the sphere simulations, the results from the FDBNL algorithm suggest a reasonable agreement for the estimation of the damping coefficient and overestimation of the added mass coefficient, compared to the body nonlinear results obtained by Hongmei (2010). The φt Exact BNL algorithm, in contrast, shows a better agreement for the added mass coefficient and | f2|. The agreement obtained for the damping coefficient is also good, provided the higher amplitude of oscillation A = 0.375 R is excluded. At this amplitude, there are issues that require further investigations. These issues are probably linked to a poor estimation of the boundary condition of the second boundary value problem, i.e. the normal derivative of the time derivative of the velocity potential. The numerical estimation of this qualtity is usually hard. The body nonlinear results for the Wigley hull undergoing forced heave oscillations brought more

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

An initial estimation of DP requirements for a non-moored operational FPSO G.E. Hearn Fluid-Structure Interactions Research Group, University of Southampton, UK

G.C. Bratu Houlder Offshore Ltd., Portsmouth, UK

ABSTRACT: This paper presents a direct estimate of the dynamic horizontal plan positioning forces and moment to keep a proposed Floating, Production and Storage Offloading system (FPSO) with open moonpool on station, without moorings, through novel application of standard frequency domain motion response analysis. Reasons for selecting frequency domain over time domain formulations presented is justified. Quality of hydrodynamic reactive quantities, required in motion response formulation, from a third party 3D boundary element based formulation is discussed in terms of expected cross term symmetry, with warning of how blind software application can be erroneous. Combining motions from standard unconstrained motion equations and global thruster loading requirements from a novel response formulation, the time-averaged DP thruster cluster power is estimated. Hence individual thruster power can be assigned recognizing the redundancy requirements of a Class 3 DP system. 1

INTRODUCTION

associated equipment based constraints. DP Class selection for a particular operation ‘should be agreed between the Owner(s) of the vessel and their respective customer based on a risk analysis of a loss of position’, whilst noting that ‘some Coastal States impose minimum DP Equipment Class requirements for activities within their domain’. Hence thruster power, its distribution and levels of redundancy must be sufficient to avoid exorbitant production downtime costs. DP Class 3 is the most stringent requirement and ‘single failure’ avoidance includes: all items listed for Class 2, with all DP components in any watertight compartment or in any one fire subdivision protected from fire or flooding. DP system single failure avoidance will be satisfied provided there is redundancy of all active Class 2 equipment components with redundancy and physical separation of all Class 3 equipment components. Additionally, MSC 645 ‘single failure criteria’ requires the safe termination of work in progress if a failure occurs. DP system design and choice of operating configuration must acknowledge that various onboard FPSO industrial activities have different termination times (DNV 2013). Each Classification Society has DP Rules with a different interpretation of redundancy. Some demand the vessel holds position with the main machinery that remains operational after the

FPSOs have worldwide acceptance in both exploration and offshore oil/gas field development. FPSOs were initially based on conversion of mothballed oil tankers. Now purpose-built FPSOs have evolved to accommodate different operational requirements, defined by the intended operational environmental and statutory health and safety requirements. FPSO design requires performance in terms of efficiency and operation in ever deeper water. Dynamic positioning system flexibility, reliability and proven robustness has allowed their application within a large category of craft, i.e. dredgers, drill-ships, FPSOs, crane, cable-laying & pipe-laying, dive support, heavy lifting, research and maritime survey vessels. Continuous development of GPS technology, coupled with consistently improved thruster propulsive proficiency, is now considered sufficient to implement automatically controlled DP systems to maintain FPSO positioning. Entire reliance on a DP system to overcome FPSO environmental forces, without moorings, is a decisive step requiring the high levels of propulsive redundancy consistent with an IMO DP Class 3 system. IMO Maritime Safety Committee Circular MSC 645 (IMO 645 1994) explains the different DP classes and their

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worst case failure, whereas others permit standby machinery being automatically brought online. Here acceptability of ‘automatic start of equipment’ requires equipment to be tested to establish its successful operation before degradation of position and heading performance (ABS 2014). The transfer should be smooth and within acceptable limitations of the operation (IMO 645 1994). A universal viewpoint of MSC 645 adopted by all classification societies is that ‘redundancy is to provide two or more items of equipment or the system is required to perform a function so that a redundant unit can take over from a failed unit without unacceptable interruption of the function’ (ABS 2014). These aspects must be addressed within the context of Worst Case Failure (WCF), corresponding to the failure that has the greatest effect on station keeping capability, and Worst Case Failure Design Intent (WCFDI) describing the minimum propulsion and control equipment remaining operational following WCF. Successful DP design corresponds to WCF achieved being always less than or equal to WCFDI. FPSOs are assumed to deploy sub-sea equipment; have a rigid riser oil production system; to offload stored crude oil to a shuttle-tanker and perform all such operations without assistance from other vessels in water depths between 100 m and 1000 m. With this background, a plausible scenario for unmoored FPSO DP control (Houlder 2009) is:

Table 1.

FPSO principal characteristics.

Length overall Breadth moulded Depth main deck Draught design Displacement Longitudinal centre of gravity Transverse centre of gravity Vertical centre of gravity Longitudinal centre of buoyancy

Figure 1.

1.1

LOA B D T Δ LCG TCG VCG LCB

153.5 m 26.0 m 18.0 m 10.0 m 31989.0 t 72.42 m 0.04 m 10.14 m 72.42 m

Aqwa®-meshed underwater section of FPSO.

Research objectives

Time domain investigations of DP or moored offshore structures is very demanding and in initial design studies is unlikely to be the most appropriate approach. For moored structures a matrix of stiffness terms is readily generated at each fairlead location, to include the mooring influences in frequency domain motion analysis. In this paper a novel modification of the frequency domain analysis is developed to determine the global magnitude of DP forces and moment required to maintain an FPSO in its preferred operational orientation. Thereafter, the forces to be developed within individual aft and fore clusters of thrusters are derived from first principles.

a. FPSO designed for unrestricted transit between worldwide operations; b. FPSO designed and equipped on the basis of Douglas Sea State definition and the associated wave height limited operational modes: − DP production (8): 9.0 to 14.0 m; − DP cargo transfer (6): 4.0 to 6.0 m; − Connection for cargo transfer (6); − Disconnection from cargo transfer (7): 6.0 to 9.0 m; − Crane operation (5): 2.5 to 4.0 m; − Crane operation transfer ship/ship (4): 1.25 to 2.5 m; c. FPSO designed to maintain position in the worst case failure with a 42.0 knot wind (with associated fully developed waves) and a 1.5 knot current acting coincidentally at +/− 45 degrees off the bow or stern.

1.2

Organization of paper

Section 2 will discuss time domain versus frequency domain motion responses using compact generalized formulations. Section 3 will present a method of identification of individual thruster cluster loads and their line of action (direction). In section 4 computations undertaken are presented and discussed with appropriate hydrodynamic data quality checks. The closure will address an improved proposed power requirement estimation methodology.

These operational characteristics determine sizing of DP system propulsive and power capacity to achieve safe station keeping. The principal characteristics of the FPSO to be investigated and the discretized form of the underwater section are provided in Table 1 and Figure 1.

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2

TIME DOMAIN VERSUS FREQUENCY DOMAIN MODELLING

because of low frequency damping (Hearn & Tong 1987, Hearn et al. 1988) and the Cummins integraldifferential equation formulation (Cummins 1962). More novel control is available via neural networks and Kalman filtering (How et al. 2010). The FPSO under consideration is a conceptual design subject to further modifications and thruster sizing specification, once possible DP system power is identified. Since the WCFDI demands of the DP system influence component location and protection, to meet DP3 redundancy issues discussed, a frequency domain approach is considered appropriate given significant demands of time dependent investigations regarding:

The governing motion response equations in each case are presented as a single generalized equation. The Kronecker delta functions permit recovery of a specific degree of freedom. Throughout m denotes FPSO mass, Iii the mass inertia terms, aii, bii the hydrodynamic mass or inertia, fluid damping coefficients, cii the hydrostatic restoring terms and si, si and  si the ith degree of freedom displacement, velocity and acceleration. 2.1

Time domain equations

1. Thruster load prediction is assumed through unknown DP system PID controller of Equation (1b); 2. Generation of sufficient number of ‘operation location’ dependent wave profile realizations to generate representative structural response behavior (Hearn et al. 1988); 3. Sufficient frequency domain calculations of FPSO hydrodynamics to provide equivalent Cummins time domain descriptions (Hearn et al. 1988).

Various nonlinear horizontal plane based time domain analysis exist (Nienhuis 1986, Chandrasekaran & Jain 2002). A simple DP vessel formulation is given by (Journée & Massie 2001):

(m + aii ) si (t )δ ii( ) + (Iiiii

(

)

aiiii )  si (t )δ iii( ) + bii si (t )

( ) + m + a jj s j (t )sk (t )δ ijijk + biiV si (t ) | si (t ) |

+ (ai

i −4

− ai

5i − 5

) si − 4 (t )si −5 (t )δ iii( 4 ) = FiW (t )

+ FiC (t ) + FiWD (t ) + Fi R + Ti (t ) : i = 1, 2 & 6

(1a)

2.2

subject to the Kronecker deltas satisfying:

{ { {

0 if i 6 1, i 1 or 2

δ ii( ) ( ) δ ijk =

δ iii( 4 ) =

δ ii( ) =

1, if j 12 211 & k 0 otherwise

{

The FPSO’s ability to weathervane and station keep would be ensured by the installed clusters of similar thrusters illustrated in Figure 2. The reference coordinate system origin and FPSO centre of gravity coincide. The required six equations of motion are recoverable from the generalized formulation:

0, if i 1 or 2 1, if i 6 6, or ijk = 612

0, if ≠ 6 1, if i = 6

msiδ ii( )

Here biiV is the quadratic viscous drag load coefficient, FiW is the wind load, FiC is the current load, FiWD is the slow-drift (not first-order) wave excitation, Fi R is the riser load and the DP thruster influences Ti satisfy: Ti +

i

I ij sjδ iij(

FEi

∑

)

)

ciijj s jδ ij( j = 3, 4,5 12 6 ( ) δ iik

FTk

)

: i 1, 2, …, 6. (2)

The Kronecker delta functions are now defined as:

{ { { {

0, if i = 4, 5, 6 1,, if i 1, 2, 3 0, if i 1 2 6 d ll j value u s δ ij( ) = 1, if i 3, 4, 5 and d indicated i di d j values 0, if i j δ ij(3) = and n i = 4, 5 & 6 1, if i j 0, if i ≠ k δ iik( 4 ) = and k = 1, 2 & 6. 1, if i k

δ ii( ) =

The τi and Ki are the time constants and gain characteristics of the PID controller. The responses Xi satisfy:

6

bij s j

j = 4,5,6

t dT Ti = − K iP X i − K iI ∫ X i dt − K iD X i : i = 1, 2, 6 0 dt

s1 cos( s6 ) s2 sin( s6 ) s1 sin( s6 ) + s2 cos( s6 ) s6

6

∑ (aiijj s j j =1

+

(1b)

X 1 X 2 X

Frequency domain equations

(1c)

Clearly δ ii( ) and δ ij( ) introduce FPSO mass or inertial terms according to recovery of a

Solutions of time domain moored structures (Hearn & Thomas 1991) are more complex

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Figure 2.

Force and moment positive sign convention and designation according to position on FPSO.

translational or rotational degree of freedom equation. δ ij( ) ensures inclusion of vertical plane only related hydrostatic restoration and δ ik( ) introduces the unknown global thruster loads required to maintain zero surge, sway and yaw motions. The excitation load is: FEi FI + FD with FI, FD corresponding to resolved incident, diffraction wave loads in ith direction; FTk is a thruster force for k = 1, 2 or yaw moment for k = 6. Thus Equation (2) is solved subject to: Case A—no thruster action to yield non DP controlled responses; Case B—subject to thruster action with vertical plane motions only. In each case we have six equations and six unknowns. That is for Case A Equation (2) becomes:

For Case B the formulation is: a13 a14 a15 ⎡ m + a11 a12 ⎢ a24 a25 ⎢ a21 m + a22 a23 ⎢ a32 m + a33 a34 a35 ⎢ a31 ⎢ a a42 a43 I 44 a44 I 45 a45 4 ⎢ 41 ⎢ a a52 a53 I554 + a54 I555 + a55 ⎢ 51 ⎢⎣ a61 a62 a63 I 664 + a64 I 65 + a65 ⎡ b11 ⎢ a16 ⎤ ⎡0 ⎤ ⎢b21 ⎥ ⎢0 ⎥ ⎢ a26 ⎥ ⎢ 3 s ⎥ ⎢ b31 a36 ⎥ 3 ⎥+ ⎢ s4 ⎥ b I 46 + a46 ⎥ ⎢⎢ 4 41 s5 ⎥ ⎢ I56 + a56 ⎥ ⎢ ⎢b I 66 + a66 ⎥⎦ ⎢⎣0 ⎥⎦ ⎢ 51 ⎢⎣ b61 6

a13 a14 a15 ⎡ m + a11 a12 ⎢ a m + a a a a 21 22 23 24 25 ⎢ ⎢ a31 a32 m + a33 a34 a35 ⎢ a42 a43 I 44 a44 I 45 a45 4 ⎢ a41 ⎢ a a a I554 + a54 I555 + a55 52 53 ⎢ 51 a a a I ⎢⎣ 61 62 63 664 + a64 I 65 + a65 a16 ⎤ a26 ⎥⎥ a36 ⎥ ⎥ I 46 + a46 ⎥ I56 + a56 ⎥ ⎥ I 66 + a66 ⎥⎦ ⎡0 ⎢ ⎢0 ⎢0 +⎢ ⎢0 ⎢0 ⎢ ⎢⎣0

0 0 0 0 0 0

0 0 c33 c43 c53 0

s1 ⎤ ⎡ ⎡ b ⎢ ⎥ ⎢ 11 s2 ⎥ b ⎢ ⎢ 21 ⎢ ⎥ ⎢ s 3 ⎢ ⎥ + ⎢ b31 ⎢ s ⎥ ⎢b ⎢ 4 ⎥ ⎢ 41 ⎢ s5 ⎥ ⎢ b51 ⎢ ⎥ ⎢b s6 ⎥⎦ ⎣ 61 ⎢⎣ 0 0 c34 c44 c54 0

0 0 c35 c45 c55 0

b12 b22 b32 b42 b52 b62

b13 b23 b33 b43 b53 b63

b14 b24 b34 b44 b54 b64

b15 b25 b35 b45 b55 b65

⎡0 ⎢ ⎢0 ⎢ ⎢0 +⎢ ⎢0 ⎢0 ⎢ ⎢⎣0

s b16 ⎤ ⎡ 1 ⎤ ⎢ ⎥ b26 ⎥⎥ ⎢ s2 ⎥ ⎢ ⎥ b36 ⎥ ⎢ s3 ⎥ ⎥⎢ ⎥ b46 ⎥ s4 ⎢ ⎥ b56 ⎥ ⎢ s ⎥ ⎥⎢ 5⎥ b66 ⎥⎦  ⎢⎣ s6 ⎥⎦

b12 b13 b14 b15 b16 ⎤ ⎥ ⎡0 ⎤ b22 b23 b24 2 b25 2 b226 ⎥ ⎥ ⎢0 ⎥ b332 b33 3 b334 b35 3 b336 ⎥ ⎢ s3 ⎥ ⎥ ⎢ s ⎥ b442 b43 4 b444 b45 b446 ⎥ ⎢ 4 ⎥ s ⎥⎢ 5⎥ b552 b53 b554 b55 5 b556 ⎢ ⎥ ⎣0 ⎥⎦ ⎥ b662 b63 6 b664 b65 6 b666 ⎦

0 0⎤ ⎡ F 1 + FT1 ⎤ ⎥ 0 0 0 0 0 ⎥ ⎡0 ⎤ ⎢ E2 2⎥ ⎥ ⎢0 ⎥ ⎢ FE + FT ⎥ 0 c33 c34 c35 0 ⎥ ⎢ s ⎥ ⎢ F 3 ⎥ ⋅ ⎢ 3 ⎥ = ⎢ E4 ⎥ 0 c43 c44 c45 0 ⎥⎥ ⎢ s4 ⎥ ⎢ FE ⎥ 5 s ⎥ ⎥ ⎢ 5 ⎥ ⎢ FE 0 c53 c54 c 0 54 55 ⎥ ⎢⎣0 ⎥⎦ ⎢⎣ FE6 + FT6 ⎥⎦ 0 0 0 0 0 ⎥⎦ 0 0

0

(4a)

⎡ FE1

⎤ s 0⎤ ⎡ 1 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢F 2 ⎥ 0 ⎥⎥ ⎢ s2 ⎥ ⎢ E ⎥ ⎢ ⎥ 3 0 ⎥ ⎢ s3 ⎥ ⎢ FE ⎥ ⎥⎢ ⎥=⎢ 4⎥ 0 ⎥ s4 ⎢ F ⎥ E ⎢ ⎥ 0 ⎥ ⎢ s ⎥ ⎢⎢ 5 ⎥⎥ ⎥ ⎢ 5 ⎥ FE ⎢ ⎥ 0 ⎥⎦ s ⎣⎢ 6 ⎦⎥ ⎢ F 6 ⎥ ⎣ E⎦

Throughout sj FEj FTj

(s + is ) e , (F + iFF ) e (F + iF ) e Re j

Im j

− iω t

jRe E

jI E

iω t

jRe T

jI T

iω t

and

for incident wave frequency ω [rad/s] and hence solution of (4a) assumes the form:

(3)

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subject to:

⎡ −ω 2 13 − iω b13 −ω 2a14 − iω b114 1 ⎢ 2 −ω 2a24 − iω b224 ⎢ −ω 23 − iω b23 2 ⎢ −ω ( m + a33 ) − iω b33 + c33 − ω 2a34 − iω b34 + c34 ⎢ −ω 2a − iω b + c −ω 2( I 44 + a44 ) −iiω b44 c44 43 43 ⎢ 2 43 −ω 2 ( I54 a54 ) − iω b54 + c54 ⎢ −ω a53 iω b53 c53 ⎢ −ω 2a −ω 2 ( 64 + 64 ) − iω b664 63 iω b63 ⎣ −ω 2 15 − iω b115 −ω 2 25 − iω b225 −ω 2 35 − iω b35 3 + c35 −ω 2 ( I 45 + 45 ) − iω b445 + c45 −ω 2 ( I55 + a55 ) − iω b55 + c55 −ω 2 ( 65 65 ) − iω b65

−1 0 0 0 0 0

0 −1 0 0 0 0

|

fwd

|

Faf2ft =

⎡s ⎤ 0 ⎤⎢ 3 ⎥ ⎥ s 0 ⎥⎢ 4 ⎥ ⎢ ⎥ 0 ⎥ ⎢ s5 ⎥ 1 ⎥ 0 ⎥ ⎢ FT ⎥ ⎢ 0 ⎥ ⎢ FT2 ⎥⎥ −1⎥⎦ ⎢ 6 ⎥ ⎣ FT ⎦

Faf1ft =

sj from (3) and FTj from (4b) will be used to estimate average power requirements over a wave period.

(5d)

Lf

(F FT1

2

( Fa2ft )2

2 2 ( F fw d)

2 FT1

(6a)

aft

aft

ˆ Transverse seffective = î sLongitudinal + Js fwd fwd fwd effective Longitudinal ˆ Transverse saft = î saft + Js aft

Figure 2 indicates two distinct clusters of thrusters. The 3 aft azimuth thrusters are 360° steerable, whereas the 3 forward thrusters are steerable and retractable. The FPSO technical specification (Houlder 2009) requires each thruster to have similar capability. Seeking the individual forces and torque attributable to individual thrusters within a cluster, necessitates making various assumptions and those investigated led to singular formulations that ignored controller influences. Hence equivalent fore and aft global cluster forces and torque are formulated as follows:

F f2wdd L f

Fa2ft La

F fwdd ⋅ seffective fwd effective Fafft ⋅ saft f

aft

FT6

La

FT6

1 ˆ 2 F fwd = î F fwd + JF fwd 1 ˆ F = î F + JF 2

IDENTIFICATION OF FORE AND AFT THRUSTER CLUSTER REQUIREMENTS

2 Fa2ft + F fw d

Lf

Using vector notation implicit in Figure 2, we have:

(4b)

FT2

(5c)

1 with first part of (5a) providing F fw d. The forward and aft thruster power required is simply determined from the vector dot cross product of resultant force and effective velocity in direction of the resultant force (at each location), that is:

Pafft

1 Fa1ft + F fw d

|

FT2 L f − FT6

2 F fw d =

Pfwd

FT1

aft

Solving complex variable based Equations (5a) subject to (5c) analytically yields:

⎡ FE1 ⎤ ⎢ 2⎥ ⎢ FE ⎥ ⎢ 3⎥ ⎢F ⎥ =⎢ E⎥ 4 ⎢ FE ⎥ ⎢F 5 ⎥ ⎢ E⎥ ⎢⎣ FE6 ⎥⎦

3

|

(6b)

Since each cluster centroid lies on FPSO centreline: sLongitudinal s1 & sTransverse s2 + L f s fwd fwd Longitudinal t Transverse saft   = s1 & saftf = s2 − La s6 f

(6c)

and therefore: Pfwd = F fwd ⋅ seffective fwd 1 ˆ 2 ⋅ îsLongitudinal + Js ˆ Transverse = îF fwd + JF fwd fwd fwd 1 2 ˆ = îF + JF ⋅ îs + Jˆ[ s + L s ]

( (

fwd

fwd

)( )(

1

f 6

2

1  2 ∴ Pfwd = F fwd s1 + F fwd ⎣⎡ s2 + L f s6 ⎦⎤

(5a)

Fa2ft La

)

) (6d)

whereas: effective Paft = Faft ⋅ saft 1 ˆ Transverse ˆ 2 ⋅ î sLongitudinal + Js = î Faft + JF aft aft aft 1 2 ˆ = î F + JF ⋅ î s + Jˆ[ s + L s ]

or, FT1

Faft cosθ a + F ffwd wd cosθ f

FT2 FT6

F fwd L f siinθ f

Faft sin i θ a + F ffwd wd sinθ f aft

( (

(5b)

aft

aft

)( )(

1

2

1 2 ∴ Paft = Faft s1 + Faft [s2 + La s6 ]

a sinθ a

a 6

)

) (6e)

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FPSO by careful mirroring of input half produced very acceptable radiation cross terms. Analysis of other geometries and comparison of hydrodynamic coefficients increased our confidence that software was being applied correctly. Permitted space prevents presentation of hydrodynamic data. The Aqwa® motion responses predicted quite large rotational responses. Therefore Equation (3) was solved independently using the MathWorks MATLAB® matrix inversion capability. Initially one solved Equation (3) as formulated, six complex simultaneous equations. Thereafter, an equivalent set of twelve real simultaneous equations were solved. The solutions were identical. However, there was little agreement with the Aqwa® predicted rotational responses. Similarly, Equation (4b) was solved as six complex simultaneous equations using the same routine. Thereafter, the equivalent 12 by 12 real coefficient matrix and the corresponding solution vector (in real form) were processed to estimate the excitation loads. These agreed with the loads used to solve the complex form of Equation (4b). Hence estimation of the thruster global loads at the centre of gravity and the initial motions has been generated independently of Aqwa® to permit average power estimates from Equations (6).

Each force component and velocity has the form: F = (F iiF s iω s ≡ ω ( s

) exp( −iωt ), is i ) exp( −iωt ).

(6f)

The average power over a wave period requires consideration of: T

1 Re{F } T ∫0

{s}dt

subject to: Re{ }Re{ } = ω {F Re s Im cos 2 (ωt ) − F Im s Re sin2 (ωt ) + (F Im s Im − F Re s Re )sin(ωt))cos( )c (

)}.

Hence: Pfwd =

{

ω Im Re ⎤ ⎡ 2 Re Im ⎡F 1Re s Im − F f1wd s1 ⎦ + ⎣F fwd s2 L f s6Im 2 ⎣ fwd 2 Im Re − F fwd ( s + L f s6Re )⎤⎦

(

}

)

(7a) and Pafft =

{

ω ⎡F 1Re s Im − Fa1ftIm s1Re ⎤ + ⎡Faf2ftRe s ⎦ ⎣ 2 ⎣ afft 1 2Im ⎤ − Fafft − ⎦ .

(

)}

(

La s

4.2 Summarised motion responses, with and without DP influences

)

Here our observations concerning the various predicted motion responses are summarised: (7b)

4

1. Surge response—dominant and comparable at wave headings of 0° and 180° as expected. Peak values reduced to zero as wave heading increased to 90° and thereafter increases as wave heading increases to 180°; 2. Sway—dominant at 90° as expected, peak value increases as heading varies from 0° to 90° and between 90° and 180° decreases in similar fashion to earlier increases; 3. Yaw response—for wave headings of 0°, 90° and 180° responses are totally insignificant, whereas the peak value for 60°, 120° and 150° is of the order of 0.012 rad/s at wavelength equal to FPSO length; 4. Heave—for wave headings of 0° and 180°, peaks occur at low frequency with DP influence increasing peak heave, but exhibiting significantly reduced influence at 180°; peak at 90° without DP influence corresponds to wavelength equal to FPSO length, whereas when DP influence is included, response amplitude increases as wavelength decreases; at 30° & 150° with DP, peak is 50% greater at shortest wavelength but for most other wavelengths is very similar; for

PRESENTATION AND DISCUSSION OF PREDICTIONS

The novelty of the suggested method and our first use of ANSYS Aqwa® made the authors very cautious in their interrogation of generated predictions. Various cross checks were carried out to ensure that intermediate results were acceptable. Each will be discussed in turn prior to presenting any predictions. 4.1

Validation and verification audits

Hydrodynamic radiation cross terms should be identical (within numerical resolution) and wave excitation loads satisfy Haskind’s relationship; proofs are available (Odabasi & Hearn 1977). The processing of the initial geometry permitted execution of different stages, but led to radiation cross terms distinctly unequal and of unacceptable magnitude. Haskind’s relationship is not checkable within Aqwa®. Generation of port-starboard image of

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heading of 60° there is no difference between with & without DP influence; at 120° variation with wavelength is similar in trend to 60° with DP presence marginally reducing response in region of wavelength equal to FPSO length; 5. Roll response—DP has negligible influence at 0° &180°; as wave direction changes from head sea of 180° to a heading of 120°, the peak with DP influences increases and the peak occurs at a longer wavelength; as heading continues to change towards following waves, the peak reduces and occurs at shorter wavelengths; 6. Pitch response—for 180° & 0° responses are comparable with and without DP thruster load; for other headings the pitch peak occurs at very long wavelengths with DP increasing amplitude significantly. 4.3

Figure 5. Pitch responses (rad/m) with and without DP influences.

Sample motion responses with and without DP

Figures 3 to 6 provide motion responses per unit wave amplitude against wave frequency (rad/sec) for a wave heading of 150°. These figures illustrate in part points summarised regarding DP influences.

Figure 6. Roll responses (rad/m) with and without DP influences.

4.4

Thruster cluster power predictions

Having an open moonpool will have some impact on the wave loading of the FPSO. Ideally, given the weathervaning nature of FPSO operation, the power predictions should be calibrated for every few degrees over a range not exceeding ±20°. Here we shall present results for just 2 of the many headings considered; namely the head seas of 150° and 180°. Figure 7 indicates the maximum power demanded by a thruster cluster, irrespective of location. In reality, at a wave heading of 180°, the peak power is essentially 48 kW for the aft cluster, with the forward cluster demanding 22 kW. For the FPSO operation with DP in sea state 8, these figures would translate into 9.6MW and 4.3MW at the aft and forward clusters. This is almost manageable with individual 3MW thruster unit ratings, although normal operation does not correspond to this upper limiting sea state. As the waves move away from a 180° heading, more power is anticipated. It is overly pessimistic to convert the peak 150° aft rating of 534 kW and the forward demand of 764 kW to the limiting sea state, as weathervaning would maintain much smaller variations around the head sea wave heading of 180°.

Figure 3. Horizontal responses (m/m or rad/m) without DP influence.

Figure 4. Heave responses (m/m) with and without DP influences.

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levels, have been considered. There is also the need to recognize practical operation of each thruster cluster. That is, effective and efficient combinations of thrusters within a cluster must recognize the practical angles of orientation to ensure minimum negative hull-thruster and thruster-thruster interaction impact. The frequency method proffered is to establish likely cluster and thence individual thruster power at the initial stage. Thereafter time domain analysis is required to check out operational survivability in the more complex and demanding environmental conditions; as set out in part (c) of earlier plausible scenario addressing the worst case failure mode. Time domain analysis would also be used to test PID characteristics. More in-depth studies are required prior to any claim that this novel approach can become an acceptable initial design tool for purely DP operated FPSOs.

Figure 7. Average power variation with wave frequency.

A more refined and restricted wave heading variation from 180° is required to provide conclusive evidence that the proposed frequency based estimation method could be included in initial design. The method has illustrated that the power demands are of a practical order of magnitude. Closure of the moonpool and existence of a shuttle tanker are to be investigated with the proposed refined wave headings. 5

REFERENCES ABS, 2014. Guide for Dynamic Positioning Systems: 9. Chandrasekaran, S. & Jain, A.K. 2002. Dynamic behavior of square & triangular offshore tensions leg platforms under regular wave loads, Ocean Engineering 29: 279–313. Cummins, W.E. 1962. The impulse response function and ship motions, Schiffstechnik, 9: 101–109. DNV, 2013. Rules for Classification of Ships: 17. Faltinsen, O.M 2005. Hydrodynamics of High-Speed Marine Vehicles. Cambridge University Press: 237. Havelock, T. 1955. Waves due to a floating sphere making periodic heaving oscillations, Proceedings of the Royal Society of London, Series A, Vol.231: 1–7. Hearn, G.E., Lau, S.M. & Tong, K.C. 1988. Wave drift damping influences upon the time domain simulations of moored structures, OTC 5632, Vol.1: 155–167. Hearn, G.E. & Thomas, D.O. 1991. The influence of practical time integration schemes upon the dynamic mooring line analysis, OTC 6604, Vol.1: 397–409. Hearn, G.E. & Tong, K.C. 1987. Wave drift damping characteristics of different offshore structures, Floating Structures and Offshore Operations Workshop, edited by Van Oortmersen, Elsevier Press: 205–218. Houlder Ltd 2009. Technical Specification for a Dynamic Positioning Production Unit: 21, confidential document. How, B.V.E., Ge, S.S. & Choo, Y.S. 2010. Dynamic load positioning for subsea via adaptive neural control, IEEE Journal of Oceanic Engineering, Vol.35, No.2: 366–375. IMO MSC Circular 645, 1994. Guidelines for Vessels with Dynamic Positioning Systems. Journée, J.M.J. & Massie, W.W. 2001. Offshore hydromechanics. Delft University of Technology: 10.22–10.23. Nienhuis, U. 1986. Simulations of Low Frequency Motions of Dynamically Positioned Offshore Structures, RINA Spring Meeting, Paper 7: 48. Odabasi, A.Y. & Hearn, G.E. 1978. Seakeeping theories – what is the choice? Trans. NECIES, 94: 53–84.

CONCLUSIONS

Using ANSYS Aqwa® for the first time requires prudence to establish some degree of verification and validation. The non-symmetry initially discovered in the radiation cross terms was overcome by not allowing automatic generation of the FPSO port-starboard image. Further critical assessments involved comparing Aqwa® predictions of hydrodynamic coefficients for: a. Havelock’s classical heaving hemisphere (Havelock 1955); b. Other geometries investigated and presented by Faltinsen (2005). The six motion responses required from solving Equation (3) was also undertaken independently of Aqwa®. Equation (4b) required independent solution by virtue of the novelty of the proposed formulation. In each case the responses from Equations (3) & (4b) were undertaken by treating each as either 6 coupled complex simultaneous equations or 12 real simultaneous equations and solutions checked as described in Section 4. Given the most stringent wave environment condition is DP production in sea-state 8, with waves in the range 9.0 to 14.0 m, the thruster power deduced per unit wave amplitude should be at least 200 times larger for this condition. The novel method has been shown to predict power demands of a practical level. However, no control system influence has been addressed and only delivered power levels, rather than installed

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

CFD based computation of bow impact loads for buckling assessment J. Oberhagemann, M. Radon & H. von Selle DNV GL SE Maritime, Hamburg, Germany

D.K. Lee Daewoo Shipbuilding and Marine Engineering Co. Ltd., Seoul, South Korea

ABSTRACT: Bow slamming impact pressure forces acting on larger areas are of primary interest for dimensioning the larger supporting bow structures of ships. Although computational fluid dynamics methods are able to reliably predict bow impact loads for a given wave situation, large uncertainty is related to the identification of relevant scenarios. These need to assure sufficient conservatism, yet shall not lead to excessively large loads. This paper compares application of conditioned wave sequences with more traditional equivalent design waves. A rough estimate of long-term expected maximum bow impact loads is made based on available simulation data. Further, structural stresses resulting from slamming pressure loads are compared with rule based bow pressure loads. Rule based results are in general exceeding the stresses found from simulations. The work was performed as part of a joint development project of DSME and DNV GL. 1

INTRODUCTION

Stokes Equations (RANSE) combined with a multiphase Eulerian approach for free-surface flows. Validation of the numerical method showed good agreement with ship model experiments in regular and irregular waves. Generation of critical wave events was based on two different strategies; more traditional equivalent regular design waves and conditioned wave sequences. The selection of the final resulting load scenario was based on maximum lateral hydrodynamic bow force. Subsequent global strength FE analyses used the computed loads to investigate the stress distribution and buckling behavior of the fore ship area. Comparison was made with the simplified slamming load method according to the guideline procedure, GL (2011). A comparative investigation used a large set of available simulation data of the same ship in random irregular waves to provide an estimate of the long-term expected hydrodynamic bow impact load, and to identify a representative wave scenario. This estimate was used to check the plausibility of impact forces resulting from simulations in conditioned wave sequences and regular design waves.

Bow slamming impacts are relevant for the dimensioning of the fore ship structure. During such events, hydrodynamic impact pressures add on the quasi-static nonlinear wave pressures that act on the flared bow regions, potentially resulting in strongly nonlinear loads of significant magnitude. The locally acting peak pressure is not of primary interest for dimensioning the larger supporting structures, such as transverse frames and stringer decks. Instead, integrated pressure forces simultaneously acting on larger areas become the key parameter. The spatial and temporal distribution of such pressure forces is difficult to linearize, so the screening process to identify relevant load cases is not a straightforward procedure. In global structural strength assessments according to the guideline procedure of DNV GL, slamming loads are included, but determined in a pragmatic way to allow easy applicability and short computation times, GL (2011). The standard procedure imposes rule design pressures on an area in the bow in such a way that the resulting vertical bending moment exactly satisfies the rule value. The present study explores alternatives that find relevant pressure loads from first-principle computations for a case study of a 14,000 TEU container carrier. The applied Computational Fluid Dynamics (CFD) method relied on a finite volume formulation of the Reynolds-Averaged Navier-

2

RANS CFD METHOD

The commercial code COMET is used for the flow simulations. COMET solves the coupled Navier-Stokes equations of mass and momentum

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conservation in their integral formulation on a discrete number of three-dimensional control volumes (cells) in the time domain with an iterative scheme. Discretisation in space and time is of second order, with a small amount of first order added to the momentum transport equation to improve numerical stability. For the given freesurface flow application, the fluids involved are air and water. They are considered viscous, isothermal and incompressible. A Eulerian multiphase approach is used for the free-surface flow. An additional scalar transport equation solves for the volume fraction of one of the fluids involved, in this case water. Fluid properties such as density are set according to the properties of the fluids and their volume fraction, so the flow equations are solved for an effective fluid. This approach allows to compute arbitrarily shaped free surfaces including breaking waves. The discretisation scheme used for this transport equation is HRIC (High Resolution Interface Capturing scheme) which blends first and second order schemes to avoid mixing of fluids and retain a sharp interface. The CFD solver is coupled to a nonlinear solver for the six degrees of freedom of rigidbody motions. The solvers are iteratively called in each time step until the coupled problem is converged. Under-relaxation stabilizes the iterative scheme. 2.1

Figure 1. Example short-term cumulative distribution of non-dimensional vertical bending moment, comparison of model experiment and computation; source: Oberhagemann et al. (2012).

Validation

Figure 2. Configuration of finite volume meshes for CFD simulations with refined regions around free surface and ship, and grid stretching towards outlet boundaries.

Various applications to wave slamming impacts demonstrated the ability of the numerical method to detect the occurrence of slamming and replicate measured impact pressures, e.g. el Moctar et al. (2005), Oberhagemann et al. (2008), el Moctar et al. (2011). Favorable agreement of integrated impact forces has been demonstrated. Further, a benchmark study of Drummen & Holtmann (2014) suggested that CFD-based methods are superior to other hydrodynamic methods in accurately computing bow slamming forces of a ferry advancing in a regular head wave. The above applications validated the numerical method with respect to slamming and impact loads using experiments in regular waves. El Moctar et al. (2011) also achieved good agreement of computed vertical bending moments with model test data of a 10,000 TEU containership in irregular waves. Oberhagemann et al. (2012) compared the vertical bending moment obtained by the coupled numerical method with model scale experiment results in irregular head waves for a cruise vessel, using a reconstruction of the measured wave elevation. Evaluation with rainflow counting provided cumulative distributions of vertical bending moments, Figure 1, indicating favorable agreement.

2.2

Numerical model

For the CFD finite volume grids, only half of the ship (the starboard side) and the fluid domain were modelled due to a restriction to head sea conditions. The volume grids were unstructured hexahedraldominant with prismatic cell boundary layers around the hull. Grid refinements were made close to the hull and at the free surface, whereas cell sizes increased far away from the ship. Cell stretching towards the rear boundary helped to numerically dampen waves and disturbances generated by the ship and ensure uniform outflow. Figure 2 shows an isometric view of the grid configuration with 930,000 cells, flow direction is from right to left. 3

SELECTION OF WAVE SCENARIOS

To allow application of CFD to practical engineering problems, the high computational effort related to transient computations necessitates a restriction of simulations to short sequences. Preliminary

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analyses with fast and efficient tools can be employed to identify situations which correspond to a given target probability level and are used for further analysis with CFD simulations. 3.1

Equivalent Design Waves

Often, simulations use regular harmonic waves. Their parameters are established based on the equivalent design wave concept, see el Moctar et al. (2005) and Kim (2011) for example applications. A simplified criterion is established which is assumed to correspond to slamming excitation. Linear statistical analysis, based on spectral moments and the narrow-band assumption, of transfer functions of this simplified criterion determines its expected long-term maximum. Then, a harmonic wave is scaled in amplitude such that the ship response to this wave corresponds to the previously found long-term maximum according to linear theory, i.e. the magnitude of the ship response to the regular wave equals the long-term expected maximum value of the response. Here, two different criteria were tested. First, the Vertical Bending Moment amidships (VBM); second, the normal (to the hull surface) component of the relative vertical velocity between the hull and the water free surface. A linear potential theory seakeeping analysis using a three-dimensional panel method provided the required transfer functions of motions and internal loads. Statistical analysis provided long-term expected maxima of the above criteria. Then, two Equivalent regular Design Waves (EDWs) were generated: EDW 1, equivalent to the long-term maximum vertical bending moment amidships; wave length λ = 338 m, wave height HW = 17.4 m. This wave was generated for later comparison with the conditioned wave sequences which were also calibrated with the long-term value of VBM. EDW 2, equivalent to the maximum normal relative vertical velocity between the bow and the free water surface; wave length λ = 193 m, wave height HW = 19.3 m. A third regular wave, EDW 3 (wave length λ = 266 m and wave height HW = 18.4 m), was tested. The parameters of this wave are chosen such that it represents an intermediate wave in between EDW 1 and EDW 2. Strictly speaking, this wave is not an equivalent design wave. The wave lengths of the first two EDWs were chosen such that they correspond to the peak of the linear transfer functions, while the wave heights were determined in a way that the linear responses to the design waves equal the long-term expected maxima of these responses. Additional constraints applied to assure the ratio of wave height to wave length did not exceed a steepness

Figure 3. Simulation snapshots of bow slamming in EDW 3 (top), and in a conditioned wave sequence (TZ = 11.5s, bottom). The edge of the force integration area in the bow region is marked with a black line.

limit, HW/λ ≤ 1/10. Figure 3 shows a bow slamming event from the CFD simulation for EDW 3. 3.2 Conditioned wave sequences Wave sequences of short duration were generated based on the concept of the Most Likely Response Wave (MLRW), Adegeest et al. (1998) and Dietz (2004). The underlying idea is a spectral moment transformation of the discretized linear response spectrum, resulting in a wave sequence that causes a given magnitude of the response at a given time instant. The nonlinear simulations in the wave sequences provide a nonlinear correction of the linear response, i.e. the nonlinear response peak at target time is interpreted as the corrected linear response. Here, the target response was again the linear long-term expected maximum of the vertical wave bending moments amidships. The instantaneous wave frequency during target time was varied in order to identify relevant wave periods for slamming. For this purpose, wave sequences were generated for Pierson-Moscowitz wave spectra with zero-upcrossing periods TZ = 8.5s, TZ = 9.5s, TZ = 10.5s, TZ = 11.5s, TZ = 12.5s and TZ = 13.5s. Figure 4 shows example time series of vertical bending moment predicted with the linear potential theory method, and the nonlinear vertical bending moment computed with the CFD method in the wave sequences. For short zero-upcrossing periods, differences were most pronounced, while linear and nonlinear time series became more similar with increasing zero-upcrossing period. This indicates that

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Figure 4. Time series of non-dimensional vertical bending moment in wave sequences with periods TZ = 8.5s (top) and TZ = 11.5s (bottom); linear (dashed lines) and nonlinear CFD (solid lines); sagging is positive. Figure 5. Time series of integrated hydrodynamic bow forces from simulations in regular design waves (top, time is made non-dimensional with encounter period Te) and conditioned wave sequences (bottom).

nonlinearities were more significant for the shorter periods. Interestingly, the largest slamming impact during these simulations was found for the simulation with zero-upcrossing periods TZ = 8.5s, during one of the first wave encounters approximately 25 s before target time, Figure 4 (top). The slamming event is characterized by a sharp spike of the vertical bending moment. Nevertheless, this event must not be considered for further analysis because the wave sequence was calibrated to the peak at target time only.

pressures were moderate for all EDWs and significantly lower than pressures found in previous investigations, e.g. Ley et al. 2013. While bow slamming impact forces were evaluated from design wave simulations after the simulation reached periodicity, the forces from conditioned wave sequences were found around the target time t0 only, Figure 5. The largest bow forces occurred for the largest zero-upcrossing periods, but were not necessarily related to the largest local pressure forces. Instead, weakly nonlinear pressure forces due to the incident wave and the bow immersion were the dominating contribution to the bow forces for longer periods. The largest pressures were observed on the lower part of the hull, while pressures on the largely flared bow regions were moderate. For the longer wave periods, the impact peak provided only a small addition to the large force impulse of the weakly nonlinear pressure forces. This contribution increased for shorter periods, and clearly dominated for the shortest zero-upcrossing period TZ = 8.5s.

3.3 Impact loads An area of 620 m2 in the bow region was selected for impact force integration. The integration region extended from the waterline at scantling draft to the upper deck, and from x/LPP = 0.95 to the forward end, where LPP is the length between perpendiculars. The longer wave EDW 1 resulted in highest total forces acting on the force integration area, but with a dominating contribution from weakly nonlinear pressures due to incident wave and bow immersion. The weakly nonlinear pressures formed the secondary peak in the time series of bow force, Figure 5, while the first peaks were caused by more dynamic impact pressures. Although the dynamic pressures from EDW 2 were larger, the total force peak was smaller. The total hydrodynamic bow force resulting from wave EDW 3 was comparable to that of EDW 1, but had a larger contribution coming from dynamic pressures. However, peak

4

PLAUSIBILITY OF IMPACT LOADS

Approximately 30 hrs of RANS CFD simulations in random irregular waves were available for this ship including a large number of slamming events,

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Figure 6. Exemplary time series of vertical bending moment from simulations in random irregular waves.

Figure 7. Cumulative distribution of non-dimensional lateral bow forces from statistical evaluation of simulations in short-term sea state conditions.

see Figure 6. These simulations were performed for various severe sea state conditions in long-crested head seas and used a flexible model of the ship, so the occurrence of whipping vibration helped to detect slamming events. The simulations focused on vertical bending moments amidships, therefore no time series of bow pressures or bow impact forces were monitored during these simulations. Instead, only sectional forces were available that included mass inertia forces and hydrodynamic forces acting on the underwater part of the hull in addition to the more relevant forces above the waterline. Nevertheless, these data were evaluated in order to obtain estimates of occurrence and severity of bow slamming events. Note that this evaluation only intends to indicate the plausibility of the severity of slamming events found with the simulations in regular waves and conditioned wave sequences. For this purpose, short-term distribution functions of peaks of the bow force were evaluated from the time series, for each sea state condition independently. Accounting for the expected longterm durations of the sea state conditions, the short-term distributions are then summed to a cumulative distribution of bow forces, conditional on the investigated sea state conditions. In a very rough estimate, it is now assumed that the sea states investigated in simulations cover, on the average, every second exceedance of the longterm expected maximum. That is, the expected number of exceedances of the long-term expected maximum in the investigated sea states is N = 0.5, so the long-term expected maximum bow force, FyLT, can be extracted from the cumulative distribution at N = 0.5, see Figure 7. In a next step, events from the time series were identified with a peak of the bow sectional force corresponding to the long-term expected maximum. Figure 6 shows the vertical bending moment from a random irregular wave simulation, where the bow force has such a peak at t = 176s. It is easily seen that this bow impact is also reflected in the vertical bending moment amidships, with a

Figure 8. Integrated bow impact force during slamming event found from re-simulation.

peak exceeding the vertical wave bending moment according to the rules. This event was re-simulated in order to monitor the hydrodynamic forces acting on the bow flare above the still water line. The re-simulation used a rigid representation of the ship. The time series of lateral force acting in normal direction on the bow area, Figure 8, indicates the occurrence of slamming at t = 176s. The peak of approximately 30,000 kN is comparable to the hydrodynamic forces on the bow observed in the equivalent design waves and conditioned wave sequences, confirming the plausibility of slamming impact forces from these simulations. 5

STRENGTH ASSESSMENT

In global strength calculations for container ships according to Germanischer Lloyd (2011), typically, a set of 40–50 wave load cases based on two loading conditions is generated. Because slamming loads are essential for containerships with excessive bow flare and stern overhang, an additional set of load cases, so called PE load cases, are generated. These PE load cases represent a simplified application of slamming pressures in order to capture the global effect of slamming on the hull structure.

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of simulations in random irregular waves. Additionally, results of this simulation provided a large contribution of strongly nonlinear impact pressure forces to the total hydrodynamic force acting in normal direction on the bow area above the still water line, compared with the contribution from rather weakly nonlinear forces due to incident wave pressures and bow immersion. EDW 3 was amongst the cases providing the largest contribution from impact pressures to the total bow forces. The resulting transient pressure loads were transferred onto a FE model of the ship for further analysis. For each selected time step, the pressure forces acting on the fluid grid boundary faces were converted into nodal forces of the finite element model. The CFD simulations are considered to provide more realistic slamming load distributions because they account for the dynamic nature of slamming: peak pressures act for short time periods only, and the area loaded by slamming pressures passes along the ship side shell. Nevertheless, the total impulse duration is much longer than natural periods of local structures, and stress evaluation can be restricted to steady state analysis. 100 time steps from the transient CFD computation, corresponding to a single slamming event, were converted to steady-state FE load cases and analyzed. This large number of loads assured not to miss relevant loads.

Typically, the standard wave load cases as well as the PE load cases are generated with the program GL ShipLoad. 5.1

PE load cases

GL ShipLoad allows a fast generation of load cases from rule-based slamming pressures pe. These pressures are applied on bow or stern areas in a way that the resulting vertical bending moment does not exceed the rule wave bending moment in sagging, see Figure 9 for an example. For this simplified approach, bow and stern areas are divided into several horizontal strips. Load cases are generated by strip-wise adding slamming pressure pe until the required vertical bending moment is reached. The pressure loads are balanced by adjusting the acceleration factors for the weight loads. Several load cases are generated so that, for each location above the ballast waterline, the pressure pe is applied. This procedure produces slamming loads for global strength analyses in a simple but reasonable way. GL ShipLoad generates these PE loads efficiently for dimensioning of fore and aft ship areas. The evaluation is limited to permissible stresses and buckling strength only. Load cases used in the present analysis are listed in Table 1. 5.2 CFD load cases Simulation results of EDW 3 were considered representative for further buckling strength assessment because the peak of the total force was comparable to the results of the approximate long-term evaluation

5.3

Next, results of the strength assessment are presented of the global FE model, loaded with pressure and inertia forces from PE and CFD load cases, respectively. Stresses resulting from both approaches were checked against the permissible stresses (yield check) and buckling. In both cases, a usage factor U = σActual /σPermissible was determined, where U > 1 corresponded to insufficient yield or buckling strength, respectively. The most critical load cases for each finite element were evaluated separately for the standard approach and the CFD generated loads, and then compared. The comparison covers the resulting stress levels as well as the buckling criteria check. Three yield criteria were checked, namely, the maximum absolute principal stress, the maximum equivalent von Mises stress, and the shear stress. Germanischer Lloyd (2014) lists the corresponding permissible stresses and the buckling criteria. The usage factors were generally lower for the pressure loads from CFD simulations. Table 2 presents the ratio of the maximum usage factors, obtained with the CFD loads, to the maximum usage factor resulting from the simplified pressure loads. Results are given for representative locations

Figure 9. Longitudinal distribution of vertical bending moments from PE load cases. Table 1. 1 2–24 25 26–49 50–56

Strength checks

Load cases. Still water, maximum hogging Still water LC1 + waves Still water, minimum hogging Still water LC25 + waves PE load cases

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6

only. The usage factors were generally lower for the pressure loads from CFD simulations. Figure 10 shows, for the CFD loads, the yield criteria resulting in the largest usage factors U for the side shell elements. Upper index indicates the criterion giving the highest usage factors (principal, equivalent or shear stress). The lower index gives the corresponding load case number which resulted in the most severe usage factor. Table 2. checks.

Buckling strength assessment under dynamic slamming loads has a high relevance for the strength assessment of modern container ships due to their large bow flare and flat aft ship lines. The key issue is to ensure consistency of the loading and the appertaining permissible stresses—both sides are directly linked to each other and must be related to an appropriate probability level. A meaningful and advanced yet practical procedure for buckling assessment under dynamic slamming loads is important for optimum structural design of container vessels. Nevertheless, magnitudes of relevant slamming impact forces are related to large uncertainty. Existing ship accident databases do not allow reliable determination of the occurrence frequency of incidents related to slamming, so numerical and experimental investigations are the only means to determine relevant loads for design. However, the uncertainty associated to such investigations is also large, and calibration of computational procedures against full-scale data is impossible.

Usage factor ratios from yield and buckling U(CFD)/U(PE)

Area

Yield

Buckling

Shell bottom Shell on SWL Shell at 22900/BL Strg. #5 at 9970/BL, #151 Strg. #5 at 9970/BL, #155 Stringer no. 3 at 15160/BL Stringer no. 2 at 17750/BL Stringer no. 1 at 20340/BL Upper deck Frame 151 at str. 3 Frame 151 at upper deck Frame 153 at str. 5 Frame 153 at str. 3

0.56 0.9–1.0 0.9–1.0 0.95 1.33 0.56 0.7 0.77 0.2–0.6 0.56 1.48 1.21 0.68

0.64 0.75–0.9 0.87–0.97 0.96 1.34 0.62 0.72 0.78 0.13–0.52 0.56 1.00 1.00 0.53

Figure 10.

DISCUSSION

6.1

Identification of relevant slamming loads

While only fast numerical tools can be used for direct long-term simulations, these fail to reliably predict the occurrence and severity of slamming. More

Sample usage factors found from yield check with CFD loads.

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data was based on several rough assumptions. The accuracy of this estimate cannot be quantified.

sophisticated methods and experiments, on the other hand, are able to accurately provide the spatial and temporal distribution of impact forces. Still, the obtained long-term distribution of bow impact loads is related to large uncertainty because of the need to restrict investigations to short time intervals. The present study intended to explore CFD simulations as alternative to the standard approach for slamming load generation used in global strength assessments. CFD simulations in selected wave scenarios obtained transient pressure distributions, which were integrated over the bow region to identify and evaluate slamming impulses. The assessment of typical areas loaded by slamming was done visually, while the comparison of time series of impacts helped to identify the most relevant wave scenario. Wave sequences conditioned on the VBM according to the MLRW concept have been used for CFD simulations of bow slamming before, Seng (2012), Ley et al. (2013). When the effect of slamming on the VBM amidships is of interest, their application may easily be justified considering the weakly nonlinear behavior of the midship VBM, Jensen (2009). Ley et al. (2013), compared a wave sequence conditioned on the VBM with a wave sequence conditioned on the relative velocity between bow and water surface, demonstrating that slamming loads resulting from the former may be more severe. Here, however, impact loads from a regular design wave simulation were used for further strength assessment. Identification of critical events was based on maximum hydrodynamic lateral bow force. The absolute values of peak pressures were not of interest, and the relevant computed pressures were all well below the rule pressures. Two different simulation strategies were tested, equivalent regular design waves and focused wave sequences. Comparison was made with an estimate of the longterm expected maximum of the lateral bow force derived from random irregular waves. This estimated long-term value should be used with care, as it is not a reliable estimate of the longterm expected hydrodynamic lateral force acting on the bow. It is merely a reference value associated with strong uncertainty, for the following reasons. The horizontal sectional force in the bow area was used in the statistical analysis to assess the occurrence and severity of bow impacts. Its applicability for this purpose needs further investigation. The correlation between sectional force in the bow and the lateral hydrodynamic bow force was established based on a single comparative computation only. Although the correlation is expected to be strong, this should be verified. Finally, the estimation of a long-term maximum of the bow sectional force from available simulation

6.2

Comparison with rule slamming loads

In a next step, hydrodynamic pressure distributions for a selected bow impact have been transferred to a finite element model of the ship. Comparison was made with a simplified procedure that used Rule slamming pressures applied to horizontal strips. The vertical extent of the strips was chosen in a way the vertical bending moment was bound by the Rule bending moment envelope. In nearly all cases, the simplified method lead to a conservative assessment and can be considered sufficient. The more advanced method resulted in larger usage factors only for a few locations. Only one single slamming impact has been used for the strength analysis from the number of available CFD calculations. It is therefore too early to draw final conclusions. A more comprehensive investigation is planned for the future. CFD simulations may also help to improve the present rule pressure application procedure with more realistic pressure distributions, i.e. the identification of typical slamming load patterns to cause characteristic loadings of bow supporting structures such as web frames, stringer decks, etc. The present rule procedure for generation of bow pressure loads assumes slamming impacts of uniform pressure and rectangular shape. These assumptions are practical simplifications for global FEM analyses, but are rather questionable for assessments of local effects.

7

CONCLUSIONS

Hydrodynamic pressure distributions for selected bow slamming impacts have been transferred to the FE model of a large container carrier. The results from an equivalent design wave simulation were most relevant with respect to pressure distribution pattern and slamming intensity, although peak pressures were moderate. A simplified procedure based on application of rule pressures is slightly conservative in comparison to the simulation-based procedure, justifying its application for global strength analyses. Nevertheless, continuation of the present studies is planned because of the limited scope of investigation.

REFERENCES Adegeest, L.J.M., Braathen, A., & Løseth, R.M. 1998. Use of Nonlinear Sea Loads Simulations in Design of Ships. Proc. 7th Int. Symp. on Practical Design of Ships and Other Floating Structures. The Hague, pp. 53–58.

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Dietz, J.S. 2004. Application of Critical Wave Episodes for Extreme Loads of Marine Structures. Ph.D. thesis, Technical University of Denmark, Lyngby. Drummen, I. & Holtmann, M. 2014. Benchmark Study of Slamming and Whipping. Ocean Engineering 86, pp. 3–10. el Moctar, O., Brunswig, J., Brehm, A. & Schellin, T.E. 2005. Computations of Ship Motions in Waves and Slamming Loads for Fast Ships using URANS. Proc. Int. Conf. on Fast Sea Transportation, St. Petersburg. el Moctar, O., Oberhagemann, J., & Schellin, T.E. 2011. Free-Surface RANS Method for Hull Girder Springing and Whipping. SNAME Transactions (119), pp. 48–66. Germanischer Lloyd (GL) 2011. Rules and Guidelines, V-Analysis Techniques, Part 1—Hull Structural Design Analysis, Chapter 1—Guidelines for Global Strength Analysis of Container Ships. Germanischer Lloyd (GL) 2014. Rules and Guidelines, I-Ship Technology, Part 1—Seagoing Ships, Chapter 1— Hull Structures.

Jensen, J.J. 2009. Stochastic Procedures for Extreme Wave Load Predictions—Wave Bending Moment in Ships. Marine Structures 22, pp. 194–208. Kim, S.P. 2011. CFD as a Seakeeping Tool for Ship Design. Int. J. Naval Architecture and Ocean Engineering 3(1), pp. 65–71. Ley, J., Oberhagemann, J., Amian, C., Langer, M., Shigunov, V., Rathje, H. & Schellin, T.E. 2013. Green Water Loads on a Cruise Ship. Proc. 32nd ASME Int. Conf. Ocean, Offshore and Arctic Engineering. Nantes. Oberhagemann, J., el Moctar, O. & Schellin, T.E. 2008. Fluid-Structure Coupling to Assess Whipping Effects on Global Loads of a Large Container Vessel. Proc. 27th Symp. Naval Hydrodynamics, Seoul, pp. 296–314. Oberhagemann, J., Ley, J., el Moctar, O. 2012. Prediction of Ship Response Statistics in Severe Sea Conditions using RANS. Proc. 31st ASME Int. Conf. Ocean, Offshore and Arctic Engineering, Rio de Janeiro. Seng, S. 2012. Slamming and Whipping Analysis of Ships. Ph.D. thesis, Technical University of Denmark, Lyngby.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Comparison of two practical methods for seakeeping assessment of damaged ships J. Parunov, M. Ćorak & I. Gledić Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb, Croatia

ABSTRACT: The purpose of the paper is to compare two simplified methods that may be used for modeling wave-induced motion and resulting structural loads of ship damaged in collision or grounding accident. The first method is the added mass method, in which it is assumed that the mass of the flooded seawater becomes integral part of the ship mass and moves with the ship. The second approach is the lost buoyancy method, where structure of damaged tanks and all of its contents are removed from the vessel. The former method is applicable for small damages while the latter is more suited for large damage extents. Response amplitude operators of vertical motions and hull girder vertical wave bending moments at amidships are calculated by the state-of-the-art linear 3D panel hydrodynamic code and compared to those for intact ship as the effect of damage is often neglected in computation of motions and wave loads of damaged vessel. An attempt of verification of procedures is done by comparison with seakeeping experiments on damaged warship, described in available literature. 1

INTRODUCTION

location. For example, Teixeira and Guedes Soares (2010) proposed a time period of one week as the voyage duration of a damaged ship to dry-dock. They concluded that the mean extreme Vertical Wave Bending Moment (VWBM) of a Suezmax tanker is about 15% lower when the exposure time is reduced from one year in the North Atlantic to one week in European coastal areas. Although research on loads on damaged ships in waves is rare, motions of damaged ships are widely covered in the literature (e.g. Korkut et al. 2004). Application of risk-based design methods that includes structural reliability of damaged ship requires rational evaluation of all pertinent random variables, including wave loads of damaged ship (Prestileo et al. 2013) that was the motivation for some of recent studies on that subject. Thus, Folsø et al. (2008) have performed seakeeping computations on a damaged ship by the 3D linear hydrodynamic method. The damage scenarios corresponded to water ingress into the forepeak and/ or the double hull ballast tanks of the ship sailing in full load. For the case of the flooded ballast tank in the midship area, they obtained Response Amplitude Operators (RAOs) of the VWBM larger than those evaluated for the intact condition. Interesting conclusion from the paper is that keeping a bow quartering encounter angle, with the higher freeboard on the weather side, minimizes VWBM. Lee et al. (2012) applied a computational tool based on a two dimensional linear method to predict the hydrodynamic loads of damaged warship.

Rational structural design of ships should consider strength of the vessel both in intact and damage condition. Damage of merchant ship may occur due to collision with another ship, grounding or some other type of human mistake. In case of such an accident, the ship strength could be significantly reduced while still water loads may increase and could become considerable cause of the structural overloading (Luis et al. 2009, Khan and Das, 2008). Not much research has been spent on wave loads of damaged ships. The main reason is that design requirements for global wave loads on damaged ship are much lower compared to the intact condition (Hirdaris et al. 2014). Thus, the IACS Harmonized Common Structural Rules (CSR-H) (IACS, 2012), are aimed at checking the hull girder ultimate bending capacity in the damaged state using partial safety factor for wave loads of 0.67, while in the intact condition this factor reads 1.1. The reason for reduced partial safety factor in damaged condition is reduced exposure time and milder environmental conditions to be taken into account. While for intact ships the North Atlantic wave environment is usually adopted, local scatter diagrams are proposed, as applicable, for the reliability assessment of damaged ships as suggested by Luis et al. (2009). Reduced exposure time to environmental conditions after damage should also be considered before salvage to a safe

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with the ship motion. Such analysis, which is presented e.g. by Santos and Guedes Soares (2008a, 2008b), Jia and Moan (2012) and Rodrigues and Guedes Soares (2014), is outside the scope of the present paper.

They obtained larger VWBM for damaged, compared to the intact ship. The global dynamic wave induced loads calculated using 2D linear method were also compared to measurements. In head and stern quartering waves, differences between computations and measurements of global dynamic wave induced load response amplitudes were reasonable. In general, however, linear strip theory overestimated measurements for both intact and damaged ship. The analysis of wave loads on damaged ship is performed also by Downes et al. (2007) where it has be shown that the RAO peak value of VWBM increases, with increasing damage size and heel angle. It can also be seen however, that there is no significant difference between the RAOs due to the effects of damage. That study indicated that the change in global hull loading may be much smaller for tankers than for Ro-Ro ferries and cruise ships. Two practical methods for modeling wave-induced motion and resulting structural loads of damaged ship are proposed in that paper. The first method is the added mass method, in which it is assumed that the mass of the flooded seawater becomes integral part of the ship mass and moves with the ship. The second approach is the lost buoyancy method, where structure of damaged tanks and all of its contents are removed from the vessel. The former method is applicable for small damages while the latter is more suited for large damage extents. The hydrodynamic interaction between the waves and the structure of the opening remains after removing the tank from the ship hull, which needs to be modeled in the lost buoyancy method. In past studies, reviewed in preceding paragraphs, only the added mass method was employed for hydrodynamic analysis of damaged ship. Results of damaged ship seakeeping assessment using the lost buoyancy method are not available from the literature. That was the motivation for the present study, aiming to investigate what could be difference if the lost buoyancy method is used instead of the added mass method. The calculations are performed on the example of the Aframax oil tanker using Hydrostar 3D panel seakeeping software (Bureau Veritas, 2012). Damage cases are generated according to the MEPC recommendations (IMO, 2003). Two mentioned methods for seakeeping assessment of damaged ship are compared and comparison is also performed with respect to the intact ship. As there is a need for validation of methods, comparison with published and well documented experiment on damaged warship is also performed (Lee et al. 2012). It should be clarified that employed methods do not consider the possibility of motion of the floodwater inside damaged tanks and coupling

2

DESCRIPTION OF SHIP AND DAMAGE CASES

The studied ship is Aframax oil tanker with main particulars presented in Table 1. Cargo hold area is divided into 6 pairs of Cargo Tanks (CT) and 6 corresponding pairs of Water Ballast Tanks (WBT) in double bottom and side. WBTs are divided into portside and starboard tanks by center line girder in double bottom. The general arrangement of the ship is shown in Figure 1, while the hydrodynamic panel model of the intact ship is presented in Figure 2. Wetted hull surface of the intact tanker is modelled with 4160 panels. Table 1.

Main particulars of the Aframax oil tanker.

Dimension

Unit (m, dwt)

Length between perp., LPP Breadth, B Depth, D Draught, T Deadweight, DWT

234 40 20 15 105000

Figure 1.

General arrangement of the Aframax tanker.

Figure 2. Hydrodynamic panel model of the intact tanker.

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damage of only one of WBTs or FP tank or engine room. Two tanks damage case means that MC simulation results in the damage of FP tank and WBT no.1, any combination of two consecutive WBTs or WBT no.6 and the engine room. For damage of more tanks applies analogous reasoning. Furthermore, it is to be noted that the collision always results in the asymmetrical damage, i.e. only starboard or portside tanks are damaged. For grounding, however, damage may be symmetrical, i.e. pairs of WBTs may be damaged together. Such conservative assumption is adopted in the present study, i.e. grounding damage is considered always as symmetrical damage. It is interesting to notice from Figure 3 that damage of single compartment has the highest probability for collision, while damage of two pairs of compartments is the most probable scenario for grounding. There is a general trend obvious from Figure 3 that grounding damage is more extensive, i.e. there is a higher probability of damage of several compartments compared to collision. For that reason, i.e. to cover the most severe cases, grounding damages are used in the comparative seakeeping assessment of damaged tanker in the present study. Two damage cases are used in the analysis: first one is the “small damage case” where grounding is assumed to damage only one pair of WBTs at amidships. The second case is the “large damage case” assuming that 3 pairs of WBTs at amidships are damaged. According to MC simulation, the small and large damage cases correspond to probability of occurrence of about 2% and 5% respectively. It is interesting to notice that the large damage case has higher probability of occurrence. That is visible also in Figure 3b where it may be seen that probability that 3 consecutive tanks along ship will be damaged is larger than that only one single tank will be damaged. Hydrostatic particulars for two damage cases are presented in Table 2. 3D panel hydrodynamic models for the lost buoyancy method and for two damage cases are shown in Figure 4. For the added mass method, the intact model (Figure 2) is used with modified mass distribution and hydrostatic particulars. It should be clarified that hydrostatic particulars (draught and trim) are the same for both methods.

As ship damage may occur in a number of ways, damage parameters are in general random quantities that may be described by probability distributions. Such probability distributions of damage size and location, for cases of the collision and grounding damages are proposed by International Maritime Organization (IMO, 2003). In order to define credible damage scenarios, Monte Carlo (MC) simulation according to IMO probabilistic models is performed. 1000 random numbers are drawn according to IMO models and events resulting in damage of certain number of compartments are counted and presented in Figure 3a and 3b for collision and grounding respectively. Figure 3 shows probabilities of damage in the longitudinal sense only, i.e. it is assumed that only WBTs are damaged, while damage does not penetrate through the inner bottom or inner hull. Fore peak tank and engine room are also considered as separate tanks in the present damage analysis. Single tank damage in Figure 3 represents case when outcome of MC simulation results in the

Table 2.

Figure 3. Probabilities of number of damaged tanks in the longitudinal direction a) collision; b) grounding. Values represent number of outcomes in 1000 simulations.

Hydrostatic particulars of damaged ship.

Damage case

Flooded mass (t)

Draught (m)

Trim (°)

Small Large

5577 16596

16.21 17.38

0.0 1.04

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Figure 4. 3D panel hydrodynamic model for two damage cases for the lost buoyancy method a) small damage case; b) large damage case.

3 3.1

Figure 5. RAOs of heave motion for a) small damage case; b) large damage case.

RESULTS OF THE ANALYSIS Comparison of ship motions

For each of two damage cases as well as for intact ship, RAOs of heave and pitch motion are compared. RAOs of ship motion for damaged vessel are determined by the added mass method and the lost buoyancy method. In all cases, head seas are assumed and constant ship speed of 5 knots. Results of the comparative analysis are presented in Figure 5 and 6 for heave and pitch respectively. It may be seen from Figures 5a and 6a that RAOs of ship motion for small damage case are almost identical for both methods of damaged ship analysis as well as for the intact vessel. The largest differences among methods may be noticed for heave motion and large damage case (Figure 5b). Differences appear mostly in the resonant region where the added mass method gives highest response, while the lost buoyancy method leads to the lowest RAO. Generally, it appears that discrepancies in pitch are lower compared to the heave motion. 3.2

Comparison of VWBM at amidships

For each damage case, RAOs of VWBM at amidships calculated for intact and for damaged ship are compared. RAOs of VWBM for damaged vessel are determined by the added mass method and

Figure 6. RAOs of pitch motion for a) small damage case; b) large damage case.

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damage, while extents of collision damages are generally lower compared to the grounding damage. That may be clearly seen from Figure 3. Secondly, while it is relatively simple to implement the added mass method for heeled ship, it is somewhat difficult to create and balance hydrodynamic model for the lost buoyancy method. That is the case especially for large heel angles. In order to assess effect of the heel of damaged ship, RAOs of VWBM for asymmetrical collision damage of the oil tanker, are calculated using the added mass method and presented in Figure 8. RAOs are compared to the results for intact ship that are shown in the same figure. Two damage cases are presented, small collision damage with damaged WBT 4 (SB) and large collision damage with damaged WBT 3 and 4 (SB). Only head seas and small forward speed of 5 knots are considered. Hydrostatic particulars of damaged ship for two cases are presented in Table 3. It may be seen in Figure 8 that differences in RAOs for intact and inclined damaged ship for head seas are almost negligible. However, even in linear calculations of heeled ship, differences could be observed between VWBM calculated for heading angles symmetrical with respect to head or following seas. Extensive analysis of such differences is presented by Folsø et al. (2008). Noticeable differences of VWBMs between waves coming from portside and starboard are reported for heel angles larger than 10°. If waves are encountered on side with higher

the lost buoyancy method. As for the ship motions, head seas are assumed and constant ship speed of 5 knots. Results of the analysis are presented in Figure 7. It may be seen from Figure 7a that for small damage case RAOs of VWBM are very similar for intact ship and for both methods for assessment of damaged ship, while maximum values of RAOs are almost identical. For large damage case, Figure 7b, there are some differences in RAOs of VWBM, while maximum value of RAO for damaged ship is larger compared to the intact vessel. Also, it may be seen that maximum RAO value for the added mass method is larger compared to the lost buoyancy method. The general trend of presented results is very similar to Figure 2 of Downes et al. (2007), where the added mass method was used for modeling of flooding water. 3.3

Influence of the heel angle on the response

The effect of inclination of ship damaged by asymmetrical damage should also be mentioned. The effect of heel angle is not considered in the comparative analysis for two reasons. First is that heel angle is in first place consequence of the collision

Figure 8. RAOs of VWBM for tanker damaged by collision damage. Table 3. Hydrostatic particulars of ship damaged by collision.

Figure 7. RAOs of VWBM for a) small damage case; b) large damage case.

Damage case

Flooded mass (t)

Draught (m)

Heel (°)

Small Large

2788 5577

15.9 16.21

3.2 6.4

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freeboard, VWBMs are much lower. Such analysis has not been performed within the present study. 4

VALIDATION

Validation of methods for prediction of wave loads on damaged ship is difficult because of the lack of experimental data. One of the rare well documented experiments including global wave loads on damaged ship is described by Lee et al. (2012). The towing tank experiments have been carried out using a model with a scale of 1/100 of a Notional US Navy Destroyer Hull 5415. The tests measured 6 degree of freedom motion responses of the stationary model without forward speed, as well as global loads in intact and damaged conditions for different headings in regular waves. Several damage cases are used, as described in details by Lee et al. (2012). Only damage case 2, having biggest consequences on dynamic response on waves is considered herein. Also, in the present paper, only VWBM at amidships for head seas are compared. 3D panel hydrodynamic models of the intact and damaged ship are shown in Figure 9. Wetted hull surface of intact destroyer is modelled with 822 panels. As in the case of tanker analysis described in Section 3, damaged hydrodynamic model is used for the lost buoyancy method, while for the added mass method intact model is used with modified mass distribution and hydrostatic particulars. RAOs of VWBM are presented in Figure 10a and 10b for intact and damaged ship respectively.

Figure 10. Transfer functions of VWBM for a) intact destroyer; b) damaged destroyer.

Figures include experimental results and also results obtained by 2D linear strip theory calculations performed by Lee et al. (2012). It may be seen from Figure 10a that transfer functions of VWBM for intact ship obtained by linear 3D panel method agrees better to experimental results compared to 2D linear strip theory. While 2D method overestimates experimental results, 3D panel method provides RAOs that are placed between measurements for small and large wave heights. Such results are in line with previous findings reported in many references that linear strip theory overestimates experimental transfer functions (e.g. Parunov & Senjanović, 2003). In Figure 10b, there is again trend that 2D results overestimate measurements. 3D results for both methods for damage modeling are in somewhat better agreement to measurements compared to 2D calculations. For most of frequencies considered, the added mass method overestimates while the lost buoyancy method underestimate measured RAOs. It is also interesting to notice that differences between measured RAOs of VWBM for

Figure 9. 3D panel hydrodynamic model of destroyer for a) intact ship; b) damaged ship.

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described in literature, experience in application of the lost buoyancy method almost does not exist. For the case of Aframax tanker, it was shown that RAOs of VWBM at amidships increase with increase of damage size. The added mass method systematically provides larger maximum RAOs compared to the lost buoyancy method. Validation of methods is performed by comparison with experiments on damaged warship, available from the published literature. Results of comparative analysis indicate that both methods result in RAOs of VWBM at amidships comparable to the measured values. However, the added mass method produce larger RAOs while the lost buoyancy method result in slightly lower values compared to the experimental RAOs of VWBM.

large and small wave heights are much smaller for damaged than for intact ship. 5

DISCUSSION

Some remarks should be put on the accuracy of the presented results. In first place it is to be noted that only linear results are considered. The main consequence of non-linearity is difference between sag and hog VWBM and these differences are more pronounced for ships with fine body lines and increase with increasing wave height. With that respect, it is interesting to notice that nonlinear effects are reduced for damaged warship (Figure 10b) compared to intact ship (Figure 10a). This is seen from smaller differences between measurements for large and small wave height for damaged than for intact ship. For tanker case, which is in the focus of the present paper, it is expected to have even smaller non-linear effects because of the large block coefficient. The next aspect disserving attention is the influence of the sloshing of the liquid in damaged compartments. There is a lot of uncertainty and lack of research in this field. Jia and Moan (2012) concluded that the effect of sloshing on vertical bending moment is small except in beam seas, while the effect of sloshing on horizontal bending moment is large, especially in beam seas. The time domain theoretical approach to the coupled problems of ship and water inside compartment motions is described by Santos and Guedes Soares (2008a). They concluded that the dynamic roll moment is much larger than the static roll moment, for high wave frequencies. Another aspect that should be mentioned and which is not included in the present analysis is the increased influence of horizontal components of global wave flexural loads. As pointed out by Folsø et al. (2008), this sums up with the presence of a horizontal component of static loads, due to the heel angle as well. Horizontal wave bending moment as well as torsional moments and other global wave load components of damaged vessel need further investigation. Finally, the effect of structure of opening on hydrodynamic interaction with the waves is also neglected and deserves further research. 6

ACKNOWLEDGEMENTS This work has been supported in part by Croatian Science Foundation under the project 8658. REFERENCES Bureau Veritas, 2006. HYDROSTAR User’s manual, Paris. Downes, J., Moore, C., Incecik, A., Stumpf, E., and McGregor J. 2007. A Method for the quantitative Assessment of Performance of Alternative Designs in the Accidental Condition, 10th International Symposium on Practical Design of Ships and Other Floating Structures, Houston, Texas. Folsø, L., Rizzuto E., and Pino E. 2008. Wave Induced Global Loads for a Damaged Vessel, Ships and Offshore Structures, Volume 3, No.4, pages 269−287. Hirdaris, S., Argiryiadis, K., Bai, W., Dessi, D., Ergin, A., Fonseca, N., Gu, X., Hermundstad, O.A., Huijsmans, R., Iijima, K., Nielsen, U.D., Papanikolau, A., Parunov, J., and Incecik, A. 2014. Loads for use in the design of ships and offshore structures, Ocean engineering, 78, pp. 131–174. IACS, 2012. Harmonized Common Structural Rules, External release, 1st July 2012. IMO Revised, 2003. Interim guidelines for the approval of alternative methods of design and construction of oil tankers under Regulation 13F(5) of Annex 1 of MARPOL 73/78, Resolution MEPC 2003;110(49), Annex 16. Jia, H., and Moan, T. 2012. The Effect of Sloshing in Tanks on the Hull Girder Bending Moments and Structural Reliability of Damaged Vessels. Journal of Ship Research, Vol. 56, No. 1, pp. 48–62. Khan, I.A., and Das, P.K. 2008. Reliability analysis and damaged ships considering combined vertical and horizontal bending moments, Ships and Offshore Structures, Volume 3, No. 4 (2008), pp. 371–384. Korkut, E., Atlar, M. & Incecik, A. 2004. An experimental study of motion behaviour with an intact and damaged Ro-Ro ship model, Ocean Engineering, Vol. 31, pp. 483–512.

CONCLUSIONS

The aim of the paper is comparison of two simplified methods that may be used for modeling waveinduced motion and resulting structural loads of ship damaged in collision or grounding accident. While the added mass method is widely used and

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Lee, Y., Chan, H.-S., Pu, Y., Incecik, A., and Dow, R. S., 2012. Global wave loads on a damaged ship, Ships and Offshore Structures, 7(3). pp. 237–268. Luis, R.M., Teixeira, A.P., and Guedes Soares, C., 2009. Longitudinal strength reliability of a tanker hull accidentally grounded, Structural Safety, Volume 31, Issue 3, pp. 224–233. Parunov, J., and Senjanović, I., 2003. Incorporating Model Uncertainty in Ship Reliability Analysis., SNAME Transactions. 111; pp. 377–408 Prestileo, A., Rizzuto, E., Teixeira, A.P., and Guedes Soares, C., 2013. Bottom damage scenarios or the hull girder structural assessment, Marine Structures; 33, pp. 33–55. Rodrigues, J.M., and Guedes Soares, C., 2014. Exact Pressure Integrations on Submerged Bodies in Waves Using

a Quadtree Adaptive Mesh Algorithm. International Journal for Numerical Methods in Fluids; 76: 632–652. Santos, T.A., and Guedes Soares, C., 2008a. Study of Damaged Ship Motions Taking Into Account Floodwater Dynamics. Journal of Marine Science and Technology; 13: 291–307. Santos, T.A., and Guedes Soares, C., 2008b. Global Loads due to Progressive Flooding in Passenger Ro-Ro Ships and Tankers. Ships and Offshore Structures; 3(4): 289–302. Teixeira, A.P., and Guedes Soares, C. 2010. Reliability assessment of intact and damaged ship structures. In: Guedes Soares, C., Parunov, J. (Eds.), Advanced Ship Design for Pollution Prevention. Taylor and Francis Group, London, ISBN:978-0-41558477-7.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Smoothed Particle Hydrodynamics (SPH) method for modelling 2-dimensional free surface hydrodynamics M.Z. Ramli, P. Temarel & M. Tan Fluid Structure Interaction Group, University of Southampton, Southampton, UK

ABSTRACT: The main goal of the current research is to implement Smoothed Particle Hydrodynamics (SPH) for the prediction of wave-induced motions and loads within the framework of 3D modelling. In this paper, the focus is twofold. First, implementation of possible additional terms to the standard Incompressible SPH (ISPH) method with reference to generating/propagating regular waves in 2D domain, using a piston wave maker. Improvements to the prediction of pressure and velocity fields are then carried out with kernel renormalization technique and shifting technique without increasing the computational cost. The arc method is employed to improve the accuracy of free surface recognition, i.e. “noise-free” free surface. In addition, the Weakly Compressible (WCSPH) is also applied to the problem of 2D regular wave generation. Comparisons of predicted free surfaces, their kinematic and dynamic characteristics between ISPH, WCSPH and analytical solutions for a range of frequencies are carried out. The second focus of the paper is the 2D radiation problem due to forced sinusoidal oscillation of a rectangular section floating on calm water. The predicted hydrodynamic actions and coefficients in sway by WCSPH are then compared against available experimental measurements. 1

INTRODUCTION

standard SPH formulations, in the case of hydrodynamic actions and coefficients that depend on the wave force and wave damping which affect the wave characteristics around a rigid body (Vugts 1968). The application of SPH to free surface flows can be dated back to 1996, when Monaghan use Weakly Compressible SPH (WCSPH) to perform 2D simulations of wave propagation onto a shallow beach, followed by comparison of SPH with published experimental results in Scott Russel wave generator. Since then, improvements have been made for WCSPH formulation including treating 2D interfacial flows with different fluids (Colagrossi & Landrini 2003), integrating Large Eddy Simulation (LES) scheme and modelling the free surface flows with consideration of complex turbulent flows. Important development of Riemann solution by Vila (1999) has the most significant impact on the free surface prediction in terms of suppressing the pressure fluctuations (Gao et al. 2012, He et al. 2013). In ISPH, other works have been done to improve accuracy near free surface boundaries by employing corrective or additional terms (Shao 2010, Li et al. 2012, Colagrossi et al. 2013). In the present work, ISPH with divergencefree velocity field is used to study the propagation of 2D waves generated by a piston wave maker into a wave tank. The wave maker is located at the upstream boundary of the tank and an artificial damping layer on the other side. The kernel summation of standard SPH formulation

Smoothed Particle Hydrodynamic (SPH) which is purely Lagrangian method developed during the seventies was an attempt to model continuum physics to overcome the limitations of finite difference methods. The Lagrangian method is a meshfree method whereby the computational domain is represented by a set of interpolation points called particles rather than grid cells. Each particle carries an individual mass, position, velocity, internal energy and any other physical quantity which evolves in time according to the governing equations. All particles have a kernel function to define their range of interaction, while the hydrodynamic variables are determined by integral approximations. These methods, where the main idea is to substitute the grid by a set of arbitrarily distributed particles, are expected to be more adaptable and versatile than the conventional grid-based methods, especially for those applications with severe discontinuities in free surface. Shao and Lo (2003) developed the ISPH method based on a strict hydrodynamic formulation and two-step semi-implicit solution process. Compared with the standard SPH, it has been demonstrated that the ISPH approach can improve the computational efficiency and pressure stability (Lee et al. 2008) and thus will be further developed for free-surface flow in this paper. However, the prediction of free surface hydrodynamics in propagating waves is very difficult for

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is observed to be insufficiently accurate in obtaining the velocity and pressure fields, thus does not perform well when simulating free surface hydrodynamics. In order to exploit fully particles inside kernel domain, the accuracy of gradient estimation is improved up to second order with kernel renormalization technique. Highly distorted particle spacing around free surface which cause instability is smoothed out by collision control and particle shifting. Then, the arc method is applied for fast and accurate boundary recognition. The numerical model built in this paper is validated through comparison between ISPH, WCSPH and potential flow solution. Comparisons of predicted free surfaces, their kinematic and dynamic characteristics for a range of frequencies are carried out. The work is then extended to investigate the 2D radiation problem due to forced sway motion of a rectangular section on calm water surface using WCSPH. The predicted hydrodynamic forces in swaying are compared against available experimental measurements.

2

Following the same integral representation and particle approximation, the derivative of a function for particle i can be written as N

∇ ⋅ f xi ) = ∑ j =1

∇i

Ω

f ( )ω(

, h ) ddx x′

(1)

where ω (x − x′, h) is a smoothing function, x′ is another arbitrary position vector in the domain of integration Ω, and h is the smoothing length. As the entire system in SPH method is represented by particles which carry individual mass and occupy individual space, integral representation in Eq. (1) can be written in the form of particle approximation. N

f x) = ∑

mj

j =1

ρj

f x j ) ω ((x x x j , h)

N

j =1

mj

ρj

f xi ) ω iij

iij

=

xi − rij

j

∂ω iij ∂rij

(5)

mj ∂ω iij D ρi = ρi ∑ v iij Dt ∂xi ρj

(6)

N ⎡ P Pj ⎤ ∂ω iij Dv i = − ∑ m j ⎢ i2 + 2 ⎥ +F Dt j =1 ⎢⎣ ρi ρ j ⎥⎦ ∂xi

(7)

2.2 Boundary conditions In SPH, three methods are, in general, widely used, the repulsive force method, the mirror particle method and the dummy particle method (Shao and Lo 2003, Lee et al. 2008). It is very difficult to build some boundary conditions, such as the homogeneous Neumann boundary for the pressure, with the repulsive boundary force method. In this paper, the dummy particles, which have the physical properties of inner fluid particles, are used for the treatment of wall boundary condition. In this 2D model, the solid boundaries are simulated by three layers of boundary particles similar to the fluid particles which balance the pressure of inner fluid particles and prevent them from penetrating the wall. These boundary particles are forced to satisfy the same equations as the fluid particles. Thus, they follow the same momentum and continuity equations. The physical properties of all boundary particles, except velocity and position, evolve with time. However, for moving boundaries, such as the solid boundaries of a moving rectangular box and the wave maker paddle, velocities and

(2)

where xj is the position vector of particle j within the support domain of x, and N is the total number of particles. For a particle i, Eq. (2) is written as f xi ) = ∑

(4)

where P is the pressure and F is the acceleration due to gravity. In ISPH, incompressibility is enforced in the projection method by a pressure Poisson equation. In this paper a quartic smoothing kernel is used for all SPH interpolations. A smoothing length of h = 1.3dx is used, where dx is the initial particle spacing.

In SPH, the approximate integral form of a function at any given position vector is

∫

f x j ) ⋅ ∇ iω iij

and rij is the distance between particle i and j. Using Eq. (3) and Eq. (4), the continuity and pressure contribution to the momentum conservation equations can be written as

SPH interpolation

f

ρj

where

SPH METHODOLOGY

2.1

mj

(3)

where mj and ρj are the mass and density of particle j, respectively, within the support domain of particle i, ωij = ω (xi − xj, h).

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is employed to better represent the proper repulsive forces near the free surface. As shown in Fig. 2, OXY is the global coordinate system, and the coordinate with origin in the center of the particle is local coordinate system. When the distance of two particles is less than βcdx, then the collision model is applied as;

positions are updated by independently solving the equations of motion for these bodies. 2.3

Arc method

Misidentification of free surface particles can take place sometimes during the simulation and this will affect the calculation of pressure directly on the free surface. In order to overcome this problem, a fast and accurate boundary recognition method is applied called the ‘arc’ method (Koh et al. 2012). The principle of the arc method is that if any arc of the circle (with radius R) around a center particle is not covered by any circle of its neighbors, the center particle is treated as a free surface particle. In other words, if the overall covered arc is more than (0,2π) without gaps, the particle is identified as inner fluid particle; otherwise it is treated as free surface particles as shown in Fig. 1. In Fig. 1, particle 3 is treated as free surface particle while particle 7 is treated as inner particle. It has been confirmed that the arc method is more effective for identifying free surface particles compared to standard SPH free surface identification (Wu et al. 2014). 2.4

mv ix + mv jx = mv ′ix + mv ′jx

where m is the particle mass, v′ix = −αcvix. Based on numerical test, the optimal combinations are αc = 0.0001, βc = 0.99 (Lee et al. 2011). 2.5

Particle shifting

The particle clustering caused by the flow can occur when ISPH with divergence-free velocity field is employed. The particle shifting technique preserves the consistency in the integration of ISPH by maintaining a uniform particle distribution, avoiding particle clustering and reducing the error caused by highly-distorted particle distribution. In this method, particles are advanced according to the projection-based method, and then are slightly shifted to a new position. Hydrodynamic variables for particles in the new position are updated accordingly by Taylor series. However, pressure is not updated due to the numerical difficulties and higher computational cost involved. Using the generic variable φ this update is

Collision control

A collision model is introduced in ISPH to prevent particles from hitting each other at high speed during simulation. The collision model was introduced by Lee et al., (2011) in the moving particle semiimplicit MPS method to simulate violent free surface motions. On the free surface where the pressure is set to zero, repulsive forces may not be effectively generated when particles accelerate and get close to each other at high speed. As a result, particles may not be identified as a free-surface particle and pressure can suddenly increase. Therefore, a simple collision model, enough to avoid collision between particles without introducing additional viscosity

Figure 1. method.

(8)

Identification of free surface particles by arc Figure 2.

Illustration of collision model method.

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φi

φ i + ( ∇φ

i

δ rii ′ + ϑ δ rii2′ )

Thus, the kernel correction term known as renormalization is applied for higher order accuracy of convergence (Khayyer et al. 2008). The expression for the kernel normalization is,

(9)

where, i and i ′ are the particle old and updated positions, respectively. The amount of position shifting, δrii′ is the distance vector between the updated and old position after shifting takes place. The shifting approach is applied to all fluid particles excluding the free surface particles where the amount of shifting calculated by Eq. (10) is added to the old position. The amount of position shifting reads, ri ′ = ri + δ ri

(10)

δ

(11)

α Ri

i

∇

ij

= L( r )∇ω iij

(14)

where Wij is the normalized kernel and ⎛ ⎜ ∑V j ( x j L(r ) = ⎜ ⎜ ⎜ ∑V j ( y j ⎝

x) x)

∂ω ij ∂x ∂ω iij ∂x

∑V ( x j

∑V ( y j

j

j

∂ω iij ⎞ ∂y ⎟ ⎟ ∂ω ij ⎟ x) ⎟ ∂y ⎠

−1

x)

(15) where, C is a constant, set to 0.04 in this study; α is the shifting magnitude, equal to the maximum particle convection distance UmaxΔt, with Umax as the maximum particle velocity, and Δt the time step. The shifting vector, Ri, is determined by Mi

Ri

ri 2

∑r

2 j =1 ij

n ij

Vj is the constant volume, mj/ρj as mass and density is same for every particles. An operator similar to that used for kernel normalization is used for the divergence.

(12)

2.7

(13)

According to Xu et al. (2009), noise on the free surface can be smoothed by artificially increasing the viscosity around free surface particles and particles adjacent to the free surface without strongly influencing the free surface predictions. This numerical treatment is called free-surface damping and is determined by

where,

ri =

1 Mi

Mi

∑r

ij

j =1

vd =

the average particle spacing in the neighborhood of particle i; Mi is the number of neighboring particles within the smoothing length h. Neighboring particles are treated slightly different if particle i is closer to the free surface. In this approach, only particles which have smaller distance to particle i than the distance of particle i to the free surface are considered. In other words, the smallest distance of particle i to the free surface particles will be assumed as the new radius replacing the smoothing length h and only particles within the new radius will be considered as neighboring particles of particle i. The shifting distance should be large enough to prevent instability and small enough not to cause inaccuracy due to Taylor series correction. 2.6

Free-surface instability damping

umax dx d Pemax

(16)

where Pemax is the global maximum Peclet number; umax is the maximum global velocity estimated as √2gh. The viscosity vd is slightly higher than the initial kinematic viscosity v = 10−6 m2/s. Peclet numbers of 7, 30 and 150 are used for the simulations in this paper. This is to avoid artificial diffusion that will influence the free surface prediction if the surface viscosity is too high and prevent insufficient artificial damping for the truncated-kernel error if the viscosity is too low. 2.8

Artificial viscosity damping layer

In order to absorb the reflected waves that reach the far (right) boundary of the tank, a damping layer of 2 m length is set at the end of the wave tank. The propagating wave is damped physically within the damping layer by modifying the momentum equation as

Higher order convergence

In 2007, Oger et al. showed that the basic gradient and divergence of the standard SPH may lead to large errors due to insufficient approximation in the discrete form. The discretized convolution approximations in the kernel domain are not sufficiently accurate for obtaining pressure and velocity fields. Fewer neighboring points used to reduce the computational cost in the kernel domain affect the accuracy in the approximation.

Dv 1 1 = g + ∇ ⋅ − ∇P − γ v Dt ρ ρ

(17)

where γ may be selected as

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Regular waves can be generated using a wave maker based on linear theory. The profile of a regular wave as a function of time t is defined as;

2

γ

⎛ x − x0 ⎞ β dω f ⎜ , x ≥ x0 ⎝ L ⎟⎠

(18)

where βd is the artificial coefficient, equal to 2.0; ωf is the circular frequency of the wave; x0 is the damping zone starting point and x is the coordinates of the free surface points. When particles are moving close to the damping layer, the velocity of each particle is forced to damp gradually.

v( ) =

3.1

2

cos ω f t

(19)

where X0 and ωf are the amplitude and frequency of the wave maker, such that X0 = H/W* and W* =

3

X 0ω f

NUMERICAL RESULTS

2(cosh (cosh kd ) sinh 2 kd + 2 kd

(20)

where k is the wave number.

Computational setup for wave generation

The detailed setup of the 2D numerical wave tank is shown in Fig. 7. The origin, x = 0 and y = 0 point is located on the still water surface at the left side of the wave tank. The initial still water depth is d = 0.6 m. The wave tank used is 6 m long with a piston wave maker located at x = 0.5 m. The incident waves are regular waves with height H = 0.05 m and H = 0.10 m. The fluid density ρ is 998kg/m3, the initial kinematic viscosity v is 10−6 m2/s and the initial particle spacing dx is 0.015 m. The number of particles is 16000 including 1700 boundary particles.

3.2 Wave surface validation Figs. 3 and 4 show the effects of the improvements on ISPH on the free surface predictions. Arc method is applied for all cases investigated. Particle shifting results are the combination of collision control and particle shifting, whilst higher order model is the combination of collision control, particle shifting and higher order convergence. From these results, it can be observed that for the smaller of the two wave heights, waves are notably non-sinusoidal and

Figure 3. Comparison of free surface predictions between different model of improvements (H = 0.05 m; λ = 1.5 m; t = 14.75 s).

Figure 4. Comparison of free surface predictions between different model of improvements (H = 0.10 m; λ = 3.0 m; t = 14.75 s).

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Figure 5.

Comparison of free surface predictions between ISPH and WCSPH (H = 0.05 m; λ = 1.5 m; t = 19.975 s).

Figure 6.

Comparison of free surface predictions between ISPH and WCSPH (H = 0.10 m; λ = 3.0 m; t = 9.75 s).

3.3

Figure 7.

Wave velocity field and pressure distribution

The influence of the aforementioned improvements to ISPH on particle velocity and pressure is shown in Fig. 8 examining the variation of these quantities with depth under wave crest and trough. The pressure variation with depth is captured well by both ISPH and WCSPH. On the other hand, even the higher order convergence ISPH model has issues with the horizontal velocity, believed to be affected by the noise in the free surface. However, particles with large horizontal velocity can be seen in Fig. 8(a)(b) even when located away from the free surface. It appears that the combination of particle shifting and higher order convergence compromise the simulation accuracy, whilst providing for simulation stability. The notion of suppressing free surface instability by artificial viscosity on free surface does help in controlling the free surface noise, but negatively affecting accuracy of velocities. On the other hand, WCSPH shows good agreement with potential flow solution. Although the pressure is slightly higher near the wave crest, the small error is assumed to be negligible.

Numerical setup for wave generation.

decay even before reaching the damping layer when using particle shifting and/or the collision models. Potential flow with increased depth is introduced to compare with higher order convergence model where the mean depth of the fluid domain appears to rise at 0.65 m within each timestep and settles to the mean depth when the wave reaches the damping layer. The reason for this rise lies in the properties of the higher order convergence on the free surface where truncated kernel results in a very noisy free surface even in the hydrostatic condition. Figs. 5 and 6 show the free surface prediction obtained using WCSPH in comparison with ISPH-higher order and ISPH-higher order with additional free surface viscosity, Pemax = 30. The WCSPH case is developed through open source software incorporating Riemann formulation between interacting particles (Vila 1999, Colagrossi & Landrini 2003). For both wave heights, WCSPH is able to maintain a stable propagating wave without the rise of mean depth and it agrees well with potential flow results. Use of free surface viscosity works well for the smaller of the two wave heights.

3.4

Convergence studies

Convergence studies are carried out setting 60 s as runtime to investigate the stability of ISPH and WCSPH. The particle distribution is refined to smaller number of dx = 0.010 m (approx. 35580 internal particles), dx = 0.015 m (15880) and

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Figure 8. Comparing horizontal velocity (a,b) and pressure predictions (c,d) under wave crest (a,c) and wave trough (b,d) (H = 0.05 m; λ = 1.5 m; t = 19.975 s). ο: prediction from ISPH with free surface viscosity Pemax = 30; x: prediction from higher order convergence model; −: prediction from potential flow; Δ: prediction from WCSPH; ◊: prediction from Xu et al. (2009).

Figure 10. Numerical setup for rectangular section in sway motion.

Figure 9. Comparison of convergence studies between (a) ISPH and (b) WCSPH at x = 3 m (H = 0.10 m; λ = 3.0 m; t = 6.0 s).

dx = 0.030 m (3980). ISPH simulation experiences instability and break down before 30 s for dx = 0.010 m while WCSPH continues to maintain stable wave propagation for all cases (Fig. 9). The Poisson equation which relates directly to the pressure calculation is sensitive to particle size which may introduce errors into the simulation. Accumulated errors within each timestep then lead to unphysical movement of particles, both in the fluid domain and on the free surface. 3.5

Hydrodynamic forces due to sway motion

Figure 11. Comparison between horizontal forces in sway motion. (a) ωf = 3.50 rads−1, (b) ωf = 5.54 rads−1 and (c) ωf = 7.00 rads−1.

In Fig. 10, the numerical wave tank is extended to 30 m long and 1.8 m deep. Damping beaches are

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located on both side of the tank and about halfway the length of the tank, a rectangular box is fixed (0.4, 0.4 m). Breadth to draught ratio is 2, namely a draught of 0.2 m. This box is subject to sinusoidal sway of amplitude 0.02 m and a range of frequencies ωf = 1 to 10 rads−1. Predictions of based on particle forces around the swaying body and experimental data by Vugts (1968) are shown in Fig. 11 for ωf = 3.50, 5.54 and 7.00 rads−1 and dx = 0.010, 0.015 and 0.020 m. Please note that added mass peaks at approximately 3.50 rads−1 while damping peaks at approximately 5.54 rads−1. The trends between predictions and measurements show reasonable agreement. However, at relatively high frequencies of oscillation, the effect of further free surface refinement requires investigation in resolving the asymmetry in the force predicted. 4

dynamics. Journal of Computational Physics, 191(2), 448–475. Colagrossi, A., Souto-Iglesias, A., Antuono, M., & Marrone, S. 2013. Smoothed-particle-hydrodynamics modeling of dissipation mechanisms in gravity waves. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 87(2). Gao, R., Ren, B., Wang, G., & Wang, Y. 2012. “Numerical modelling of regular wave slamming on subface of open-piled structures with the corrected SPH method.” Applied Ocean Research, 34, 173–186. He, M., Ren, B., Jiang, F., & Ma, C. 2013. Simulation of dynamic coupling between waves and a free-floating rectangular box by smoothed particle hydrodynamics. ASME 2013 32nd International Conference on Ocean, Offshore and Arctic Engineering, OMAE 2013, American Society of Mechanical Engineers. Khayyer, A., Gotoh, H., & Shao, S.D. 2008. Corrected incompressible SPH method for accurate watersurface tracking in breaking waves. Coastal Engineering, 55(3), 236–250. Koh, C.G., Gao, M., & Luo, C. 2012. A new particle method for simulation of incompressible free surface flow problems. International Journal for Numerical Methods in Engineering, 89(12), 1582–1604. Lee, E.S., et al. 2008. Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method. Journal of Computational Physics, 227(18), 8417–8436. Lee, B.H., Park, J.C., Kim, M.H., & Hwang, S.C. 2011. Step-by-step improvement of MPS method in simulating violent free-surface motions and impact-loads. Computer methods in applied mechanics and engineering, 200(9), 1113–1125. Li, J., Liu, H.X., Gong, K., Tan, S.K., & Shao, S.D. 2012. SPH modeling of solitary wave fissions over uneven bottoms. Coast Eng, 60, 261–75. Monaghan, J.J. 1996. Gravity currents and solitary waves. Physica D., 98, 523–533. Oger, G., Doring, M., Alessandrini, B., & Ferrant, P. 2007. An improved SPH method: Towards higher order convergence. Journal of Computational Physics, 225(2), 1472–1492. Shao, S., & Lo, E.Y.M. 2003. Incompressible sph method for simulating newtonian and non-newtonian ows with a free surface. Advances in Water Resources, 26, 787–800. Shao, S.D. 2010. Incompressible SPH flow model for wave interactions with porous media. Coast Eng., 57, 304–16. Vila, J.P. 1999. On particle weighted methods and smooth particle hydrodynamics. Mathematical models and methods in applied sciences, 9(02), 161–209. Vugts, J.H. 1968. The hydrodynamic coefficients for swaying, heaving and rolling cylinders in a free surface. No. Report No. 194. Wu Qiao-rui, Tan, M., & Xing, J.T. 2014. An improved moving particle semi-implicit method for dam break simulation Journal of Ship Mechanics, 18(9). Xu, R., Stansby, P., & Laurence, D. 2009. “Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach.” Journal of Computational Physics, 228(18), 6703–6725.

CONCLUSIONS

The current studies have been focusing on understanding the fundamental mechanism of SPH and how to incorporate ISPH in simulating free surface wave propagation. Standard SPH is not the best option for such simulation without additional terms. Therefore, efforts focused on investigating various aspects of ISPH for suitability in maintaining a good wave profile. The arc method is effective in preventing any misjudgment of free surface particle and collision control has proven to be effective between high speed particles without the need for additional fluid viscosity. The fundamental theory of SPH demonstrates that uniform distribution particle could reduce the approximation errors. By incorporating particle shifting and higher order convergence method, simulations are able to maintain a good wave profile. Additional viscosity is shown to suppress noisy configuration of fluid particles on the free surface. The general agreement for wave profiles between ISPH and potential flow are acceptable, accounting for the rise in mean depth, the pressure are predicted well, though velocity distribution is not consistent with potential flow result. WCSPH shows a good agreement in wave profile, velocity and pressure distribution along the water depth. WCSPH is employed to predict the hydrodynamic force due to sway motion. Preliminary results indicate that prediction of forces around a moving boundary depend on refinement in the free surface vicinity, which will be further investigated.

REFERENCES Colagrossi, A., & Landrini, M. 2003. Numerical simulation of interfacial flows by smoothed particle hydro-

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Wave-induced responses of a bulk carrier in heading and following seas X.L. Wang China Ship Scientific Research Center, Wuxi, Jiangsu, P.R. China

R.M. Liu China Classification Society, Beijing, P.R. China

J.J. Hu China Ship Scientific Research Center, Wuxi, Jiangsu, P.R. China

ABSTRACT: A bulk carrier is introduced in this paper to study its low- and high-frequency responses both in heading and following seas. Effects of four loading conditions on wave-induced responses are investigated by a 3-D hydroelasticity method. Moreover, experimental results of two loading conditions are also presented to test and verify the theoretical results of the 3-D method. It is shown that loading conditions have great influences on the wave-induced responses. Springing and whipping behaviors will be influenced obviously by high forward speeds. Small draft in ballast condition will cause high peak in 2-node vertical vibrations. 3-D hydroelasticity theory can make up the limitations in experiments because of the high cost and the capacity of facility. Results of this paper show the necessity of taking into adequate consideration of high-frequency vibrations on strength and fatigue damage of ship structure in the bulk carrier’s design stage. 1

INTRODUCTION

theoretical methods during the past decades. The effects of many factors such as loading conditions, hull girder stiffness and ship form on wave-induced responses had been studied (Wang 2012, Wang et al. 2014). Ultra large ore carrier and large LNG carrier were introduced to studied these effects by 2-D and 3-D theoretical methods and model tests. They found the difference of wave-induced responses of these ships with the consideration of the above effects. Segmental model test is an important and useful method to study the hydrodynamic responses of a ship travelling in waves. It can be used to get the distribution of wave loads in each segmental crosssection. Furthermore, it can also be used to confirm the validity of theoretical results. So this kind of segmental test has an important significance to analysis the wave-induced responses including springing and whipping behavior by a ship model. However, the high cost of segmental model test of a ship will limit the number of test cases. So more and more theoretical calculations will be performed to investigate the wave-induced responses. Gu et al. (2000) had investigated the high-order harmonious component in vertical bending moment of a ship travelling in regular waves. He found nonlinear springing behavior will happen if the frequency of high-order component of the vertical bending moment equals to the eigen frequency of a hull

The rapid development of science and technology and the rapid increase of the requirement of bulk carriers in the world make deadweight of bulk carrier being larger and larger. Bulk carriers have already become the main ships forms of bulk cargo transportation both from the existing size and shipping order all over the world. The increasing trend of bulk carriers scale and the widely use of high strength steel to reduce the structural weight results in the hull girder being relative flexible. Springing phenomenon is presented frequently in these bulk carriers when they are travelling in waves. This kind of high frequency vibration will result in the rapid increase of stress cycle number, and then it will result in serious structural fatigue damage of bulk carriers in their travelling courses even in small and moderate sea states. On the other hand, there is another kind of high frequency vibration, whipping, which is caused by slamming events when sea states are being heavy. Whipping will not only have effects on the fatigue damage but also the ultimate strength of ship structure. Thus the safety of ship structure is deeply influenced by springing and whipping, and they will result in the growth of ship maintenance costs. Many scholars have done the investigations of springing and whipping by experimental and

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Wu (1984) had put forward the generalized fluidsolid interface condition and had developed a 3-D hydroelasticity theory which can be used in the performance analysis of dynamic responses of a 3-D arbitrary and flexible structure under inner and outer wave excitations. A computer algorithm THAFTS had been developed based on the above theory in CSSRC. The following is the brief depiction of 3-D linear hydroelasticity theory which is used in this paper. The dry structure of ship hull girder can be predigested as a Timoshenko beam with varied sections and no any restrictions in both ends. During the analysis of symmetry and dissymmetry modes of dry structure by FEM method, the concentrated force of a FEM node in beam can be calculated at each section in different modal shape. The equation of motion may be written in the generalized matrix form:

girder. As to traditional ships, Jensen et al. (2002) mentioned the small bending stiffener, high travelling speed and heavy nonlinear excitation will result in severe springing behavior. Zhu et al. (2011) employed a flexible backbone to investigate the wave load responses of large container ship with Lpp, 290 m. They studied the vertical, horizontal and tortional vibrations of the open ship in regular and irregular waves. It was found that the corresponding wave length was very short though the first order flexible mode can result in a high peak value of the RAO of wave load. In fact, this kind of short wave will have little energy to excite the springing behavior in real sea states. They found a few linear springing behaviour in this model test, but they found nonlinear springing behavior frequently which was accompanied by whipping with the increasing of wave heights. Wang (2012) found the relationship between the frequency of highorder component of vertical bending moments and the wave encounter frequency was an important factor to springing behavior from model test results. If the relationship was an integral multiple, then this frequency of high-order component will usually equal to the eigen frequency of a hull girder. Furthermore, he found some other frequencies appeared in frequency spectrum curves between the frequency of high-order component of vertical bending moments and the wave encounter frequency. These frequencies also agreed very well with the integral multiple relationship between them and the wave encounter frequency. All these frequencies with the multiple relationships can result in springing behavior. As one of the three main ship forms in the world, bulk carrier has attracted many investigations from ship designers and researchers on its characteristic of the wave-induced responses. A 3-D hydroelasticity theory was employed to investigate the wave-induced responses of a bulk carrier with displacement more than 200,000 DWT. Segmental model test results were also presented and compared with theoretical results about the flexible modes, the RAOs and the high-order responses in two loading conditions. Meanwhile, the influence of loading condition on wave-induced responses was discussed in this paper. 2

)  p( ) (t ) + ( +(

) p ( ) (t )

) p (t ) = Z ( ( )

)

( )

()

(1)

where a, b and c are respectively the generalized modal inertial, modal damping and modal stiffness of the dry structure. A, B and C are the matrices of generalized hydrodynamic added mass, damping and restoring g coefficients, while Z ( ) and Ξ( ) {Ξ1( ) , Ξ(2 ) , ..., Ξ(m) } are respectively the generalized steady-state and the first-order generalized wave exciting force. Their elements can be found in Wu (1984).

3

THEORETICAL PREDICTION AND COMPARISON WITH EXPERIMENTAL RESULTS

In this section, wave-induced responses of the bulk carrier by employing the 3-D hydroelasticity method will be presented. Some of the numerical results will be compared with the experimental data both in heading and following seas. The responses here including heave and pitch motion, and VBMs in heading and following seas. The experiment had been done in a towing tank located at China Ship Scientific Research Center (CSSRC). The tank is 474 m long, 14 m wide and 7 m deep, and the wavemaker is able to produce both regular and irregular waves. For the convenience of comparison with theoretical results, all experimental data used in this paper are presented in full scale.

THEORETICAL BACKGROUND

In the past thirty years, many 3-D hydrodynamical methods which were used to analyze the seakeeping problems of large floating offshore structure had been rapid developed with the appearance of mainframe computer (Hirdaris et al. 2003, Wang et al. 2012, Hu et al. 2012). By the integration of 3-D navigability theory and 3-D structural dynamics,

3.1

Main particular

With a length LOA 300 meters, the bulk carrier is a kind of large commercial ship of the three main

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ship forms in the world. Usually, there will be more than dozens of loading conditions of the bulk carrier which can be found in its loading manual. Effects of different loading conditions on waveinduced responses are important to decide the design wave loads of ship structure in its design stage. Four loading conditions, F-1, F-2, B-1 and B-2, are employed to studied the effects in numerical prediction. Where F-1 and F-2 are full-load conditions, whilst B-1 and B-2 are ballast conditions. Model tests results of F-2 and B-1 are available in consideration of the cost of experiments. The main particular of the bulk carrier is listed in Table 1. Figure 1 shows the body plan of the bulk carrier. No-bulb-bow ship form is employed. Figure 2 shows the weight distributions of the bulk carrier in four loading conditions. 3.2

Figure 2. Weight distributions of the bulk carrier in four loading conditions.

Mode analysis

A whole-ship FEM model and a Timoshenko beam can both be used as dry structure in the calculation of hydroelasticity theory. The former needs the data of ship inner structure in detail, so the workload of data preparation and modeling is huge (Tian et al. 2009, Mockler, 2009). A satisfied Table 1.

Main particular of the bulk carrier. Full-load conditions

Ballast conditions

Physical quantities

F-1

F-2

B-1

B-2

LOA (m) LBP (m) B (m) H (m) Tf (m) Ta (m) Δ (t) Ta (m) Xg (m)

300.0 295.0 50.0 24.7 17.76 18.68 235610.2 10.16 5.37

300.0 295.0 50.0 24.7 17.71 18.73 235610.1 8.03 5.23

300.0 295.0 50.0 24.7 6.73 7.50 85422.6 12.77 9.74

300.0 295.0 50.0 24.7 8.42 12.82 129980.1 12.45 2.11

Figure 1.

Figure 3. The 2-node, 3-node and 4-node VBM modes of the dry structure in vacuum in four loading conditions.

precision of springing prediction can also be carried out by using a Timoshenko beam with varied sections as the dry structure in 3-D hydroelasticity calculations. It results in the great decrease of data preparation and modeling and great increase of work efficiency when the detail structural data are unavailable in early stage of ship design (Hu et al. 2012). Figure 3 shows the vertical bending vibration modes of 2-node, 3-node and 4-node of the dry structure of the bulk carrier in four loading conditions in vacuum. Where, the abscissa is the station whilst the ordinate is the normalized displacements. It shows the similar characteristic of the Vertical Bending Moment (VBM) modes of the dry structure except the 3-node mode in B-1 condition. It states that the loading conditions will influence the dry modes of vibration of ship structure. The corresponding wet-structure models of

Body plan of the bulk carrier.

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y FEM, ω0 is wet-structure frequency by theory, by ω*n is wet-structure frequency by model test (in full scale), Δω = (ωn−ω0)/ω0, Δω* = (ω*n−ωn)/ωn. Added mass and damping of ship hull in water will cause the decrease of the wet-structure frequencies comparing to the dry-structure frequencies. Table 3 presents good accuracy of theoretical prediction. The 4-node frequencies from theoretical prediction were not very conspicuous, so the 4-node mode were not listed in Table 3.

this ship in the 3-D hydroelasticity calculation is shown in Figure 4. The dry structure of each loading condition is different because of the different weight distributions which are presented in Figure 2. But node number and beam-element number are same. The wet structure in each loading condition is also different because of the different draft, Figure 4 shows the different wet structure which will be used in the following 3-D hydroelasticity analysis. The node number and the cell-element number of the wet structure in each loading condition is different. Table 2 lists the node number, the beam-element number and the cell-element number of each dry and wet structure. According to the hydrodynamic calculation in the 3-D hydroelasticity code, comparison of the dry and the wet modes is given in Table 3. Where, ω0 is the dry-structure frequency

3.3 Heave and pitch Three forward speeds were considered in theoretical calculations. However, only one forward speed 14.8 knots was taken into consideration in model test. So the comparison between theoretical and experimental results is presented at this forward speed. On the other hand, in order to state the characteristic of wave-induced responses in following seas, the results of calculations at zero forward speed were also compared with the experimental data at this forward speed 14.8 knots. Figure 5 shows the heave and pitch of each loading condition at different forward speeds in heading seas. Generally speaking, features of these curves present the steady trend of wave-induced heave and pitch motions with the increasing of forward speeds of a ship travelling in waves. Motion curves at low forward speed 3.7 knots have not changed much comparing to those at zero speed. A remarkable peak is appeared in both heave curves in full-load conditions when the forward speed is 14.8 knots, and there is a shift of the heave curve

Figure 4. Wet-structure models of the bulk carrier in four loading conditions. Table 2.

The node number, beam element and cell number of each dry and wet structure. Dry structure

Wet structure

Loading conditions

Node number

Beam-element number

Node number

Cell-element number

F-1 F-2 B-1 B-2

21 21 21 21

20 20 20 20

1216 1194 746 916

1060 1042 614 776

Table 3.

Comparison of the dry and the wet modes.

Loading cond.

F-1

F-2

B-1

B-2

Mode

2-node

3-node

2-node

3-node

2-node

3-node

2-node

3-node

ω0 (Hz) ωn (Hz) ωn* (Hz) Δω Δω*

0.63 0.46 – −26.98% –

1.64 1.19 – −27.44% –

0.67 0.48 0.49 −28.36% 2.08%

1.67 1.20 1.15 −28.14% −4.17%

0.98 0.62 0.67 −36.73% 8.06%

2.51 1.57 1.50 −37.45% −4.46%

0.74 0.62 – −16.22% –

2.04 1.35 – −33.82% –

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Figure 6. Comparison of heave and pitch between experimental and theoretical results (U = 14.8 knots, heading seas).

Figure 5. Heave and pitch of each loading condition at different forward speeds (heading seas).

towards high-value direction. Relatively speaking, no remarkable peaks can be found in heave curves in ballast conditions at the forward speed 14.8 knots. It can be seen that the pitch curves move to low-value direction when ω0 is about less than 4.0 rad/s at this speed. However, there is also a shift of the pitch curve towards high-value direction when ω0 is more than 4.0 rad/s. This kind of shift of motion curve is similar in all four loading conditions. Comparison of heave and pitch between experimental and theoretical results at forward speed 14.8 knots in heading seas is presented in Figure 6. Two loading conditions were included. Some discrepancy between calculation and experiment is existing though, the agreement is good as a whole. Figure 7 shows the comparison of heave and pitch between experimental and theoretical results in following seas. No suitable result at a forward speed was available by the 3-D hydroelasticity code

Figure 7. Comparison of heave and pitch between experimental and theoretical results (following seas).

because of the minus encounter wave frequency in following seas. Relative comparison of calculation results and experimental data at different forward speeds was presented in this figure. It is interesting that good agreement can be seen is the figure both in heave and pitch curves. It indicates that forward speed has little influence on heave and pitch of the bulk carrier in following seas. 3.4 VBMs Wave-induced load is one of a key factor which will cause the damage of ship structure by the interaction between wave and ship hull. VBM is the main response and will be considered sufficiently in the

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will cause high peak of 2-node vertical vibration which is very important to designers if taking into consideration of an issue on strength and fatigue damage of ship structure. Experimental results are available when the forward speed was 14.8 knots in heading seas. Figure 9 presents the comparison of VBMs between experimental and theoretical results in F-2 and B-1 conditions. The peak of experimental results is about 13 percents higher and 9 percents lower than that of theoretical prediction in F-2 and B-1 conditions respectively. As a whole, this discrepancy is supposed acceptable in the early stage of a ship design. On the other hand, the experimental data almost concentrate on the frequency range 0.0∼1.0 rad/s. It is the limitation of wave-making ability of the facility. Making high-frequency wave with sufficient energy needs a larger towing tank than the existing one in CSSRC. When talking about the difference of VBMs in heading and following seas, Figure 10 presents the responses of F-2 and B-1 conditions at a forward speed 14.8 knots. As a matter of fact, seldom references had investigated the characteristic of VBMs in following seas. So here the comparison has a significance to reveal the discrepancy of the responses by experimental method. It can be seen that the responses in heading seas were high than those in following seas in most range of the wave frequencies. The highest values in heading seas were about 1.36 and 1.20 times the corresponding values in following seas. As a whole, without consideration of the wave condition in following seas in the bulk carrier’s early stage is conservative and acceptable. On the other hand, if the comparison of the VBMs in different loading conditions is discussed, the

design stage by ship designers. High-frequency vibration are main coming from the contribution of 2-node vertical vibration of a hull girder in waves. It will result in a large number of stress number. So when talking about the structure problem of a large ship in waves, the fatigue damage caused by springing and whipping is an key point which cannot be avoided in its design stage (Wang, 2012, Storhaug, 2007, Drummen et al. 2008). Figure 8 shows the comparison of VBMs of each loading condition with different forward speeds in heading seas. It presents the feature of VBMs to wave frequencies. Generally speaking, the components of wave frequencies have no remarkable difference, it means they have similar amplitudes whether in full-load and ballast conditions. However, the components of high frequencies at low forward speeds in four loading conditions are all smaller than those at the forward speed 14.8 knots. It demonstrates that those high-frequency parts which could result in high-frequency vibrations including springing and whipping will be influenced obviously by high speeds. The encounter wave frequency will be higher at a forward speed in heading seas. So the increasing of high-frequency components in VBMs was influenced by the encounter wave number in a same time interval. When talking about the influence of loading conditions on VBMs, it can be found from Figure 8 that 2-node components of VBMs are higher in ballast conditions especially the forward speeds are low. 3-node components of VBMs are also higher in two ballast conditions especially at the forward speed 14.8 knots. Highorder components of VBMs more than 3-node is not discussed in this paper because the 3-D hydroelasticity results did not present the clear feature of high-order modes more than 3-node. When comparing the VBMs in B-1 and B-2 condition, it can be found that small draft in B-1 condition

Figure 9. Comparison of VBMs between experimental and theoretical results (heading seas, U = 14.8 knots).

Figure 8. Comparison of VBMs of each loading condition at different forward speeds (heading seas).

Figure 10. Comparison of VBMs of experimental results between heading and following seas (U = 14.8 knots).

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calculations, but also the experiments. Usually, ship designers believe that experiments can offer more accurate and believable values for their design efforts. However, the cost of experiments will restrain the scale of model, number of wave conditions, loading conditions and many other factors. So, the designers will usually take into consideration of all valuable methods to decide the design wave load of a ship.

highest peak in F-2 condition is about 1.69 times that in B-1 condition. So F-2 condition should be paid more attention than B-1 condition when using the VBMs to check the structural strength of the bulk carrier. Similar to the motion responses, the comparison of VBMs of each loading condition in heading and following seas at zero speed by theoretical method is presented in Figure 11. No remarkable discrepancy was found from these four loading conditions. So the similar conclusion can be making that without consideration of the wave condition in following seas in the bulk carrier’s early stage is acceptable. Comparison of VBMs between experimental and theoretical results in following seas is presented in Figure 12. In F-2 condition, high value was presented in model test at a forward speed 14.8 knots. However, almost same peak value was presented in B-1 condition both form theoretical calculation and experimental data. Though the comparison was performed in figure, the forward speed is different. So if taking into account of the VBMs in following seas, the model test results will give the conservative results. As a matter of fact, many new ships’ design will employ not only the classification rules, theoretical

4

CONCLUSION AND REMARKS

Bulk carrier is one of the main ship form which is used in the international shipping market, it will be developed to its large scale trend because of the imperative seeking of the energy conservation and operational efficiency. This trend results in the rapid increasing of the high-frequency component in total wave loads of a ship travelling in waves. This kind of high frequency component is springing in small sea state. If the sea states become moderate and heavy, the component will be the superposition of springing and whipping. Generally speaking, springing behavior will not result in the problem of ship structural strength. However, it will give an increase in the number of stress cycle of ship structure which will bring about the problem of fatigue damage. On the other hand, whipping will not only bring about the problem of fatigue damage, but also the problem of structural strength. In such a way that the superposition of springing and whipping will definitely add the complexity of structural damage mechanization and make the research more difficult. Based on the 3-D hydroelasticity theory and experimental method at different loading conditions in heading and following seas, the characteristic of the waveinduced responses of a bulk carrier was presented in this paper. Some conclusions can be listed as follows: 1. A Timoshenko beam can be used as the dry structure in the 3-D hydroelasticity analysis of the bulk carrier. Wet modes with good accuracy can be achieved by using the dry modes and different wet structures for the 2-node and 3-node vertical bending modes. 2. At the forward speed 14.8 knots, remarkable peaks are appeared in the two heave curves of the full-load conditions, and there is a shift of both the heave curves towards high-value direction. No remarkable peak can be found in heave curves in ballast conditions at the same forward speed. And the pitch curves move to low-value direction when ω0 is about less than 4.0 rad/s. However, there is also a shift of the pitch curves towards high-value direction when ω0 is more than 4.0 rad/s. This kind of shift of motion

Figure 11. Comparison of VBMs of each loading condition in heading and following seas (U = 0.0 knot).

Figure 12. Comparison of VBMs between experimental and theoretical results (following seas).

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curve is similar in all four loading conditions. Good agreement is achieved according the comparison of theoretical and experimental data of the pitch and heave motions. 3. The high-frequency components of VBMs at low forward speeds in four loading conditions are all smaller than those at the forward speed 14.8 knots. Springing and whipping behaviors will be influenced obviously by high speeds. The increasing of high-frequency components in VBMs was influenced by the encounter wave number in a same time interval. The 2-node components of VBMs are high in ballast conditions especially the forward speeds are low. 3-node components of VBMs are also higher in two ballast conditions especially at a high forward speed. Small draft in ballast condition will cause high peak of 2-node vertical vibration. It is very important for the ship designers to take into consideration of the issue on strength and fatigue damage of ship structure. 4. The wave-induced responses of the bulk carrier especially the VBMs are higher in ballast conditions than in full-load conditions. 3-D hydroelasticity theory is an useful method to analyze the feature of the responses of the ship travelling in waves. And it can make up the limitation of experiments because of the high cost and the capacity of the facility. The 3-D code towards advanced direction with sufficient nonlinear factors is developing in CSSRC to be a better tool in ship design.

Gu. X.K., Shen. J.W. & Moan. T. 2000. Experimental and theoretical investigation of higher order harmonic components of nonlinear bending moments of ships. Journal of Ship Technology Research, Schiffstechnik, 4:143–154. Hirdaris. S.E., Price. W.G. & Temarel. P. 2003. Two- and three-dimensional hydroelastic modeling of a bulker in regular waves. Marine Structures, 16(8):627–658. Hu. J.J., Wu. Y.S., Tian. C., Wang. X.L. & Zhang. F. 2012. Hydroelastic analysis and model tests on the structural responses and fatigue behaviors of an ultralarge ore carrier in waves. Proc. of the Institution of Mechanical Engineers, Part M: Journal of Engineering of the Maritime Environment, 226(1):1–21. Jensen, J.J. & Vidic-Perunovic, J. 2002. On springing of mono-hull ships, DNV Workshop on Fatigue Strength Analysis of Ships, Finland. Mockler. S. 2009. Wave induced loads and motions on a container ship in regular waves. Individual project of ship science, the University of Southampton. Storhaug. G. 2007. Experimental investigation of wave induced vibration and their effect on the fatigue loading of ships. Ph.D thesis, Norwegian University of Science and Technology. Tian. C., Wu. Y.S. & Chen. Y.Q. 2009. The hydroelastic responses of a large bulk carrier in waves. Proc. of the 5th International Conference on Hydroelasticity, Southampton, UK. Wang, X.L. 2012. Investigations on Springing and Fatigue Damage of Ship Structures. Ph.D thesis, China Ship Scientific Research Center. Wang. X.L., Gu. X.K., Hu. J.J. & Xu. C. 2012. Experimental Investigation of Springing Responses of a Large LNG Carrier. Shipbuilding of China, 16:1–12. Wang. X.L., Hu. J.J., Gu. X.K., Ding. J. & Zhao N. 2014. Wave loads investigation of a VLCC by experimental and theoretical methods. Proc. of the ASME 2014 33rd international conference on ocean, offshore and arctic engineering, San Francisco, California, USA. Wu. Y.S. 1984. Hydroelasticity of floating bodies. Ph.D thesis, Brunel University. Zhu. S.J., Wu. M.K. & Moan. T. 2011. Experimental investigation of hull girder vibrations of a flexible backbone model in bending and torsion. Applied Ocean Research, 33:252–274.

ACKNOWLEDGEMENTS The authors would like to thank China Ship Scientific Research Center and China Classification Society for their kind supports. The first author also appreciates Prof. Yousheng Wu regarding his 3-D hydroelasticity computer code. REFERENCES Drummen. I., Storhaug. G. & Moan T. 2008. Experimental and numerical investigation of fatigue damage due to wave-induced vibrations in a containership in head seas. Journal of Marine Science and Technology, 13:428–445.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Numerical simulation of the dynamics of a large moored tanker Songsong Zhang, Qiu Jin, Jianjian Xin & Tingqiu Li School of Transportation, Wuhan University of Technology (WUT), Wuhan, Hubei, China WUT-UoS High Performance Ship Technology Joint Centre, Wuhan, China

Pandeli Temarel & W. Geraint Price Fluid Structure Interactions Group, University of Southampton (UoS), Boldrewood Southampton, Southampton, UK WUT-UoS High Performance Ship Technology Joint Centre, Wuhan, China

ABSTRACT: In this paper, the dynamics of a large moored tanker, in wind, waves and current, is investigated using the commercial software AQWA. Two typical mooring types are implemented, namely I-type and V-type, to examine the motion response of the moored tanker and the relevant cable tension under different environmental conditions. The results show that the V-type configuration reduces surge, sway and cable tension. The offloading system, with the tanker in tandem behind the FPSO, is further investigated using the V-type mooring configuration and comparing it to the conventional offloading system, in which the tanker is attached to the FPSO only with a hawser. The investigations show that the V-type configuration has advantages in terms of motion response and cable tension over the hawser connection. 1

INTRODUCTION

Naturally, the tugs used as part of the offloading system are useful during offloading operations, but are not considered in the simulations undertaken in this paper. It may be conjectured that the tandem offloading operation using only a hawser is not adequate for maintaining the position of vessels in the offloading system effectively. Thus, there is a need to design an appropriate mooring system for the shuttle tanker, to enhance its stability and safety. In this paper, two configurations of typical mooring systems are considered for the study of the dynamics of the tanker, excluding the actually complex mooring system together with the effects of the undersea pipes from the FPSO. The ANSYS-AQWA software, with the coupling method in time domain, is implemented for investigating the dynamics between ships and mooring system in wind, waves and currents during the oil offloading operation. The software is first validated through comparisons with experimental measurements for a floating moored breakwater. Two simple mooring configurations, V-type and I-type, are applied for analysis of the dynamics of the moored tanker alone, for a prescribed set of environmental conditions. The more efficient V-type configuration is implemented for the tandem offloading case and compared with the traditional hawser type offloading system. The investigations clearly show the advantages of the V-type tandem offloading system, in terms of motion response and cable tension.

With the development of offshore oil exploration, design and research of systems comprising large tanker and Floating Production Storage and Offloading (FPSO) is becoming the subject of more investigations. The crude oil extracted is preprocessed and instantaneously stored in the FPSO, which has to be offloaded onto the shuttle tanker. In most cases, the tanker in this system is moored by a hawser in tandem configuration, whilst the gas in the LNG-FPSO system is offloaded in side-byside configuration. Normally, the tandem offloading configuration is subject to more severe effects from the marine environment. Besides, it has the advantage of fast disconnection. An elucidation of the more detailed performance of the tandem arrangement is, therefore, essential. Only the tandem configuration is studied and discussed in this paper. According to a recent study by Wilkerson & Nagarajaiah (2009), the FPSO is offloaded on average every 10.5 days. A whole cycle covering loading, demobilizing and unloading has a duration of, at least, 8 days. During this process, the offloading takes about 75% of the whole time. In addition, increases in FPSO and shuttle tanker sizes result in increasing the offloading time. It is, therefore, necessary to predict the dynamics of the large moored tanker, to ensure its safety, as in this case the tanker will face more complex environmental effects.

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2 2.1

BACKGROUND TO COMPUTATIONS

The local axes are initially parallel to the spacefixed axes and it is assumed that body motions are small. Therefore, any Eulerian angular displacements coincide with the angular displacements about the space-fixed axes. Please note that only three degrees of freedom, namely surge, sway and yaw are used in this paper.

Computational process

The computations are carried out using the ANSYS-AQWA suite of software following the process illustrated in Figure 1. In the first step, the tanker and the FPSO are modeled in the aforementioned software. The frequency domain analysis solution, using potential flow theory, provides the radiation and wave excitation related components, as well as the motion response of the ships in a range of regular waves. In the second step, the overall response of the system, comprising both ships and the selected mooring system, in irregular waves is carried out using a time domain analysis. This step makes use of AQWADRIFT, whereby the relevant forces at each time step are obtained using the position, velocity and acceleration of each ship from the previous step and the equation of motion solved. Finally, the time record of motion responses and cable force are analyzed to obtain motion statistics and maximum tension. 2.2

2.2.2 Frequency and time domain methods The frequency domain analysis is carried for regular waves of unit. Laplace’s equation [L] is used as the governing equations. Boundary conditions are set up for the linearized free surface boundary condition [F], the impenetrable body surface condition [S], the bottom condition at depth h [B] and the radiation condition at infinity [R], as shown in Equation 1, namely

[L ] [F ] [S ] [B ] [R ]

Theoretical background

2.2.1 System of coordinates Each ship hull is treated as a rigid body, with six degrees of freedom. Three different systems of coordinates are set up, as shown in Figure 2. OX0Y0Z0 is the space-fixed axis system with origin on the calm water free surface. The other two are local body-fixed axis systems Gixiyizi, i = 1, 2, with origin at the bodies’ centre of gravity Gi.

Figure 1.

where nj and Uj denote the normal velocity of jth body. The total potential function, Φ, in Equation 1 can be expressed as follows: Φ( x, y, z, t ) φ ( x, y, z )e − iω t 6 ⎡ ⎤ − iω t (2) = ⎢(φ I d ) + ∑φ jxj ⎥ e j =1 ⎣ ⎦ where φI: incident potential function, φd: radiation potential function, φj: velocity potential function caused by motion in j direction, xj: motion response in j direction and ω: incident wave frequency. In the frequency domain analysis, the motion RAOs of each ship obtained using Equation 3, as follows:

Computational process.

[

Figure 2.

∇ 2 ( x, y z, t ) 0, In the fluid domain; ∂2Φ ∂Φ +g = 0, z = 0; 0 ∂z ∂ 2t ∂Φ = U j n j , at object plane; (1) ∂n ∂Φ i ∇Φ = 0; z h = 0 or lim z→ ∞ ∂n infinity y radiation n condition

Coordinate systems.

2

(M s + M a ( )))

]X ( ) = F ( )

RAO =

X( ) A

where

A: wave amplitude, Ms: ship mass matrix, Ma: added mass matrix, B: damping matrix, K: hydrostatics stiffness matrix, F: wave excitation vector and X: motion response vector.

(3) (4)

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The modified total wave force is calculated, as shown in section 2.2.3, and added to the other forces applicable to this problem to form the equation for drift and wave frequency motions. The equations of motion in the time domain are set up using the convolution integral method, namely:

[M

M d ] X (t ) + KX (t (t ) ∫ h(t )X )X ( )dτ 0 = Fsv (t ) + Fc (t ) + Fw (t ) + Fm (t ) Fh (t ) + Fd (t ) + Fwf w (t )

Thus simplifying Equation 7 to: N

i =1 j =1

i

j

t+

i

−

j

}.

(9)

t

S

FXw

(5)

FYw

acceleration vector, added mass matrix at drift frequency, slowly varying drift force vector, current drag force vector, wind drag force vector, mooring force vector, hydrostatic force vector, damping force vector, the total wave frequency force vector (i.e. diffraction and Froude-Krylov) and h(t): acceleration impulse response matrix.

M XYw

N

)

FXc

N

∑ ∑ {P

− ij

i =1 J =1

cos[ −(

i

j

)t + (

i

j

)]

N

FYc

}

+ P cos[ −(ω i + ω j )t (ε i + ε j )] + ij

N

{

+ ∑ ∑ Qij i [ (ω i i =1 J =1

+ Qiij+ si s n[ −((

ω j )t + (ε i i

j

M XYc

ε j )]

)t + (

i

j

N

N

∑ ∑ {P

ij

i =1 J =1 N N

cos[ −(

{

i

+ ∑ ∑ Qij sin[ −(ω i − i =1 J =1

j

)t + ( +

i

j

−

} (7)

where ωi and ωj are the frequencies of each wave pair, ai and aj are the corresponding amplitudes; εi and εj are the radiation phase angles. According to Newman (1974): ⎛ P − Pjj− ⎞ 1 ai a j ⎜ ii2 + 2 ⎟ , 2 aj ⎠ ⎝ ai Qij− = 0 Pij

1 CXc ρcVc2 LBPT 2 1 CYc ρcVc2 LBPT 2 1 CXYc ρcVc2 L2BPT 2

(11)

2.2.5 Mooring line drag and added mass force The mooring force Fm in Equation 5 is calculated using nonlinear catenary theory and updated at each time step based on the vessel’s position. The coupled responses of the structure are obtained solving Equation 5. The mooring line drag and added mass force must be accounted. The definitions of in-line or tangential and normal drag forces (per unit length) are as follows:

)]

}

(10)

where FXc, FYc and MXYc are surge and sway current forces and yaw current moment, respectively. CXc, CYc and CXYc are the surge and sway current force coefficients and the yaw current moment coefficients, respectively, T is the draught and V is the current velocity.

}

)] (6)

where Pij and Qij are the in-phase and out-of-phase components of the time independent transfer function. Assuming the sum-of-frequency contributions can be ignored, Equation 6 becomes: )

1 CXw ρwVw2 AT 2 1 CYw ρwVw2 AL 2 1 CXYw ρwVw2 AL LBP 2

where FXw, FYw and MXYw are the surge and sway wind forces and yaw wind moment, respectively. CXw, CYw and CXYw are the surge and sway wind force coefficients and the yaw wind moment coefficients, respectively. AT and AL are the transverse and longitudinal areas. LBP is the length between perpendiculars and Vw is the wind velocity. The resultant current forces and moment on moored vessels are calculated by the following equations:

2.2.3 Calculation of second order wave excitation The near field solution of second order wave excitation forces is:

Fsv(

{

2.2.4 Wind and current loads The resultant wind force and moment on moored vessels are calculated by the following equations:

where X : Md: Fsv: Fc: Fw: Fm: Fh: Fd: Fwf:

Fsv(

N

Fsv (t ) = ∑ ∑ Pij

FDt FDn

(8)

1 ρCDt dV V 2 1 ρCDn dV V 2

(12)

where CDt and CDn are the tangential and normal drag coefficients, respectively, d is the ‘nominal’

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diameter for chain, or wire diameter, and V is the velocity associated with the mooring line. The added mass force (per unit length) is defined as follows: Fa

Ca Aa

Table 2. Comparison of calculated and experimental results for surge RAOs for the floating moored breakwater.

(13)

where A is the Equivalent Cross Sectional Area, defined as πd 2/4 and Ca is the added mass coefficient. 3

Period (s)

Calculated

Experimental

Error

4.3 4.9 5.7 6.4 6.9

0.7 0.9 1 1.3 1.6

0.73 1 0.99 1.26 1.62

4.11% 10.11% 1.01% 3.17% 1.23%

EXPERIMENTAL VALIDATION

In order to validate the accuracy of the numerical calculation, the moored floating breakwater, in the shape of a rectangular box, experimented on by Dong et al. (2009) is used. The arrangement for the breakwater mooring system is shown in Figure 3. The calculations were carried out using full-scale dimensions, shown in Table 1 for the tank, the breakwater and the mooring lines. The experimental and calculated results for surge RAOs of this breakwater in regular waves of varying periods are compared in Table 2 and Figure 4. The surge RAOs were obtained from analysis of calculated time domain motion records. The comparison shows that calculation results agree well with the experiments.

Figure 4. Comparison between calculated and experimental surge RAOs for the floating moored breakwater.

This indicates that modeling errors and settings of parameter, when using the software, are acceptable and the simulation method is feasible and reliable. 4

MODELS AND PARAMETER SETTING

4.1 Environmental parameters All cases are calculated under marginal offloading environmental conditions, according to Wang et al. (2010). The marginal environmental parameters are shown in Table 3. 4.2 Ship and mooring line dimensions Figure 3. Rectangular box floating breakwater mooring system.

Two vessels, namely a tanker and an FPSO, are modelled in this paper. Their principal dimensions are shown in Table 4. The water depth used in the simulation is 1000 m. The simulation method for the mooring lines is the non-linear catenary method; thus, detailed mooring line section parameters are required, shown in Tables 5 and 6.

Table 1. The principal dimensions of floating breakwater system—full-scale. Name

Parameter

Value

Tank

Length (m) Width (m) Depth (m) Width (m) Length, L (m) Depth, D (m) Designed draft, T (m) Length (m) Number

100 14 12 9 13.5 5.4 4.05 13.0 4

Breakwater

Mooring line

4.3 Coefficients of wind and current The low frequency excitation caused by the random waves and wind loading results in resonant motion responses. This may lead to high mooring line forces, according to Brown et al. (1999). Therefore, the low frequency forces, wind loading, as well as, current loads are taken into consideration in the calculation. The Oil Companies International

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Table 3.

Definition of environmental conditions.

Wave

Wind Current

Table 4.

Wave spectrum Gama Sig. wave height (m) Peak wave period (s) Wind spectrum 1 hr. mean speed (m ⋅ s−1)

JONSWAP 3.3 3.1 9.65 API 12

Surface speed (m ⋅ s−1)

0.4

The particulars of the tanker and FPSO.

Main particulars

Tanker

FPSO

Length LPP. (m) Breadth (m) Depth (m) Design draft (m) Displacement (m3) Number of cables

316 60 29.7 19.2 300000 2

302 59 31.3 20.8 320000 3*4

Table 5.

Table 7. Coefficients for wind and current in different directions. CXw

CYw

CXYw

0 45 90 135 180

0.75 0.45 0.04 −0.55 −0.96

0 0.58 0.72 0.5 0

0 −0.166 −0.112 −0.06 0

Direction (°)

CXc

CYc

CXYc

0 45 90 135 180

0.036 −0.012 0.01 0.005 −0.034

0 0.4 0.6 0.38 0

0 −0.16 −0.01 0.05 0

Table 8.

Mooring line characteristics for the tanker.

Main particulars

Section 1

Section 2

Section 3

Length (m) Mass (kg/m) Area* (m2) Stiffness EA (MN) Max Tension (MN)

1000 150 0.0158 794.0 75.0

500 42 0.0158 690.0 75.0

700 170 0.0158 794.0 75.0

Chain Wire

4.4

Mooring line characteristics for the FPSO.

Main particulars

Section 1

Section 2

Section 3

Length (m) Mass (kg/m) Area* (m2) Stiffness EA (MN) Max Tension (MN)

121.92 164.84 0.01 794.0 75.0

1127.8 42 0.01 690.0 75.0

45.7 164.84 0.01 794.0 75.0

Chain and wire coefficients. CDt

CDn

Ca

0.6 0.2

3.2 1.8

1.6 1.02

Mooring line hydrodynamic coefficients

Mooring line hydrodynamics are greatly affected by the selection of mooring line hydrodynamic coefficients; thus, requiring caution when making the selection. According to Brown et al. (1999), the mooring line hydrodynamic coefficients for the chain and wire are shown in Table 8. The calculation method of the drag and added mass forces was detailed in subsection 2.2.5.

*Equivalent Cross Sectional Area, A. Table 6.

Direction (°)

5

TANKER MODEL

In the first instance the moored tanker is considered on its own. To maintain the position of tanker, as well as minimize the tanker mooring system’s effects on the FPSO mooring system when the two are in tandem, two simple mooring types are applied to the tanker. For the, so called, I-type configuration the mooring line is attached at the bow and stern of the tanker, respectively, as shown in Figure 5(a). For the V-type configuration the mooring line is attached at port and starboard sides in the bow region, as shown in Figure 5(b). The other ends of all cables are fixed onto the sea bed. Calculations are carried out for both configurations and for two cases involving wind, irregular waves and current acting in different directions, as shown in Table 9. Low frequency drift is also taken into account. Calculations were carried out for 10800 s, i.e. 3 hours, in both cases.

*Equivalent Cross Sectional Area, A.

Marine Forum (OCIMF 1994) published guidance for the computation of wind and current loads on VLCCs, i.e. tankers in the 150,000 to 500,000 DWT range. Wind and current coefficients are presented in non-dimensional form for a moored vessel. Coefficients for wind and current forces for the tanker and FPSO are obtained from these data using interpolation and are shown in Table 7. The same coefficients for wind and current forces are applied to both the FPSO and the tanker as they have similar dimensions. The calculation method for wind and current loads was detailed in subsection 2.2.4.

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Figure 5.

Mooring types. (a)I-type,(b) V-type.

Table 9. Direction of wind, waves and current conditions. Direction (°)

Case 1 Case 2

Wind

Wave

Current

180 180

180 150

180 210

Figure 6. Motion response of the moored tankers using I-type and V-type configurations. (a) Case 1, (b) Case 2.

The maximum motion responses for both cases shown in Table 9, are presented in Figure 6. In the relatively calm environmental conditions, depicted by the marginal offloading condition, the motion responses of the moored tanker are quite small. The predicted yaw is nearly the same for either configuration at each case considered. However, the surge resulting from the I-type mooring configurations is larger than that of the V-type, for both cases considered. The same observation is also applicable for the sway, especially for case 2. The predicted cable forces are shown in Figure 7. The load on both cables of the V-type configuration is symmetrical, with the same maximum value in case 1 and very close maximum values in case 2. On the other hand the load on the fore cable i.e. cable 1, is always greater than that of the stern cable in the I-type configuration. The stern cable, i.e. cable 2, is underutilized making the whole mooring system inefficient and resulting in overall larger cable tension for the I-type configuration. In conclusion, the V-type configuration, being more efficient, is adopted for further investigations. In the V-type configuration, changing the distance between the two anchor points results in different angle between the two mooring lines when

Figure 7. Cable force of the moored tanker using I-type and V-type configurations. (a) Case 1, (b) Case 2.

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tanker behind the FPSO with cables, as shown in Figure 10(b). The FPSO has single point mooring, rotating with wind and current, thus, always against them. Calculations, i.e. solving Equation 5, are carried out only for case 1 (see Table 9), i.e., directions of wind, wave and current are all 180°. The shadow effects of wind and current are ignored. The duration of simulation is set to 24000s, approximately 6.7 hours. The maximum motion response of the tanker, for surge, sway, and yaw, are shown in Figure 11, for both offloading systems shown in Figure 10. As can be seen, using the traditional way of offloading, Figure 8. Variation of the mooring angle with distance between the two anchor points of the V-type configuration.

Figure 9. Variation of maximum motion responses with mooring angle for the moored tanker, case 1.

the system reaches balance, as shown in Figure 8. This angle can be used as the parameter for seeking the optimal form of the V-type mooring system. The predicted maximum motion responses of the moored tanker for different mooring line angles, under the environmental conditions of case 1, are shown in Figure 9. As can be seen from Figure 9 yaw and sway decrease with increasing mooring angle, however surge, the most important factor for evaluating the safety of the offloading system, first decreases and then increases with increasing mooring angle, reaching a minimum when this angle ranges between 80 to 90 degrees. The value of sway and yaw is quite small in this range. Therefore, it is the optimal range for tanker cables and is applied to the offloading system. 6

Figure 10. Offloading system forms. (a) Tanker attached to FPSO with a hawser, (b) Tanker moored with two cables.

OFFLOADING SYSTEM

Two different ways of offloading are discussed. One is the traditional way namely, attaching the tanker to FPSO at the stern with a hawser, as shown in Figure 10(a). The other is mooring the

Figure 11. Maximum motion responses of the tanker for the offloading systems shown in Figure 10.

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Maximum tension of hawser nearly reaches fracture tension limit when the distance is greater than the hawser initial length, which does occur occasionally as seen from Figure 12. The hawser may snap apart on these occasions. By contrast, the tension of the cable offloading system is more even and with a much smaller maximum tension value. Additionally, one has to bear in mind that the tanker will begin to yaw when it is moored behind the FPSO for a substantial period of time. The flow is disturbed by the FPSO, resulting in increasing yawing of tanker. This also follows the conclusions by Fucatu et al. (2001), namely that the shadow effect generates sway force and yaw moment which do not otherwise exist in the undisturbed condition.

with the hawser, results in much greater motions than the case where the tanker is moored. Furthermore, the maximum responses of the tanker for the latter case, i.e. when moored with cables, are even smaller than the tanker moored on its own (see Figure 6(a)). This observation agrees with the conclusion by Fucatu et al. (2001). They pointed out that the surge force can decrease to half of its undisturbed value due to the shadow effect from the presence of the FPSO, as observed in their experiment. Following Morandini et al. (2002), we conclude that the most important design variables of the offloading procedure are as follows: • distance between the vessels, to avoid collision, • hawser tension, to avoid failure of the connections, • relative vessel heading, to minimize the effects of yaw instability should fishtailing occur.

7

Accordingly, the distance between the tanker and the FPSO, as well as the maximum tension of hawser or cable, need to be monitored. The variation of the distance between the tanker and FPSO against time is shown in Figure 12, for both offloading ways. The corresponding maximum hawser tension values are shown in Table 10. Initial distance between tanker and FPSO is 50 m. Initial hawser length is 55 m. Examining Figure 12 it can be seen that the distance between tanker and FPSO is controlled so that it is not less than 30 m for either offloading way. However, it is evident that the cable moored offloading provides for more stable operations.

Figure 12. FPSO.

A systematic set of calculations simulating the dynamics of a large moored tanker has been carried out, focusing on mooring configurations. All the cases are calculated under marginal offloading environmental condition, with the effects of wind, current, wave loads and the low frequency drift force included. Two types of mooring configurations are proposed and compared. It is clear from the results of the comparative study that the V-type is much better than the I-type due to smaller ship response and more even cable tension. Compared with the traditional way of connection with a hawser, mooring the tanker behind the FPSO with cables arranged in V-type configuration, is shown to be more effective. From the comparison studies, it can be seen that both ship motion responses and the cable tension is smaller for the case when cables are used, under the same environmental conditions. During the simulation study, it was observed that the shadow effect in the tandem offloading system will cause the tanker to yaw. When the yaw response continues to build up to a certain extent, the forces on the tanker would increase rapidly and cause the tanker to drift backwards, which may result in significant damage to the oil pipelines. Thus, the tanker has to be adjusted to alter the impact of the shadow effect, if the offloading process takes a long time. Furthermore, when the distance between the two vessels is less than 20 m, hydrodynamic interference also results in sudden yawing of the tanker, with resultant rapid increases in wind and wave forces causing the tanker to drift backwards. Thus, the two vessels cannot be too close to each other in the offloading process. A relatively safe value, for the size of tanker considered in this study, is of the order of 50 m.

Variation of distance between tanker and

Table 10. Maximum tension of hawser and Cable.

Max tension (kN)

Cable

Hawser

1262

63810

CONCLUSIONS

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The optimization of mooring condition discussed in this paper is a preliminary study. A more detailed analysis will be put forward for further research.

breakwater hydrodynamic characteristics. Fishery Modernization 3: 7–11. Fucatu, C.H., Nishimoto, K., Maeda, H. & Masetti, I.Q. 2001. The shadow effect on the dynamics of a shuttle tanker. Proc. 20th Int. Conf. on Offshore Mechanics and Arctic Engineering, Rio de Janeiro, Brazil. Morandini, C., Legerstee, F. & Mombaerts, J. 2002. Criteria for analysis of offloading operation. Proc. Annual Offshore Technology Conference, Houston, USA, 302–307. Newman, J.N. 1974. Second order slowly varying forces on vessels in irregular waves. Int. Symp. on the Dynamics of Marine Vehicles and Structures in Waves, University College London, UK. OCIMF. 1994. Reduction of Wind and Current Loads on VLCCs. 2nd Edition, Witherby & Co.Ltd, England. Wang, Q., Sun, L.P. & Ma, S. 2010. Time-domain analysis of FPSO-tanker responses in tandem offloading operation. Journal of Marine Science and Application 9(2): 200–207. Wilkerson, S.M. & Nagarajaiah, S. 2009. Optimal offloading configuration of spread-moored FPSOs. Journal of Offshore Mechanics and Arctic Engineering 131(2): 021603.

ACKNOWLEDGMENTS The present work is supported by the National Science Foundation of China (No. 51179142, 51139005), 111 Project (No B08031), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20130143110014). REFERENCES Brown, D.T. & Mavrakos, S. 1999. Comparative study on mooring line dynamic loading. Marine Structures 12(3): 131–151. Dong, H., Wang, Y., Hou, Y. & Zhao, Y. 2009. Experimental study on the rectangular box floating

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Hydroelasticity

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Slamming impact loads on high-speed craft sections using two-dimensional modelling J. Camilleri, D.J. Taunton & P. Temarel Fluid Structure Interactions Group, University of Southampton, Southampton, UK

ABSTRACT: The slamming impact loads experienced by high-speed craft travelling in rough seas are numerically investigated by analysing the idealized problem of a two-dimensional rigid wedge impacting an initially calm water surface. We are using the commercial CFD software Star CCM+, which is based on the Finite Volume (FV) method and an interface capturing scheme of the Volume-of-Fluid (VOF) type. The set-up of the numerical model is described in detail. The influence of mesh size, time step, and other numerical parameters such as inner iterations and relaxation factors, as well as three-dimensional modelling, on the solution is studied in a systematic manner. Comparisons against published experimental data show favorable agreement. 1

INTRODUCTION

Zhao et al. (1996) applied a nonlinear Boundary Element Method that accounts for flow separation to study the water entry of a wedge and a bow section. This 2D model was found to overpredict the pressures for the wedge section. The authors were able to quantify the three-dimensional (3D) effects and successfully correct their results. Muzaferija et al. (1998) used a FV method with a free-surface capturing model of the VOF type. This method is well capable of predicting the large free surface deformations, including overturning and breaking waves. However, a fine grid is needed to capture all the details of the jet. The numerical model was validated against the experimental results of Zhao et al. (1996). The size of the numerical domain (tank) and three-dimensional effects were found to have a significant influence on the results. Sames et al. (1999) adopted a similar approach to predict the impact loads on the bow section of a containership. The motion history of water entry was found to have a significant influence on the pressures. The authors conclude that coupling with a rigid body motion solver is required to achieve realistic design pressures. Similar conclusions were reached by Reddy et al. (2002). Recently developed modelling techniques applied to the water entry problem include Smoothed Particle Hydrodynamics (SPH) methods, e.g. see Oger et al. (2006), Veen & Gourlay (2012) and explicit Finite Element methods, e.g. see Stenius et al. (2006), Wang & Guedes Soares (2012). Brizzolara et al. (2008) applied a wide range of potential and viscous CFD methods to assess their suitability for simulating a bow section impacting at various speeds and heel angles.

Slamming in rough seas is of concern for a wide range of ship and offshore structures, particularly for small high-speed craft, such as patrol, military and rescue craft, which often have to travel at the highest speed possible. The craft frequently launches off waves, emerging from the water, and then violently impacts onto the free surface with high relative velocity. This may lead to crew injuries and structural failure, either due to fatigue loading or catastrophic failure during an extreme event. Slamming induced loads constitute a significant portion of the design loads and, thus, need to be carefully considered for safe, reliable, and efficient structural design. An accurate prediction method for the loads and responses is, therefore, crucial. Traditionally, hull-water impacts have been investigated by analysing the idealized problem of a twodimensional hull section such as a bow or V-shaped section impacting an initially calm water surface. Significant amount of research has been reported in the literature on the two-dimensional (2D) water entry problem. Generally, the structure has been assumed rigid and the hydrodynamic impact loads and structural response obtained without coupling, neglecting the effect of the flexibility of the structure on the fluid loading. Pioneering research was carried out by Wagner (1932) who applied potential flow theory to estimate the pressure distribution on rigid wedges for constant velocity impact. Since Wagner (1932), the problem has been approached using a wide range of analytical and numerical methods.

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Air and water are considered as two immiscible components of a single fluid, separated by an interface called the free surface. The location of the free surface (spatial distribution of air and water) is defined in terms of a scalar variable called the volume fraction, found by solving an additional transport equation. The fluid properties are calculated using the physical properties of the two phases and their volume fractions. In Star CCM+, the VOF multiphase model is used to model the free surface and its deformations. To complete the mathematical model, initial and Boundary Conditions (BC) are specified. Initial conditions describe the state of the flow at start time, whereas boundary conditions describe the state of the flow on the domain boundaries. The latter can be either Dirichlet or Neumann type. Dirichlet conditions specify the actual value of the flow variable (inlet and wall BCs) whereas Neumann conditions specify its gradient (outlet and symmetry BCs). The above equations close the mathematical model of flow with free surfaces computed using moving grids. The motion of rigid bodies is assumed to be governed by Newton’s laws of motion. The external forces acting on the body consist of flow-induced forces (pressure and shear contributions) obtained by solving the mathematical model of fluid flow, and other body forces such as the gravitational force. In Star CCM+, the Dynamic Fluid Body Interaction (DFBI) model is used to simulate rigid body motion. The spatial solution domain is first divided into a finite number of contiguous CVs, collectively known as the ‘mesh’ or ‘grid’. Each CV surrounds a point at which the dependent variable is evaluated. The CVs can be of any polyhedral shape, allowing for local refinements in the regions of interest. The time interval of interest is also subdivided into a finite number of small time steps. Volume and surface integrals in the governing equations are approximated for each CV using the mid-point rule. Linear interpolation and linear shape functions are generally used to compute cell face values and gradients from known information at the CV centers. All of these approximations are of second order accuracy. An implicit time-marching scheme is used as it allows larger time steps to be used and provides better stability compared to explicit schemes. Implicit schemes, however, require solutions of system of coupled non-linear equations, which are computationally expensive. The time derivative is approximated using either first-order Euler or second-order three-time-levels scheme, with the latter being more accurate but for flows with free surfaces it has a constraint on the time step size, as discussed in section 3.2.1.

The 2D slamming problem has also been investigated experimentally. Aarsnes (1996) carried out drop tests using a wedge and a bow section from different heights and at different roll angles to investigate the pressure distribution and resulting impact force. Yettou et al. (2006) studied the influence of deadrise angle, wedge mass and drop height on the pressure distribution and dynamic behavior of a free-falling wedge. All three parameters were found to have a significant influence on the wedge velocity, and thus, the impact loads. Tveitnes et al. (2008) measured the vertical hydrodynamic force on rigid wedge-shaped sections during constant velocity entries. Lewis et al. (2010) presented a comprehensive set of high-quality experimental data for a free-falling 25° deadrise wedge for two drop heights and wedge mass values. The results are in line with those of Yettou et al. (2006) namely, pressure increases with increasing drop height and wedge mass. A detailed experimental uncertainty analysis is presented making the data highly suitable for validation of numerical models. The present paper investigates modelling of a 2D rigid wedge impacting an initially calm water surface using the commercial CFD software Star CCM+. The influence of mesh size, time step, and other numerical parameters, as well as the influence of 3D effects, on the solution is studied in a systematic manner. The numerical results are compared against the experimental drop test data of Lewis et al. (2010), showing good agreement for different drop heights and wedge mass. 2

NUMERICAL SOLUTION METHOD

The computations presented in this paper are performed using the commercial CFD software Star CCM+. It uses the FV method to transform the continuous governing equations into a system of algebraic equations that can be solved numerically. The free surface is modeled using an interface capturing method of the Volume-of-Fluid (VOF) type. Motion of the rigid body i.e., the wedge, is determined as part of the solution by solving the rigid body equations of motion and the mathematical model of fluid flow in a coupled manner. The starting point for the computations of incompressible viscous fluid flows is the NavierStokes (NS) equations i.e., the mass and momentum conservation equations. In the present approach, a single grid that extends over the entire computational domain and moves with the rigid body (moving-grid approach) is adopted. Thus, the space conservation law, describing the conservation of volume when the Control Volumes (CVs) change their shape or position with time, must also be satisfied.

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3

The discretization of the convective part of the volume fraction equation should neither produce numerical diffusion nor unbounded values of volume fraction. In Star CCM+, the HRIC convection discretization scheme of Muzaferija et al. (1998) is used to achieve a sharp interface and avoid unphysical solutions. The numerical solution of the system of coupled non-linear equations is obtained in a segregated iterative manner following the SIMPLE algorithm. The linearized momentum equations are solved first using prevailing pressure and mass fluxes to yield an approximate velocity field. The pressurecorrection equation, derived from the discretized continuity equation, is then solved to correct the pressure and velocity fields such that continuity is satisfied. Additional conservation equations, such as for volume fraction are then solved. The process is repeated until all the nonlinear and coupled equations are satisfied within an acceptable tolerance (inner iterations) after which the solution advances to the next time step. In the SIMPLE algorithm, relaxation techniques, governing the extent to which the old solution is supplanted by the newly computed solution are employed. The choice of number of inner iterations and underrelaxation factor values (for pressure, velocity, and other scalar variables) can have a significant influence on the convergence and stability of the solution. The greater the under-relaxation factor used the more of the new solution is used in the calculation. Low under-relaxation factors help maintains numerical stability in the solution at the expense of slower solution convergence rates and vice-versa. Optimal under-relaxation factor values are problem dependent and systematic studies should be performed to determine these values. The number of iterations should be such that the solution is converged within each time step. Small physical time steps imply that the solution is not changing much from one time step to the other, thus, requiring less inner iterations per time step. However, for strongly coupled problems, such as free surface flows with body motion, more inner iterations (around 10) are generally needed to achieve convergence. This can be judged by monitoring the magnitude of the residuals drop within each time step. For coupled simulations of fluid flow and flowinduced body motion the iteration loop is extended to update the body position. The updated flow fields are used to estimate the flow-induced forces on the body. The governing equations of rigid motion are then solved to find the new body position and adapt the grid accordingly. Since the boundary conditions also change position with time, they need to be adapted at each time step. For instance, the hydrostatic pressure on the bottom boundary increases as the wedge is falling.

WATER ENTRY OF A RIGID WEDGE

3.1 Experimental drop tests of Lewis et al. (2010) Lewis et al. (2010) carried out a detailed experimental investigation into the impact of a free falling 25° deadrise angle rigid wedge with water. A schematic diagram of the experimental set-up is shown in Figure 1, and the tank and wedge dimensions are given in Table 1. The wedge was dropped from two different heights, 0.5 m and 0.75 m and for each drop height two wedge masses were tested, 23.3 kg and 33.3 kg. The wedge impact velocities, calculated using the high speed camera images are 2.78 m/s and 3.58 m/s for 0.5 and 0.75 m drop height, respectively. The time-varying pressure distribution on the wedge bottom during impact is measured using six pressure transducers evenly distributed as shown in Figure 2. Wedge acceleration and position are also measured. The experimental pressure and acceleration time histories for a 0.5 m drop height and a 23.3 kg wedge mass are shown in Figure 3. It is important to note that the pressure and acceleration data is low pass filtered using a cut-off frequency of 1000 Hz and 250 Hz respectively to remove any high frequency noise in the signals. A high-speed camera is also used to capture the impact and subsequent formation of the jet. 3.2

Model generation

In this paper, only one test configuration is analysed in detail, a 23.3 kg wedge dropped form a

Figure 1.

Experimental set-up (Lewis et al. 2010).

Table 1. Tank and wedge dimensions; L–length; W–width; D–depth. Tank L × W × D (m)

Wedge L × W × D (m)

5.8 × 0.75 × 0.59

0.944 × 0.735 × 0.22

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Figure 2. 2010).

Position of sensors on the wedge (Lewis et al.

Figure 4. 2D model of wedge and tank, before freefall; air depicted by the dark shaded region; medium grid (Table 4).

Figure 3. Pressure and acceleration time history for a 0.5 m drop height with a wedge mass of 23.3 kg (Lewis et al. 2010). Figure 5. Views of the mesh illustrating (a) local refinements and (b) prism layers; medium grid (Table 4).

height of 0.5 m. However, the peak pressures for the other configurations are also presented. The 2D discretization of the wedge and tank is shown in Figure 4. Only half of the wedge is considered, and a symmetry boundary condition is imposed on the geometrical symmetry plane of the wedge. The size of the numerical tank is extended to limit the influence of the domain boundaries on the solution. The half-width of the tank is 4.0 m (y-direction), the height is 5.0 m (z-direction) and the initial water depth is 2.36 m. The geometrical model has an in-plane thickness (x-direction) of 10 mm with a symmetry condition applied at the front and back faces to ensure 2D flow. A noslip wall condition is applied on the wedge walls

and the (right-hand-side) tank wall. The bottom boundary is set to velocity inlet and the top boundary to pressure outlet with prescribed velocity and pressure profiles. The single mesh extending over the whole computational domain is trimmed Cartesian, with local refinements in the vicinity of the wedge bottom and on the free surface to accurately resolve the highly localized peaked pressure distribution and the free surface profile, respectively shown in Figure 4 and Figure 5. Orthogonal prismatic cells (prism layers) are used near the wedge bottom surface to accurately resolve the near wall flow features such as jet

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If the free surface moves by more than half a cell per time step, the HRIC scheme can lead to local disturbance in the flow and even divergence. A systematic study has been performed to find an optimum time step size for the original mesh developed i.e., the medium grid shown in Figure 4. Three different time steps are studied. Their influence on the maximum Courant number (CFL), CPU time, pressure and vertical force are presented in Table 2, Figure 6 and Figure 7 respectively. It is important to

formation and flow separation, which can have a significant influence on the pressure and velocity fields (see Figure 5b). The refined mesh extends well below the wedge apex to accurately capture the free surface, during time interval of interest. The mesh gets coarser towards the tank walls as shown in Figure 4. This is achieved by setting up volumetric controls with varying levels of refinement. Preliminary simulations with a 0.5 m drop height were found to overestimate the impact velocity. This is most likely to be due to mechanical friction between the bearings and cylindrical posts in the experiments (see Figure 1), which is not accounted for in the simulations. The initial position of the wedge keel was thus set to 0.44 m above the water surface to match the experimental impact velocity. The flow is assumed to be viscous and incompressible. Furthermore, a laminar flow model is employed, meaning that the Navier-Stokes equations are solved rather than the ReynoldsAveraged-Navier-Stokes (RANS) equations. Turbulence effects are believed to be small compared to the large slamming impact pressures which occur during the initial stages of water entry (Piro & Maki 2013). The free surface is modelled using the VOF multiphase and VOF wave models. The VOF wave model is used to simulate surface gravity waves on a light fluid-heavy fluid interface. Here a flat wave with zero fluid velocity is used to represent the calm water surface. The VOF wave model provides the necessary information to initialize the volume fraction, pressure and velocity fields (initial conditions), and to describe the flow at the domain boundaries (boundary conditions). The DFBI model is used to simulate the motion of the rigid body and grid in response to the flow induced forces and gravity. In the computations, motion is restricted to the negative z-direction only (vertical). All computations were carried out with an implicit unsteady time-stepping scheme and a constant time step size. The choice of time step, grid size and other numerical parameters such as inner iterations and under relaxations factors has been performed in a systematic manner as discussed in the following.

Table 2. Effect of time step on Courant number and CPU time; medium grid. Time step size (s)

Courant number

CPU time* (h)

0.0001 0.00005 0.00002

≈ 2.0 ≈ 1.0 ≈ 0.5

0.8 4.7 11.4

*Simulations were run using 48 cores of the IRIDIS 4 supercomputer at the University of Southampton running at 2.6GHz.

Figure 6. Effect of time step size on the pressure time history for all the pressure sensors; medium grid (Table 4).

3.2.1 Effect of time step size The choice of time step size is mainly governed by the time scales of the physical phenomena of interest i.e., the highly localized (in time) peaked pressure distributions. Furthermore, the time step size must also be chosen in relation to the grid size, to avoid instabilities in the solution. The Courant number is a helpful indication for selecting the time step size. It describes how far the fluid travels during one time step relative to the mesh size. For flows with free surfaces, and a second-order time integration scheme, the Courant number has to be less than 0.5.

Figure 7. Effect of time step size on the time history of vertical force on the wedge; medium grid (Table 4).

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relatively fine, the influence of the relaxation factor values and number of inner iterations on the pressures was found to be small. The chosen values were, therefore, such that the solution is converged within each time step, i.e. the residuals drop by at least one order of magnitude, and stability in the pressure and velocity fields is attained.

note that the computed pressures are surface averaged over a circular area 9 mm in diameter; equivalent to the diameter of the pressure sensors used in the experiments. Furthermore, the presented vertical force time histories are filtered using a 10-point moving average to remove high frequency noise, particularly during the initial stages of water entry, to enable better comparison and analysis. The pressure time histories show good agreement indicating that the chosen time steps are sufficiently fine to resolve the highly localized impact pressures, except for the peak pressure at P1. Sensor P1 is located very close to the impact region (see Figure 2) where the fluid is suddenly accelerated to very high velocities. That is to say, in this location the pressure is high and rising very sharply, which is difficult to capture in the experiments, as well as the simulations. The Courant condition might also be violated in this region leading to local flow instability. The force plots also show good agreement both in terms of shape and magnitude. Further simulations are thus carried out using a 0.00005 s time step size even though the Courant condition is violated. However, this occurs in a very small number of cells (around 10) and the solution is stable. Even though a 0.0001 s time step size provides reasonably good predictions, it is important to ensure that the Courant condition is not violated (CFL ≈ 2.0) to avoid instabilities. Furthermore a 0.00002 s time step size is too fine and requires significant computational power.

3.2.3 Effect of mesh size A sufficient grid resolution is needed, particularly near the wedge bottom, to accurately capture the highly localized (in space) impact pressures during water entry. Furthermore, the mesh needs to be sufficiently fine such that further refinement won’t change the solution (grid independent). A systematic study is performed to assess the influence of the mesh size on the pressures and vertical force. The principal parameters of the three grids tested are presented in Table 4. Refinements were applied to the whole domain and the number of prism layers was adjusted to obtain a smooth transition between the near wall mesh and the core mesh. The grid used in the previous studies (time step and numerical parameters), is referred to here as the ‘Medium’ grid. The time step size is also adjusted by the same factor (of two) to maintain the same Courant number. The computed pressure and vertical force time histories on the three grids are shown in Figure 8 and Figure 9 respectively. The coarse grid is not Table 4.

3.2.2 Effect of numerical parameters As discussed in section 2, the numerical stability and convergence of the solution depends on a number of parameters, such as temporal discretization, relaxation factors and inner iterations. The influence of these parameters on the solution is studied in a systematic manner, by changing their value, one at a time, and noting the effect on the results. The parameters studied, values tested and those chosen are presented in Table 3. Simulations with a 1st order scheme were found to show negligible difference indicating that the violation of the Courant condition in a few number of cells for a 0.00005 s time step does not affect the stability of the solution. As the time step chosen is

Particulars of the grids tested.

Grid

Cell count

Δxmin* (mm)

Δtmin* (mm)

Time step size Δt (s)

Coarse Medium Fine

8534 24174 111365

5 2.5 1.25

5 2.5 1.25

0.0001 0.00005 0.000025

*Dimensions of the smallest cell located along the wedge bottom; longitudinal (x-) and tangential (t-) directions.

Table 3. Numerical parameters tested in the systematic study; medium grid (Table 4). The chosen values are also included. Parameter

Tested

Chosen

Temporal discretization Under-Relaxation: Pressure Under-Relaxation: Velocity Maximum inner iterations

1st/2nd Order 0.3, 0.4, 0.5, 0.6 0.7, 0.8, 0.9, 1.0 8, 10, 15

2nd order 0.4 0.9 10

Figure 8. Pressure time histories showing the effects of systematically refining space and time discretisations.

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Figure 9. Vertical force time histories showing the effects of systematically refining space and time discretisations.

sufficiently fine to capture the maximum vertical force. The computations on the medium and fine grids agree well, both in terms of shape and magnitude indicating that the influence of the mesh size on the solution is small. The same conclusions can be reached from the pressure plots. The medium and fine grids show good agreement apart from the slight differences in peak values at sensors P1 and P2 of 1.5 kPa and 2 kPa respectively, due to reasons discussed earlier. The medium grid is thus chosen for further analysis.

Figure 10. Predicted and experimental pressure and acceleration time histories and peak values; medium grid (Table 4). Table 5. Numerical and experimental peak pressures and time of peak after impact, medium grid (Table 4). Experiments

3.2.4 Validation of numerical model The computed pressure and acceleration time histories, using the medium grid with a 0.00005 s time step size, are shown in Figure 10. The results are low pass filtered with a Butterworth filter built in MATLAB using the same cut-off frequencies as in Lewis et al. (2010). Here, zero time is when the wedge apex reaches the water surface, determined from the high-speed camera images in the experiments and the wedge position plot in the numerical simulations. The initial highest peak is recorded by sensor P1, after which sensors P2 to P6 record progressively lower peaks due to wedge deceleration. Also the shape of the pressure distribution changes from a sharp peak at P1 to a more rounded one at P6. The experimental peak pressures at the time after initial contact with water are also included in Figure 10 for comparison (connected black dots). Compared to the experimental results presented in Figure 3 the computed pressures and acceleration show good agreement in terms of overall shape. The numerical and experimental peak pressures and time of peak are presented in Table 5. Numerical peak pressures recorded by sensors P3 to P6 agree well with the experiments (difference of less than 0.6 kPa). However, P1 and P2 are over estimated. There are a number of possible reasons for this discrepancy. For instance, friction between the linear bearings and the vertical posts can have

Simulations

Sensor

Pressure (kPa)

Time (ms)

Pressure (kPa)

Time (ms)

P1 P2 P3 P4 P5 P6

25.5 19.8 16.6 9.96 6.16 3.78

8.8 15 21.8 30.9 42.8 57.1

33.6 26.4 16.6 10.4 6.7 3.9

5.3 10.9 17.8 26.5 37.1 50.9

a significant influence on the wedge velocity, after the initial impact, and, thus, the impact pressures. Unfortunately, the wedge velocity was not measured during the experiments, making it difficult to draw solid conclusions. Nevertheless, the acceleration time histories show some differences (see Figure 10), indicating that the wedge dynamics are not accurately predicted. Secondly, 2D simulations are known to overpredict the impact pressures and forces (Muzaferija et al. 1998). Hydroelastic effects are not believed to have a significant influence, the rigid wedge construction and conditions tested make any influence from elastic effects unlikely. The time of peak pressure after impact shows less favorable agreement. All numerical sensors record the peak values earlier in time. This difference can be attributed to the uncertainties associated in determining the point of impact in both the experiments and numerical simulations. For instance, Lewis et al. (2010) reports that the

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Figure 12. tank.

Figure 11. Experimental and numerical peak pressures showing the effects of drop height and wedge mass.

accelerometer begins to be affected by the impact 2.5 ms after the high speed camera images show the wedge touching the water. During this time the wedge travels approximately 10 mm. The experimental and numerical peak pressures for the other test configurations, using the medium grid (Table 4) are presented in Figure 11. The drop height has been adjusted for all cases to match the experimental impact velocity. As can be seen, peak pressure increases with increasing impact velocity, as also observed by Wang & Guedes Soares (2012). Generally speaking, the results for a 33.3 kg wedge show better agreement with the experiments. This further suggests that friction has a significant influence on the results. In this case, gravitational force is much larger than the frictional forces and the latter will therefore have less influence on the wedge velocity.

Three-dimensional model of the wedge and

Figure 13. Numerical and experimental peak pressures showing the three-dimensional effects; 0.5 m drop height, 23.3 kg mass.

about 520,000 cells. The rigid body velocity time history is the same for the 2D and 3D simulations. The predicted peak pressures for the 3D case are shown in Figure 13. Two-dimensional results and experimental measurements are included for comparison. Three-dimensional effects have a significant influence particularly during the early stages of water entry. The predicted pressures for sensors P1 and P2 are lower than those obtained with a two-dimensional grid and tend to compare more favorably with the experimental results (difference of 3.4 kPa and 3.7 kPa respectively).

3.2.5 Three-dimensional results Three-dimensional simulations were carried out to investigate the influence of these effects on the predictions. Due to symmetry, only one quarter of the experimental tank and wedge is modelled, as shown in Figure 12. The distance between the wedge end and the tank wall (x-direction, width) is 7.5 mm, as in the experiments. The boundary conditions, physics models, numerical parameters and time step are the same as in the 2D simulations. The medium mesh settings were adopted for the three-dimensional grid, with additional refinement in the narrow gap between the wedge end and the tank wall to accurately resolve the flow in this region. The dimensions of the cells are the same in all directions (isotropic mesh refinement). The mesh has

4

CONCLUSIONS

The impact of a rigid wedge with water has been successfully investigated using the commercial CFD software Star CCM+. The influence of mesh

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size, time step and other numerical parameters on the solution has been studied in a systematic manner. This gives more confidence in the numerical model. Sufficient grid and time step resolution is needed to accurately capture the highly localized impact pressures. Results show that a 0.00005 s time step size and a cell size of 2.5 mm (along the wedge bottom) is needed to accurately capture the impact pressures. For a sufficiently fine time step, numerical parameters such as relaxation factors and iterations were found to have a secondary effect on the results. The chosen values were such that the solution is converged within each time and stable. The predicted impact pressures were compared with the experimental measurements of Lewis et al. (2010). The numerical model is well capable of predicting the impact event, although the peak pressures at P1 and P2 are slightly over estimated. Three-dimensional simulations were found to provide better agreement with the experiments indicating that some three-dimensional effects are present. However, slight discrepancies still exist, which are believed to be due to mechanical friction in the experiments. Nevertheless, it is concluded that the numerical model is sufficiently accurate to allow it to be coupled with a structural solver for further study on hydroelastic impacts.

Muzaferija, S., Peric, M., Sames, P. & Schellin, T., 1998. A two-fluid Navier-Stokes solver to simulate water entry. Proc. 22nd Symp. Nav. Hydrodyn., pp. 638–651. Oger, G., Doring, M., Alessandrini, B. & Ferrant, P., 2006. Two-dimensional SPH simulations of wedge water entries. J. Comput. Phys., 213(2), pp. 803–822. Piro, D. & Maki, K., 2013. Hydroelastic analysis of bodies that enter and exit water. J. Fluids Struct., 37, pp. 134–150. Reddy, D.N., Scanlon, T. & Kuo, C., 2002. Prediction of Slam Loads on Wedge Section using Computational Fluid Dynamics (CFD) Techniques. In 24th Symp. Nav. Hydrodyn. Fukuoka, JAPAN, pp. 706–718. Sames, P.C., Schellin, T.E., Muzaferija, S. & Peric, M., 1999. Application of a two-fluid finite volume method to ship slamming. J. Offshore Mech. Arct. Eng., 121(1), pp. 47–52. Stenius, I., Rosén, A. & Kuttenkeuler, J., 2006. Explicit FE-modelling of fluid-structure interaction in hullwater impacts. Int. Shipbuild. Prog., 53, pp. 103–121. Tveitnes, T., Fairlie-Clarke, A.C. & Varyani, K., 2008. An experimental investigation into the constant velocity water entry of wedge-shaped sections. Ocean Eng., 35(14–15), pp. 1463–1478. Veen, D. & Gourlay, T., 2012. A combined strip theory and Smoothed Particle Hydrodynamics approach for estimating slamming loads on a ship in head seas. Ocean Eng., 43, pp. 64–71. Wagner, H., 1932. Phenomena associated with impacts and sliding on liquid surfaces. Z. Angew. Math. Mech. Wang, S. & Guedes Soares, C., 2012. Analysis of the water impact of symmetric wedges with a multi-material Eulerian formulation. Int. J. Marit. Eng., 154(A4), pp. 191–206. Yettou, E.-M., Desrochers, A. & Champoux, Y., 2006. Experimental study on the water impact of a symmetrical wedge. Fluid Dyn. Res., 38(1), pp. 47–66. Zhao, R., Faltinsen, O. & Aarsnes, J., 1996. Water entry of arbitrary two-dimensional sections with and without flow separation. In Proc. 21st Symp. Nav. Hydrodyn. pp. 408–423.

REFERENCES Aarsnes, J., 1996. Drop test with ship sections—effect of roll angle. MARINTEK report. Brizzolara, S., Couty, N., Hermundstad, O., Ioan, A., Kukkanen, T., Viviani, M. & Temarel, P., 2008. Comparison of experimental and numerical loads on an impacting bow section. Ships Offshore Struct., 3(4), pp. 305–324. Lewis, S.G., Hudson, D.A., Turnock, S.R. & Taunton, D.J., 2010. Impact of a free-falling wedge with water: synchronized visualization, pressure and acceleration measurements. Fluid Dyn. Res., 42(3), pp. 1–30.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Non-linear hydroelastic and fatigue analyses for a very large bulk carrier B. Cristea, C.I. Mocanu & L. Domnisoru Naval Architecture Faculty, University “Dunarea de Jos” of Galati, Galati, Romania

ABSTRACT: The paper is focused on the ship hull dynamic and fatigue analyses, in order to predict the preliminary design ship life. The ship dynamic analyses are carried out with own program codes package DYN, based on linear and non-linear hydroelasticity theory. As study case there is considered a very large double hull bulk carrier. The numerical approach includes three main steps: the global-local strength analysis on equivalent quasi-static head waves, the hydroelastic response analysis on head irregular wave and the cumulative damage factor fatigue analysis. Based on 3D/1D-FEM models, the stress 3D/1D correlation factors on structural details are obtained. The 2D/1D dynamic analyses outline the extreme wave loads, induced by slamming-whipping and springing hydroelastic responses. The study is focused on the influence of the hydroelastic responses on the fatigue strength assessment, making possible to obtain a more realistic ship preliminary structural service life prediction for a very large bulk carrier. 1

INTRODUCTION

on Femap/NX Nastran (Femap 2010) program, for the 3D-FEM structural models, and on several own codes (Domnisoru 2006) for the hydroelastic analysis and fatigue prediction.

In the last years the shipbuilding industry became a very competitive field, so that the ship design companies are required from the early design steps to carry out the assessment of the ship hull strength including also the fatigue based preliminary ship service life prediction (DNV 2008, GL 2004, 2013, IACS 2012, Lotsberg 2006, Jang & Hong 2009). This study is coupling the following numerical analyses: the global-local strength analysis on equivalent quasi-static head waves, based on 3D FEM and 1D equivalent beam models (Hughes 1988, Lehmann 1998, Mansou & Liu 2008, Hughes & Paik 2010, Rozbicki et al. 2001), the linear and non-linear hydroelastic response analysis in head irregular waves, based on 2D hydrodynamic and 1D equivalent beam models (Bishop & Price 1979, Domnisoru 1998, Guedes Soares 1999, Tao & Incecik 2000, Xing & Price 2000, Hirdaridis & Ge 2005, Park & Temarel 2007), the fatigue analysis, based on Palmgren-Miner, and the preliminary ship structure service life prediction, taking into account the hydroelastic response (Domnisoru et al. 2008, Knifsund & Tesanovic 2012, Fricke & Kahl 2005, Garbatov & Soares 2005). The numerical analyses are carried out on a very large bulk carrier, with double shell on sides and transversal bulkheads panels. There are considered full cargo and ballast loading cases. The main objective of this study is to analyse the influence of waves induced vibration response on long term fatigue strength. The numerical analyses are based

2

THE LARGE BULK CARRIER SHIP 1D AND 3D FEM NUMERICAL MODELS

The numerical analyses in this study are carried on a very large bulk carrier ship of 162000 tdw, with the hull offset lines presented in Figure 1 and the main dimensions in Table 1. The bulk carrier hull structure material is selected according to Germanischer Lloyd’s Rules (GL 2013), being considered isotropic steel Table 1. Main dimensions of the 162000 tdw bulk carrier ship. LOA [m] LBP [m] BMLD [m] DMLD [m]

Figure 1. carrier.

289.87 279.00 45.00 24.00

TMLD [m] cB vs [knots] Dw [t]

15.20 0.805 16 162000

Offset lines of the 162000 tdw large bulk

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(Table 2). The ship hull initial structural design is developed according to Germanischer Lloyd’s Rules (GL 2013) and CSR Common Structural Rules for Bulk Carriers (IACS 2012), taking into account the prescribed net scantling, at ICE ICEPRONAV Galati. 2.1

The 162000 tdw bulk carrier ship equivalent 1D-beam model Figure 2a. full cargo.

In this section are presented the characteristics of the 1D-equivalent beam of 162000 tdw bulk carrier model, used for hydroelastic numerical analyses in section 4. In Table 3 are presented the amidships structural characteristics, in the vertical plane, of the large bulk carrier ship: moment of inertia Iy [m4], transversal section area A [m2], equivalent shearing area Afz [m2], maximum shearing tangential stress in the neutral axis coefficient Kτn−n [m−2], deck and bottom bending modules WD, WB [m3]. In Table 4 are presented the 162000 tdw bulk carrier loading conditions, full cargo and ballast, used as study cases, where: Tm, Taft, Tfore are the ship draughts, medium, aft peak, fore peak; vs is the ship design speed. The mass diagrams for the two loading cases are presented in Figures 2a,b. Based on 1D-FEM approach, there are obtained the natural hull oscillation and vibration mode frequencies, dry and wet hull conditions, presented in Table 5.

Figure 2b. Mass diagram of 162000 tdw bulk carrier, ballast case. Table 5. Natural modes frequencies f [Hz] of the large bulk carrier.

Table 2. Material characteristics and steel grades (GL 2013). E [N/m2] G [N/m2]

2.06 1011 7.92 1010

Steel grade ReH [N/mm2] Rm [N/mm2] σadm [N/mm2] τadm [N/mm2]

A 235 400 190 120

ρm[t/m3] ν AH32 315 440 244 154

Table 4. study.

674.73 2.78 1.66

Tm [m]

Full cargo 14.79 Ballast 8.92

Oscillations

Vibrations

No. Load case

0

1

2

3

4

1

– 0.094 – 0.101

– 0.103 – 0.115

0.744 0.543 0.963 0.663

1.408 1.029 1.837 1.248

2.605 1.505 2.681 1.822

2

AH36 355 490 264 167

2.2

0.60 51.53 59.03

Taft [m]

Tfore [m]

vs [Knots]

15.36 8.14

15.07 8.51

16.00 16.00

Full cargo dry wet Ballast dry wet

The 162000 tdw bulk carrier ship structural 3D CAD/FEM model

In this section is presented the structural 3D CAD/ FEM model (Rozbicki et al. 2000, Hughes & Paik 2010, IACS 2012) for the 162000 tdw bulk carrier, used for numerical analyses in section 3. Figure 3a presents the overall 3D-FEM model of the ship hull, one sided taking into account that the waves are only on head condition. Figures 3b,c present details of the 3D-FEM model form cargoholds. The 3D FEM model (Figs. 3a-c) is developed by Femap/NX Nastran (Femap 2010) program, using mapped mesh technique, with the following finite elements: quadratic and triangle membrane and thick plate (Mindlin) elements for shells and main longitudinal and transversal girders (Domnisoru 2006). The 3D beam elements (Timoshenko) for ordinary stiffeners are used. The mesh size is between 600–800 mm, according to the ordinary stiffeners spacing.

Large bulk carrier loading cases for numerical

No. Load case 1 2

Kτn−n [m−2] WB [m3] WD [m3]

Modes

7.8 0.3

Table 3. Amidships section characteristics of the large bulk carrier. Iy [m4] A [m2] Afz [m2]

Mass diagram of 162000 tdw bulk carrier,

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the ship hull, in the vertical plane, under onboard masses gravity loads and equivalent quasi-static head wave pressure. The global strengths analysis, based on the 1D and 3D-FEM models, includes the following steps: − 1D-beam equivalent ship girder model; The global ship strength analysis with 1D model is based on own SW_EQW code (Domnisoru 2006). Besides the shear forces and bending moments diagrams, there are obtained the ship & equivalent quasi-static wave vertical equilibrium position parameters.

Figure 3a. The 3D-FEM model full extended over the ship length.

− 3D-CAD ship hull offset lines and external shell surface generation, using MaxSurf (2009) program; − 3D-CAD ship hull structure model generation, using the geometric module from Femap (2010); − 3D-FEM ship hull structure model generation, using the mapped mesh approach from Femap/ NX Nastran (Femap 2010) program, with details in section 2.2; − 3D-FEM model boundary conditions; The boundary conditions applied on the 3D-FEM bulk carrier full extended hull model, are modelling the symmetry in the ship centre line and both ends control master nodes (Table 6, Fig. 3a).

Figure 3b. 3D-FEM.

− 3D-FEM model onboard masses gravity load, except the cargo and ballast which are modelled as local pressure loads; − 3D-FEM model external shell pressure, which is modelling the still water and quasi-static head wave conditions, taking as reference the shipwave vertical equilibrium parameters, based on the 1D-beam equivalent model. So, the balance of the 3D-FEM model is ensured and the vertical reaction forces at the two control nodes (Table 6, Fig. 3a) become zero.

Amidships cargo hold, large bulk carrier,

Based on Germanischer Lloyd’s Rules (GL 2013), the statistical reference height of the equivalent quasi-static wave, for the 162000 tdw bulk carrier, is hw = 10.65 m. At the numerical hull strength analyses, based on 1D and 3D models, there are considered the sagging and hogging wave conditions, for both loading cases, full and ballast Figure 3c. Typical transversal bulkhead, with front shell removed, and typical web frame of the large bulk carrier, 3D-FEM model.

3

Table 6. Boundary conditions on the 3D-FEM bulk carrier model.

THE 3D-FEM SHIP STRENGTH ANALYSIS

Boundary condition

Translation

Rotation

Ux Uy

Rx

Uz

Ry Rz

Symmetry in CL – Fix – Fix – Aft control node NDaft Fix Fix Fix Fix – Fix Fix Fix – Fore control node NDfore –

Based on the 1D and 3D FEM structural models of the 162000 tdw bulk carrier ship (section 2), in this section is included the strengths analysis of

– – –

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(Table 4), with the wave height range hw = 0 to 12 m (1 m step).

Table 7. The maximum ship vertical deflection wzmax [m], taking as reference wave height hw = 10.65 m.

− 3D-FEM and 1D models numerical results. Figures 4a,b present the equivalent quasi-static wave pressure acting over the ship hull external shell, with rules based wave height hw = 10.65 m, for full cargo load case on sagging wave condition and ballast load case on hogging wave condition. Figures 5a,b present the maximum deck shell equivalent von Mises stress σvonDmax diagrams,

Full cargo

Figure 4a. sagging.

Ballast

Load case

Sagging

Hogging

Sagging

Hogging

Criterion

0.412 0.71

0.310 0.53

0.305 0.53

0.271 0.37

max. wz[m] wzmax/wadm < 1

based on 3D-FEM model, for head wave height hw = 0–12 m, for full cargo load case on sagging wave condition and ballast load case on hogging wave condition. They are pointing out the significant stress hot-spots induced around the cargo hold hatch details (Fig. 3b) and that in the other parts of the deck panel the stress values are similar to the 1D model (those diagrams not included in the paper). Table 7 presents the maximum ship girder vertical deflection, wz[m], based on the 3D-FEM model, with the admissible value wadm = L/500 = 0.580 m, taking as reference hw = 10.65 m wave height. Tables 8, 9 present the maximum stress at hotspot structural details, deck and bottom shells von Mises stresses, tangential stresses in the neutral axis, for wave height reference hw = 10.65 m. In the case of 1D model, in the extreme fibres of the equivalent beam, in deck and bottom shells, the von Mises stress is equal to the longitudinal normal stress. In order to perform the fatigue analysis (section 5) for the 3D-FEM model, there is used the stress 3D/1D correlation coefficient k3D/1D with values in Table 13, for the hot-spot structural details in Figure 10, based on the following expressions:

Wave pressure distribution hw = 10.65 m, full,

Figure 4b. Wave pressure distribution hw = 10.65 m, ballast, hogging.

⎧ σ vonD 3 D sagg ⎫ gg σ vonD 3 D hogg k3 D 1D = max ⎨ , ⎬ ⎩ σ xD 1D sagg σ xD 1D hogg ⎪⎭

Figure 5a. sagging.

Where: σvonD 3D sagg and σvonD 3D hogg are the 3D-FEM model deck von Mises stresses; σxD 1D sagg and σxD 1D hogg are the 1D-beam model deck normal stresses, on sagging and hogging conditions, at the structural detail with stress hot-spots.

3D-FEM, deck σvonD[N/mm2], full cargo,

4

Figure 5b. hogging.

(1)

THE LINEAR AND NON-LINEAR HYDRO-ELASTIC RESPONSE OF BULK CARRIER

In order to obtain the dynamic response in irregular head waves of the 162000 tdw bulk carrier, taking into account that the natural vibration frequencies are low 0.543–0.663 Hz (Table 5), the numerical analyses in this section are based on the hydro-elasticity theory.

3D-FEM, deck σvonD[N/mm2], ballast load,

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Table 8. The maximum stress [N/mm2] at hot-spot structural details, full cargo load case, sagging and hogging wave conditions. Sagging

3D-FEM

3D/1D

1. Deck von Mises stresses |σD-max| σadm-AH36 = 264 N/mm2 10.65 200.65 251.24 1.25 89.25 0.760 0.952 0.338 σmax/σadm

98.83 0.374

1.11

2. Bottom von Mises stresses |σB-max| σadm-AH32 = 244 N/mm2 10.65 177.44 179.95 1.01 70.83 0.727 0.738 0.290 σmax/σadm

91.92 0.377

1.33

hw[m]

1D-beam

Hogging 3D-FEM

3D/1D

1D-beam

3. Tangential stresses in neutral axis |τ nn-max| τadm-AH32 = 154 N/mm2 10.65 39.09 64.22 1.64 59.32 62.60 0.254 0.417 0.385 0.406 τmax/τadm

1.06

Table 9. The maximum stress [N/mm2] at hot-spot structural details, ballast load case, sagging and hogging wave conditions. Sagging hw[m]

1D-beam

Hogging 3D-FEM

3D/1D

1D-beam

3D-FEM

3D/1D

1. Deck von Mises stresses |σD-max| σadm-AH36 = 264 N/mm2 10.65 125.01 174.29 1.39 122.89 0.474 0.660 0.465 σmax/σadm

156.18 0.592

1.27

2. Bottom von Mises stresses |σB-max| σadm-AH32 = 244 N/mm2 10.65 105.99 135.03 1.27 95.46 0.434 0.553 0.391 σmax/σadm

152.34 0.624

1.60

59.84 0.389

1.45

3. Tangential stresses in neutral axis |τ nn-max| τadm-AH32 = 154 N/mm2 10.65 46.82 47.26 1.01 41.21 0.304 0.307 0.268 τmax/τadm

Figure 6a. Longuet-Higgins wave time record, h1/3 = 10.65 m, x/L = 0.5.

Figure 7a. Time record of vertical displacement, nonlinear hydroelastic analysis, h1/3 = 10.65 m, x/L = 0.5, full cargo case.

Figure 7b. Amplitude FFT spectrum of vertical displacement, non-linear hydroelastic analysis, h1/3 = 10.65 m, x/L = 0.5, full cargo.

Figure 6b. Longuet-Higgins wave FFT spectrum, h1/3 = 10.65 m, x/L = 0.5.

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Figure 8a. Time record of bending moment, linear hydroelastic analysis, h1/3 = 10.65 m, amidships x/L = 0.5, full cargo load case.

Figure 10. analysis.

Deck structural details considered at fatigue

− 1D-FEM ship hull model, based on the 1D-beam equivalent ship girder model (section 2.1); − multi-parametric conformal transformation method applied for ship off-set lines modelling; − 2D potential flow model applied for the hydrodynamic excitation forces, according to the hydro-elasticity and strip theories; − bottom slamming pressure, at fore and aft peak, are modelled according to the Ochi formulation and side slamming according to momentum approach (Bishop & Price 1979); − the ship dynamic response is decomposed, using the modal analysis technique, on ship oscillation (0.094–0.115 Hz) and vibration (>0.543 Hz) modes; − the irregular head waves are based on the Longuet-Higgins model, with first order ITTC spectrum (Price & Bishop 1974) and second order wave interference components.

Figure 8b. Amplitude FFT spectrum of bending moment, linear hydroelastic analysis, h1/3 = 10.65 m, amidships x/L = 0.5, full load.

The dynamic ship response obtained with DYN program (Domnisoru 2006) includes the following components: the linear and non-linear oscillation response, the springing phenomenon, linear and non-linear steady state vibration response, due to the ship structure-wave resonance, and the whipping phenomenon, induced transitory vibration response due to the occurrence of bottom and side slamming. The steady state linear ship dynamic response is based on a solution in the frequency domain for each wave component. The transitory non-linear ship dynamic response is based on a solution in the time domain, using a combined iterative and implicit integration β-Newmark scheme, for 80 s simulation time, with time step 0.01 s (triggering frequency 100 Hz). The hydroelastic response analyses in irregular waves of the 162000 tdw bulk carrier are carried out for first order head wave spectra ITTC, with short term significant wave height h1/3 = 0–12 m (step 0.5 m). The ship speed is constant in this study, being considered a conservative approach. The following numerical results are presented:

Figure 9a. Time record of bending moment, non-linear hydro-elastic analysis, h1/3 = 10.65 m, amidships x/L = 0.5, full cargo load.

Figure 9b. Amplitude FFT spectrum of bending moment, non-linear hydroelastic analysis, h1/3 = 10.65 m, x/L = 0.5, full cargo.

Based on the linear and non-linear hydroelastic ship dynamic model (Bishop & Price 1979, Domnisoru 1998, Xing & Price 2000, Hirdaris et al. 2003, Hirdaris & Ge 2005, Perunovic & Jensen 2005, Park & Temarel 2007) there has been developed an own program code DYN (Domnisoru 2006), under the following hypothesis:

− Figures 6a,b present the time record and the amplitude spectrum of Longuet-Higgins head

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wave model, with first order ITTC spectrum h1/3 = 10.65 m, at amidships section x/L = 0.5; − Figures 7a,b present the time record and the amplitude FFT spectrum of vertical displace-

Table 12. The maximum significant normal deck and bottom shells, tangential neutral axis stresses [N/mm2] on first natural vibration and oscillations hull modes, full cargo and ballast, h1/3 = 10.65 m, 1D equivalent beam model, without stress hot-spots.

Table 10a. The ratio between the significant deformations (vibration) and displacements (oscillation), full cargo case, h1/3 = 10.65 m.

Stress values σmax < σadm

Full cargo case

σmax_LIN deck σmax_LIN+ bottom τmax_LIN n−n σmax_NL deck σmax_NL bottom τmax_NL n−n

62.14 56.77

11.22 10.25

66.16 60.45

7.20 6.58

36.62 63.03 57.59

5.98 28.73 26.25

37.18 66.31 60.58

3.57 23.78 21.72

37.99

12.50

37.26

9.79

x/L

%vib/osc %vib/osc Bottom Side Green linear non-linear slamming slamming sea

0.00 0.50 1.00

6.00 4.80 6.34

6.09 4.98 6.44

Average 5.71

5.84

Yes >5.75 >11.50 – – – Yes >12.00 >10.65 h1/3 wave references

Table 10b. The ratio between the significant deformations (vibration) and displacements (oscillation), ballast case, h1/3 = 10.65 m. x/L

%vib/osc %vib/osc Bottom Side Green linear non-linear slamming slamming sea

0.00 0.50 1.00

3.16 3.34 3.48

3.27 3.37 3.55

Average 3.33

3.40

−

Yes No >0 – – Yes No >8.50 h1/3 wave references

−

Table 11a. The maximum ratios for the significant bending moments and shearing forces, on first natural vibration mode and hull oscillation modes, full cargo case, h1/3 = 10.65 m.

−

%vib/osc linear

%vib/osc non-linear

x/L

BM

SF

BM

SF

0.25 0.50 0.75 Average

13.74 17.54 16.08 15.79

14.40 10.41 16.99 13.93

30.71 44.68 34.72 36.70

31.62 Linear: High 14.90 reduced 38.53 Non-linear: small 28.35

Springing

Whipping

−

Table 11b. The maximum ratios for the significant bending moments and shearing forces, on first natural vibration mode and hull oscillation modes, ballast case, h1/3 = 10.65 m. %vib/osc linear

%vib/osc non-linear

x/L

BM

SF

BM SF

0.25 0.50 0.75 Average

10.95 10.94 9.59 10.49

10.92 9.73 10.02 10.22

34.26 35.67 26.10 32.01

Springing

5

Ballast case

Oscillation Vibration Oscillation Vibration

ment, wave significant height h1/3 = 10.65 m, x/L = 0.5; Figures 8a,b & 9a,b present the time record and the amplitude FFT spectrum, of the linear and non-linear bending moment, wave significant height h1/3 = 10.65 m, at amidships section x/L = 0.5; the ratio between the significant deformations, on the first natural vibration mode and the ship rigid displacement on oscillation modes (Tables 10a,b); the limit state for bottom & side slamming and green sea occurrences are included based on the vertical significant displacements diagrams (not included). the maximum ratios for the significant bending moments and shearing forces, on first natural vibration and hull oscillation modes (Tables 11a,b); the maximum short term significant normal deck and bottom shells, tangential neutral axis stresses on first natural vibration and oscillations modes (Table 12), according to the spectral decomposition, based on the 1D equivalent beam model, without hot-spots.

LONG TERM STRENGTH FATIGUE ANALYSIS AND SHIP SERVICE LIFE PREDICTION

Based on the long term fatigue analysis, according to the methodology from Germanischer Lloyd’s Rules (2004, 2013), in this section is obtained the preliminary ship service life prediction of the 162000 tdw bulk carrier ship. The fatigue analysis is carried on the structural elements of the ship hull having the highest stress values. From Tables 8, 9 & 12 (sections 3 & 4)

Whipping

32.68 Linear: High 17.92 reduced 28.78 Non-linear: small 26.46

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results that for the large bulk carrier at the deck shell are recorded the maximum stress values, being the reference structural details for fatigue analysis (Fig. 10). In order to evaluate the ship fatigue strength criterion, according to the Germanischer Lloyd’s Rules (GL 2004, 2013) methodology, the PalmgrenMiner cumulative damage ratio D method, based on steel standard design S-N curves, is applied. From section 4, based on the short-term prediction analysis of the ship hydroelastic response, there are obtained the significant stresses σ1/3 on oscillation and vibration components, for waves significant height h1/3. The structural model for hydroelastic analysis is 1D-FEM. In order to perform the fatigue analysis on the 3D-FEM model of the 162000 tdw bulk carrier, based on the stress 3D/1D correlation coefficient k3D/1D (Equation 1), there are obtained the hydroelastic short-term stresses for the structural details: 3 D 1D σ vonM 1 3 osc ,vib

σ 1xD1 3 osc ,vibib ⋅ k3 D 1D

Figure 11. Word Wide Trade wave significant height h1/3 histogram.

(2) Figure 12. North Atlantic wave significant height h1/3 histogram.

Figure 10 presents the main deck selected structural details, which are considered at fatigue analysis and preliminary ship service life prediction, with the correlation coefficients k3D/1D from Table 13. The fatigue analysis of the large bulk carrier is considered with reference time R = 20 years. The cumulative damage ratio D, based on short term hydroelastic stresses from section 4 and correlation coefficients from Table 13 (Fig. 10), has the following expression: Dos oscc + Dvib ; ni _ osc ,vib i = pi nmax_ osc ,vib m ni _ osc ,vibi Dosc ,vib = ∑ ; Ni _ osc ,vibi fSN Δσ i _ osc ,vib i =1 Ni _ osc osc ,vib i nmax_ osc ,vib = 3.1536 1 ⋅10 107 R ⋅ fosc ,vib ; Δσ i _ osc ,vibi = 2 ⋅ σ 1/ 3i _ osc ,vib

cycles resulting from the steel S-N curves for a stress range Δσi_osc,vib. Figures 11 & 12 present the significant wave height h1/3 Word Wide Trade and North Atlantic long term histograms (Price & Bishop 1974, DNV 2008, GL 2004, 2013, Domnisoru 2006), taking as reference period Tb = 1 year. For the 162000 tdw bulk carrier ship, with a simplified navigation scenario, the full cargo and ballast loading cases are considered with the same occurrence probability, resulting for the PalmgrenMiner cumulative damage ratio D the following expression:

D

(

)

(3)

D

where: fosc,vib are the natural frequencies on oscillation and vibration modes (Table 5); nmax_osc,vib are the maximum number of cycles on oscillation and vibration modes; pi(h1/3i ), i = 1,m the probabilities of wave significant height h1/3 long term histogram (Figs. 11 & 12); ni_osc,vib the number of stress cycles for wave h1/3i; Ni osc,vib the number of endured stress-

Table 13. details.

Zone 1

Zone 2

Zone 3

k3D/1D (full) k3D/1D (ballast)

1.381 1.384

1.252 1.271

1.293 1.408

1 L = R/D

(4)

Where: L [years] is the ship service life prediction. For the initial fatigue analysis there are considered the longitudinal butt weld joints of the ship main deck shell, type B1 details (GL 2013), with 125 N/mm2 fatigue stress reference value corresponding to steel S-N curve at 2⋅106 cycles. Tables 14a,b and Tables 15a,b present the fatigue criteria assessment for Word Wide Trade and North Atlantic wave significant height histograms, taking the maximum cumulative damage D factors for the three deck structural details (Figure 10), based on 1D-beam model significant stresses σx1/3 (section 4) and 3D&1D model significant stresses σvonM1/3 (Equation 2), for the combined navigation scenario of full cargo and ballast loading cases (Equation 4).

Stress 3D/1D correlation coefficients for deck

Detail

0.5 D full + 0.5 ⋅ Dballast

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Table 14a. Fatigue criterion, oscillations short term stresses, long term Word Wide Trade wave significant height h1/3 histogram. Loading cases

Full

Ballast

Combined

Analyses

DSN_full

DSN_ballast

DSN

L [years]

0.203 0.254

>35 >35

3D & 1D model, σvonM1/3 deck maximum equivalent von Mises stress Linear 0.449 0.319 0.384 Nonlinear 0.577 0.344 0.461

>35 >35

1D-beam model, σx1/3 deck maximum normal stress Linear 0.310 0.096 Nonlinear 0.404 0.104

Table 14b. Fatigue criterion, hydroelastic short term stresses, long term Word Wide Trade wave significant height h1/3 histogram. Loading cases

Full

Ballast

Combined

Analyses

DSN_full

DSN_ballast

DSN

L [years]

0.204 0.264

>35 >35

3D & 1D model, σvonM1/3 deck maximum equivalent von Mises stress Linear 0.451 0.319 0.385 Nonlinear 0.603 0.353 0.478

>35 >35

1D-beam model, σx1/3 deck maximum normal stress Linear 0.311 0.096 Nonlinear 0.422 0.106

Table 15a. Fatigue criterion, oscillations short term stresses, long term North Atlantic wave significant height h1/3 histogram. Loading cases

Full

Ballast

Combined

Analyses

DSN_full

DSN_ballast

DSN

L [years]

0.615 0.740

32.5 27.0

3D & 1D model, σvonM1/3 deck maximum equivalent von Mises stress Linear 1.286 1.092 1.189 Nonlinear 1.592 1.157 1.375

16.8 14.6

1D-beam model, σx1/3 deck maximum normal stress Linear 0.887 0.344 Nonlinear 1.113 0.366

Table 15b. Fatigue criterion, hydroelastic short term stresses, long term North Atlantic wave significant height h1/3 histogram. Loading cases

Full

Ballast

Combined

Analyses

DSN_full

DSN_ballast

DSN

L [years]

0.617 0.779

32.4 25.7

3D&1D model, σvonM1/3 deck maximum equivalent von Mises stress Linear 1.291 1.093 1.192 Nonlinear 1.690 1.194 1.442

16.7 13.9

1D-beam model, σx1/3 deck maximum normal stress Linear 0.890 0.344 Nonlinear 1.182 0.376

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− For the long term Word Wide Trade significant wave height histogram (Fig. 11), the fatigue criterion based on the damage cumulative factor is satisfied on all cases, D = 0.203–0.478 < 1 (Tables 14a,b). There result no restrictions from fatigue criterion, having a predicted service life over L > 35 years. − Based on significant stresses from 1D-beam model and long term North Atlantic significant wave height histogram (Fig. 12), the fatigue criterion is satisfied, D = 0.615–0.779 < 1, so that the predicted service life results over L > 25 years (Tables 15a,b). − Taking into account the 3D-FEM model stresses and long term North Atlantic significant wave height histogram (Fig. 12), from fatigue analysis results on linear dynamic response D = 1.189– 1.192 > 1, with a predicted service life of L = 16.7– 16.8 years and on non-linear dynamic response D = 1.375–1.442 > 1, with a predicted service life of L = 13.9–14.6 years. On hydroelastic response (oscillations and vibrations) results higher restrictions, on fatigue criterion, compare to standard seakeeping response (oscillations), (Tables 15a,b). The structural details with damage cumulative factor D >1 have to be considered for redesign. − Based on the numerical results from Tables 14a,b and Tables 15a,b results that the North Atlantic navigation conditions are harder than the averaged synthesized World Wide Trade navigation conditions.

The influence of the vibration dynamic response is more significant in the case of the North Atlantic wave histogram, where the wave with larger significant height have higher probability to occur. In this case the slamming occurrence is higher inducing a more significant whipping transitory response. 6

CONCLUSIONS

Based on the ship hull strength analysis, under equivalent quasi-static head waves (section 3), the following conclusions result: − The maximum global deflection satisfy the deformation criterion wzmax/wadm = 0.37–0.71 < 1 (Table 7). − The maximum stresses satisfy the stress criterion (Tables 8 & 9) σ,τmax1D/σ,τadm = 0.254–0.760 < 1 (1D-beam), σ,τmax3D/σ,τadm = 0.290–0.952 < 1 (3D-FEM). The maximum 3D-FEM stress values (Figs. 5a,b) are significant around the transversal bulkheads domain, where the 1D-beam model has no sensitivity. − The selection of the steel grade AH36 for deck and AH32 for the other structural panels is justified by the recorded maximum stresses (Tables 8 & 9). Based on the hydroelastic linear and non-linear dynamic response analysis, under head irregular waves (section 4), the following conclusions result: − Vibration deformations (deflection) are small compare to oscillation displacements 3.33– 5.84% (Tables 10a,b, Figs. 7a,b), so that the seakeeping parameters can be assessed only based on rigid hull motions. − The hydroelastic numerical results (Tables 10a,b) point out that bottom slamming occurs at aft peak on both loading cases, with higher probability on ballast (h1/3 > 0), and at fore peak occurs only in ballast case (h1/3 > 8.5 m). Side slamming occurs on both loading cases, due to flare ship extremities shape. Green sea can occur only on full load case (h1/3 > 10.65 m). − The springing phenomenon occurs with small intensity compare to the whipping phenomenon induced by slamming. The non-linear model is recording higher vibration response, mainly due to whipping phenomenon (Figs. 8a,b & Figs. 9a,b). The bending moments and shearing forces short-term vibration and oscillation significant values ratio are on linear analysis 9.59–17.54% and on non-linear analysis 14.90– 44.68% (Tables 11a,b).

The numerical results from this study are pointing out that the very large bulk carrier ships are sensitive to the transitory whipping hydroelastic loads induced by slamming, so that in order to obtain a more realistic prediction of the ship service life based on the fatigue strength criterion, the significant stresses of ship hull structure must be obtained by non-linear hydroelastic response analysis in irregular waves. Further studies will be focused on the influence of the ship speed changes, according to the sea state, on very large bulk-carries hydroelastic and fatigue responses in irregular waves.

ACKNOWLEDGEMENTS This study has been accomplished in the frame of the EMSHIP project Integrated Advanced Ship Design Ref. 159652-1-2009-1-BE-ERA MUNDUS (Coordinator University of Liege, ANAST Naval Architecture and Transportation System Analysis) at “Dunarea de Jos” University of Galati, Naval Architecture Faculty, Department of Ship Structures, and ICE ICEPRONAV Galati, Department of Structural Analysis.

Based on the fatigue analysis and the preliminary ship service life prediction (section 5), the following conclusions result:

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REFERENCES

Hughes, O.F. & Paik, J.K. 2010. Ship Structural Analysis and Design. N.J: The Society of Naval Architects and Marine Engineers. IACS. 2012. Common Structural Rules for Bulk Carriers. London: International Association of Classification Societies. Lehmann, E. 1998. Guidelines for strength analyses of ship structures with the finite element method. Hamburg: GL. Jang, C.D. & Hong, S.Y. (editors) 2009. Proceeding of the 17th international ship and offshore structures congress (ISSC). Volumes 1 & 2. Seoul National University. Knifsund, C. & Tesanovic, A. 2012. Global and Detailed Local Fatigue Assessment of a Container Vessel. Report X-12/281. Gothenburg: Chalmers University of Technology. Lotsberg, I. 2006. Assessment of fatigue capacity in the new bulk carrier and tanker rules. Marine Structures 19: 83–96. Mansour, A. & Liu, D. 2008. Strength of ships and ocean structures. New Jersey: The Society of Naval Architects and Marine Engineering. MaxSurf. 2009. MaxSurf program user guide. FormSys. Park, J.H. & Temarel, P. 2007. The influence of nonlinearities on wave-induced motions and loads predicted by two-dimensional hydroelasticity analysis. ABSPRADS 1–5 Oct. 2007, Houston (1): 27–34. Perunovic, J.V. & Jensen, J.J. 2005. Non-linear springing excitation due to a bidirectional wave field. Marine Structures 18: 332–358 Price, W.G. & Bishop, R.E.D. 1974. Probabilistic theory of ship dynamics. London: Chapman and Hall. Rozbicki, M., Das Purnendu, K. & Crow, A. 2001. The preliminary finite element modelling of a full ship. International Shipbuilding Progress Delft 48(2): 213–225. Tao, Z. & Incecik, A. 2000. Nonlinear Ship Motion and Global Bending Moment Predictions in Regular Head Seas. International Shipbuilding Progress 47(452): 353–378. Xing, J.T. & Price, W.G. 2000. The Theory of Non-linear Elastic Ship-water Interaction Dynamics. Journal of Sound and Vibrations 230(4): 877–914.

Bishop, R.E.D. & Price, W.G. 1979. Hydroelasticity of ships. Cambridge: University Press Cambridge. DNV. 2008. Classification Note No. 30.7 Fatigue Assessment of Ship Structure. Hövik: Det Norske Veritas. Domnisoru, L. & Domnisoru, D. 1998. The unified analysis of springing and whipping phenomena. Transactions of the Royal Institution of Naval Architects London 140(A): 19–36. Domnisoru, L. 2006. Structural analysis and hydroelasticity of ships. Galati: University “Lower Danube” Press. Domnisoru, L., Dumitru, D. & Ioan, A. 2008. Numerical methods for hull structure strengths analysis and ships service life evaluation, for a LPG carrier. OMAE 15–20 June 2008, Estoril: 509–518. Fricke, W. & Kahl, A. 2005. Comparison of different structural stress approaches for fatigue assessment of welded ship structures. Marine Structures 18: 473–488. FEMAP. 2010. FEMAP/NX Nastran FEM, Siemens PLM Software Corporation. GL. 2004. Guidelines for fatigue strength analyses of ship structures. Hamburg: Germanischer Lloyd. GL. 2013. Germanischer Lloyd’s Rules. Hamburg. Garbatov, Y., Tomasevic, S. & Guedes Soares, C. (2005). Fatigue Damage Assessment of a Newly Built FPSO Hull. Maritime Transportation and Exploitation of Ocean and Coastal Resources, Guedes Soares, C. Garbatov Y. & Fonseca N., (Eds.), Taylor & Francis Group, London, UK, pp. 423–428. Guedes Soares, C. 1999. Special issue on loads on marine structures. Marine Structures 12(3): 129–209. Hirdaris, S.E., Price, W.G. & Temarel, P. 2003. Two and three-dimensional hydroelastic modelling of a bulk carrier in regular waves. Marine Structures 16: 627–658. Hirdaris, S.E. & Ge, C. 2005. Review and introduction to hydroelasticity of ships. Report 8. London: Lloyd’s Register. Hughes, O.F. 1988. Ship structural design. A rationallybased, computer-aided optimization approach. New Jersey: The Society of Naval Architects and Marine Engineering.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Hydroelastic analysis of a flexible barge in regular waves using coupled CFD-FEM modelling P. Lakshmynarayanana, P. Temarel & Z. Chen Fluid Structure Interactions Group, University of Southampton, Southampton, UK

ABSTRACT: The aim of this paper is to investigate the wave-body interaction of flexible floating bodies by coupling RANS/CFD and Finite Element software. A combination of overset and morphing approaches and finite volume solution to allow for the motion of a barge at the free surface is used. Results are presented for the motion response of the Three-Dimensional (3-D) barge, treated both as rigid and flexible body, in regular head waves using STAR-CCM+, the latter carried out by a two-way coupling between Star-CCM+ and Abaqus. To illustrate this application, the structure of the flexible barge is modelled as a beam, in line with the flexible backbone model used in experiments. The RAOs of vertical displacements, at a number of positions along the barge, calculated using this coupling technique are compared against experimental measurements and Two-Dimensional (2-D) linear hydroelasticity predictions. 1

INTRODUCTION

omit some important fluid-structure interactions. A two-way coupling method will be more suitable in such cases. Few investigations employing twoway coupling have shown promising results (Paik et al. 2009) (Kim & Kim 2009), but need further investigations. In this study a two-way coupling between a finite volume CFD method, using Star-CCM+ (version 8.04), and a Finite Element Method (FEM), using Abaqus (version 6.13–1), is applied to assess the hydroelastic effects of a flexible barge in regular head waves. Only symmetric distortions of the barge for a number of wave frequencies are employed and compared against experimental measurements (Remy et al. 2006). 3-D computations are first carried out treating the barge as a rigid body to establish the influence of domain size and mesh refinement along the free surface and the barge. A two-way coupling is then established between Star-CCM+ and Abaqus to investigate the hydroelastic response. The coupling takes place through exchanging pressures and displacements between Star-CCM+ and Abaqus more than once every time-step, namely implicit scheme. The structure is modelled as a non-uniform Timoshenko beam with properties, such as stiffness and mass distribution, as per the model test data. Numerical predictions are also obtained using 2-D hydroelasticty (Bishop et al. 1977). RAO of vertical displacements at various locations along the barge is compared against experimental measurements and numerical predictions to validate the coupling method used.

Modern seakeeping computations of ships are carried out using a variety of techniques ranging from Two-Dimensional (2-D) strip theory using potential flow methods to solving fully nonlinear unsteady RANS (Reynolds-Averaged NavierStokes equations) methods. Application of CFD (Computational Fluid Dynamics) to study wavebody interactions of ships and offshore structures using RANS have increased over the years due to increase in computational power (ISSC. 2012). Traditionally, seakeeping analysis of ships is carried out treating them as rigid bodies. However, the ever increasing size of ships and offshore platforms have resulted in ‘softer’ or flexible hulls which require hydroelastic effects to be taken into account when predicting fluid-structure interactions (Bishop & Price 1979). Such investigations are predominantly experimental, using flexible backbone models, or numerically using potential flow solvers (Bishop et al. 1977). Although partial nonlinear potential flow methods are also used, RANS/CFD can fully take into account the freesurface and body nonlinearities as well as viscous effects, making it more efficient and realistic for some problems (Brizzolara et al. 2008). Presently the majority of investigations using RANS/CFD and FEA are carried out using oneway coupling. When the deformations of the structure are large enough to significantly affect the flow field around it, one-way coupling would

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2 2.1

fall outside the refined region of the grid, resulting in high numerical diffusion before the wave reaches the body. To avoid this problem, an overset or overlapping grid can be used. In this case two regions, background and overset are created, where the background grid is adapted to the free surface. The overset grids are attached to the floating body and move with it freely depending on the motion response. For rigid body simulations of the barge in head waves, grid adaptation has been carried out using overset grids. In the case of two-way coupling, both morphing and overset is applied to move the grid to follow the barge motions. Morphing condition is set to the boundaries of the barge and it deforms due to the nodal displacements supplied by Abaqus. Floating condition is set as the Morpher boundary condition for the overset, so that it moves freely in accordance with the grid deformation applied by the morpher.

NUMERICAL METHOD Finite volume method

The CFD software used for all computations in this paper is Star-CCM+. Here we present only a brief description of the numerical method implemented and a detailed theoretical background is provided by Ferziger & Peric (2003). The numerical method used in Star-CCM+ is a Finite Volume (FV) method in which the flow is assumed to be governed by RANS equations. The RANS equations reduce to the well known Euler equations for the case of inviscid flow. First, the spatial fluid domain is discretized into a finite number of Control Volumes (CVs) or cells. The integral form of conservation equation, with the initial and boundary conditions, is then applied to cell centers and simplified into an algebraic system of equations. The governing equations not only contain surface and volume integrals but also time and spatial derivatives. They are solved using a segregated iterative algorithm, called SIMPLE. All integrals are computed using midpoint rule. The Hybrid Gauss-Least Square gradient method is used to solve the transport equations. Free surface flows are implemented using the Volume of Fluid (VOF) tracking method. In order to account for the position of free surface in multiphase flows and allow for its arbitrary deformation, an additional equation is solved for the volume fraction c. When the motion of a body at a free surface is involved, the position of the body is updated at each iteration. The equations of motion of the body are solved to obtain the velocities and, hence, update the displacements and rotations. The fluid grid is adjusted at every outer iteration to follow the updated position of the moving body. 2.2

2.3

Coupling scheme

The coupling schemes control the sequence of data exchanges in the simulations. There are two major coupling schemes in Star-CCM+. For a loosely coupled problem an explicit scheme can be chosen. In this scheme, the data or field exchange takes place once every time step. An implicit scheme is chosen when strong coupling is sought as it is more stable than the former, but at a higher computational cost. In the implicit scheme, field exchanges between the software take place at every single iteration within a time step. The explicit scheme was tested with even very small time steps but resulted in pressure divergence. 2.4

Field data exchange

The coupling is implemented by exchanging pressure and nodal displacements, the so called field data, between Star-CCM+ and Abaqus. The geometry of the floating body must have the same dimensions and coordinates in both software; otherwise the co-simulation will fail due to inconsistency in topology. The response of the fluid to the structural deformations is expressed through the grid flux term. It represents the ratio of volume swept due to the movement of cell face from one time step to the next time step. Grid flux intensity can be calculated by the product of normal velocity of the face and its area. In transient simulations, the initial conditions are far from realistic. For the simulation to settle down during the initial phase, it is recommended to relax the grid flux term by lowering the grid flux URF (Under Relaxation Factor). Large fluctuations in pressure at the fluid-structure interface can

Grid adaptation in FSI

In the present study, grid adaptation to follow the motion of body is implemented by two different methods, namely morphing and overset grids, the choice depending on the problem solved. In the case of a two-way coupling, the nodal displacements imported from Abaqus redistribute the mesh vertices by generating an interpolation field throughout the domain. The deformation of the fluid grid must conform to the body and also maintain a good quality of finite volume grid. The arbitrary motion of the mesh vertices is taken into account when solving the fluid transport equations. Star-CCM+ uses a “space conservation law” to balance the volume of a CV as a function of time and the motion of the surface. Morphing could create problems in the case of large body motions and waves. The deformation of the entire grid could result in the free surface to

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be decreased using a lower value of grid flux URF. However, a value of URF less than 0.5 will lead to an unrealistic, time-inaccurate solution, especially for problems requiring dynamic accuracy. 2.5

Table 1.

3.1

2.445 m

Length of barge, LPP (caissons + clearance) Beam Depth Draft KG Total mass (caissons + equipment) Length of each caisson Mass of each caisson (except bow) Mass of bow caisson Moment of inertia of rod Bending stiffness of rod Young’s modulus of rod

2-D hydroelasticity analysis

Generalised coordinates for rigid and flexible motions of the barge is calculated using the 2-D hydroelasticity method by Bishop et al. (1977). In brief, the strip theory is used to calculate the hydrodynamic properties of the barge, using Lewis form representation. The barge structure is modelled as a Timoshenko beam. Modal summation is employed to represent vertical displacement, bending moment and shear force at a specified location. The resultant unified equations of motion in regular waves provide the requisite generalized coordinates for a range of wave frequencies. 3

Barge characteristics.

Table 2.

NUMERICAL SIMULATIONS OF A BARGE IN REGULAR WAVES Barge characteristics

The experimental model of a flexible barge consisting of 12 connected caissons is considered for validation of the present numerical method (Remy et al. 2006). Each caisson is clamped to a steel rod which is placed at 57 mm above deck level. The rod has a square cross-section of 1 cm × 1 cm. All the caissons are rectangular sections, except for the bow caisson, which has a bevelled shape. The towing tank dimensions are 30 m × 16 m × 1 m. The main characteristics of the barge and the flexible rod are given in Table 1. Vertical, horizontal and torsional bending was allowed for the barge model and tests were conducted in regular and irregular waves of varying headings. In the case of regular wave, the barge motions were measured at 6 different locations, as shown in Table 2 (x, y and z measured from stern, centerline and the keel of the barge) for a number of wave periods.

0.6 m 0.25 m 0.12 m 0.163 m 172.5 kg 0.19 m 13.7 kg 10 kg 8.33 × 10−10 m4 175 N m2 2.1 × 1011 N/m2

Measuring points from aft of barge.

Location

x (m)

y (m)

z (m)

1 3 5 7 9 12

2.445 2.035 1.625 1.225 0.805 0.19

0.0 0.0 0.0 0.0 0.0 0.0

0.25 0.25 0.25 0.25 0.25 0.25

Table 3.

Test conditions.

Wave period (s)

Wave frequency (rad/s)

Wave length (m)

Wave height

1.8 1.6 1.2 1.0 0.9

3.490 3.926 5.235 6.283 6.981

5.058 3.996 2.248 1.561 1.264

100 mm 100 mm 100 mm 100 mm 100 mm

3.2 Test conditions

2. Response of the barge, treated as a flexible body, in head regular waves is calculated for the same periods. A two-way coupling technique is implemented between Star-CCM+ and Abaqus to simulate hydroelastic response of the barge. Comparisons were made against experimental measurements and hydroelastic predictions.

Two types of simulation were carried out in the present study.

3.3 Computational domain

1. Response of the barge, treated as a rigid body, in head regular waves is calculated for the five wave periods, shown in Table 3. The objective is to obtain insight into domain size, damping zones, motion of rigid body and mesh refinement around the body and free surface for each wave period. The rigid body responses calculated using CFD is compared against 2-D predictions.

A 3-D domain is used for all CFD calculations, with x along the barge and y and z in the athwartships and vertical directions, respectively. The lengths of the domain in the inlet-outlet and side wall directions are generally calculated based on LPP or wave length (λ) based on similar ship-wave interaction studies (Peric et al. 2007) (Seng et al. 2012). In the present study, the wavelength to barge length (λ/L) ratio varies from 2 to 0.4. For λ/L ≥ 1, the inlet

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Table 4. Summary of domain sizes against wave lengths. Wave length (m)

Location of inlet (m)

Wake region (m)

Damping zone (m)

Side wall* (m)

Water depth (m)

5.058 3.996 2.248 1.561 1.264

7.6 6.0 5.0 5.0 5.0

10 8.0 5.0 5.0 5.0

7.6 6.0 3.5 2.5 2.0

8.0 8.0 8.0 8.0 8.0

4.0 4.0 4.0 4.0 4.0

*Port side only.

and wake region is located at about 1.5 λ and 2λ, respectively, from the barge. The length of inlet and wake region for cases λ/L ≤ 1 is 2.0 LPP for both. A numerical beach is provided at the outlet to damp the waves and prevent any reflections. The length of this damping region is set to 1.5 λ. In the CFD simulations, the length of the side wall (y-direction) is fixed as 8 meters (same as the tank) on one side of the barge for all cases. Initially, a reduced length of the side wall (6 meters) was tested for a few frequencies. They showed evidence of wave reflections from the side walls after 4–5 wave periods. The domain sizes selected for each wave frequency, for both rigid body and coupled simulations, are shown in Table 4. Symmetry condition is used for rigid body simulations, whereas full domain is modelled for the co-simulation cases. 3.4

Figure 1. Mesh refinement around the body and the free surface, for wave period of 1.2 s.

zone around the barge using volumetric controls. A typical mesh, corresponding to 1.2 s wave period, is shown in Figure 1. Mesh refinement was carried out based on the disturbed wave contour around the body. After testing a few cases it was noted that the waves radiated out of the body in a circular pattern. Hence, refinement around the body was also carried out in a manner so as to capture the disturbed wave pattern. In the free surface region, 45–60 cells are placed per wavelength and 12–15 cells per wave amplitude. Around 320 cells per wavelength and 160 cells per wavelength are clustered in the near body region (bow, stern and around body) and wave radiation zone, respectively. The global mesh count for the co-simulation case varied from 2.3 million to 13.6 million. Boundary conditions were selected so that they mimic the conditions of a towing tank. At the velocity inlet boundary the kinematics of the wave, i.e. the position of the free surface and velocity of the first-order wave as field functions are prescribed. At the outlet boundary, the outlet pressure and the position of free surface is prescribed. The pressure at the outlet is set to the hydrostatic pressure of the wave. All other boundaries are set to no-slip wall condition.

Meshing strategy & boundary conditions

Star-CCM+ provides various volume meshing models. In all the present investigations, a combination of trimmer, extruder and overset mesh is used. For the same wave frequency, the mesh refinement used for both rigid body simulations and co-simulations is identical. Trimmer mesher is a robust and efficient method of producing high quality hexahedral meshing with minimum cell skewness. Once the core mesh is created, the extruder mesh produces orthogonal extruded cells for user specified boundaries only. The mesh is extruded from the specific boundary in the normal direction based on the user specified extrusion parameters (i.e. number of layers, stretching ratio and extrusion magnitude). The side wall, in y-direction, and the outlet in all models is extruded using appropriate extrusion parameters. Not only does this aid in saving the global cell count but also dissipates the waves in the far field due gradual coarsening of grid size. Nevertheless, the mesh growth in the extruder region was kept under 1.1 to prevent any numerical reflections arising due to sudden change in grid sizes between adjacent cells (STAR-CCM+ 2012). The mesh was refined along the free surface region, near the barge and in the wave radiation

3.5

Rigid body simulations

This section describes the numerical setup and settings specific to the rigid body barge simulations. Due to symmetry of this problem, the computational domain only extends to the port side of the barge. After an initial orientation of the body is specified by the user, Star-CCM+ automatically creates a new Cartesian coordinate system which is updated showing the position and orientation of the barge throughout the simulation. Body release and ramp time were specified for all simulations which are calculated on the basis of the time step.

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It is best to allow the fluid flow to initialise and become steady before the calculation of body motions commences. A typical value of 50 time steps is specified as a release time. At the release time, forces and moments are suddenly applied on the body, and can cause shock effect. To minimise this and facilitate a more robust solution by reducing oscillations, a ramp time equal to 10 times the release time is specified. The VOF under-relaxation factor is decreased from the default value of 0.7 to 0.6. This implies that a fraction of the newly computed solution will be supplied to the old solution, and has been done to increase stability. Consequently, the number of inner iterations was raised to reach a convergent solution. Computations were carried out using inviscid fluid model with an implicit unsteady solver. The 2nd order scheme is chosen for temporal discretisation and convection of segregated flow solver and VOF solver. Time step for each simulation was chosen such that the Courant number on the free surface at all times is less than 0.5.

Figure 2. Finite element mesh with the beam and dummy surface linked using kinematic coupling.

This section details the fluid solver settings specific to the coupled simulations. The boundary conditions are those used in the rigid body simulations. Co-simulations were also carried out using an inviscid flow model using implicit unsteady solver. The implicit coupling scheme was chosen for all simulations. The grid flux URF is lowered from a default value of 1.0 to 0.8, implying that the fluid response to structural response is slightly reduced by a fraction commensurate to the URF. This was done to provide stability as many attempts with a higher grid flux URF resulted in quick pressure divergence. For some cases, where pressure peaks were observed even when the morpher was running smoothly, the URF for pressure was lowered to 0.3 from a default value of 0.4. Similar to the rigid body cases, all simulations were run using 2nd order temporal and convective schemes. When carrying out co-simulations using Star-CCM+ and Abaqus, the FSI boundary has to be defined explicitly. The barge boundary is set as the FSI boundary in the fluid mesh.

The elements chosen are the 2-node linear beam element B31 and 4-node quadrilateral surface element SFM3D4, the latter representing the barge surface. B31 is a Timoshenko beam element allowing for transverse shear deformation. Abaqus automatically calculates the transverse shear stiffness values required in the formulation of element. Please note for this investigation zero structural damping was used, since the frequency range in experiments and CFD/FE simulations was below the first resonance. Surface elements have no inherent stiffness but may have mass/unit area, though none is specified in this case. They can be used to transmit only in-plane forces and have no bending or transverse shear stiffness. The dummy surface elements are linked to the nodes on the beam elements using kinematic coupling constraints. A large number of nodes or surfaces can be constrained to the rigid body motion of control nodes (in this case the beam element nodes) using kinematic coupling. All six degrees of freedom are constrained in the kinematic coupling of beam nodes and the dummy surface, in the sense that the beam deformations are imparted on to the barge hull. The total mass of the barge is distributed on the beam elements. For the 2-D hydroelasticity analysis the barge is represented as a non-uniform beam element divided into 48 sections, to achieve consistency with the FE model. The mass distribution, moment of inertia for each segment, is similar to the FE model. No rotary inertia is specified for the beam elements.

3.7

4

3.6

Co-simulation of the barge

FE model

The structural or finite element mesh is modelled in Abaqus. When beam elements are used to represent structural models, they have to be linked to surface elements that define the actual wetted surface of the body. In this study, the flexible barge is represented using a 2-D beam model, with 48 beam elements. All material and geometric properties are modelled in line with model test data, shown in Table 1.

RESULTS AND DISCUSSION

4.1 Modal analysis Modal analysis was performed in Abaqus using Block Lancozs eigen value extraction method. The natural frequencies and mode shapes obtained in Abaqus were compared against calculations performed using the finite difference method applied to a non-uniform beam (Bishop et al. 1977), to ascertain the accuracy of the modelling. The dry

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Table 5. Symmetric dry hull natural frequencies (rad/s). Mode

Abaqus

2-D hydroelasticity

2-node 3-node 4-node 5-node 6-node

6.01 16.43 32.00 52.66 78.426

6.03 16.49 31.10 52.73 78.34

hull natural frequencies for the first 5 flexible modes are shown in Table 5. 4.2

Figure 3. of barge.

Heave, pitch RAOs and principal coordinates

Simulation results

The motion responses of the barge treated as rigid and flexible body in regular head waves are presented in this section. Predictions using the inviscid flow option in Star-CCM+ were compared against experimental measurements by Remy et al. (2006) and numerical calculations using 2-D linear hydroelasticity, the latter providing both rigid and distortional displacements along the barge. Wave elevations were recorded at one Lpp in front of the barge. Time history of the wave elevations revealed very little wave dissipation, and maximum decrease in wave height was around 5–6% as the simulation progressed. Average wave amplitude over 8–10 wave periods was used to calculate the RAOs. The heave and pitch RAOs obtained from the rigid body CFD simulation, denoted by STAR, and the corresponding 2-D hydroelasticity rigid as well as distortional principal coordinate amplitudes, denoted by MARS, are shown in Figure 3. The rigid body response of the barge, obtained from STAR-CCM+ and denoted by STAR_ RIGID, is compared with the 2-D potential flow predictions, denoted by MARS_RIGID, in Figure 4. It can be seen that both numerical results predict the rigid motion response with relatively good accuracy, although discrepancies were noted especially towards the forward end of the barge, e.g. points 1 and 3. At the forward section of the barge, strong bow waves were seen to develop in the CFD simulations from a wave frequency of 3.926 rad/s and upwards. In addition, with increase in frequency, diffraction becomes more dominant resulting in strong localized bow waves. The instantaneous wave pattern around the rigid barge, for frequencies of 3.926 and 6.0 rad/s, is shown in Figures 5 and 6, respectively. In general, very good agreement is noted at all frequencies at locations from around amidships towards the stern of the barge. At point 12 near the stern of the barge (0.19 m), the RAO of the vertical displacement predicted by CFD at 3.926 rad/s

is however larger to that of the linear 2-D linear potential flow analysis. The instantaneous wave contour plot at this frequency shows strong bow and stern waves influenced by pitch motion, as seen in Figure 5. A peak in pitch response can be seen at around 4 rad/s in Figure 3. The reason for the discrepancy between CFD and 2-D linear analysis predictions at the forward and stern sections of the barge is mainly due to the influence of strong localised wave systems, not very well predicted using a linear potential flow theory. The predictions using CFD is considered more reliable in this case since it accounts for the nonlinear interactions between wave-body. Strong 3-D effects and the bevel shape of the bow could also be one of the influencing factors for the differences observed in the two numerical results for the rigid body analysis. RAOs of vertical displacements, for the barge treated as a flexible body, are also shown in Figure 4. Those obtained by 2-D hydroelasticity and the two-way CFD/FEM coupling are denoted as MARS_FLEX and STAR_FLEX, respectively. Very good agreement can be seen between the CFD/FEM coupling method and experimental measurements. Comparisons made between the 2-D hydroelasticity numerical prediction and the CFD/FEM coupling also show fair agreement, with some discrepancies observed at higher frequencies. The reason for these differences is attributed to the strong diffraction effects at higher frequencies also prevalent in the flexible body motions. From the amplitude of the 2-node principal coordinate, shown in Figure 3, resonance appears to occur at around 7.5 rad/s, where computations were unfortunately not carried out. When the rigid and the flexible body responses are compared with each other, vertical displacement of flexible barge is lower, at lower frequencies, towards the bow and stern part of the barge. A clear difference between the two can be observed

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Figure 4. RAOs of vertical displacements along the barge, point 12 near the stern and point 1 near the bow, for both rigid and flexible body analyses.

Figure 5. Instantaneous wave contour around the rigid body at a 11.5 s wave frequency of 3.926 rad/s.

Figure 6. Instantaneous wave contour around the rigid body at 9.5 wave frequency of 6 rad/s.

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in the wave contours at 3.926 rad/s by comparing Figure 5 and Figure 7. Strong bow and stern waves are developed at this frequency for the rigid body motions, whereas the flexible body tends to deform following the wave resulting in weak localized waves. This is the reason for larger vertical displacements in the rigid body simulation when compared to the flexible body. At higher frequencies diffraction effect dominates, which results in a similar wave contour around the body for both rigid and flexible body approaches (see Figures 6 and 8), leading to a small difference in predicted vertical displacements. There are differences between the coupled CFD/ FEM simulation results and the experimental measurements in some cases. In general, the predicted response of the flexible barge from amidships to the forward end agrees better with the corresponding measurements. The RAOs are slightly over predicted towards the aft end. It is thought that these differences can be improved by using more refined grids and fine tuning the numerical parameters used in the coupling. The difference in mesh resolutions between Star-CCM+ and Abaqus could also be a source of

Figure 7. Instantaneous wave contour around the flexible body at 11.5 sec t for wave frequency of 3.926 rad/s.

Figure 8. Instantaneous wave contour around the flexible body at 9.5 sec for wave frequency of 6 rad/s.

instability. The finite element mesh is much coarser when compared to the Star-CCM+ mesh. Investigations using a finer mesh need to be carried out to study grid convergence. In the coupled simulations constant co-simulation time of 0.005 s is set for all cases. The grid size close to the body is quite small and the deformations of the barge are normal to these thin cells. It creates a scenario where the FSI boundary could possibly move more than the thickness of the cell near the boundary. A combination of expected motion and the thickness of cells near the FSI boundary define the time step and co-simulation time. Possibility of using different time steps and their effect on the solution has to be further investigated. 5

CONCLUSIONS

The time domain hydroelastic investigation is carried out using commercially available software Star-CCM+ and Abaqus. The field equations are coupled using an iterative implicit scheme, and a Full Newton solution technique is used for solving the structural dynamics. The numerical solution is compared to experimental measurements and 2-D linear hydroelastic predictions. Calculations are carried out for both rigid body and flexible structural idealisations. Very good agreement is achieved between time domain predictions and experimental measurements in most cases, with some exceptions especially at the forward section of the barge. Although the comparisons for rigid body motions between the two numerical methods agreed well overall, large differences were observed in the bow and stern regions of the barge at lower frequencies. This is believed to be due to strong bow and stern waves systems mainly influenced by the pitch, which are captured in CFD. CFD solves nonlinear Navier-Stokes equations, even when an inviscid flow model is selected which makes it more realistic than the linear potential flow code. Nevertheless, it should be noted that the rigid body approximation is not suitable for this very flexible barge. Comparisons made between the present coupling technique and the experimental measurements showed very few discrepancies. Very good agreement was observed at relatively low frequencies, but slight differences were noted at higher frequencies. Strong diffracting wave systems are developed at these higher frequencies and they may be influencing the motion at the bow and stern sections. It is thought that the predictions at these relatively high frequencies can be improved by appropriate mesh refinement of fluid and structural models and coupling parameters (such as time step, URF’s etc). This will be studied in detail

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in future investigations. Of the two hydroelasticity numerical methods, predictions using the coupled CFD method showed a far better agreement with experiments as it allows for nonlinearities. Influence of flexibility is clearly seen in the relatively low frequencies as the body deforms with the wave resulting in weak stern and bow waves, hence; lower vertical displacements when compared to the rigid body approach. The results show that the coupling technique investigated is reliable and compares well with experimental measurements. The next stage of the investigations will involve applying the coupling technique to predict the forces and bending moments of the barge. Special attention will be focused at higher frequencies where resonance occurs for the 2-node vertical bending mode.

REFERENCES Bishop, R.E.D. & Price, W.G., 1979. Hydroelasticity of Ships. Cambridge University Press. Bishop, R.E.D., Price, W.G. & Tam, P.K.Y., 1977. A unified dynamic analysis of ship response to waves. Trans. R. Inst Nav. Architects 119: 363–390.

Brizzolara, S., Couty, N., Hermundstad, O., Ioan, A., Kukkanen, T., Viviani, M. & Temarel, P., 2008. Comparison of experimental and numerical loads on an impact bow section. Ships and Offshore Structures 3(4): 305–324. Ferziger, J. & Peric, M., 2003. Computational Methods for Fluid Dynamics 3rd Edition. Berlin: Springer. ISSC., 2012. Report of the Technical Committee I.2 on Loads. Proc. 18th Int. Ship and Offshore Structures Congress: 79–150. Kim, Y. & Kim, K.H., 2009. Analysis of Hydroelasticity of Floating Shiplike Structure in Time Domain Using a Fully Coupled Hybrid BEM-FEM. J. Ship Res 50(1): 31–47. Paik, K.J., Carrica, P.M., Lee, D. & Maki, 2009. Strongly coupled fluid–structure interaction method for structural loads on surface ships. Ocean Engng 36: 1346–1357. Peric, M., Zorn, T., El Moctar, O., Schellin, T. & Kim, Y.S., 2007. Simulation of Sloshing in LNG-Tanks. In OMAE. Remy, F., Molin, B. & Ledoux, A., 2006. Experimental and numerical study of wave response of a flexible barge. In 4th Int. Conf. on Hydroelasticity in Marine Technology: 255–264. Seng, S., Andersen, I.M.V. & Jensen, J.J., 2012. On the influence of hull girder flexibility on the wave induce bending moments. In 6th Int. Conf. on Hydroelasticity in Marine Technology: 341–353. STAR-CCM+, 2012. STAR-CCM+ version 8.04 manual.

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Vibrations

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Simplified method for natural frequency analysis of stiffened panel E. Avi, A. Laakso, J. Romanoff & I. Lillemäe Aalto University, Espoo, Finland

ABSTRACT: Vertical vibration of decks is typically the most significant mechanical vibration affecting passenger comfort in ships. In order to save modelling and calculation time, the global finite element model is often created using equivalent elements, where secondary stiffeners are incorporated into the plate or shell formulation in a way that it results in equivalent stiffness. However, the existing equivalent shell elements do not consider the local plate vibration between the stiffeners. This limitation causes the largest errors in areas of thin decks, where the natural frequencies of plates are closer to the global frequencies of the stiffened panel. Proposed approach corrects this error by combining the natural frequencies of the equivalent element with the local plate frequencies, which are pre-calculated using analytical formulae or sub-modelling technique. The combination is performed based on the assumption that the separate plates and stiffeners act as springs and masses in series. The method is applied for stiffened panels with pinned and clamped boundary conditions and good agreement with 3D fine mesh analysis is observed. 1

INTRODUCTION

In the design of modern ships and other large thinwalled structures, new optimized structural solutions are being utilized. This means that advanced numerical methods are needed to accurately evaluate global, i.e. primary, and local, i.e. secondary and tertiary response (Gudmunsen 2000). Generally two types of global response analysis are performed: equivalent quasi-static and vibration. In the static analysis, the stresses and deformations of the hull girder due to hogging, sagging, racking, torsion and docking load conditions are obtained (DNV 2013, 2007) and the aim is to guarantee structural safety. A thorough assessment of the vibration and noise is essential for the passenger comfort. For ships that are dynamically loaded by waves and machinery, vibration assessment has become more and more time-consuming. Therefore, computational methods are needed which allow easy modifications to the structural dimensions and yet are fast enough for every day design work. Recently Avi et al. (2013) developed a method to assess static and vibration response of cruise ships using Equivalent Single Layer (ESL) theory. They considered decks stiffened by Holland profiles and system of web-frames and girders. In terms of static response the agreement with fine mesh 3D Finite Element (FE) models was excellent. However, for the vertical vibrations larger differences were observed in the modes where the plate between the stiffeners vibrates and therefore interacts with the global mode. This causes the largest

errors in areas of thin decks, where the natural frequencies of plates are closer to the global frequencies of the stiffened panel. The problem is due to the fact that in ESL the homogenized stiffness properties are used, mathematically only the average response of the panel is considered and most importantly the behavior between the stiffeners is neglected. For static case this limitation can be corrected using superposition principle (Avi et al. 2013) but in the vibration problem the issue is far more complex. The aim of this paper is to extend the ESL to the cases where the global and local vibration modes of the stiffened deck panels interact. This is done by deriving the equations of motion of ESL plate, homogenizing these for certain local mode shapes and derivation of the localization scheme from which the eigenfrequencies can be extracted. 2 2.1

THEORY Notations

The stiffened panel is considered as a three layer laminate element, where the first layer represents the plate, the second layer the stiffener web and the third the stiffener flange. Plate layer has the thickness of the plate tp, web layer has the thickness of the web height hw and the flange layer has the thickness of the flange height hf. The cross section of the flange is idealized as a rectangle, which dimensions result in equal mass and second moment of area as the original HP-profile, Figure 1.

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[E ] f

Figure 1. Stiffened panel division into three layer laminate element and the notations used in this paper.

2.2

⎡D ⎡⎣ DQ ⎤⎦ = ⎢ Qx ⎣ 0

(1)

[B ]

hf

∫ [E ]

h/

z

f

− h/2 − h/

hp

w

− h/

hf

∫ [E ] z f

− h/2

h/2

∫ [E ] dz

z+

h/

hp

∫ [E ] zdz

hp

h/2

w

− h/

(2)

p

hf

h/

∫ [E ] dz ,

hf

∫ [E ] zdz , p

h/

[D ]

hf

∫ [E ]

h/ f

z z+

− h/2

hp

∫ [E ] z dz ∫ [E ] w

p

(5)

The web and flange layers are described as 2D orthotropic shell elements, where the components of the elasticity matrix ([E]w and [E]f) are found by applying the Rule of Mixtures:

[E ]w =

⎡E tw ⎢ 0 s ⎢ ⎣0

0 0⎤ 0 0⎥ , ⎥ 0 0⎦

k=

z dz .

h / 2 − hp

The plate layer is described as a 2D isotropic shell element, where ε3 = γ13 = γ23 = 0. According to the Hooke’s law, the elasticity matrix [E] of the plate layer is: E νE 0 ⎤ 1 ⎡⎢ = ν E E 0 ⎥. (1 − 2 ) ⎢ 0 0 G ( 1 − ) ⎥⎦ ⎣

)

(9)

tw , s bf Gs . s Gs

(10)

kxz is the shear correction factor in the xz-plane. Shear correction factor k relates the maximum shear stress (τxz)max, i.e., the shear stress at the centroid of the cross section, to the average shear stress (τxz)avg:

2

(4)

[E ]p

(

Gf

h/2

hf

(8)

hp

2

− h/

0 ⎤ . DQy ⎥⎦

where Gp is the shear modulus for the plate layer and Gw and Gf are the shear moduli for web and flange layers, respectively. These can be obtained from the following relations:

(3) − h/

(7)

kxz G pt p + Gw hw + G f h f ,

DQx

Gw

The [A], [B], and [D] stiffness matrices for stiffened panel are obtained from following expressions (Avi et al 2013): − h/

0 0⎤ 0 0⎥. ⎥ 0 0⎦

The shear stiffness DQx can be written as:

Based on ESL theory presented by Reddy & Ochoa, (1992), the relationship between homogenized internal forces, strains and curvatures can be summarized as a single matrix form:

[ A]

bf ⎡E ⎢0 s ⎢ ⎣0

The out-of-plane shear stiffness matrix [DQ] contains shear stiffness in stiffener direction DQx and transverse to stiffener direction DQy:

Relationship between homogenized internal forces and strains and curvatures

⎧ N ⎫ ⎡A B 0 ⎤ ⎧ε ⎫ ⎥⎪ ⎪ ⎪ ⎪ ⎢ ⎨M ⎬ = ⎢ B D 0 ⎥ ⎨κ ⎬ . ⎪ Q ⎪ ⎢ 0 0 D ⎥ ⎪γ ⎪ ⎩ ⎭ ⎣ Q⎦ ⎩ ⎭

=

(

xz )avg

(

xz )max

(11)

.

The average shear stress can be obtained by using the well-known approach by dividing the shear force by the effective shear area: (

xz )avg

≈

Aw

Qz . twt p tw h f

(12)

The maximum shear stress (τxz)max in stiffened panel is defined as: (

xz )max

=

Qz ⎡⎣ Ap (

na

p ) + tw ( na

2 I ztw

p)

2⎤

⎦,

(13)

where zna is the distance from the neutral axis and Iz is the second moment of area. The laminate shear stiffness transverse to the stiffener direction DQy can be written as:

(6) DQy

(

)

kyz G pt p ,

(14)

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where the shear correction factor kyz 5/6 is derived from the plate shear energy and is included in Reissner-Mindlin plate theory (Whitney & Pagano 1970). The density of the ESL is defined with rule of mixtures and results in weight per area, which consists of structural and non-structural mass. The resulting equations of motion are given in Reddy (2004, pp. 139–141).

by analytical methods. In this paper FE-analysis is used. The obtained local plate angular frequencies are referred to as follows: ωcp for clamped-pinned plate, and ωcc for clamped-clamped plate. The angular frequencies practically represent stiffness of the individual plates as the mass is assumed to be equally divided along the plate. 2.4

2.3 Sub models of plates between stiffeners Plate vibration between the stiffeners, see Figure 2, is defined separately by using sub-modelling technique as one assumption in homogenization is that s/l approaches asymptotically zero; l is the characteristic length of vibration half waves. As the assumption is not valid, correction is needed. The plate between the stiffeners is assumed to have clamped-clamped boundary conditions due to continuity of the structure. At the location where the half wave of the global mode changes direction, the plate is assumed to have pinned-clamped boundary conditions, see Figure 2. These boundary conditions for the local models are presented in Figure 3a and Figure 3b, respectively. The angular frequency of the lowest flexural vibration mode is calculated either by standard FE-modal analysis or

Combination of the eigenfrequencies

The global vibration modes received from the laminate model are combined with the local modes of the plate. Single degree of freedom springmass systems are used to represent each stiffener and plate as presented in Figure 4a. For practical

Figure 2. Local vibration between stiffeners in two half wave mode.

Figure 3. Boundary conditions of local models for (a) pinned and for (b) clamped panel.

Figure 4. (a) Local vibration as part of global modes of the panel and (b) the simplified model.

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vibration. Hence, it represents a situation where ks in Figure 4b is infinitely stiff. The remaining part is thus a single DOF oscillator with mass md and stiffness kd + kw. For such oscillator the relation between mass, stiffness and angular frequency is following:

calculations the systems are simplified as shown in Figure 4b. The mass ms represents the mass of all stiffeners and md represents the mass of all plates without stiffeners. The spring ks represents the bending of stiffeners, kd represents the bending of the plate relative to stiffeners and kw bending of the plate relative to the boundaries. First, the local model results are combined to find the average local angular frequency for different global modes. The combination is done by taking weighted average of local plate angular frequencies ωcp and ωcc so that the weighting is in line with their appearance in the considered global mode. The resulting angular frequencies for different global modes and number of stiffeners n are presented in Table 1. The angular frequency of the panel ωs given by the ESL model represents a situation of the Figure 4b, where the spring kd is infinitely stiff, as the local plate deformation does not exist in the theory. The remaining part is thus a single DOF oscillator with mass ms + md and stiffness ks + kw. For such oscillator the relation between mass, stiffness and angular frequency is following: ks

kw = ( md + ms )ω s2 .

kw = md ω d2 .

kd

(16)

One additional equation is needed to solve three unknowns. Stiffness ratio between the springs kd and kw is approximated with the envelope method. The kw is proportional to stiffener spacing s and kd is proportional to stiffener length L. Following relation is obtained: kw =

(

(

s

L s ) kd 1

ks−1

)

(17)

.

The spring constants can now be solved from group of Eq.-s (15), (16) and (17): ⎧ ks kw = ( md + ms )ω s2 ⎪ kd kw = md ω d2 . ⎪ ⎨ s ⎪k = ⎪ w ( L s ) k 1 k −1 d s ⎩

(15)

(

The average angular frequency from the deck plate models ωd includes only the local plate

(18)

)

Table 1. Averaged local plate frequencies ωd for different global modes and number of stiffeners n for (a) pinned panel and (b) clamped panel. Plate BC:s Global half waves

ωcp

a) Pinned panel 1 2 2

4

ωcc

ωd

n−1

2ω cp

n−3

Range of applicability 1)ω cc

n≥1

4ω cp

n +1 ( 3)ω cc

n≥3

n +1 ( 5)ω cc

n≥5

(

3

6

n−5

6ω cp

N

2N

n + 1 − 2N

n +1 2 Nω cp ( n 1 22N N )ω cc

n ≥ 2N − 1

n +1 b) Camped panel 1 0 2 2

n+1 n−1

ωcc 2ω cp

3

4

n−3

N

2N − 2

n + 3 − 2N

1)ω cc

n≥1 n≥3

4ω cp

n +1 ( 3)ω cc

n≥5

(

n +1 )ω cp

(

(

)ω cc

n ≥ 2N − 1

n +1

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Solving the group of Eq. (18) leads to the following spring coefficients: md + ms

ks

2 s

kw ,

(19)

⎛ Lmd ω d2 + L ( md + ms )ω s2 − ⎜ 2 2 1 ⎜ −4 Lmd ( md + ms ) sω d ω s + kw = 2 2 2 ⎜ md ( md + ms ) s ω d ω s + 4 L s ⎜ − +m ⎜ + Lmd ω d2 L ( md ms )ω s2 ⎝

(

⎞ ⎟ ⎟ ⎟, 2⎟ ⎟ ⎠

(

)

)

(21) The squared angular frequency of the stiffened panel ω2panel can be found from the lowest eigenvalue of mass normalized stiffness matrix (Weaver et al. 1990): ⎡[M ]−1 [ K ]⎤ , ⎣ ⎦

(22)

where the mass matrix [M] is: ⎡m

[M ] = ⎢ 0s ⎣

0 ⎤ , md ⎥⎦

(23)

− kd ⎤ . kd + kw ⎥⎦

⎡ ks kd ⎣ − kd

2π

.

(27)

CASE STUDY

The presented method is applied for natural frequency analysis of the stiffened panel, where first three half-wave modes are considered, see Figure 5. The scantlings represent typical deck in passenger ship, Figure 6a. The panel width B is 6.12 m and length L is 2.5 m. The thickness t of the deck plate varies from 3 mm to 20 mm and it is stiffened with HP100 × 6 profiles. The stiffener spacing s varies from 204 to 1020 mm. The panel is made of steel with Young’s modulus of 206 GPa, Poisson ratio of 0.3 and mass density of 7850 kg/m3. The mass matrix is formulated using the lumped scheme, where the element mass is simply divided between the nodes and a diagonal mass matrix is built, which provides higher computational economy and better accuracy in coarse mesh analysis (Avi et al 2013). The FE analysis is carried out using NX Nastran 7.1 software. The pre- and post-processing has been done with FEMAP 10.3. 3.2

Laminate element model

The stiffened panel is modelled using laminate element as described in Avi et al. (2013). Each layer is described as CQUAD4 (quadrilateral) shell

and the stiffness matrix is:

[K ] = ⎢

ω 2ppanel

3.1 General

)

(

f panel =

3

(20) ⎛ md ( L − s ) d2 L md + ms ω s2 ⎞ ⎜ ⎟ 2 2 1 ⎜ −4 Lmd ( md + ms ) sω d ω s + ⎟ kd = ⎜ ⎟. md ( md + ms ) s 2ω d2ω s2 + 4 L s ⎜ − +m ⎟ ⎜ + Lm 2 + L m + m ω 2 2 ⎟ ( ) d d d s s ⎝ ⎠

⎡~⎤ ⎢K ⎥ ⎣ ⎦

The final eigenfrequency in Hz of stiffened panel is then:

(24)

By inserting Eq. (23) and Eq. (24) into Eq. (22): ⎡ kd ks ⎢ ⎡ ~ ⎤ ⎢ ms K = ⎢ ⎥ ⎢ k ⎣ ⎦ d ⎢ − ⎣ md

kd ⎤ ms ⎥ ⎥. kd kw ⎥ ⎥ md ⎦ −

(25)

The smallest eigenvalue ω2panel of matrix (25) are found from the following relation: ⎛ ks md + kw ms + kd ( md + ms ) − ⎞ ⎜ 2 ⎟ (kd ks ) md (kd + kw ) ms − ⎟ ⎜ ⎜ − −4 k k + k + ) md ms ⎟⎠ s w d( ⎝ = . (26) 2 md ms

(

ω 2panel

(

)

)

Figure 5. First three half wave modes for (a) fine mesh model and (b) for coarse laminate model, h = 612 mm.

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element, which has four nodes and 24 Degrees of Freedom (DOF). The mesh size is h = 612 mm, see Figure 5b, which is four elements per web spacing and 10 elements per panel width, Figure 5b. The models consist of 40 elements and 330 DOF. In some cases the coarse mesh laminate model is additionally compared to fine mesh laminate model, where the mesh size is 85 mm. 3.3

3D fine mesh model

The deck plate and stiffener webs are modelled using four-node shell elements (QUAD4). The flanges of the stiffeners are described as offset beam elements, see Figure 6b. The mesh size is 85 mm and stiffeners have one element in height (z) direction. 3.4

Figure 8. Eigenfrequencies of two half waves mode of the simply supported panel with varying deck plate thickness.

Pinned panel results

The eigen frequencies for one, two and three halfwave modes as a function of deck plate thickness are presented in Figure 7, Figure 8 and Figure 9.

Figure 9. Eigenfrequencies of three half waves mode of the simply supported panel with varying deck plate thickness.

Figure 6. (a) The dimensions of the stiffened panel (mm); (b) mesh density in fine mesh model.

Figure 7. Eigenfrequencies of one half wave mode of the simply supported panel with varying deck plate thickness.

In thinner plates the plate vibration between the stiffeners is more dominant and thus the averaged eigenfrequency of sub-models is similar to the one of the panel. When the deck plate thickness increases the averaged eigenfrequency of the sub-models gets higher and does not influence the panel solution anymore. At the same time the ESL theory without the correction starts to follow the fine mesh results better. Between both extremes the developed correction method predicts the panel eigenfrequencies with very high accuracy, see Figure 10. In three half wave mode, the coarse mesh results start to deviate slightly from the 3D-FE solution. The error comes from too coarse mesh, since only 10 elements are used to describe three half waves. When fine mesh laminate model is used, the error decreases to less than 5%. Figure 11 shows natural frequencies of one half wave mode for different stiffener spacing. Here the sub-model result is closer to panel response with large stiffener spacing and ESL without the correction with smaller stiffener spacing. ESL with

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proposed correction method evaluates the natural frequencies with good accuracy for all considered cases, Figure 12. However, here also the error is somewhat larger at higher modes if too coarse mesh for describing the half waves is used.

Figure 10. Difference in eigenfrequencies between fine mesh and ESL with correction as a function of deck plate thickness.

Figure 11. Eigenfrequencies of one half wave mode of the pinned panel with varying stiffener spacing.

3.5

Clamped panel results

The eigenfrequencies for clamped panel as a function of plate thickness is shown in Figure 13 and as a function of stiffener spacing in Figure 15. Again good accuracy is obtained, see Figure 14

Figure 13. Eigenfrequencies of one half wave mode of the clamped panel with varying deck plate thickness.

Figure 14. Difference in eigenfrequencies between fine mesh and ESL with correction as a function of deck plate thickness.

Figure 12. Difference in eigenfrequencies between fine mesh and ESL with correction as a function of stiffener spacing.

Figure 15. Eigenfrequencies of one half wave mode of the clamped panel with varying stiffener spacing.

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the method to deck structure with girders and web frames seems possible and left for the future work. Because of the low computational cost of this method, the utilization in optimization process of ship deck against vibrations is attractive. Future work may also be carried out for including other vibration modes and tertiary effects of stiffeners. ACKNOWLEDGEMENTS

Figure 16. Difference in eigenfrequencies between fine mesh and ESL with correction as a function of stiffener spacing.

The research work carried out in this paper was funded by LOURA (Lounaisrannikkoyhteistyö) and Meyer Turku. The financial support is gratefully appreciated. REFERENCES

and Figure 16. The error is the highest (10%) when the natural frequencies of plates are closer to the global frequencies of the stiffened panel. This could be explained by the assumptions made in sub-modeling approach, i.e., by the boundary conditions, averaging the local vibration modes and neglecting tertiary effects of stiffeners. Nevertheless, considering the plate thicknesses and stiffener spacings used in ship design, the method gives less than 5% error.

4

CONCLUSIONS

The presented equivalent shell element with the correction method allows accurate and fast calculation of eigenfrequencies of the stiffened panel. None of the existing equivalent elements used for stiffened (e.g. Hughes, 1983, Satish Kumar & Mukhopadhyay, 2000) or sandwich panels (Lok & Cheng, 2000, 2001, Romanoff & Varsta, 2007) are including these tertiary vibration effects. In present case studies the equivalent shell element models with 330 DOF were used and less than 10% difference compared to 3D fine mesh results was obtained. The mesh sensitivity analysis showed that four 4-noded elements per wave shape is a minimum mesh density to obtain the eigenfrequencies with very good accuracy. In current paper the developed method was applied to stiffened panel with pinned and clamped boundary conditions. The extension of

Avi, E., Lillemäe, I., Romanoff, J. & Niemelä, A. 2013. Equivalent shell element for ship structural design, Ship and Offshore Structures. DOI:10.1080/17445302.2013. 819689. Det Norske Veritas. October 2007. Direct Strength Analysis of Hull Structures in Passenger Ships. DNV Rules for Classification of Steel Ships. January 2013. Gudmunsen, M.J. 2000. The Structural Design of Large Passenger Ships, Lloyd’s Register of Shipping: 1–16. Hughes, O.F. 1983. Ship Structural Design: A RationallyBased, Camputer-Aided, Optimization Approach. John Wiley&Sons. Lok, T.S. & Cheng, Q.H. 2000. Free vibration of clamped orthotropic sandwich panel. Journal of Sound and Vibration. 229(2):311–327. Lok, T.S. & Cheng, Q.H. 2001. Free and forced vibration of simply supported, of orthotropic sandwich panel. Computers and Structures. 79:301–312. Reddy, J.N. & Ochoa, O.O. 1992. Finite Element Analysis of Composite Laminates. Kluwer Academic Publishers. Reddy, J.N. 2004. Mechanics of Laminated Composite Plates and Shells—Theory and Analysis, Second Edition. CRC Press Romanoff, J. & Varsta, P. 2007. Bending response of web-core sandwich plates. Composite Structures. 81:292–302. Satish Kumar, Y.V. & Mukhopadhyay, M. 2000. Finite element analysis of ship structures using a new stiffened plate element, Applied Ocean Research. 22:361–374. Weaver, W., Timoshenko, S. & Young, D. 1990. Vibration Problems in Engineering. John Wiley & Sons, Inc. Whitney, J.M. & Pagano, N.J. 1970. Shear deformation in heterogenous anisotropic plates. Journal of Applied Mechanics. 37(4):1031–1036.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Investigation of vibrational power flow patterns in damaged plate structures for damage localisation P. Boonpratpai & Y.P. Xiong Fluid Structure Interactions Research Group, Faculty of Engineering and the Environment, Boldrewood Campus, University of Southampton, Southampton, UK

ABSTRACT: Investigation of changes in vibrational power flow patterns in plate structures when damage appears is presented in this paper. The case studies are performed with thin rectangular plates which are widely used in marine applications. Damage on the plates is simulated using Finite Element Method (FEM) as a through-thickness crack. The high stress region is located at the vicinity of the crack tips. The power flow is generated by a unit harmonic point force placed at the specific locations on the plate. The patterns of the power flow are demonstrated in the form of vector fields. When the damage arises on the plate, the patterns of the power flow at the location close to the damage change significantly. These changes at the plates show that the location of the damage is effectively detected by the power flow. The detected location of the damage in this case will be presented visually using only the power flow vector plots. 1

INTRODUCTION

Damage may occur on structures due to many reasons. The presence of the damage will increase the deflection of structures under loading and change the structural dynamic properties such as natural frequencies and mode shapes. It is worthwhile to discover the damage and repair before it is able to cause hazardous structural failure. This is very crucial for marine structures since they are required to have the ability to firmly withstand environmental loadings such as waves and winds at the time of operations. In this paper, patterns of vibrational power flow on damaged plates are studied. The basic concepts of the power flow were rigorously discussed in the published works of Goyder & White (1980). The linear mathematical model of the power flow in structures was introduced by Xiong et al. (2003). The power flow measurement techniques for plate structures were proposed by a number of researchers, including Noiseux (1970), Gavric & Pavic (1993), and Cieslik & Bochniak (2004). The damage for the present work is modelled as a through thickness crack with high stress region around its tips. The Finite Element Method (FEM) implemented by the commercial finite element package, ANSYS 12.1, is employed to complete the simulation of the damaged plates and the power flow computations. The results show that the location of the damage can be discovered due to the changes in the power flow patterns, even if the

damage is small when compared to the plate size. The vibrational power flow is potentially utilised as a damage index for damage detection. 2 2.1

POWER FLOW COMPUTATION Vibration response of plate

The FEM is applied to determine plate vibration responses, which will then be used in vibrational power flow calculations. The general equation of motion of a structure excited by an external force using in FEM is written as (Petyt 1990). [

]{ } + [C ]{ } + [ ]{ } = { }

(1)

where [M], [C] and [K] are the structural mass matrix, damping matrix and stiffness matrix, respectively; { } is the nodal acceleration vector; { } is the nodal velocity vector; {u} is the nodal displacement vector and {F} represents the external force vector. Since the system contains damping, the displacement vector, {u}, will be in the form of complex numbers. Equation (1) can then be given as

([

] Ω[

] + i [C ]){

φ

}e iΩ

Fe } e {Fe i

i Ωt

(2)

where Ω is the radian frequency of the excitation force; i equals to ; −1; φ and ψ are the displacement and force phase shifts, respectively; u denotes

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the displacement amplitude and F represents the amplitude of excitation force. The term e i t of both sides of the equation can then be cancelled out. 2.2

Power flow formulation

The instantaneous power flow per unit area at each position on plate structures is a dot product of stress and velocity vectors in arbitrary directions. It is given mathematically as (Gravric and Pavic, 1993) I k (t ) = −σ kl (t )vl (t ); k , l = 1, 2, 3, …

(3)

where σkl denotes the kl th stress component and vl is the lth velocity vector. For dynamic analysis, it is more convenient to express power flow as a time-average value. For thin plate structures, the stress components in Equation (3) can be transformed to force and moment components, and the time-average power flow per unit length can subsequently be calculated using (Cieslik, 2004) Px

Ω ℑ ⎡ N x u* + N xyv* Q x w * 2 ⎣

 θ* − M  θ* ⎤ M x y x xyy x ⎦

Py

Ω ℑ ⎡ N yv * + N yx u* Q yw * 2 ⎣

 θ* − M  θ* ⎤ M y x yyx x y⎦

Poisson ratio and density are 0.3 and 2100 kg/m3. The structural damping ratio of 0.005 and the point force of 1 N with the frequencies 14 Hz and 128 Hz are applied to the plate at the centre. The two short edges of the plate are simply supported, while the other two are kept free. The results of the power flow patterns are displayed by the vector plots in Figure 1. The heads of the arrows show the directions of the power flows generated by 12 and 28 Hz forces. These results agree well with those shown in the work of Li and Lai (2000). 2.4 Model of damaged plate For the case studies in this paper, the damaged steel plates are modelled using eight-node shell elements. The plates have 1.2 m length (L), 1.0 m width (W) and the thickness (H) of 0.01 m. The shorter sides are parallel to the coordinate y axis of the plate models, whereas the longer sides are on the coordinate x axis. The origin of these coordinates is at the bottom left corner of the plate models. The Young’s modulus, Poisson ratio and density of the plates are 210 GPa, 0.31, and 7800 kg/m3, respectively (Cremer et al. 2005).

(4)

(5) where Ω is a frequency of the excitation force; I represents an imaginary part; N x and N v are complex in-plane normal forces per unit length; p N xv N vx is complex in-plane tangential force per unit length; Q x and Qv are complex transverse  and M  are shear forces per unit length; M x v complex bending moments per unit length; p   M M xv vx is complex twisting moment per unit length; u* , v* and w * are complex conjugate of displacement in x, y and z directions; and θx* and θv* are complex conjugate of rotational displacement about x and y directions. To be more convenient, the power flow will be used for the time-average power flow per unit length from now on in the paper. 2.3

Validation of finite element modelling

The power flow computation is implemented by the commercial finite element package, ANSYS 12.1. To confirm the validity of the computation method, some parts in the published work of Li and Lai (2000) is repeated here. The intact plate in this published work is modelled using 560 eightnode shell elements. This plate has the length, width and thickness of 0.707 m, 0.5 m and 0.003 m, respectively. Its Young’s modulus is 70 GPa. The

Figure 1. Power flow vector plot in intact plates excited by unit force with excitation frequencies 14 Hz (Top) and 128 Hz (Bottom).

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The plate contains the structural damping ratio of 0.005. The damaged plates used later in the case studies of the paper are assumed to have the same aforementioned structural properties. The damage on the plate is in the form of a small through-thickness crack. At the vicinity of the crack tips, the finite element mesh density is increased to represent the region of high stress (Vafai et al. 1999). High order of mesh refining at the crack tips is able to reveal more stress and deflection details than a coarse mesh. The geometries of the crack are given in the next section. 3

CASE STUDIES

All case studies to investigate the power flow patterns are performed in the all-edge clamped plates which are more practical than the plates with simply supported boundaries. The sizes of the elements out of the damage region are 0.02 m × 0.02 m. The source of the power flow is a vertical point force with the magnitude of 1 N placed on the plate at x = 0.5 m and y = 0.2 m. The excitation frequencies are fixed at 20 Hz and 100 Hz. Initially, the study is carried out in the plate containing the through-thickness crack having its length (l) of 0.04 m parallel to the coordinate y axis of the plate models. These crack dimensions can be written in the dimensionless forms by dividing the crack length with the length of the plate as l/L = 0.0333. Its centre is located at x = 0.9 m and y = 0.5 m. The detection is then extended to operate in the plates with the crack having its dimensionless size reduced to l/L = 0.0083 located at the same location. This smaller crack is modelled using the same order of mesh refining at the crack tips as

Figure 2. ANSYS finite element mesh of damaged plate with crack size l/L = 0.0083 located at x = 0.9 m and y = 0.5 m.

that of the previous one. The finite element mesh of the smaller crack is demonstrated in Figure 2. The investigation is then moved to the plate containing the same crack with its orientation changed to be perpendicular to the coordinate y axis of the plate model. The crack used in this case is the smaller one, l/L = 0.0083, and its centre is located at the same place as before. The last case study is carried out when the position of the excitation force is altered, while the crack still locates at the same location. The width of the cracks is very small when compared to the lengths mentioned formerly (Anderson 2005). There are two crack tips in each crack. For the crack parallel to y axis, the crack tips will be labelled as the upper and lower crack tips hereinafter. The crack perpendicular to y axis will have the left crack tip which is the one closer to the input force and the other which is designated as the right crack tips. The real and imaginary parts of the maximum deformed shapes when the damaged plate shown in Figure 2 is subjected to the 100 Hz force at the location x = 0.5 and y = 0.2 are presented through the contour plots in Figure 3. The element sizes are

Figure 3. Maximum deformed shape of damaged plate (l/L = 0.0083) when loaded by 100 Hz force: real part (top); imaginary part (bottom).

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displayed by the lines on the backgrounds of the plots. This can show that the sizes of the elements are sufficiently fine to capture the data from the generated waves. 4 4.1

RESULTS Crack parallel to y axis

The vector plots of the power flow in the plates containing the crack having the size l/L = 0.0333 parallel to the coordinate y axis are shown in Figures 4 and 5. From the figures, the powers transmit outwards from the location of the input forces, and they are then dissipated by the structural damper as can be seen from the lengths of the vector arrows. However, it should be noted that the scales of the arrows used in the vector plots are not identical due to the huge different of the power flow magnitudes when the plates are excited by 20 and 100 Hz forces. Thus, only the lengths of the arrows in the same plot can be compared to each another. The patterns of the power flow at the areas around the cracks of the plates change noticeably. For each plot, the lengths of the vector arrows become longer at the crack locations. This means that the power flow magnitudes are huge at the areas around the cracks. The vector plots in Figures 4 and 5 are zoomed in at the damage locations, and illustrated in Figures 6 and 7, respectively. The zoomed-in versions of the plots clearly presented that at the vicinity of the cracks the power transmits towards the crack tips which are the locations of the high stress concentration. It can also be observed that the power at the other locations near the cracks

Figure 5. Vector plot of power flow in damaged plate excited by 100 Hz force (crack size l/L = 0.0333 parallel to y axis).

Figure 6. Zoomed version of power flow vector plot at damage location of the plate in Figure 4.

Figure 4. Vector plot of power flow in damaged plate excited by 20 Hz force (crack size l/L = 0.0333 parallel to y axis).

Figure 7. Zoomed version of power flow vector plot at damage location of the plate in Figure 5.

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besides the crack tips propagates away from the crack lines. The zoom-in level is varied according to the sizes of the power flow vector arrows around the crack of each case study. This level can be known from the scales of the plate dimensions at the borders of each plot. The power flow approach still works well to detect damage location of the smaller crack (l/L = 0.0083) as shown in Figures 8 and 9. The zoomed-in versions of the plots for the smaller-crack cases are given in Figures 10 and 11. The patterns of the power flow in these zoomed plots are similar to those of the plates with the bigger crack. The magnitudes of the power transmitting through the cross sections at the crack tips are computed by integrating the time-average power Figure 10. Zoomed version of power flow vector plot at damage location of the plate in Figure 7.

Figure 8. Vector plot of power flow in damaged plate excited by 20 Hz force (crack size l/L = 0.0083 parallel to y axis).

Figure 11. Zoomed version of power flow vector plot at damage location of the plate in Figure 8. Table 1. Magnitude of power flow around crack tips of the cracks parallel to y axis.

Figure 9. Vector plot of power flow in damaged plate excited by 100 Hz force (crack size l/L = 0.0083 parallel to y axis).

l/L = 0.0333

l/L = 0.0083

Frequency

Upper (dB)

Lower (dB)

Upper (dB)

Lower (dB)

20 Hz 100 Hz

−1.1027 33.4627

−1.1027 33.4627

−4.2824 30.2821

−4.2824 30.2821

flows per unit length at the points around those crack tips (Gravric et al. 1980). These power flow magnitudes of the large and small cracks are given in Table 1. The unit of the power is dB with the reference power equals to 10–12 Watt. The power flows around the upper and lower crack tip of the plate loaded by each excitation frequency are comparable

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due to the symmetry of the excitation force and crack locations. The power flow at the crack tips of the plate excited by 100 Hz force is bigger than that of the one subjected to 20 Hz force. 4.2

Crack perpendicular to y axis

The orientation of only the smaller crack (l/L = 0.0083) is changed to be perpendicular to the coordinate y axis of the plate model. The results when the cracked plate is loaded by the unit force with the excitation frequencies of 20 and 100 Hz are illustrated in Figures 12–15. Similar to the first case study, the patterns of the power flow change considerably at the location of the damage. The power propagates in and out Figure 14. Zoomed version of power flow vector plot at damage location of the plate in Figure 12.

Figure 12. Vector plot of power flow in damaged plate excited by 20 Hz force (crack size l/L = 0.0083 perpendicular to y axis).

Figure 15. Zoomed version of power flow vector plot at damage location of the plate in Figure 13. Table 2. Magnitude of power flow around crack tips of the cracks parallel and perpendicular to y axis. Parallel to y axis Upper Frequency (dB) 20 Hz 100 Hz

Figure 13. Vector plot of power flow in damaged plate excited by 100 Hz force (crack size l/L = 0.0083 perpendicular to y axis).

Lower (dB)

Perpendicular to y axis Left (dB)

Right (dB)

−4.2824 −4.2824 −9.9966 −12.6627 30.2821 30.2821 25.2623 21.7908

of both crack tips. It is found that the equality of the power flow patterns at two crack tips vanishes in this case. There is no power propagating against the crack fronts. The magnitude of the power propagating through the cross section at the left crack tip which is closer to the location of the input force is higher than that of the top one (Table 2).

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Comparing the data of the cracks parallel and perpendicular to y axis in Table 2 show that the magnitudes of the power flows around the crack tips of the crack parallel to y axis are greater than those of the crack perpendicular to y axis. The magnitudes of the power flows are proportional to the excitation frequency. 4.3

Effect of change in excitation force location

All of the results presented formerly are of the power flows generated by the unit force located at x = 0.2 m and y = 0.5 m. For this section, the force is moved to the new position at the coordinates (x, y) = (0.6 m, 0.5 m). Only the small crack, (l/L = 0.0083), is used in this case study. The source of the power flow is generated by the 20 Hz unit force. The vector plots of the power flow when the source is moved close to the damage are displayed in Figures 16–19. When the excitation force is moved to the new position closer to the crack, the power flow magnitude at each point around the crack and the crack fronts is larger. The diversity of the power flow magnitudes at the left and the right crack tips of the crack normal to y axis are increased as we can see when comparing Figure 19 to Figure 14. The patterns of the power flows around the cracks are analogous with those in the former cases. Tables 3 & 4 present the magnitudes of the power flowing through the cross sections at the crack tips when the 20 Hz unit excitation force is located at two different positions on the plates. It can be observed that the power flow magnitudes are increased when the input force is moved towards the cracks.

Figure 17. Vector plot of power flow in damaged plate excited by 20 Hz force located at x = 0.5 m and y = 0.6 m (crack size l/L = 0.0083 perpendicular to y axis).

Figure 18. Zoomed version of power flow vector plot at damage location of the plate in Figure 16.

Figure 16. Vector plot of power flow in damaged plate excited by 20 Hz force located at x = 0.5 m and y = 0.6 m (crack size l/L = 0.0083 parallel to y axis).

Figure 19. Zoomed version of power flow vector plot at damage location of Figure 17.

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Table 3. Magnitudes of power flows around crack tips of the cracks parallel to y axis when plate is excited by force at two different positions.

Crack parallel to y axis

Force (0.2, 0.5)*

Force (0.6, 0.5)*

Upper (dB)

Lower (dB)

Upper (dB)

Lower (dB)

−4.2824

−4.2824

8.4271

8.4271

*Position of excitation force (x, y) in metre. Table 4. Magnitudes of power flows around crack tips of the cracks perpendicular to y axis when plate is excited by force at two different positions.

Crack normal to y axis

Force (0.2, 0.5)*

Force (0.6, 0.5)*

Left (dB)

Right (dB)

Left (dB)

Right (dB)

−9.9966

−12.6627

−0.7350

−9.4676

4.4

5

CONCLUSIONS

The changes in the vibrational power flow patterns in the plate structures, when the damage exists, were studied. The damage was simulated as a through thickness crack implemented by ANSYS. The power flow was generated by a point unit force. The results from the case studies displayed by the vector plots showed that the location of the damage can be effectively detected using the changes in the power flow patterns at the location of the damage. The patterns and characteristics of the power flow at the region closed to the damage are summarized as follows:

*Position of excitation force (x, y) in metre.

Figure 20. crack tips.

for both plates. The magnitude of the power flow at the crack tip of the crack perpendicular to y axis is higher than those of the other two. These results show that the location of the damage will be displayed on the power flow vector plot most clearly when using the input excitation force having the frequency identical to one of the resonant frequencies of the plate. However, it should be noted that the shape of the plate will highly be deformed at the resonance. This may increase the extent and severity of the damage.

Trends of power flow magnitudes around

Trends of power flows around crack tips

The magnitudes of the power flows around the crack tips of the cracks parallel and normal to y axis are computed when the plates are subjected to the excitation frequencies 0–150 Hz at x = 0.2 m and y = 0.5 m. The results are displayed through the graph in Figure 20. The magnitudes of the power flows are in dB. The reference power is equal to 10–12 Watt. It can be seen from Figure 20 that the magnitudes of the power flows propagating through the cross sections at the crack tips of all cases reach their maximums at the first and second resonant frequencies which are approximately 77 and 139 Hz

• When the location of the excitation force is fixed, the higher frequencies can created larger power flow magnitudes at the crack tips and the crack fronts. • The magnitudes of the power flow at the crack tips are reversely proportional to the distance between the damage and the excitation force. • The orientation of the damage can affect the magnitude of the power flow around the damage. • The power flow is more sensitive to the crack when its orientation is perpendicular to the direction of the power flow, as can be seen from the higher power flow magnitudes at the tips of the crack parallel to y axis than those of the crack perpendicular to y axis. • The maximum power flow magnitude at the crack tips of each mode is generated when the plate is excited by the resonant frequency. The advantages of the use of the power flow in damage localisation found from the presented case studies are that the damage can be localised effectively using only a unit force with a low-range frequency; the power flow can be measured directly without the need of doing modal analysis; and the damage location can be visually inspected through vector field plots. The future work is to apply the simulations in the presented case studies to the practical damage detection technique.

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REFERENCES Anderson, T.L. 2005. Fracture mechanics fundamentals and applications. Boca Raton: Taylor & Francis. Cieslik, J., & Bochniak, W. 2004. Vibration energy flow in rectangular plates. Journal of Theoretical and Applied Mechanics 42(1): 195–212. Cremer, L., Heckl, M. & Petersson, B.A.T. 2005. Structureborne sound. Berlin: Springer-Verlag Berlin Geidelberg. Goyder, H.G.D. & White, R.G. 1980. Vibration power flow from machines into built-up structures. Journal of Sound and Vibration 68(1): 77–96. Gravric, L. & Pavic G. 1980. A finite element method for computation of structural intensity by the normal mode approach. Journal of Sound and Vibration 164(1): 29–43.

Li, Y. J. & Lai, J.C.S. 2000. Prediction of surface mobility of a finite plate with uniform force excitation by structural intensity. Applied Acoustics 60: 371–383. Noiseux, D.U. 1970. Measurement of power flow in uniform beams and plates. J. Acoust. Soc. Am 47(1): 238–247. Petyt, M. 1990. Introduction to finite element vibration analysis. Cambridge: Cambridge University Press. Vafai, A. & Estekanchi, H.E. 1999. A parametric finite element study of cracked plates and shells. ThinWalled Structures 33: 211–229. Xiong, Y.P., Xing, J.T. & Price, W.G. 2003. A general linear mathematical model of power flow analysis and control for integrated structure-control systems. Journal of Sound and Vibration 267: 301–334.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Influence of the sea action on the measured vibration levels in the comfort assessment of mega yachts E. Brocco, L. Moro, P.N. Mendoza Vassallo & M. Biot University of Trieste, Italy

D. Boote & T. Pais University of Genoa, Italy

E. Camporese Cantieri Navali Benetti, Livorno, Italy

ABSTRACT: Understanding the methods to evaluate the comfort on board ships through direct measurements of the vibration levels is crucial for obtaining accurate results, and so a clarifying explanation of the methods required by the ISO 6954:1984 would be useful. The checking is performed using a Maximum Repetitive Value (MRV) that is not clearly defined, nor specified in which circumstances or for which dimension of the ship it should be used or not. Neither has an explanation been given to the use of a conversion factor that inherently represents time dependent vibration phenomena, modulation, which depends on the Sea State. The Crest Factor represents these modulations, and depends on the Sea State and ship dimensions. It is important to understand better the parameters that affect the Crest Factor to evaluate better the comfort on board ships. Some Classification Societies give guidelines on how to use the ISO standard in question, but they do not always cover certain aspects that must be considered. In the paper, suggestions on how to use the standard are given, approaches from Classification Societies are analyzed and a procedure to obtain MRA and the Crest Factor generated. Different influencing factors to the Conversion Factor and the MRA, such as time length and position of the measurement during testing and filter type, have also been analyzed to obtain a more accurate value of the Crest Factor. 1 1.1

INTRODUCTION Problem definition

The procedure for the validation of the global vibration levels of merchant ships proposed within the ISO 6954:1984 normative is done with reference to the Maximum Repetitive Amplitude (MRA) that can be obtained during the measurement procedures and not with the Root Mean Square (RMS) value obtainable from an analysis in the frequency domain as it is the normal practice in the mechanical vibration analysis. Within the standard no definition is given concerning the Maximum Repetitive Value (MRV) other than the self-explaining name, and no further information is given in the ISO 4867 or ISO 4868 standards where, as severity of vibration, reference is made to the peak value during stationary vibrations. This last affirmations might mislead the reader into the conclusion that a stationary phenomenon can be treated as a sum of purely sinusoidal signals, obtaining the peak value from an effective value multiplied by a factor of 2 . This last definition seems scarcely compatible with what

is mentioned in ISO 6954. Another critical point in the standard in question is the absence of a motivation for the adoption of the Maximum Repetitive Value. It is affirmed rather simply that such value can be obtained by multiplying the effective value, at the same time obtainable from a frequency analysis of the signal, by a conversion coefficient that can be obtained in direct way or by approximating it to 1.8; it is also mentioned in the ISO standard that a value of the conversion coefficient equal to 1 can be used when the vibrational phenomenon is purely sinusoidal. From this last affirmation we can deduce that this conversion coefficient, which is actually the Conversion Factor, is used with means to describe vibrational phenomenon that are not purely sinusoidal, that is signals whose amplitude is variable in time regardless their frequency remains constant like for example signals modulated in time. On board a ship, this modulation phenomenon can occur due to the fact that the propeller, being submersed in a sea condition with waves, perceives the variations of orbital speed that the

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fluid experiments during the wave motion, and also because of the turbulences present in the wake flow, generated by the ship’s own surge motion, in which the propeller works. The consequence is that we have a continuous variation of the thrust and the amplitude of the pressure oscillations in the stern area on above the propeller. The Crest Factor presents itself then as a way for taking into account this modulation even when doing the signal analysis in the frequency domain inside which, by definition, the decomposition of the signal is done using perfect sine functions. 1.2

State of the art

Classification societies such as the American Bureau of Shipping (ABS) and the Lloyd’s Register (LR) give some guidelines for taking measurements on board by following the ISO 6954:1954 standard. The LR affirms that the stresses generated by the propeller undergo a modulation due to the variation in the forces generated by the propeller, but does not correlate explicitly the variation of the forces generated by the propeller to the effect of the sea. This modulation is not detectable by performing an analysis of the measurements in the frequency domain, in this domain in fact the temporal signal is approximated as a sum of pure sinusoidal signals, which is why when using such type of analysis there is a risk of underestimating by much the real amplitudes of the modulated signal. A methodology for calculating the MRA is proposed by the LR, which consists in the use of paper-rolls and precision filters to isolate the source frequency to analyze and on this, calculate the value of the MRA. Once this parameter is defined the LR points out that when results in the frequency domain are used the ISO standard proposes the use of a Crest Factor equal to 2.5 that takes into account a factor of 2 (to convert from RMS value to peak value for sinusoidal signals) to which we sum a factor (the Conversion factor) of 1.8 to take into account the normal modulations of the vibrational signals in a Sea State 3, which is still a tentative value that can differ for ships of different dimensions and for different states of the sea. The LR, regarding the Sea State in which to perform the measurements, suggests to follow the ISO 4867:1984 and the ISO 4868:1984 standards, where only in the last one is specified that the preferable conditions for the test are of Sea State 3 or lower. The LR proposes a registration time for the measurement of 2 minutes. The LR also recommends a procedure for the analysis where for the exciting forces of the engine suggests to use a Crest Factor of 1.4, which means signals without modulation, whereas for the

exciting forces from the propeller suggests to use a Crest Factor of 2.5, and in case of proximity to the limit, a procedure for a more thorough analysis that takes to the direct calculation of the MRA is suggested. The ABS, diversely from the LR, gives a good definition of the MRA, presenting it as the maximum repetitive value that is presented designing the envelope of the peaks of the modulated signal. For the execution of the testing they propose limitations on the Sea State that depend on the dimensions of the ship. Regarding the analysis of the measurements, the ABS remains more ambiguous than the LR, in fact they don’t specify for which exciting force the Crest Factor should be used and speaks generally of a Crest Factor of 2.5 that approximates relatively good the example shown without mentioning a direct dependence between Sea State and Crest Factor value. For the validation of the MRA a measurement time of 3 minutes is proposed. 2 2.1

PROPOSED METHODOLOGY Procedure

According to the indications given by both the LR and the ABS, a procedure has been set up. First, the time domain signal must be filtered with a band-pass filter with a range such as to consider a single exciting source. The choice of the filter bandwidth is done by studying the signal spectre obtained with a Hanning window. As bandwidth individuation parameter, it has been decided to consider the frequencies in which the amplitude falls below the 20% of the maximum peak amplitude. This procedure is easy to apply when the peak is well defined, which is in the low frequencies and in zones relatively close to the source. For higher frequencies, like for example the second order of the propeller frequencies, the signals turn out more disturbed, also because of the local resonances that cause the peaks readable from the spectre to be more dispersed turning the application of the criteria more difficult and arbitrary. After this a numerical integration of the signal must be done, to pass from acceleration to velocity (the amount requested for frequencies superior to 5 Hz). This is possible due to the fact that the low frequencies, that produce deviations in the numerical integration process, have been filtered. Now it is possible to constitute the positive and negative envelope curves of the integrated signal. The result curves obtained should come as shown, partially, in Figure 1. With this curves, we can now identify the Maximum Repetitive Amplitude, both positive and negative, through the construction of histograms representing the number of peaks

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Figure 2. taken.

Benetti Yacht in which measurements were

Table 1. Figure 1. Positive and negative envelope curves of the filtered signal (partial) of the first propeller harmonic. Position: Sun Deck.

of the envelope curve for the analyzed amplitude interval. The MRA value is defined as the mean of the maximum repetitive amplitude positive and negative values. These magnitudes are much influenced by the number of intervals with which it is desired to discretize the found amplitudes, which is why it is recommended not to use an excessively high number of discretization intervals. Roughly 20 to 25 per temporal tracks of 5 minutes. At this point, we have to calculate the mean RMS amplitude through spectral analyses done on the filtered signal. Regarding acceleration, it is recommended not to consider in the analysis the flat trait generated by the late response induced from the filter application. The spectre in acceleration is integrated to obtain the vibrational velocity spectre from which the RMS value of the exciting source being studied will be read. Finally, we can obtain the Conversion Factor (CF) with Equation 1:

Measurement points.

#

Measurement point

1 2 3 4 5 6 7

Shell plating above the propeller Sun deck Main deck pt. 35 Main deck pt. 38 Main deck pt. 39 Main deck pt. 41 Upper deck pt. 104

This kind of analysis is done of course for validating the comfort of the ship, which is why the main interest points correspond to public areas and main source areas, as shown in Table 1. 4 4.1

RESULTS AND ANALYSIS Sensitivity analysis

(1)

4.1.1 Filter order The order of the filter has a scarce influence on the MRA values and on the calculation of the conversion factor. As it can be seen on Table 2, on analyses 1, 2 and 3, there has been only a variation in the filter order, resulting in variations of the CF of 1–4%.

For the application and study of this method, a measurement campaign has been done, in collaboration with the Benetti Shipyard. The measurements were all done on board a yacht, shown in Figure 2, with L = 54 m, while navigating at a speed of 15 knots with a Sea State 4. For the measurements, piezoelectric mono-axial accelerometers have been used, connected to a 12 channel data acquisition system that transforms the analogic signal into digital signal and to analyze the signal in real time to verify the quality of the signal being measured.

4.1.2 Length of the temporal record The length of the temporal record has a very high influence in the determination of the MRA value and in consequence, of the conversion factor as well. The amplitude of the wave motion is a stochastic phenomenon and it is conceivable that also the interaction between wave motion and vibration sources on board the ship are of the same nature. This is why it is highly probable that a long measurement would take us to observe a higher amplitude value of MRA respect to a shorter measurement. On Table 2, analysis number 7 and 10, differ only in the length of the measurement time, and we can appreciate how the MRA and consequently the conversion factor vary significantly, of approximately 50%. Briefly, it can be said that a short

CF =

3

MRA 2RMS

APPLICATION

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Table 2. Analysis made on vibrational time measurements to obtain the Conversion Factor. All measurements were taken with the ship navigating at a speed of 15 knots, with a Sea State 4.

Case

Lower Interest frequency point (Hz)

Upper frequency (Hz)

Filter Filter width order (Hz)

MRA RMS value from average envelope value

CF

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1 1 1 1 1 1 1 1 1 1 2 2 2 3 4 5 6 6 7

37.125 37.125 37.125 36.125 39.875 35.9 35.9 71.7344 71.8 35.9 34.3672 68.57 13.57 34.6641 34.9531 34.4375 34.375 34.375 34.5156

20 30 10 20 20 30 10 10 10 10 10 10 10 10 10 10 10 10 10

2.90E-03 2.80E-03 3.20E-03 1.40E-03 4.20E-03 7.31E-04 7.92E-04 1.10E-03 9.99E-04 5.49E-04 3.45E-04 5.60E-05 – 4.43E-04 1.54E-03 6.76E-04 1.54E-04 2.05E-04 1.17E-04

5.15 5.09 5.35 2.46 6.91 2.52 2.43 3.97 3.55 1.26 2.18 3.27 1.11 4.89 6.70 3.34 1.80 2.39 3.38

34.75 34.75 34.75 35.375 32.125 35.7 35.7 71.4766 71.4 35.7 34.1641 68.25 13.55 33.9688 33.8672 34.1563 34.2188 34.2188 34.1406

2.375 2.375 2.375 0.75 7.75 0.2 0.2 0.2578 0.4 0.2 0.2031 0.32 0.02 0.6953 1.0859 0.2812 0.1562 0.1562 0.375

measurement shows only a partial representation of the phenomenon where there could be missing the extremity components of the excitations, taking us to an underestimation of the MRA index. 4.1.3 Width of filter The width of the filter is a significant parameter and is also an object of some uncertainties. Within the guidelines given by ABS and LR it is recommended to use precision filters able to show only the vibrational component due to the examined source. This recommendation comes simple to apply in the case in which, when analyzing the spectre generated by the temporal series, the exciting source can be identified with one low dispersed peak and with a very narrow base (in the order of 0.1 Hz). When analyzing the spectre in areas of the ship sufficiently distant from the vibration source, there can be noticed several disruptions that will lead to force the widening of the band-pass frequency range used to filtrate the signal (following the 20% amplitude rule). Such disruption will generally generate a higher “energetic” content of the filtered signal with consequent higher values of the MRA value. A similar phenomenon can be noticed also for the vibrations generated by the higher frequencies of the propeller where disruptions can be noticed probably due to the local structural resonance phenomena. The influence of the filter width on the conversion factor can be noticed if in

3.98E-04 3.89E-04 4.23E-04 4.02E-04 4.30E-04 2.05E-04 2.30E-04 1.96E-04 1.99E-04 3.08E-04 1.12E-04 1.21E-05 1.93E-04 6.41E-05 1.62E-04 1.43E-04 6.06E-05 6.06E-05 2.45E-05

Notes

Propeller first harmonic Propeller first harmonic Propeller second harmonic Propeller second harmonic Test time = 150 seconds Propeller first harmonic Propeller second harmonic Engine first harmonic

Table 2 we compare the analysis case 1, 2 or 3 with the case 7, where the only difference is the width of the filter and the difference in conversion factor is of between 52% and 54%, or even more if we compare case 4 with case 5, where the conversion factor difference is of 180%, and the same for case 2 against case 6, where the difference is again of 50%. 4.2

Conversion factor variability analysis

4.2.1 Position of the ship There seems to be no great influence regarding the position on the ship. When comparing the measurements, shown in Table 2, relative to the stern of the ship above the propellers, case 6 or 7, sun deck, case 9, and main deck, case 19, the values for the conversion factor differ nearly by 5%. This deduction is valid if the width of the band-pass filter used in the validation of the conversion factor remains within a range of values that can be compared with each other. 4.2.2 Exciting source order When increasing the order of the exciting source, for example in Table 2 when comparing case 6 to case 9, we can notice an increase in the value of the conversion factor, of between 2.5 to nearly 3.5, even maintaining an amplitude for the filtering relatively limited, from 0.3 to 0.4 Hz due to the

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disruptions of the present signals. In case of exciting sources of superior orders a filter amplitude identification problem has been found. In the case of the yacht being studied, the second order of the exciting source of the propeller has a frequency of 71 Hz, in which frequency local resonance phenomenon can be verified. This phenomenon introduces disruptions in the response of the shell plating and turn complicate the discernment of which frequency contribution is effectively due to the vibrational source and which due to the resonances. It must be noted that a local resonance can also produce a reduction in the comfort perception on board, which is why in these cases it must be clarified whether it is right or not to consider the whole resonance area or only the more steep part of the peak where the exciting source is located. 4.2.3 Effect on the engine exciting sources The engine exciting sources have a very long modulation, especially the well-defined ones that present a very narrow spectre. In Figure 3 we can see a temporal record of the vibration signal of the first harmonic of the propeller measured in the Sun Deck, where the modulation is clearly noticeable. In Figure 4, we can see a temporal record of the vibrational signal of the engine’s first harmonic measured in the same point. Here the measurements show that the effect of the sea on the modulation is secondary and not as determining as for the propeller. This is why it is impossible to define a conversion factor value with a measurement time of 5 minutes. With such long measurement it is only possible to identify a local maximum that leads to an approximate definition of the conversion factor. However, the LR suggests not to apply it on the engine exciting sources when in presence of a diesel electric propulsion system, and to verify

Figure 3. Temporal record (partial) of the propeller first harmonic modulated signal, measured at the Sun deck, where the modulation caused by the propeller’s presence in a sea condition with waves and the influence of the wake flow turbulences.

Figure 4. Temporal record (partial) of the engine first harmonic modulated signal, measured at the Sun deck.

it when in presence of gearboxes or couplings that can have a certain degree of torsional vibration decoupling between propeller and engine. 4.3

Observations

When analyzing the ISO 6954:1984 standard it is not clear the role and the motivations that should take to the application of a conversion factor of a tentative value of 1.8. The standard does not specify what type of exciting source this value should be applied to and, given these lack of information and or explanation, it could be said that the conversion factor can have a safety coefficient role regarding the quality of the measurement and the variability of the peak values that present a certain degree of variability, even in stationary cases. If this was the case, a conversion factor of 1.8 would be unjustified since it would be too high. To solve the doubts on the nature of the conversion factor, trust must be put in the LR’s guidelines that, even if not in a clear way, confirm that the introduction of the conversion factor has been made to consider the effects of the sea on the propeller. In fact within the guidelines it is suggested not to apply the conversion factor for the engine exciting sources while the same factor must be used for the propeller exciting sources, particularly the exciting sources created by the passing of the propeller blades below the stern. A similar deduction can also be obtained from the ABS guidelines, where the mean amplitude value of the exciting sources is defined as equal to the RMS amplitude value obtained from the spectre of the temporal record multiplied by the square root of 2, which is equivalent to considering the vibrations obtained during sea trials with calm seas as sine waves. Within the standard, it is explained that the rules can be applied to ships with length superior to 100 metres, but within the standard there are no references made to

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specific sea trial testing conditions. To find a limit regarding the Sea State for being able to perform a measurement, the ISO 4868:1984 standard, which is made referenced within the ISO 6954:1984 standard, must be analyzed. There it is mentioned and established that Sea State 3 is the maximum Sea State in which measurements can be done. The LR does not propose any additional limitations, whereas the ABS subdivides the ships in three groups and, depending on the ship size, gives different maximum Sea State limits. For example for yachts it can be taken as Sea State 1. The use of the conversion factor can be broadened and also understood as a factor to guarantee the comfort during navigation with Sea State equal or superior to 3. 5

CONCLUSIONS

The ISO 6954:1984 standard is not very clear regarding certain aspects. Within the rules, reference is made to a conversion factor that depends mainly on the Sea State, but it is not clear to which vibration source it should be applied. Only the guidelines given by the classification societies clarify that this conversion factor must be applied only to the propeller exciting source. Before applying the standard it is recommendable to specify if the comfort must be guaranteed in heavy seas or in calm seas and in case of heavy seas, the Sea State must be defined. In case the previous conditions are not specified it is advisable to perform the measurements with calm sea assuming a conversion factor of 1. From the performed calculations it has been verified that an important parameter in order to define the conversion factor is the time of the measurement, which is why it is recommended to

define the duration of the measurement previously in order to be able to provide with more satisfying results. FUTURE DEVELOPMENTS With the aim of providing more general indication regarding the use of the conversion factor, further developments must focus on the definition of the probability density distribution of the amplitudes of the modulation. In fact, when using long enough measurements it would be possible to arrive to a definition of the probability density distribution of the amplitudes accurate enough, and at this point it would be possible to define, for every modulated amplitude, a “probability of exceedance”. At this point the conversion factor could be defined in a more scientific and less empirical manner, by fixing a conversion factor value linked to the probability of exceeding a given limit of wave amplitude.

REFERENCES Guidance notes on Ship Vibrations. American Bureau of Shipping. 2014. ISO 6954:1984 (E): Mechanical vibration and shock— Guidelines for the overall evaluation of vibration in merchant ships. ISO 6954:2000 (E): Mechanical vibration—Guidelines for the measurement, reporting and evaluation of vibration with regard to habitability on passenger and merchant ships. ISO 4867: Code for the measurement and reporting of ship-board vibration data. ISO 4868, Code for the measurement and reporting of local vibration data of ship structures and equipment. Ship and Vibration Noise—Guidance Notes. Lloyd’s Register. 2006.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

A computational framework for underwater shock response of marine structures C. Diyaroglu, D. De Meo & E. Oterkus University of Strathclyde, Glasgow, UK

ABSTRACT: Composite structures have extensive area of practice in engineering disciplines. One of the application areas is to use them under extreme loading conditions. However, our understanding of their behaviour under shock loading is rather limited. As a result of this, current designs of composite structures are very conservative which significantly reduce the weight saving advantage. In order to improve our understanding, experimental studies are essential but they can be prohibitively costly. On the other hand, computer simulations can be a good alternative. Hence, the main objective of this study is to investigate underwater shock response of marine composite structures by using a new theory called peridynamics. Numerical approach based on peridynamics is used to predict the failure modes in marine composite structures. The evaluated results are validated by comparing against the available data in literature which demonstrates the capability of peridynamics for such complex problems. 1

INTRODUCTION

In recent years, composite structures have found extensive application area in marine field. Especially, they have received a lot of interest in military applications, such as in naval ships, submarines and torpedoes (Mouritz 2001), because these structures may possibly be exposed to extreme loading conditions in their life time and their design must satisfy such conditions without any compromise from its weight. Composite structures have many superior characteristics, such as high strength/weight ratio, high corrosion resistance and low noise transmission (Kalavalapally et al. 2006). However, they constitute from varied materials and each of which possesses very different mechanical properties. Hence, they are rather complex and understanding dynamic behavior of composite materials is not easy especially under extreme loading conditions. Therefore, various studies concentrate on investigating dynamic deformation and/or failure characteristics of composite structures under shock loading conditions. Recently, several numerical studies were carried out for complex structures and detailed damage evolution process was studied extensively in composite structures for underwater shock loading by using Finite Element (FE) analysis. Batra & Hassan (2007) developed an FE code with rate-dependent damage evolution equations and it was used for analysing a laminated composite plate under shock loading. Moreover, LeBlanc (2011) performed FE

analysis incorporating progressive damage and numerical results were compared with experiments. Wei et al. (2013a) proposed a progressive degradation model in order to analyse different damage mechanisms in composite structures and they compared their results with experimental observations obtained from underwater shock tube. Later on, these results were improved by considering strainrate effects on mechanical behavior of constituents of composites (Wei et al. 2013b). Experiments are also carried out in order to gain better understanding on dynamic and damage behaviour of composite structures under shock loading. Mouritz (2001) showed the effect of stitching in Glass/vinyl ester composites with prototype-scale experiments. It was shown that damage characteristics and especially delamination damage can be improved effectively. Arora et al. (2012) carried out large scale field experiments and investigated failure mechanisms of E-Glass fiber reinforced sandwich panels and laminated tubes. As mentioned earlier, numerical studies, which have been carried out to date, only used FE analysis technique and its equation of motion is based on local continuum theory which needs extra kinematic relations and/or damage evolution equations for the damage prediction analysis. In this study, a novel nonlocal continuum theory called, peridynamics, is used for analysing underwater shock responses of composite structures. The governing equations of Peridynamics naturally incorporate damage into structure and no

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additional equations are needed. Moreover, its numerical implementation is done by meshless approach. In this paper, implementation of peridynamic formulation for composite structures is briefly explained. Then, peridynamic analysis of a 4-ply composite structure subjected to underwater shock loading is considered and results are compared against a previous study done by Batra & Hassan (2007). 2

PERIDYNAMIC THEORY FOR COMPOSITE MATERIALS

Peridynamic Theory (PT) was introduced by Silling (2000) at Sandia National Laboratories. It is the state-of-art technique which is especially promising tool for failure analysis of structures. PT can be defined as a generalization of classical continuum mechanics theory introduced by the French mathematician Augustin Louis Cauchy. The governing equation of PT is in integral form and it does not contain any spatial derivatives. As a consequence, the formulation remains valid everywhere regardless to the presence of discontinuities in the domain. Therefore, peridynamics is a powerful tool for predicting complex material failure mechanisms such as crack nucleation, crack propagation, crack branching, coalition of multiple cracks and crack arrest. 2.1

Where c and s represent the peridynamic material parameter and the stretch, between material points x and x′, respectively. The stretch, s, is defined as s=

cF =

Silling (2000) reformulated the continuum mechanics equations and replaced the spatial derivatives of stresses in the classical equation of motion with an integral term, which leads to the new form continuum mechanics formulation: x u ′ u ) dH + b ( x

)

(3)

Since the fiber-reinforced composite lamina is an anisotropic material, the directional dependency is included in the peridynamic formulation with two different peridynamic material parameters, which is shown in Figure 1 for a fiber-reinforced composite lamina with a fiber orientation of θ. In this figure, the material point q is in the direction of fibers and p represents material points in any arbitrary direction. They are all family members of the material point of interest, i. Each interaction between material points is named as bond in peridynamics. The peridynamic parameters or bond constants concerning the interaction of material points in the fiber direction only and in any arbitrary direction can be defined as cF and cA, respectively. As indicated by Oterkus & Madenci (2012), the peridynamic parameters can be expressed in terms of the engineering material constants E1, E2, G12 and ν12 by equating strain energy densities of a material point based on the classical continuum mechanics and PT for simple loading conditions. Explicit expressions for bond constants are given by Oterkus & Madenci (2012) as

Peridynamic formulation for a composite lamina

( x) u ( x ) ∫ f ( x′

y′ − y − x ′ − x x′ − x

cA =

(1)

2 E1 ( E − E ⎛ E ⎝ 1

⎛ E ⎝ 1

)

Q ⎞ 1 ⎞⎛ E2 ⎜ ∑ ξqqi Vq ⎟ 9 ⎠ ⎝ q =1 ⎠

8 E1 E2 1 ⎞ E π hδ 3 9 2⎠

(4)

(5)

H

where, ρ, u and b defines density, displacement and body force of a material point, x. In Equation 1, material point x interacts with other material points inside its domain of influence, horizon, H. The interaction force or peridynamic force between material points x and x′ can be expressed as f(x′ − x, u′ − u) and it is along the same direction of the relative position of these material points in the deformed configuration, i.e., y′ − y = (x′ + u′) − (x + u). For a fiber-reinforced composite lamina, the peridynamic force can be expressed as f = cs

y′ y y′ y

(2)

Figure 1. Peridynamic horizon of material point i for a lamina with a fiber orientation of θ and peridynamic bonds.

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Where ξqi is the reference distance between material points q and i, Vq is the volume of a material point q, h is the thickness of each ply and δ represents horizon radius of a material point i. The critical parameters or critical stretches, which define the failure of these bonds under tension and compression, can be determined based on the experimental measurements, given by Oterkus et al. (2012). Failure in PT can be defined by terminating the interactions between material points or breaking the bonds. Thus, the peridynamic force relation given in Equation 2 is further modified by introducing the failure parameter μ(x′ - x, t) f

μ ( x′ − x

)c s

y′ y y′ y

(6)

where μ is a history-dependent parameter. If the stretch of a bond exceeds its critical value μ takes 0 value otherwise it is equal to 1. Moreover, local damage, ϕ(x, t), at a material point can be defined as

∫ μ(x

ϕ ( x, t ) = 1 − H

x, t ) dH

∫ dH

cN =

Em hV

(8)

cS =

2 Gm 1 πh ⎛ 2 ⎛ h2 ⎞ ⎞ 2 ⎜ δ + h ln ⎜ 2 ⎟⎟ ⎝ δ + h2 ⎠ ⎠ ⎝

(9)

where Em and Gm are the elastic and shear moduli of the matrix material, respectively. Force density of shear bonds is different than others and it can be written as y′ y y′ y

DAMAGE EVOLUTION IN A LAMINATED COMPOSITE

3.1 Definition of the problem

Oterkus & Madenci (2012) further extended the formulation of a composite lamina to represent composite laminate. In this regard, two additional bonds in the thickness direction, as shown in Figure 2, are defined between plies of laminate. As shown in Figure 2, interlayer (cN) and shear bond constants (cS) are defined and they are responsible to resist transverse normal and shear deformations. Oterkus & Madenci (2012) defined PT interlayer and shear bond constants as

)2

where Δx is the spacing between material points on the plane of the lamina and ϕ is the shear angle of the crossing shear bonds. Failure of interlayer and shear bonds represent Mode-I and Mode-II delamination damage, respectively, and their critical stretch values are obtained by using mode-I (GIc) and mode-II (GIIc) fracture toughness values. Formulations of their critical stretch values are given by Oterkus & Madenci (2012). 3

Peridynamic formulation for a composite laminate

f = cS ϕ (

Peridynamic bonds in a composite laminate.

(7)

H

2.2

Figure 2.

(10)

Damage evolution in a 4-ply composite laminate subjected to underwater shock loading is analysed by using PT and compared against a previous study done by Batra & Hassan (2007). 4-ply composite laminate has all plies oriented along the fiber direction ([0/0/0/0]). The laminate has the same properties and loading conditions which was used in Batra & Hassan (2007). In their analysis, Batra & Hassan (2007) assumed that 64 kg of TNT is placed 10 m below the ship surface. When the explosive charge is detonated under the water, pressure (P) rapidly increases to high values in a very small time range, which may be less than 10–7 s, and it is followed by an exponential decay (Liang & Tai 2006). Batra & Hassan (2007) used empirical exponential decay function in order to find pressure profile on the ship surface or on the representative composite plate and it is given as P (t ) = 17.678e −t /

.424

(11)

Furthermore, they assumed non-linear pressure distribution over the plate, as shown in Figure 3. Laminated composite plate used in this analysis was clamped from all edges. This boundary condition is implemented in peridynamic analysis by using fictitious region along the boundaries, which

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3.2

Figure 3.

Shock distribution over the plate.

Table 1.

Material properties of AS4/PEEK.

Poisson’s ratio Young’s Modulus (GPa) Shear Modulus (GPa) Mass Density

PEEK

AS4

0.356 6.14 2.264 1.44

0.263 214 84.7 1.78

may be assumed with a depth of δ (Madenci & Oterkus 2014). The length, width and thickness of composite plate are specified as L = W = 22 cm, and t = 10 mm, respectively. The composite material used is AS4/PEEK with fiber volume fraction (Vf) of 0.6 and material properties of PEEK matrix and AS4 carbon fiber, are given in Table 1 (Batra & Hassan 2007). If matrix and carbon fibers are homogeneously distributed over the plate, the generalized properties of each uni-directional lamina takes the values of E1 = 130.86 GPa, E2 = 14.70 GPa, G12 = 4.962 GPa and ν12 = 0.3 (Hassan 2005). Bond constants for PT are calculated from Equations 4, 5, 10 & 11 for a given mechanical and geometrical properties of each lamina. Moreover, peridynamic parameters used in analysis are taken as Δx = 1 × 10–3 m, δ = 3.015Δx. In this study, the explicit time integration scheme is used and stability condition is chosen based on von Neumann analysis (Silling & Askari 2005). In this regard, time step size and total time steps are chosen as Δt = 7.12 × 10–8 s and 7023, respectively. Both fiber and arbitrary direction bonds are only allowed to fail in tension. Arbitrary direction and fiber bonds critical stretch values are chosen based on the failure strain graphics of matrix and fiber materials given by Hassan (2005). So, arbitrary direction and fiber bonds critical stretch values are determined as sA = 0.0081 and sF = 0.01075, respectively. Interlayer and shear bonds critical stretch values are calculated by using mode-I (GIc) and mode-II (GIIc) fracture toughness values, which are GIc = 1.7 kJ/m2 and GIIc = 2.0 kJ/m2, respectively (Cytec Engineering 2012).

Results

The total analysis time is 230 μs as in Batra & Hassan (2007) before the damage values reach very high values and composite plate fails completely. Transverse displacement values (UZ) of a material point which is at the centre of plate are plotted against time and it is found that peridynamic results are very well comparable with FE results which is based on additive damage evolution equations, as shown in Figure 4. In PT, matrix damage starts first at clamped edges of top plies (1st and 2nd plies from the top) at about 108 μs and it propagates towards the edges, which is perpendicular to fiber direction, but it is arrested nearly at 160 μs, as shown in Figure 5a. Batra & Hassan (2007) (Fig. 5b) also observed a similar damage behavior. Although the path which crack follows seems different in both approaches, the place of its emergence is the same which is at the centre of edges along the fiber direction. Moreover, matrix damage emerges at the centre of bottom ply at about 148 μs. This is slightly later than observed time in Batra & Hassan (2007) which is about 108 μs. However, it then propagates very fast along fiber direction as shown in Figure 6a and as in FE analysis (Fig. 6b). It is seen in this figure that the behaviour of damage propagation is similar to that of Batra & Hassan (2007).

Figure 4. Comparison of transverse displacements of centre point in time.

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Figure 5.

Figure 6.

Figure 7.

Bottom section matrix damages at 160 μs.

Figure 8.

Matrix damages of all plies at 230 μs.

Top section matrix damages at 160 μs.

Propagation of matrix damages.

Matrix damage also occurs at the centre of middle ply which is the second ply from the bottom, at about 160 μs as shown in Figure 7. As mentioned in Batra & Hassan (2007), similar matrix cracking damage emerges at the middle surface at about 125 μs. Also in both approaches, matrix damage region is quite different for bottom and top plies

at 160 μs, even if the loading is circular over the top plate, as shown in Figures 5a & 7a for PT and in Figures 5b & 7b for FE method. This is probably as a result of wave reflection from the bottom ply where compressive wave is transformed to a tension wave (Batra & Hassan 2007). Based on comparisons of damage characteristics of top and bottom plies with FE analysis, it is obvious that damage propagation has the same behaviour in both approaches. Furthermore, central matrix cracking reaches top surface at a very later time in both theories and this behaviour can also be explained as a result of wave reflection from the bottom surface which reaches the top surface at a later stage in the analyses. As shown in Figure 8, central matrix damage reaches top ply nearly at 230 μs in PT. In the current peridynamic model, the damage of arbitrary bonds incorporates both matrix damage and fiber/matrix debonding damage. Comparing the occurrence region and propagation direction

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of fiber/matrix debonding damage in FE approach with the damage pattern in PT can allow us to distinguish failure modes. As shown in Figure 9a, debonding damage in PT can be observed at the

Figure 9.

edges, which is perpendicular to fiber direction and it propagates towards the centre of the plate along fiber direction as indicated by Batra & Hassan (2007) (Fig. 9b). This conclusion seems reasonable since the debonding between fibers and matrix occurs along the fibers rather than perpendicular to them. Fiber damage first develops at the edges of 2nd ply from the bottom at about 160 μs and then it grows rapidly through the edges of same ply. When the time reaches 188 μs, damage characteristic is the same with top ply as shown in Figures 10a & b. A similar fiber breakage damage pattern is also observed in FE results at the bottom surface with a very small value at 160 μs (Fig. 10c). Most importantly, fiber damage develops at the centre of bottom ply and below the top ply (3rd ply from the bottom) at a later time about 220 μs, as shown in Figure 11a. Peridynamic fiber damage occurrence

Top section damage patterns.

Figure 11. Fiber damages at the centre of a plate in PT.

Figure 10.

Comparison of fiber damages at the edges.

Figure 12. Fiber damage at the centre of a plate in FE method (Batra & Hassan 2007).

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region is similar to the FE approach (Fig. 12a) but later in time, as well as the fiber damage at the top surface emerges earlier in time in FE approach, at about 160 μs (Fig. 12b). However, it did not grow as quick as in the bottom ply until nearly 200 μs, so it can be comparable with PT results of the 3rd ply at 220 μs and 300 μs, shown in Figures 11a & b. Based on the comparisons, it can be concluded that peridynamic fiber damage formation region and its evolution are very similar and comparable with FE analysis obtained in Batra & Hassan (2007). PT can further distinguish mode-I delamination and mode-II or shear delamination damages between each other. On the other hand, there is no distinction in delamination failure modes in FE analysis. As shown in Figure 13a, shear delamination (mode-II) damage emerges at nearly 130 μs with very small values, which is lower than 0.5, which means that plies are not totally delaminated and until 188 μs no significant change is observed in damage values. However, after this time, plies delaminate from the edges, which is perpendicular to fiber

direction and later in time at 220 μs delamination occurs in a very different region which is at the centre of all plies, as shown in Figures 13b & c. Batra & Hassan (2007) (Fig. 14) provided transverse shear stresses at t = 160 μs and it can be interpreted that delamination may initiate from the edges which show very high stress values. The time and place of delamination initiation are very close to what is observed in PT. Mode-I delamination damage emerges in similar regions as in the mode-II damage at 188 μs and 220 μs as shown in Figures 15a & b. The final delaminated areas are given in Figures 16a & b for the top ply shear delamination and for the 3rd ply (from the bottom ply) mode-I delamination damages at 220 μs and they are very well comparable with FE delamination damage results (Fig. 16c).

Figure 14. Transverse shear stresses for top (left) and bottom (right) surfaces at 160 μs in FE method (Batra & Hassan 2007).

Figure 13. Evolution of shear delamination damage in PT.

Figure 15. in PT.

Evolution of mode-I delamination damage

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that PT can be used for more complex composite structures and loading conditions, where FE approach may suffer to give detailed damage evolution in the structure. Hence, PT can give us better understanding on damage behaviour of complex composite structures under extreme and/or complicated loading conditions.

REFERENCES

Figure 16. 220 μs.

4

Comparison of delamination damages at

CONCLUSION

Peridynamic analysis of a laminated composite plate, in which all plies are oriented in fiber direction, is performed for underwater shock loading. Capability of PT is shown by doing comparisons with FE analysis, which uses additive progressive degradation damage model. PT damage results are generated without any additive equations in the theory as in the FE analysis and it is found that they are very well comparable with FE results. In PT, crack initiation and propagation occurs spontaneously and cracks are initiated without triggering or introducing pre-cracks. The results show

Arora, H. Hooper, P.A. & Dear, J.P. 2012. The effects of air and underwater blast on composite sandwich panels and tubular laminate structures. Experimental Mechanics 52(1): 59–81. Batra, R.C. & Hassan, N.M. 2007. Response of fiber reinforced composites to underwater explosive loads. Composites Part B: Engineering 38(4): 448–468. Cytec Engineering 2012. APC-2 PEEK Thermoplastic polymer technical data sheet. Hassan, N.M. 2005. Damage development in static and dynamic deformations of fiber-reinforced composite plates. Virginia Polytechnic Institute and State University. Kalavalapally, R. Penmetsa, R. & Grandhi, R. 2006. Multidisciplinary optimization of a lightweight torpedo structure subjected to an underwater explosion. Finite Elements in Analysis and Design 43(2): 103–11. LeBlanc, J. 2011. Dynamic response and damage evolution of composite materials subjected to underwater explosive loading: an experimental and computational study. University of Rhode Island. Liang, C.-C. & Tai, Y.-S. 2006. Shock responses of a surface ship subjected to noncontact underwater explosions Ocean Engineering 33: 748–772. Madenci, E. & Oterkus, E. 2014. Peridynamic theory and its applications. New York: Springer. Mouritz, A.P. 2001. Ballistic impact and explosive blast resistance of stitched composites. Composites Part B: Engineering 32(5): 431–39. Oterkus, E. & Madenci, E. 2012. Peridynamic analysis of fiber-reinforced composite materials. Journal of Mechanics of Materials and Structures 7: 45–84. Oterkus, E. Madenci, E. Weckner, O. Silling, S. Bogert, P. & Tessler, A. 2012. Combined finite element and peridynamic analyses for predicting failure in a stiffened composite curved panel with a central slot. Composite Structures 94: 839–850. Silling, S.A. 2000. Reformulation of elasticity theory for discontinuities and long-range forces. Journal of the Mechanics and Physics of Solids 48(1): 175–209. Silling, S.A. & Askari, E. 2005. A meshfree method based on the peridynamic model of solid mechanics. Computers & Structures 83: 1526–1535. Wei, X. Tran, P. Vaucorbeil, A.de Ramaswamy, R.B. Latourte, F. & Espinosa, H.D. 2013a. Threedimensional numerical modeling of composite panels subjected to underwater blast. Journal of the Mechanics and Physics of Solids 61(6): 1319–1336. Wei, X. Vaucorbeil, A.de Tran, P. & Espinosa, H.D. 2013b. A new rate-dependent unidirectional composite model—Application to panels subjected to underwater blast. Journal of the Mechanics and Physics of Solids 61(6): 1305–1318.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Liquid sloshing analysis for independent tank with elastic supports Wenfu Liu, Hongxiang Xue & Wenyong Tang State Key Laboratory of Ocean Engineering, Shanghai, China Collaboration Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai, China

ABSTRACT: The sloshing characteristics are investigated in the independent tank with elastic supports. A 2D coupled algorithm involving the Hilber-Hughes-Taylor implicit method for the tank motion and the VOF method for the fluid is proposed and then validated by comparing with the experimental results. By conducting coupled calculation and comparing the numerical results with those obtained by ignoring the elastic impact, obvious differences of the tank motion and liquid sloshing are found between the two models. The stiffness of elastic support has a significant impact on the sloshing load. Precisely, with the increase of the stiffness, the sloshing load keeps growing dramatically and reaches its peak at the resonant point, before dropping continually and eventually leveling off to a constant value, which is coincident with the sloshing load without considering the elastic influence. An efficient and effective method is finally presented for forecasting the sloshing load of independent tank with elastic supports. 1

INTRODUCTION

As the natural gas is widely used as a clean and efficient energy source around the world, the need for the transportation and storage of the gas arises, various LNG carriers and offshore floating LNG plant concepts have aroused wide concern. The International Gas Carrier (IGC) code defines membrane tanks as well as three type categories for self-supporting tanks. Instead of constituting a part of the hull, the self-supporting tanks are independent of hull structure, so they are also called the independent tanks, such as the IMO Type B tanks including SPB (Self-supporting, Prismatic IMO Type B) by IHI and Moss spherical tank by Moss Maritime. Sloshing is of particular importance as the LNG cargo containment systems must be designed to withstand dynamic loads and sloshing pressure of the LNG when the tank is partially filled. Faltinsen (1974, 1978) obtained steady-state solutions of the nonlinear sloshing in a rectangular tank based on Moiseev’s theory which constructs normal mode functions and characteristic numbers by integral equations in terms of Green’s function of the second kind (Ibrahim et al. 2001). However, when the tanks are equipped with internal structures and the sloshing is radically violent, the influence of viscosity of fluid must be considered and the analytical method could no longer be applicable, experimental method and numerical techniques have become the main means. The most important task in the analysis of liquid sloshing is how to determine the location of the free surface. At present, a series of

numerical techniques have tracked the free surface successfully. There are numerical techniques based on grid methods such as MAC (Marker and Cell) method, VOF (Volume of Fluid) method, Ls (Level-set) method and CIP (Constrained Interpolation Profile) method, and those base on non-grid methods such as SPH (Smoothed Particle Hydrodynamics) method, MPS (Moving Particle Semifinal Implicit) method, MLPG (Meshless Local Petrov-Galerkin) method (Zhu et al. 2013). Zhu (2002) adopted VOF method to simulate the liquid sloshing in rectangular tank. Shen et al. (2009) introduced a partial cell parameter to improve the original VOF method in uniform grid for simulating the sloshing in the prismatic tanks. Nagashima (2009) applied FEM combined with Level-set method to study the tank sloshing in different fill ratio and compared the results with the theory and experiment. Cao et al. (2014) used SPH method to study the tank sloshing. To oil tanker and LNG carrier with membranetype containment system, the tank wall as an integral part of hull structure is regarded as a rigid boundary. The loads and forces from ship motion excitation are directly transferred to the rigid tank wall, which is called non-elastic support tank in this paper. As the support of the independent tank belongs to elastic structure, the ship motion excitation needs to be passed to the tank through elastic support structure so as to induce the liquid sloshing, which in turn feeds back to the elastic support structure. Accordingly, the motion of the independent tank is the combined results of elastic support passing the external excitation and inner tank liquid

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sloshing, which should be regarded as a dynamic problem considering the coupled effect of the elastic support function and tank sloshing loads. This paper deals with tank sloshing considering the effect of elastic support, the tank form refers to Type B independent tanks. By conducting coupled calculation, the tank motion and the sloshing loads are analyzed. The method established in this paper can be applied to the general consideration of tank sloshing considering the elastic support effect. 2 2.1

MATHEMATICAL FORMULATION Tank motion function

An analytical model of tank sloshing with elastic supports is shown in Figure 1, and the roll dynamic function of the tank can be described as follows: I 0θ Cθθ + Kθθ

M ex (t ) M fluid (θ θ θ t ) M g (θ ) (1)

where I0 [kg ⋅ m ] is the polar moment of inertia of the rigid tank with respect to the rotation axis; Cθ [N ⋅ m/(rad/s)] is the roll damping due to the elastic support; Kθ [N ⋅ m/rad] is the roll stiffness due to the elastic support; θ [rad/s] is the rotational displacement of tank with respect to the rotation axis; Mex [N ⋅ m] is the external torque acting on the tank with respect to the rotation axis; Mfluid [N ⋅ m] is the torque acting on the tank due to the liquid with respect to the rotation axis; Mg [N ⋅ m] is the torque created by the tank gravity with respect to the rotation axis. Applying sinusoidal displacement excitation to the hull, the external torque is transformed to the torque acting on the tank: 2

M ex (t ) = ϕ 0

2 Cθ2ω ex

Kθ2

sin(ω ext + β )

(2)

where ϕ0 is the amplitude of displacement excitation; ωex is the circular frequency of displacement excitation; and β is the phase difference, β = arctan (Cθ ωex/Kθ). The liquid torque acting on the tank is due to the integral of the pressure through the tank wall and inner baffles if exist. The torque caused by the tank gravity is defined as: Mg ( )

Mgg ( H o M

H g ) sin( in(θ )

(3)

where Ho [m] is the height of the rotation axis; Hg [m] is the height of center of tank gravity. By substituting the results of θ and its first and second time derivatives at time t into Mfluid and Mg at time t, Equation 1 which is a nonlinear second order differential equation is simplified into a linear second order differential equation, then HilberHughes-Taylor (HHT) method (Hilber et al. 1997) is applied to discrete the tank motion equation. A pre cursor of HHT method is the Newmark method, in which a family of integration formulas that depend on two parameters β and γ is defined. The major drawback of the Newmark family of integrators was that it could not provide a formula that was A-stable and second order and displayed desirable level of numerical damping. The HHT method came as an improvement because it preserved the A-stability and numerical damping properties, while achieving second order accuracy when used in conjunction with the second order linear Ordinary Differential Equations (ODE) problem. 2.2

Governing equations of liquid sloshing

The free-surface flow is simulated by the VOF method (Hirt et al. 1981), which introduces a fluid fraction function F(x,y,t) representing the volume fraction of a cell being occupied by the fluid. In order to treat irregular boundaries, including arbitrary bottom topography and internal obstacles, a partial-cell parameter λ is introduced to the original VOF method. The variable of λ is dependent of time and has a value of 0 to 1 depending on how much volume the fluid can occupy the cell. Assuming the fluid is incompressible, the equations governing the flow are as follows: Continuity equation: ∇ ⋅ (λ ) = 0

(4)

Momentum equation: Figure 1. analysis.

Liquid tank with elastic support for sloshing

∂u + (u ⋅∇ ) ∂t

1 ∇ p + Fb ρ

μ 2 ∇ u ρ

(5)

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The F transport equation: D( λ F ) =0 Dt

(6)

where u is the velocity vector, defined in the tankfixed coordinates; ρ, p, Fb, μ are the fluid density, pressure, external force vectors and coefficient of kinetic viscosity, respectively; and D/Dt indicates material derivative. The external force consists of gravitational force, translational and rotational inertia forces taking the following form: Fb

dΩ d( − ) × ( R − R ) − 2Ω × dt dt − Ω × [ Ω × ( R R )]

(7)

where g and Ω are the gravitational vector and rotational velocity vector; In addition, r and R are the position vectors of considered point and motion origin. The second term of the right-hand side is the translational inertia, while the third, fourth and fifth terms are due to rotational motion, i.e. the angular acceleration, Coriolis, and centrifugal forces. It should be noticed that these forces are defined in the tank-fixed coordinated system. For viscous fluid, it satisfies no slip condition and impenetrable condition on rigid tank wall: u et = ub et

(8)

u en = ub en

(9)

where et and en are unit tangential vector and unit normal vector; ub is tank velocity respect to bodyfixed coordinate system, which is 0 while considering no tank deformation. At the free surface, the fluid must satisfy the kinematic boundary condition and the kinetic boundary condition:

− p + 2μ

∂un = − p0 ∂n

3 3.1

g−

∂un ∂u ∂u + t =0 ∂t ∂n

the cell, while the velocity is defined in the center of boundary of the cell. The pressure gradient term and the diffusion term are discretized with the central difference scheme. To ensure the stability and accuracy, the strong nonlinear convection terms is discretized with mixed scheme of the upwind scheme and the central difference scheme. The Youngs method (Jin et al. 2012) which has a relatively high accuracy is adopted to reconstruct the free surface. EXPERIMENTAL VALIDATION Experimental model

The essence of the liquid sloshing analysis for independent tank with elastic supports is a coupled problem of tank motion and liquid sloshing. As no experiment exists for the model of the independent tank with elastic supports shown in Figure 1, an experiment based on a model with the same principle is chosen to validate the algorithm. The experiments were conducted with the tank testing device of the CEHINAV-UPM research group (Bulian et al. 2010), and it treats the coupling of the motion of a sloshing tank and a Single Degree Of Freedom structural system (SDOF), which is generally denoted as a Tuned Liquid Damper (TLD). The whole dynamic system is composed of the shifting mass, the moving part of the sloshing rig including the empty tank but excluding the shifting mass and the fluid, as shown in Figure 2a,b. An analytical model of the SDOF structural system used in the experiments is needed in order to have it incorporated into the structure part of the numerical code. It was obtained by rigorously analyzing the dynamics of the system and by obtaining the coefficients after carefully analyzing a set of tests with the empty tank and thereafter

(10) (11)

where un and ut are the normal velocity and tangential velocity at the free surface; p0 is the design vapor pressure. The SIMPLE algorithm (Ingram et al. 2003, Griffith 2009) is employed to solve Equation 4 and Equation 5, and we adopt the finite difference method for discretization and staggered grids to mesh the computational domain. The pressure P and volume fraction F are defined in the center of

Figure 2a.

The sliding device.

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Figure 2b. The experimental sloshing flow sample image.

Figure 3a.

Figure 3b. Analytical model of the tank in the experiment.

Tank dimensions of the experiment.

finding a data-consistent damping term model, as shown in Figure 3a, b. The analytical model used to describe the behavior of the system is, in general, as follows: [

ξm2 (t)] (t)] )] φ 2m mξm ( )ξm ( ) ⋅ φ − g SG ⋅ sin( i (φ ) + mg gξm ( ) ⋅ cos(φ ) = M friction + M fluid / mass

0

M friction

Bφ φ − K df sign(φ )

(12) (13)

where φ [rad] is the roll angle; g [m/s2] is the gravitational acceleration; I0 [kg ⋅ m2] is the polar moment of inertia of the rigid system with respect to the rotation axis; m [kg] is the mass of the moving weight; ξm (t ) [m] is the instantaneous (imposed) position of the excitation weight along the linear guide (tank-fixed reference system); ξm (t ) [m/s] and ξm (t ) [m/s2] are the first and second time derivatives of ξm (t ) [m]; SG=MR ⋅ ηG [kg ⋅ m] is the static moment of the rigid system with respect to the rotational system; MR [kg] is the total mass of the rigid system; ηG [m] is the (signed) the center of the gravity of the rigid system with respect to the rotation axis (tank-fixed reference system);

M friction Bφ φ − K df sign(φ ) [N ⋅ m] is the assumed form of roll damping moment comprising: A dry friction term −K K df sign(φ ) with K df [N ⋅ m] being the dry friction coefficient; A linear damping term −Bφ φ with Bφ [N ⋅ m/(rad/s)] being the linear damping coefficient. The values of these coefficients together with an assessment of the quality of the model investigated using free decay and forced motion empty tank tests have been included in Bulian’s tests. It is worthwhile to note that the difference of the experimental model and the independent tank model is the tank movement mode shown in Equation 1 and Equation 12, but they are both the problem of tank motion and liquid sloshing. 3.2

Experimental results

The natural period of the system T0 is approximately 1.927 s. The liquid depth (92 mm) was chosen so as to match the first resonance period T0 of the structural system. Cases around the resonance were considered for a range of mass motion amplitudes A. We describe briefly the case of resonance one with different mass motion amplitudes and refer the reader to reference. As is shown in Figure 4, the empty tank roll angle of the present simulation matches well with the experiment. As for the partially filled tank, we choose the resonance case and set the moving weight amplitude A as 100 mm for two liquids (water and oil), and Figure 5 shows that the simulation results in the real coupled calculation also have a relatively good agreement with the experiment except for the phase difference, which may be attributed to the inertia delay of the moving mass at the beginning of the experiment. One variable that was discussed in Bulian’s work is the reduction ratio in amplitude between the

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Table 1. Amplitude reduction ratio (amplitude partially filled tank/amplitude empty tank).

Figure 4. ωex=ω0.

Roll angle of the empty tank, A = 100 mm,

A

50 mm

100 mm

150 mm

Experiment Water Oil SPH Water & oil VOF Water Oil

6.4 13.1 10.4 9.5 11.3

23.6 31.4 32.2 29.3 34.5

46.7 51.4 45.5 51.1 53.9

be acquired by taking the maximum values of the partially filled tank roll angle and the empty tank roll angle. The values of this ratio are presented in Table 1, including the experimental and numerical results. The SPH results are derived from Bulian’s calculation, while the VOF results are obtained by applying the algorithm in this paper. The results show that the algorithm proposed in this paper is applicable to the coupled issue between the motion and liquid sloshing.

4

NUMERICAL CALCULATION

4.1 Physical model

Figure 5a. Roll angle of tank with water inside, A = 100 mm, ωex=ω0.

Figure 5b. Roll angle of tank with oil inside, A = 100 mm, ωex=ω0.

partially filled tank and the empty tank roll angles (percentage) in a specific period of time. A ratio close to 100% means that the liquid has no dampening effect, while a ratio close to 0% means that the resulting amplitude is very small. The ratio can

A model which takes the form of a typical section of an independent type B LNG prismatic tank without inner baffles is selected as reference, and the support structures between the hull and the tank are equivalent to springs with corresponding stiffness. A two-dimensional coupled calculation procedure of tank sloshing with elastic support is set up. The numerical model is shown in Figure 6. The fill ratio is 30%, and the points of P1 and P2 are the pressure monitoring points located on the free surface. The natural period of ship roll motion is 10.63 s. Structural grid is set as 120 × 80, and the cargo density is 500 kg/m3. The other parameters can be found in Table 2. The external excitation is applied on the hull, which amounts to a forced harmonic displacement motion acting on the support of the springs. The overall torque of elastic supports acting on the tank with respect to the rotation axis varies with the relative angle between the hull and the tank, and it can be linearized by least square estimation, from which equivalent roll stiffness is obtained. The equivalent roll damping is set in the same way. In the case of a constant dimensionless damping coefficient, the equivalent roll stiffness is varied to set three typical conditions, which is shown in Table 3. The Case 1 corresponds to a weak stiffness, while the Case 3 to a strong stiffness. Case 2 refers to the resonance condition.

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Figure 6. Numerical model for the simulation of sloshing (the different color shows the pressure distribution).

Table 2.

Time histories of tank roll angle.

Figure 8.

Time histories of tank roll angular velocity.

Parameters of numerical model.

Quantity

Value

Units

Polar moment of inertia of the rigid tank I0 Height of the center of tank gravity Hg Height of sloshing center H0 Tank mass M Excitation period Tex Excitation amplitude ϕ0 Tank size (breadth × Height)

1.86 × 107

kg ⋅ m2

12.65

m

13.93 7.4 × 104 10.63 12 38 × 27

m kg s deg m

Table 3.

Case 1 Case 2 Case 3

4.2

Figure 7.

Parameters of design conditions. Equivalent roll stiffness Kθ N ⋅ mm/rad

Equivalent roll damping Cθ N ⋅ mm ⋅ s/rad

1.856 × 106 6.671 × 106 1.856 × 107

1.958 × 106 3.713 × 106 6.192 × 106

Elastic effect on the tank motion

In the resonance case of Case 2, we compare the roll angle, roll angular velocity and roll angular acceleration with those in the tank without elastic support in 30% fill ratio and those in the empty tank with elastic support (same as Case 2), as shown in Figure 7 to Figure 9. The values of the parameters of external excitation are the same as those shown in Table 2. The analysis reveals that the amplitude of the tank with elastic support is larger than the one without elastic support, and

Figure 9. Time histories of tank roll angular acceleration.

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the phase lag occurs between the two conditions. On the other hand, the motion amplitudes of the tank with elastic support in 30% fill ratio and the empty tank with elastic support are 25.97° and 34.38°, respectively, from which we can see that the liquid sloshing restrains the tank motion. The time histories of excitation torque, gravity torque and liquid torque acting on the tank wall in Case 2 are shown in Figure 10. The gravity torque and liquid torque weaken the effect of the excitation torque due to the phase lags, which results in a reduction of the tank angle amplitude compared with the empty tank motion. That is exactly the design principle of Tune Liquid Damper (TLD). Figure 11 shows the time histories of tank motion in the three cases. In Case 1, the tank response appears to be an envelope shape, and it is smaller than the case without elastic support. The reason for this phenomenon might be that

Figure 10.

the stiffness of the elastic support in Case 1 is not sufficient to transfer the external excitation to the tank. In Case 2, the resonance of tank motion and the external excitation occurs so that the tank motion is the most dramatic. The tank motion in Case 3 is larger than that without elastic support, but it is smaller than that in Case 2, although the stiffness of Case 3 is larger. The behavior of Case 3 indicates that with the increase of the stiffness after the resonance point the tank movement tends to be consistent with the hull movement, and we can suppose that when the stiffness is infinite the hull and the tank will move in the same rhythm. 4.3

Elastic effect on the sloshing loads

The snapshots of the liquid surface in Case 2 and the case without elastic support at two same moments are shown in Figure 12 and Figure 13.

Figure 12a. Liquid surface motion simulation at t/T = 7.26, tank without elastic support.

Time histories of various torques.

Figure 11. Time histories of tank roll motion in various conditions.

Figure 12b. Liquid surface motion simulation at t/T = 7.26, tank with elastic support (Case 2).

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Figure 13a. Liquid surface motion simulation at t/T = 7.44, tank without elastic support.

Figure 14. Liquid surface motion simulation at t/T = 7.44, tank with elastic support (Case 2).

Figure 13b. Liquid surface motion simulation at t/T = 7.44, tank with elastic support (Case 2).

Figure 15. Liquid surface motion simulation at t/T = 7.44, tank with elastic support (Case 2).

Obvious difference of the free surface profile can be found in these two cases at the same moment. The liquid in the tank with elastic supports displays a more dramatic sloshing phenomenon than that without elastic supports. By comparing the time histories of the pressure on P1 (See Fig. 14), we can see that the pressure in Case 1 is smaller than that in the case without elastic support. On the contrary, in Case 2 the pressure is twice as much as that in the case without elastic support, that is to say the elastic support strengthens the effect of the sloshing loads. In Case 3, although the stiffness is larger than the one without elastic support, the pressure is far less than that in Case 2. From above analysis, we can conclude that the elastic support’s effect on the sloshing loads keeps consistent with that of the tank motion. Assuming the deformation of the elastic support is unlimited, the pressure of P1 is calculated in the tank with a series of stiffness of elastic

support, and the extreme value in the stable period is acquired to plot a curve, shown in Figure 15. In the initial phase when the stiffness is relatively small, the pressure of P1 with elastic support is smaller than that without elastic support. With the increase of the stiffness, the sloshing load keeps growing dramatically and reaches its peak at the resonant point, before dropping continually and eventually leveling off to a constant value, which is coincident with the sloshing load without considering the elastic influence. It is worth mentioning that the stiffness of elastic support in real tanks locates the later part of the curves in Figure 15, in this case, the sloshing load only increases slightly compared with the case without elastic support. 5

CONCLUSION

In this study, the independent prismatic LNG tank is selected to investigate its sloshing characteristics

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with elastic support. The numerical results can be summarized as follows: 1. There is a clear distinction between movements of the tank with and without elastic support, and the tank motion reaches the most dramatic when the excitation frequency equals the resonant frequency of the tank. 2. The stiffness of the elastic support has a significant impact on the sloshing loads. Precisely, with the increase of the stiffness, the sloshing load keeps growing dramatically and reaches its peak at the resonant point, before dropping continually and eventually leveling off to a constant value, which is coincident with the sloshing load without considering the elastic influence. An efficient and effective method is finally presented for predicting the sloshing load of independent tank considering the elastic support effect, and it provides a reference for the material selection and design of the elastic support. Further work remains still to be done relating to an experiment based on the independent tank model with elastic supports. REFERENCES Bulian, G., Souto-Iglesias. A., Delorme, L, Botia-Vera, E. 2010. SPH simulation of a tuned liquid damper with angular motion. Journal of Hydraulic Research 48: 28–39. Cao, X.Y., Ming, F.R. & Zhang, A.M. 2014. Sloshing in a rectangular tank based on SPH simulation. Applied Ocean Research 47: 241–254. Faltinsen, O.M. 1974. A nonlinear theory of sloshing in rectangular tanks. Journal of Ship Research 18(4): 224–241.

Faltinsen, O.M. 1978. A numerical nonlinear method of sloshing in tanks with two-dimensional flow. Journal of Ship Research 22(3): 193–202. Griffith, B.E. 2009. An accurate and efficient method for the incompressible Navier Stokes equations using the projection method as a preconditioner. Journal of Computational Physics 228: 7565–7595. Hilber, H.M., Hughes, T.J.R. & Taylor, R.L. 1997. Improved numerical dissipation for time integration algorithms in structural dynamics. Earthquake Engineering and Structural Dynamics 5: 283–292. Hirt, C.W. & Nichols, B.D. 1981. Volume of fluid (VOF) method of free boundaries. Journal of Computational Physics 39: 201–225. Ibrahim, R.A., Pilipchuk, V.N. & Ikeda, T. 2001. Recent advances in liquid sloshing dynamics. Applied Mechanics Reviews 54(2): 133–199. Ingram, D.M., Causon, D.M. & Mingham, C.G. 2003. Developments in Cartesian cut cell methods. Mathematics and Computers in Simulation 61: 561–572. Jin, J., Xue, H.X., Tang, W.Y. & Zhang, S.K. 2012. Calculation procedure of sloshing loads for largescale depot ships. Chinese Journal of Ship Research 7(6): 50–56. Nagashima, T. 2009. Sloshing analysis of a liquid storage container using level set X-FEM. Communications in Numerical Methods in Engineering 25(4): 357–379. Shen, M., Wang, G. & Tang, W.Y. 2009. Sloshing analysis in prismatic tanks based on VOF method. Ship Building of China 50(1): 1–9. Zhu, R.Q. & Wu, Y.S. 2002. A numerical study on sloshing phenomena in a liquid tank. Ship Building of China 43(2): 15–28. Zhu, R.Q., Ma, H.X., Miao, Q.M. & Zheng, W.T. 2013. Prediction of pressure induced by liquid sloshing for LNG carrier. Journal of Ship Mechanics 17(1–2): 42–48.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Numerical simulation of the dynamic behaviour of resilient mounts for marine diesel engines L. Moro, E. Brocco, P.N. Mendoza Vassallo & M. Biot University of Trieste, Trieste, Italy

H. Le Sourne ICAM, Nantes-Angers-Le Mans University, France

ABSTRACT: On board ships, to avoid high levels of structure-borne noise due to marine four strokes diesel engines, the latter are usually resiliently mounted. In order to improve the effectiveness of the decoupling between the diesel engines and the ship structures, it is important to have available tools and numerical models that simulate the dynamic behaviour of the isolator system as well as of the receiving structures. In the paper, after an introduction to the basics for the characterization of passive resilient mounts in high frequency range, an FE non-linear dynamic model of the isolator is presented. Such model takes into account the dynamics of the cast-iron parts, the contact between the top and base casting with the rubber core and the non-linear behaviour of the constitutive law of rubber. A procedure to achieve the characteristics parameters of rubbers is presented. The simulations have been then validated by experimental tests. 1

INTRODUCTION

The medium speed diesel engines which are installed on ships as both, prime movers or gensets, are dominant sources of vibrations and structure-borne noise. The vibration energy, generated in the audible frequency range by the engines, is transmitted to the diesel engine foundation and, spreading through the ship’s structures, generates noise. This phenomenon can decrease the comfort levels on-board ships. In order to reduce the radiated vibration energy, the marine diesel engines are usually resiliently mounted. This solution guarantees the decoupling between the source and the receiving structures but, to optimize its effectiveness, a proper design of the resilient elements, of the diesel engine foundation and of the receiving elements should be made. For this reason, it should be useful to have available tools and numerical models to simulate in the high frequency range, the dynamics of the decoupling systems and of the receiving structures. It is well known that the structure borne noise due to the marine diesel engine is function of three mechanical quantities: the velocity levels measured at the diesel engine feet, the mechanical impedance of the resilient mounts, and the mechanical mobility of the diesel engine foundation (Nilsson et al. 1998, Cremer et al. 2005, Moorhouse et al. 1993). Once all these three quantities are allowable, the

structure-borne noise can be calculated according to the so called single-point approach (Biot et al. 2013). The velocity levels at the diesel engine feet can be measured during the testing of the diesel engines in the test rooms. As for the mobility of the diesel engine foundations, it can be achieved by measurement campaigns carried out on board of ships (Petersson & Plunt 1982) or simulating it by FE dynamic linear analysis. With regards to the estimation of the dynamic impedance of the resilient mount, special laboratory tests are to be carried out. The passive resilient mounts which usually are installed aboard ships are formed by a top and a base casting which are made of steel or cast iron, and allow the isolator to interface with the diesel engine and with the ship structures (see Figure 1 and Figure 2). The core of the resilient mount is made of rubber and it is the passive isolator element of the whole resilient mount. The rubber core is characterized by a non-linear behaviour that depends on the static preload. Moreover at high frequency range, standing waves are present in the rubber and these alter the dynamic response of the resilient mount. For these reasons, the dynamic behaviour of rubber isolators is strongly dependent on frequency. The tests to measure the dynamic characteristics of resilient mounts are carried out using testrigs that are specifically designed for this purpose.

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Figure 1. Conical passive resilient mount for marine diesel engine.

Figure 2. Components of conical passive resilient mount for marine diesel engine.

These allow performing measurements for acquiring data according to the ISO 10846 Standard (ISO 2008 and ISO 2002). The latter is considered a sound reference for this kind of experiment in both, the low and the high frequency range. In last years, several research activities have been developed to improve these kinds of experimental tests and to completely characterize the isolators. Dickens and Norwood (1997 and 2001) have presented the design of a new test-rig for the characterization of resilient mounts and to achieve their four-pole parameters at several static pre-load (see also Dickens 2000). Vermeulen et al. (2001) have presented a practical method to actively control the unwanted input vibrations due to the dynamic behaviour of the test-rig structure and of the moving masses which are used to control the input driving excitation. Moro et al. (2013) have studied the coupling between the top-casting of a conical isolator for marine diesel engines and the masses of the test rigs and they have investigated the influence of the resonances of the top-casting of such resilient mount on its dynamic stiffness.

With regards to analytical solutions or numerical methods to simulate the dynamic stiffness of the resilient mounts, good results have been obtained only in the case of isolators with simple geometric shapes. Dickens (2000b) have investigated a dynamic model as a function of the compression ratio of its rubber element. The dynamic model has been validated with data acquired by laboratory tests. Kari (2003) has studied the non-linear, preload-dependent dynamic stiffness of an isolator made of two circular plates (top and bottom interfaces) and a rubber cylindrical core. Also in this study, the results of the model have been compared with the experimental data. Beijers and de Boer (2003) have proposed an iterative procedure for carrying out FE simulations of a cylindrical shaped passive isolator at different static pre-loads. They have also tuned another iterative procedure to identify the material parameters of the rubber core (Beijers et al. 2004) if those are not available for the definition of the numerical model. At the Ship Noise and Vibration Laboratory of the University of Trieste and at the ICAM of Nantes, a joint research activity is being done with the aim of defining a numerical procedure to achieve the mechanical impedance of a conical resilient mount, which has been designed for large marine diesel engines. In the paper some relevant results of this research activity are presented and discussed. First of all, a numerical model that takes into account the non-linear behaviour of the material and the non-linearity due to the contact between the top casting and the rubber core of the resilient mount has been created. Then, using the outcomes of experimental tests carried out on the isolator, an iterative procedure has been developed to obtain the parameters of the hyper-elastic model which describes the mechanical behaviour of the rubber. Once the iterative procedure has been carried through, a second series of numerical simulations has been performed with the aim to simulate the dynamic behaviour of the resilient mount in the high frequency range (100 Hz–800 Hz). 2 2.1

THEORETICAL BACKGROUND Four-pole terms and dynamic transfer stiffness of a resilient mount

The dynamic behaviour of an isolator is usually described using the four-pole parameters approach. According to such theory, the resilient mount can be described by a linear lumped mechanical element. This is characterized by four parameters which allow to correlate the input variables of the system to the output ones.

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transfer impedance of the resilient mount Z21 has to be measured. In a most general case, the input velocity and the force transmitted to the receiver are characterized by 6 orthogonal terms: vs = {vx, vy, vz, γx, γy, γz} and Fr = {Fx, Fy, Fz, Mx, My, Mz}. Hence, the Equation 1 can be rewritten as: Figure 3. Four-pole parameters for the characterization of a lumped mechanical system.

Figure 3 shows a mechanical lumped system subject to a force F1 at the side 1 which moves at the velocity v1. The side 2 moves at velocity v2 and transmits the force F2. All these variables are frequency dependent. The forces F1 and F2 are related to the velocities v1 and v2 as follows:

{

F1 F2

Z11v1 Z12v2 Z21v1 Z22v2

(1)

where the parameters Zij are the four-pole parameters defined as: Z11 = Z21 =

F1 v1 F2 v1

Z12 = v2 0

Z22 = v2 0

F1 v2

v1 = 0

F2 v2

v1 = 0

(2)

Hence, the Zij parameters are the mechanical driving point impedances (for i = j) and the transfer blocked mobility functions (for i ≠ j). In the case of a resilient mount for marine diesel engines, the input force F1 is the force transmitted from the engine to the isolator, while the force F2 is the force mutually exchanged between the isolator and the receiving structures, i.e. the diesel engine foundation. The force F2 is obtained as follows by the Equation 1: Fr

F2 =

Z21 v Z s 1 + 22 Zr

(3)

where Fr is the force at the receiver, vs is the source velocity, and |Zr = Fr/vr| is the mechanical driving point impedance of the receiving structures. In the case that |Z22| « |Zr|, as usually happens for the resilient mounts installed on board ships, the Equation 3 can be approximated as follows: Fr

Z ⋅ vs

(4)

that means that for a complete characterization of a resilient mount in each direction, the mechanical

⎧ {F1} ⎫ ⎡ ⎡⎣Z1,1 ⎤⎦ ⎡⎣Z1,2 ⎤⎦ ⎤ ⎧ { ⎥ ⎨ ⎨ ⎬=⎢ ⎩{F2 }⎭ ⎢⎣ ⎡⎣Z2,11 ⎦⎤ ⎡⎣Z2,2 ⎤⎦ ⎥⎦ ⎩{

}⎫ }⎬⎭

(5)

where Zij is a 6 × 6 matrix, and the global mechanical impedance matrix is 12 × 12. So, to completely describe a resilient mounting system of N elements, 144 × N transfer functions are needed. Owing of symmetry, many elements of the mechanical impedance matrix are equal to zero and some non-zero elements are equal in magnitude. In practical cases, few diagonal elements are enough to describe the isolator behaviour in the translational directions (Verheij, 1982). 2.2 Method for the experimental characterization of rubber isolators The mechanical impedance of a resilient mount can be obtained by special experimental tests. Depending on the frequency range to be investigated, three different experimental methods are proposed in the ISO 10846 standards “Acoustics and vibration— Laboratory measurement of vibro-acoustic transfer properties of resilient elements”. According to the ISO standard, the mechanical impedance Z21 of the resilient mount under investigation is obtained by the following equation: Fr

Z ⋅ vs

k21 ⋅ us

(6)

where us = u1 is the displacement of the source. Hence, Z21 can be easily calculated once the dynamic transfer stiffness is achieved in laboratory. In the audio-frequency range, the dynamic transfer stiffness of resilient elements is determined using the so-called indirect method (ISO 2002). Such method is characterized by the fact that the blocking force at the output of isolator under test is not directly measured but it is derived from acceleration measurements performed on the blocking mass which is dynamically decoupled from the frame of the test-rig. Figure 4 shows the layout of the test-rig set to carry out laboratory tests according to the indirect method. The resilient mounting is not directly coupled with the vibration source, as a compact mass, called excitation mass, is interposed. The excitation mass function is to provide the condition of contact point at the input side of the resil-

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Figure 4. Scheme of the test-rig for dynamic characterization of the resilient mounts using the indirect method.

a suitable constitutive model to describe the rubber material has to be used. Rubber behaves as a visco-elastic material and so it combines the viscosity of the liquids with the elasticity of the solid materials. When rubber undergoes deformation, it exhibits time-dependent strain. Moreover the material behaviour strongly depends on temperature, frequency range of the excitation and on pre-deformation. Several models have been developed to describe the behaviour of rubber. The visco-elastic models usually take into account both the viscous and the elastic behaviour of the resilient mount, while the hyper-elastic models provide a mean to model the non-linear stressstrain relationship of the rubber. The Yeoh model (Yeoh, 1993) is a hyper-elastic model that is widely used to simulate the rubber stress-strain relationship. According to this model, if the material to be simulated is nearly incompressible, its strain energy density function W depends on the first strain invariant I1 as follows: 3

ient mounting and, as the blocking mass, it is dynamically decoupled from the test rig structure using soft isolators. Under the resilient mount, the so-called blocking mass is placed. It provides a high-stiffness contact point at the isolator output side, so that the forces between the isolator output side and the receiving mass are approximately equal to the blocking forces. The blocking mass must have a high inertia, both translational and rotational, whereas its decoupling isolators should have a suitable low stiffness so as to keep low the resonance frequencies of the 6 rigid motions of the mass. An actuator is used for applying a static preload, so that the resilient mounting is tested in working condition. In the indirect method test, the acceleration (or displacement or velocity) of the excitation mass and the acceleration (or displacement or velocity) of the blocking mass are measured and the dynamic transfer stiffness is derived as: k21 ≈

F2 = −((2 f u1

2

m2

u2 u1

(2 f

2

m2

a2 a1

(7)

where the dynamic inertia is usually referred to as a dynamic transmissibility, and so denoted as T2b,1(f ), in consideration that it is an output to input force ratio where the input force is referred to a unit mass. 2.3

Dynamic behaviour of rubber

In order to properly simulate the dynamic behaviour of rubber in the high frequency range,

W

∑C

i0

( I1 − 3 ) i

(8)

i =1

To completely characterize the rubber, three material coefficients are needed: C10, C20 and C30. For the consistency condition with linear elasticity, it can be shown that the first material coefficient C10 is equal to half of the initial shear modulus μ. The material coefficients are usually obtained carrying out laboratory tests on rubber standardized specimen for the achievement of the stressstrain curve. Unfortunately, these data are often not available or they are strictly confidential. The dynamic material parameters which characterize rubber like materials are the storage modulus E’ and the loss modulus E’’. The former measures the stored energy, representing the elastic portion of the material, the latter, measures the energy which is dissipated as heat and so it represents the viscous portion. These parameters can be measured performing the Dynamic Mechanical Analysis (DMA) tests. Tests facilities can usually perform this kind of measurement in a low frequency range, and these results cannot be extrapolated at higher frequency range. 3

OUTLINE OF THE NUMERICAL PROCEDURE

In order to set a numerical model which simulates the dynamic behaviour of a resilient mount, the characteristic material data of the rubber core are needed. These data can be obtained by DMA tests, but they are usually confidential and so they are often not available to properly identify the material

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coefficients of the rubber part of the resilient mount. Beijers (2004) set a procedure to identify the Yeoh coefficient C10, C20 and C30 starting from static force-deformation compression tests. This procedure is based on static non-linear simulations and it has been tested in the case of a small resilient mount with a cylindrical shape. Then, a second set of numerical simulation were carried out to characterize the dynamic behaviour of the rubber, and so to obtain the loss modulus E’’ of the rubber, starting from the experimental curve of the dynamic transfer stiffness of the same resilient mount. The procedure defined by Beijers has been reused by the authors and generalized to be applied in the more general case of a conical resilient mount for large marine diesel engine. The procedure can be divided into several steps. First of all, non-linear static simulations are carried out to find the static coefficients that characterize the rubber material C10, C20, C30. Such simulations reproduce the static compression test of a resilient mount and a Force-Displacement curve is obtained as result. The latter are to be compared with an experimental force-displacement curve, obtained as results of a compression test carried out on the resilient mount under investigation. The first step is to fit the numerical curve with the experimental curve in the interval of the latter where the non-linear behaviour is negligible. In this way the linear coefficient of Equation 7, i.e. C10, is set. Then the fitting procedure carries on in order to find the non-linear terms of Equation 7, i.e. C20 and C30. The procedure terminates when the quadratic error between the measured reference curve and the simulated one (both have been sampled at constant intervals) is minimized. Once the static non-linear simulations are completed and all the three coefficients of the Yeoh models are set, the second step of the procedure can begin. The resilient mount is preloaded with the work static load and then a series of non-linear dynamic simulations is carried out. Such simulations take into account the non-linearity due to the contact between the resilient mount top casting and the rubber core, and between the latter and the resilient mount base casting. The viscous behaviour of rubber affects the dynamic response of the resilient mount. Since the rubber material has been modelled using the Yeoh constitutive law, i.e. a hyper-elastic model, the damping is taken into account defining a damping coefficient of the rubber part ξ which strongly depends on frequency. A procedure has been developed to define such coefficient. In this procedure, a measured curve of the dynamic stiffness of the resilient mount has been taken as reference. For each frequency step of about 20 Hz, a harmonic acceleration a1 in the vertical direction has been imposed to the flange

of the resilient mount top casting. The constraint force that results at the resilient mount base casting constraint, in the vertical direction, is the output force F2 to be used in the calculation of the simulated dynamic transfer stiffness, once the displacement of the top casting u1 is obtained from a1. The value of the damping coefficient to be used in the model is tuned minimizing, for each frequency step, the quadratic error between the calculated values and the measured ones. 3.1

The finite element model

The procedure presented has been tested on a conical resilient mount designed for medium speed marine diesel engines of a rated power up to 16 MW. The resilient mount is made of three different parts: the top casting, the base casting and the rubber core. In Figure 5 the FE model of the resilient mount is shown and in Figure 6 the FE model has been sectioned by a vertical plane of symmetry. The resilient mounting top holds a flange for the coupling with the engine foot, which flange allows the adjustment of resilient mounting in height, by means of a ring nut, for a better engine alignment.

Figure 5. tigation.

FE model of the resilient mount under inves-

Figure 6. Cross section of the FE model of the resilient mount under investigation.

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The rubber part is a conical-shaped toroid body (the light grey part in Figure 6) which is coupled by a conical-shaped seat with the resilient mounting base. The base is bolted to the diesel engine foundation on the ship inner bottom. In Figure 6, the border lines between the rubber part and the top and bottom casting belong to the contact surfaces between the parts. The top and the base casting of the resilient mount are made of cast iron while the isolator part is made of carbon filled rubber. The resilient mount has been modelled using constant stress solid elements. Penalty contact interfaces have been defined between the rubber part and the top and base steel parts. The mesh size has been set in such a way that the lowest wave length λ of the excitation signal can be properly simulated in each part of the isolator. Since the upper frequency range that has been investigated in this study is 800 Hz, the mean mesh size is about 6 mm. 3.2

Experimental tests

At the Ship Noise and Vibration Laboratory of the University of Trieste an extensive experimental campaign has been carried out to obtain the data to be used as reference in the numerical procedure. First of all the resilient mount has been statically compressed. The test has been performed increasing the compression static force at fixed steps. For each force step, the measurement of the deflection of the resilient mount has been done after 24 hours, so that the relaxation of the rubber had not altered the measured data (i.e. the test has been performed maintaining constant the compression force during the relaxation phenomenon and measuring the displacement when the rubber relaxation has been considered finished). Once the force-displacement curve was obtained, the dynamic tests to measure the dynamic transfer stiffness of the resilient mount have been carried out. In Figure 7 the test-rig for the characterization of passive elastomeric isolators is shown. The parts are: (A) crossheads for shaker support placed on the top of the (B) heavy and rigid main frame of the test rig, (C) trunnion hanged by isolators to the crossheads, (D) electrodynamic vibration generator, (E) mobile crosshead and static load upper support plate, (F) moving system between upper and lower isolator beds, (G) static load lower support discs and (H) static force transducers and cylinder for hydraulic preload. The tests have been performed to characterize the isolator in the high frequency range (100 Hz–1000 Hz). To do this, the so called indirect method has been applied. The resilient mount has been undergone to its nominal static load. The blocking mass and the exciting mass has been properly design. In particular, the exciting mass has

Figure 7. Test-rig for the dynamic characterization of the resilient mount according to the indirect method.

been designed to uniformly distribute the excitation signal generated by the electro-dynamic shaker to the resilient mount top casting. In turn, the blocking mass has been designed to behave as a rigid body in all the frequency range of the experiment. The soft isolators, used to decouple the moving part of the test-rig from the rigid frame, have been chosen to lower as much as possible the rigid body resonances of the oscillating masses. The excitation signal has been controlled with a closed loop control system, while the unwanted input vibrations, i.e. vibrations of the exciting mass acting in directions orthogonal to the driving one, have been checked. The latter check is very important in order to ensure that the measurement of the transfer function k21 is not affected by terms out of the matrix diagonal. To verify the proper dynamic behaviour of the moving part of the test-rig several studies have been performed in past experimental campaigns. (Moro et al. 2013, Moro & Biot 2013). 3.3 Numerical simulations The numerical simulations have been performed using LS-Dyna software at the mechanical engineering department of ICAM Nantes Campus. All the series of runs have been carried out according to the presented procedure. The first series

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of simulations aimed to obtain the characteristic parameters of rubber simulating the compression test, and the second series of simulations have been done with the aim to obtain the damping coefficient and so to completely define a dynamic nonlinear model of the resilient mount. 4

RESULTS

In Figure 8 some relevant curves of the first series of simulations are shown. It is worth pointing out that the first part of the experimental curve (dashed curve), in the neighbourhood of the graph origin is almost linear and that its non-linear behaviour arises at higher loads. That curve has been used as reference in the numerical runs that simulates such test. The curve fitting carried out in the first series of runs leads to the individuation of the first characteristic parameter C10 of the Yeoh model. The point-dashed curves are the outcome of some runs and the continuous curve is obtained with the coefficient C10 that has been obtained once the optimization process has been accomplished. The values of the coefficients obtained once the iterative procedure has been completed, are shown in Table 1.

Figure 9 and Figure 10 show some force-displacement curves obtained by runs done to achieve the second and the third parameters, C20 and C30 respectively. The Figure 11 shows the force-displacement curve obtained by a numerical simulation in which the rubber part of the model is characterized by the three Yeoh coefficients obtained as results of the optimization process. The Figure 12 shows the experimental curve obtained as outcome of the experimental tests for the measurement of the dynamic transfer stiffness k21. The excitation levels of the tests are shown in Figure 13. Those are shown in terms of

Figure 9. Force-displacement curve obtained by compression experiments (dashed curve) and by simulations (point-dashed and continuous curves) with different values of the C20 coefficient. The continuous curve has been achieved with the coefficient obtained at the end of the fitting procedure.

Figure 8. Force-displacement curve obtained by compression experiments (dashed curve) and by simulations (point-dashed and continuous curves) with different values of the C10 coefficient. The continuous curve has been achieved with the coefficient obtained at the end of the fitting procedure. Table 1. Yeoh coefficient of the rubber part of resilient mount. Coefficients

Estimated values

C10 C20 C30

9236 2371 1678

Figure 10. Force-displacement curve obtained by compression experiments (dashed curve) and by simulations (point-dashed and continuous curves) with different values of the C30 coefficient. The continuous curve has been achieved with the coefficient obtained at the end of the fitting procedure.

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Figure 11. Force-displacement curve obtained by compression experiments (dashed curve) and by simulations (continuous curves) carried out with the three different coefficients achieved once the numerical procedure has been completed.

Figure 14. Measured (dashed curve) and simulated (continuous curve) dynamic transfer stiffness.

accelerations. In the same figure the accelerations of the unwanted vibrations are shown to give evidence about the effectiveness of the results. The same vibration levels have been applied to the top casting of the FE model to reproduce, in the simulation, the excitation condition of the experimental tests. The Figure 14 shows the results of the non-linear dynamic simulations, once the damping coefficient has been set for each frequency step. 5

Figure 12.

CONCLUSIONS

The study presented in this paper aims to define a numerical model to simulate the dynamic behaviour of a resilient mount for marine diesel engine at the high frequency range. As benchmark for the validation of the numerical model, two experimental tests have been carried out: a static compression test to achieve a load-displacement curve and a dynamic test for the measurement of the dynamic transfer stiffness of the resilient mount. In the numerical model of the resilient mount, the rubber part has been modelled using the Yeoh constitutive law. The three parameters which characterize that law have been obtained applying an iterative procedure. Once those parameters have been identified, a second iterative procedure allowed to tune the numerical model in order to simulate the dynamics of the isolator.

Measured dynamic transfer stiffness k21.

REFERENCES

Figure 13. Accelerations levels of the excitation mass in x, y and z directions measured during the experimental tests.

Beijers, C., de Boer, A. 2003. Numerical modelling of rubber vibration isolators. Proceedings of the 10th Internation Congress on Sound and Vibration (ICSV 10), Stockholm, Sweden. Beijers, C., Noordman, B., de Boer, A. 2004. Numerical modelling of rubber vibration isolators: identifica-

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tion of material parameters. Proceedings of the 11th Internation Congress on Sound and Vibration (ICSV 11), St. Petersburg, Russia. Biot, M., Mendoza Vassallo, P.N., Moro, L. 2014. Prediction of the structure-borne noise due to marine diesel engines on board cruise ships. Proceedings of the 21st Internation Congress on Sound and Vibration (ICSV 21), Beijing, China. Cremer, L., Heckl, M., Petersson, B.A.T. 2005. Structureborne sound, structural vibrations and sound radiation, Berlin: Springer. Dickens, J.D. 2000a. Methods to measure the four-pole parameters of vibration isolators, Acoustics Australia, vol. 28 (no. 1), pp. 15–21. Dickens, J.D. 2000b. Dynamic model of vibration isolator under static load. Journal of Sound and Vibration, vol. 236 (no 2): pp. 323–337. Dickens, J.D., Norwood, C.J. 1997. Design of a test facility for vibration isolator characterization, Acoustics Australia, vol. 25 (no. 1): pp. 23–28. Dickens, J.D., Norwood, C.J. 2001. Universal method to measure dynamic performance of vibration isolators under static load, Journal of Sound and Vibration, vol. 244 (no. 4): pp. 685–696. ISO 10846-1:2008 Acoustics and vibration—Laboratory measurement of vibro-acoustic transfer properties of resilient elements—Part 1: Principles and guidelines, Geneve. ISO 10846-3:2002 Acoustics and vibration—Laboratory measurement of vibro-acoustic transfer properties of resilient elements—Part 3: Indirect method for determination of the dynamic stiffness of resilient supports for translatory motion, Geneve. Kari, L. 2003. On the dynamic stiffness of preloaded vibration isolators in the audible frequency range: Modelling and experiments. Journal of the Acoustical Society of America, vol. 113 (no. 4): pp. 1909–1921.

Moorhouse, A.T. & Gibbs, B.M. 1993. Prediction of structure-borne noise emission of machines: development of a methodology. Journal of Sound and Vibration, vol. 167 (no. 2): pp. 223–237. Moro, L., Biot, M., Mantini, N., Pestelli, C. 2013. Solutions to improve accuracy in experimental measurement of the dynamic response of resilient mountings for marine diesel engines. 4th International Conference on Marine Structures, MARSTRUCT 2013, Espoo, Finland. Nilsson, A., Kari, L., Feng, L., Carlsson, U. 1998. Resilient mounting of engines. Proceedings of the 16th International Congress on Acoustics, vol. 4: 2373–2374. Seattle, U.S.A. Petersson, B., Plunt, J. 1982. On effective mobilities in the prediction of structure borne sound transmission between a source structure and a receiving structure, Part I: Theoretical background and basic experimental studies, Journal of Sound and Vibration, vol. 82 (no. 4): pp. 517–529. Petersson, B., Plunt, J. 1982. On effective mobilities in the prediction of structure borne sound transmission between a source structure and a receiving structure, Part II: Procedures for the estimation of the mobilities, Journal of Sound and Vibration, vol. 82 (no. 4): pp. 531–540. Verheij, J.W. 1982. Multi-path sound transfer from resiliently mounted shipboard machinery, Delft: Technisch Physische Dienst TNO-TH, PhD Thesis. Vermeulen, R., Lemmen, R., Berkhoff, A., Verheij, J. 2001. Active cancellation of unwanted excitation when measuring dynamic stiffness of resilient elements, Proceedings of the 2001 International Congress and Exhibition on Noise Control Engineering (INTER NOISE 2001), The Hague, The Netherlands. Yeoh, O.H. 1993. Some forms of the strain energy function for rubber. Rubber Chemistry and Technology, vol. 66 (no. 5): pp. 754–771.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Characteristic of low and middle frequency underwater noise of a catamaran F.Z. Pang, Y.Z. Xue, D. Tang & S. Li College of Shipbuilding Engineering, Harbin Engineering University, Harbin, P.R. China

Y.P. Xiong Faculty of Engineering and the Environment, University of Southampton, Southampton, UK

ABSTRACT: This paper examines the characteristic of low and middle frequency underwater noise of a catamaran. Based on acoustic-structure coupling method, the underwater noise radiations excited by both the mono-hull and double hulls are studied, respectively. Calculation method of underwater noise of twin-hull craft is also discussed. Study shows that due to the existence of free surface and twin hulls of the craft, the underwater sound radiation varies especially in the near field. The sound directivity changes severely when standing wave appears between the region of twin hulls if the distance of bilateral hulls and sound wavelength satisfies certain conditions at certain frequency. However, if the standing wave effects do not occur between twin hulls the principle of linear superposition is approximately satisfied. 1

INTRODUCTION

With twin-hull technology matures Catamaran quickly becomes quite popular for its excellent performances of large deck area, low resistance and reliable stability. However, its underwater noise characteristic has not been explored deeply yet. Underwater noise can be extremely harmful, for civilian ships it often adversely affects marine environment, marine fish’s migration, reproduction and other life activities; for military vessels such as destroyers, aircraft carriers and stealth warships it often leads to great risk to their stealth and vitality. Obviously, research on catamaran underwater noise radiation is of great significance. Currently, on the research of underwater noise radiation characteristic, scholars have made considerable achievements with the following methods in the field of mono hull craft, including statistical energy analysis (Yao et al., 2011; Pang, 2012), finite element method (Zou et al., 2003; Zou et al., 2004), boundary element method (Wang et al., 2010; Wang et al., 2012; Zhou, 1996), acousticstructure coupling method (Jin et al., 2011; Miao et al., 2009; Miao et al., 2012; Pang et al., 2012), while in the field of catamaran still rarely reported. Describing fluid as acoustic medium, simulating the propagation of sound wave in fluid field with acoustic equation, then with the help of boundary impedance method or acoustic infinite element method we can better simulate sound wave propagation characteristic in infinite waters. Based on

acoustic-structure coupling method, researching characteristic of low and middle frequency underwater noise radiation of a catamaran, this paper provides a reference for the acoustic design of a catamaran.

2 2.1

THEORETICAL BACKGROUND Acoustic equation

For compressible, adiabatic fluid, considering the loss of flow momentum, the equation of small amplitude motion is shown as below: ∂p + γ ( x θ i )u f + ρ f ( x θ i )u f = 0 ∂x

(1)

where p = fluid overpressure; x = spatial coordinate; u f = fluid particle velocity; u f = is fluid particle acceleration; ρf = fluid density; γ = volume drag force (ratio of force and speed on per volume element); θi = field variable which is independent with fluid particle position but may be associated with ρf and γ. For inviscid, linear and compressible fluid, the dynamic pressure of acoustic medium is closely related to its volume modulus and volume strain: p

K f (x

∂

i

) ∂x ⋅ u f

(2)

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where Kf = volume modulus of fluid: p

K f εV

(3)

where εv = volume strain

εV = ε11 + ε 22 + ε 33 2.2

(4)

Simulation of infinite fluid field

Usually, we simulate infinite fluid field in two ways, namely: boundary impedance method and acoustic infinite element method. The fundamental of the boundary impedance method is a Non-Reflecting Boundary Condition (NRBC), which is realized by preventing acoustic energy reflecting on the interface. However, a large enough fluid field is usually required to ensure calculation accuracy in this method. Acoustic infinite element method covers unlimited units on the boundary to realize the simulation of infinite fluid field. These elements can be directly applied on the boundaries of structural or acoustic finite element fluid field, so that we can reduce the fluid field model according to different requirements, thereby reducing the cost of modeling. Both methods are capable to achieve nice results with little difference (Cipolla, 2002; Grote & Keller, 1995; Lu et al., 2005). Through the two methods above, the fluid field consists of acoustic medium meets the Sommerfield condition on the infinite boundary of fluid field: ⎛ ∂pp ⎞ lim r ⎜ + jkp k ⎟ =0 ⎝ ∂r ⎠

r →∞

(5)

Applying Galerkin method, multiplying the sound pressure variation δp and then integrating (6) in the fluid field volume V, we can be able to achieve the equation below: 1 ∂2 p T δ p ∫∫∫ c2 ∂t 2 dV ∫∫∫ L δ p ( Lp) dV V V ⎛ ∂ 2U ⎞ = ∫∫ ρ f δ ppnT ⎜ 2 ⎟ dS ⎝ ∂t ⎠ S

(

LT = ∇ ⋅ ( L = ∇(

∇2 p =

1 ∂2 p c 2 ∂t 2

(6)

where c = sound velocity in the fluid medium; p = instantaneous sound pressure.

⎡ ∂ ⎣ ∂x

)=⎢

∂ ∂y

∂⎤ ⎥ ∂z ⎦

)

(8) (9)

After discretizing the fluid equation into finite elements and other operations we can fully achieve fluid-structure coupling vibration equation: 0 ⎤ ⎧U ⎫ ⎡CS 0 ⎤ ⎧U ⎫ ⎡ MS ⎢ ρ R M ⎥ ⎨  ⎬ + ⎢ 0 C ⎥ ⎨  ⎬ f ⎦ ⎩P ⎭ ⎣ f ⎦ ⎩P ⎭ ⎣ f ⎡K +⎢ S ⎢⎣ 0

RT ⎤ ⎧U ⎫ ⎧FS ⎫ ⎥⎨ ⎬ = ⎨ ⎬ K f ⎥⎦ ⎩ P ⎭ ⎩ 0 ⎭

(10)

where Ms = structural mass matrix; Cs = structural damping matrix; Ks = structural stiffness matrix; Mf = fluid mass matrix; Cf = fluid damping matrix; Kf = fluid stiffness matrix; R = fluid-structure coupling matrix; U = node displacement vector; P = sound pressure vector; Fs = structural load vector.

Fluid-structure coupling vibration equation

As structural motion on the fluid-structure interface produces fluid load and sound pressure generates an additional force on the structure at the same time, we must calculate structural dynamic equation and wave equation of fluid field simultaneously. With the help of discrete model, both wave equation and motion equation can be solved to obtain displacement and sound pressure value on the fluid-solid interface. Assuming that fluid is an ideal acoustic medium, acoustic wave equation is:

(7)

where u = displacement vector on S surface;

3 2.3

)

3.1

PREDICTION MODEL OF CATAMARAN UNDERWATER NOISE RADIATION Catamaran model

The catamaran hull model used in this paper has geometric dimensions as below: breadth B = 20 m; draught T = 5 m; centerline between left and right bodies D = 15 m; length L = 60 m. Structure material properties: elastic modulus = 2.1*1011 pa, Poisson’s ratio λ = 0.3, density ρ = 7800 kg/m3. 3.2 Prediction model of catamaran underwater noise radiation In order to fully reflect the coupling between the wet surface of the outer hull plates and fluid field, usually we need to determine the minimum radius of the fluid field (Pang, 2012): Rf

max( D / + .2 λ , D )

(11)

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Table 1.

Calculation cases for analysis.

Calculation cases Remarks

Figure 1.

1 2 3 1 2 3

Unilateral excitation of left body Unilateral excitation of right body Bilateral excitations of double bodies Unilateral host engine starts Unilateral host engine starts Bilateral host engines start

Catamaran sketch.

Figure 3. Examination sections and points for low and middle underwater noise prediction.

3.3

Figure 2.

Internal and external fluid fields.

where D = the maximum diameter of the structure; λ = the sound wavelength corresponding to the predicted frequency. Here we make 20 Hz as the lower limit of frequency with a corresponding wavelength of 5.92 m. The width of bodes is about 5 m. Then the minimum radius of the fluid field is calculated as 10 m according to (11), which is applied. The fluid field is divided into internal and external fluid fields as shown in Figure 2. The internal one has a radius of 3 m and length of 66 m, which contains two semi-cylinders and two hemispheres; the external one consists of semi-cylinders and hemispheres with a radius of 10 m which is coupled with the internal one by coupling tie. Fluid material properties: sound velocity c = 1460 m/s, density = 1000 kg/m3.

Excitations, calculation cases and examination points settings

In this paper, two host engines are selected as excitation sources and three working cases are examined. The excitations of host engines depend on the vibration characteristic of the device itself, as well as the efficiency of anti-vibration device. The analysis will not consider the effect of any anti-vibration device, but will apply the vertical vibration acceleration directly to the corresponding positions of the hull structure. Calculation cases as shown in Table 1. Since sound pressure distribution varies more obviously along length than width of the craft, 5 sections along length are selected and 25 points with equidistance of 1.5 m (edge two points except) are arranged on each section. The points are 7.5 m away from the free surface (z = 5.2). Examination points and sections settings are shown in Figure 3. 4

UNDERWATER NOISE RADIATION OF CATAMARAN

4.1 Phenomenon of “passive excitation source” under unilateral excitation 4.1.1 Characteristic of underwater noise radiation of catamaran under unilateral excitation Calculation analysis shows that even under unilateral excitation the characteristic of catamaran

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underwater noise radiation still changes greatly as a result of its boundary conditions, which is obviously different from a mono hull ship. The results show that at some specific frequencies (e.g. 63 Hz), the unilateral excitation will excite a strong sound field in the other body. To make it clear, we make a comparison in the section x = 0 m between 63 Hz (strange phenomenon) and 20 Hz (ordinary phenomenon) as below:

The unilateral excitation produces a strong sound field in another body, which may be the result as below: At some specific frequencies, when the sound wave excited by the unilateral excitation propagates to the surface of the other body, it will excite the body to vibrate with the same frequency. Thus the body becomes a “passive excitation source” and then strong sound field is generated in its near field by itself. Therefore, at some specific frequencies catamaran sound field in near field under unilateral excitation can be seen as a superimposition of two sound fields with the same frequency but different phases and amplitudes. The centerline between the two bodies and sound frequency will determine the appearance as well as the strength of the “passive excitation source”.

and 125 Hz. For example, we make a comparison in the section x = −15 m between 125 Hz (standing wave) and 20 Hz (ordinary) as below:

Multiple sound sources within a limited distance will produce standing wave phenomenon (He & Zhao, 1981). As can be seen from Figure 5 that standing wave appears in near field at 125 Hz while 20 Hz not. The reason may be that when the sound wave excited by the unilateral excitation propagates to the surface of the other body, reflected sound wave forms and interferes with the previous wave when their path difference satisfies certain conditions. Thus standing wave appears. The centerline between the two bodies and sound frequency will determine the number and position of the standing wave. Then we suspect that standing wave will be more significant when the catamaran is under bilateral excitations. Law of unilateral excitation in the opposite body is substantially the same as described above, which is not mentioned here. 4.2

Characteristic of underwater noise radiation of a catamaran under bilateral excitations

4.1.2

Standing wave in near field under unilateral excitation Results show that standing wave will initially appear at some specific frequencies such as 50 Hz

4.2.1 Characteristic of underwater noise radiation in near field under bilateral excitations After calculations and analysis we can find that at some specific frequencies such as 80 Hz and

Figure 4. Phenomenon of "passive incentive source" under unilateral excitation (x = 0 m).

Figure 5. Phenomenon of standing wave under unilateral excitation (x = −15 m).

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125 Hz, horizontal-transverse interference phenomenon appears in near field between left and right bodies, which leads to standing wave and sound pressure strengthened there. To make it clear, we respectively make comparisons between 80 Hz as well as 125 Hz (standing wave) and 20 Hz (ordinary) as below:

in near field of a catamaran which is under bilateral excitations don’t satisfies linear superposition principle, but significantly strengthened in the antinodes and reduced in the knots of the standing wave instead. 4.2.2 The sound directivity in near field under bilateral excitations Further analysis shows that at specific frequencies such as 25 Hz and 125 Hz vertical directivity in near field is quite apparent. For example, we make a comparison in the section x = −15 m between 25 Hz as well as 125 Hz (apparent vertical directivity) and 20 Hz (ordinary) as below:

As can be seen from Figures 6 and 7, horizontaltransverse standing wave appears at 80 Hz and 125 Hz in the fluid field between left and right bodies, due to the special structure and boundary conditions of a catamaran. The number and position of standing wave are determined by sound frequency and the distance of the two excitations. This suggests that the sound pressure distribution

Figure 6. Horizontal-transverse standing wave in near field under bilateral excitations (80 Hz, x = 0 m).

Figure 8. Sound’s vertical directivity in near field under bilateral excitations (25 Hz, x = −15 m).

Figure 7. Horizontal-transverse standing wave in near field under bilateral excitations (125 Hz, x = −15 m).

Figure 9. Sound’s vertical directivity in near field under bilateral excitations (125 Hz, x = −15 m).

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Visibly, sound directivity in near field changes drastically when a catamaran is under bilateral excitations. The reason may be that the near field can be seen as a semi-closed zone with two elastic boundaries, a free surface and infinite waters in the remaining space. Thus the dissipation of energy is greatly reduced due to the isolation effect from the boundaries. Thus the energy of sound wave concentrates below the centerline of the hull and an “energy collection zone” forms. While leaving the semi-closed zone, sound energy dissipates dramatically and energy collection zone quickly disappears. This makes the sound pressure distribution in the near field below the centerline of catamaran hull shows significant vertical directivity and quickly disappears in the far field. 4.2.3

Characteristic in far field of underwater noise radiation of a catamaran In order to investigate the sound pressure distribution in the far field, we superimpose the results of

unilateral excitation on different examination sections in left and right bodies and compare it with the results of bilateral excitations condition. For example, we make a comparison between them in section x = −30 m and section x = 30 m as shown in Figure 10. Although the degree of agreement of results in different examination points varies, most examination points basically satisfy the linear superposition principle. Therefore we can draw the conclusion that in far field the relationship of the sound pressure level of catamaran underwater noise between the overall and each component generally satisfies the following formula: ⎛ n ⎞ Loa = 10 × lg ⎜ ∑10 Li / 10 ⎟ ⎝ i =1 ⎠

5

(12)

CONCLUSIONS

In this paper, based on acoustic-structure coupling method, research on characteristic of low and middle frequency of a catamaran is developed; analysis about low and middle frequency underwater sound field distribution of a catamaran under both unilateral excitation and bilateral excitations is also studied. We can draw the following conclusions from all above:

Figure 10. Typical comparison of results between unilateral excitation superposition and bilateral excitations.

1. Characteristic of underwater noise radiation of a catamaran under unilateral excitation are quite different from that of a mono hull. A “passive excitation source” may forms and standing wave may appears, which are determined by sound frequency and the distance between the centerlines of the two bodies. 2. At specific frequencies obvious horizontaltransverse standing wave will form in the near field of a catamaran when under bilateral excitations, where sound pressure strengthened obviously. The number and position of standing wave are determined by sound frequency and the distance between two excitations. 3. Significant vertical directivity will appear in the near field when a catamaran is under bilateral excitations. “Energy collection zone” will form in the region just below the centerline of the catamaran, the position of which is determined by sound frequency and the distance between two excitations. 4. The sound pressure distribution of a catamaran in the far field substantially satisfies the linear superposition principle, that is, the influence of the twin-hull on the sound pressure distribution can be ignored when sound field is far away from the catamaran.

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ACKNOWLEDGEMENTS The authors gratefully acknowledge the cooperation between University of Southampton and Harbin Engineering University, the financial support from the National Natural Science Foundation China (No. 51209052), Heilongjiang Province Natural Science Foundation (QC2011C013), Harbin Science and Technology Development Innovation Foundation of youth (2011RFQXG021), Fundamental Research Funds for the Central Universities (HEUCF40117), High Technology Ship Funds of Ministry of Industry and Information Technology of P.R. China, Opening Funds of State Key Laboratory of Ocean Engineering of Shanghai Jiaotong University (No. 1307), funded by China Postdoctoral Science Foundation (NO. 2014M552661). REFERENCES Cipolla, J.L., 2002. Acoustic Infinite Elements with Improved Robustness. Proceedings of the ISMA2002. September. 2002:16–18. Leuven: Belgium. Grote, M. & Keller. J., 1995. On Nonreflecting Boundary Conditions. Journal of Computational Physics. vol. 122:231–243. He, Z.Y. & Zhao, Y.F., 1981. Foundation of Acoustics Theory. Press of National Defense Industry. Beijing. Jin, G.W., Zhan, L.K., Miao, X.H., Jia, D. and Wan, X.R., 2011. Vibration Transmissibility of A Submerged Cylindrical Double-Shell Based on Reconstructing Velocity Field. Journal of Vibration and Shock. 05:218–221. Lu, Y.C., D’Souza, K. & Chin, C., 2005. Sound Radiation of Engine Covers with Acoustic Infinite Element Method. SAE Paper. No. 2005- 01-2449.

Miao, X.H., Qian, D.J., Yao, X.L. and Huang, C., 2009. Sound Radiation of Underwater Structure Based on Coupled Acoustic-Structural Analysis With ABAQUS. Journal of Ship Mechanics. 02:319–324. Miao, X.H., Wang, X.R., Jia, D., Jin, G.Y. and Pang, F.Z., 2012. A Numerical Simulation Method For Predicting Sound and Vibration Characteristics of Big and Complex Cylindrical Structures. Chinese Journal of Computational Mechanics. 01:124-128-34. Pang, F.Z., 2012. Numerical Research on Truncated Model Method of Ship Structural Borne Noise Prediction. Harbin Engineering University. Pang, F.Z., Yao, X.L., Mia, X.H. and Jia, D., 2012. Research On The Exciting Force Of Equipment to Ship Structure and Its Application. Engineering Mechanics. 07:283–290. Wang, L.C., Zhou, Q.D., JI, G., Xie, Z.Y. and Mo, D.Y., 2010. Approximate Method for Acoustic Radiated Noise Calculation of Sub Cabin Model in Replacing Full-Scale Model. Chinese Journal of Ship Research. 06:26–32. Wang, L.C., Zhou, Q.D. and Ji, G., 2012. Effect of Longitudinal Beams on Acoustic Radiation of Cylindrical Shell. Journal of Naval University of Engineering. 02:87–92. Yao, X.L., Wang, X.Z., Sun, L.Q. and Pang, F.Z., 2011. The Hybrid Method for Vibro-Acoustic Problem of The Complex Structure. Journal of Vibration Engineering. 04:444–449. Zou, C.P., Chen, D.S. and Hua, H.X., 2003.Study on Structural Vibration Characteristics of Ship. Journal of Ship Mechanics. 02:102–115. Zou, C.P., Chen, D.S., Hua, H.X., 2004. Study on Characteristics of Ship Underwater Radiation Noise. Journal of Ship Mechanic. 01:113–124. Zhou, Q.D., 1996. Coupled Finite Element-Boundary Integral Method for acoustic Radiation from Slender Shell. Journal of Naval Academy of Engineering. 02:35–44.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Simulation of the vibration characteristics of a propulsion system excited by hull deformations Z. Tian School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, China Fluid Structures Interaction Research Group, Faculty of Engineering and the Environment, University of Southampton, Southampton, UK

X.P. Yan & C. Zhang School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, China

Y.P. Xiong Fluid Structures Interaction Research Group, Faculty of Engineering and the Environment, University of Southampton, Southampton, UK

ABSTRACT: This paper aims to investigate the vibration characteristics of the propulsion system subjected to hull deformation. Excited forces caused by severe sea waves have considerable effects on the hull deformation which could have great impact on the shaft propulsion system. On the contrary, the operation quality of ships and the durability of machines are threatened by the malfunctions of shaft propulsion system. This paper establishes a numerical model of the large ship propulsion-hull coupling system to analyze the vibration characteristics of the ship propulsion system subjected to hull deformation. In different sea conditions, the hull deformation was obtained as the exciting forces which are used on the coupling system. As a result, the vibration characteristics of the ship shaft are obtained. Based on the result, suggestions are proposed to ensure the normal operation of the propulsion system in different sea conditions. 1

INTRODUCTION

Nowadays more and more large vessels are built to meet the rapid requirement of the marine transport trades. However, the large vessels always receive many excited forces which are caused by sea waves or other factors during its operations. Ship hull deformation caused by the sea wave will have an effect on the vessel’s propulsion system through the support bearings (Yan & Li, 2013). The propulsion system which is suffered from the ship hull deformation excitations threats the operation quality and the durability of the machines in the vessels. In 2005, the Swedish Club published a 6-year study of the accident statistics report from 1998 to 2004 (The Swedish Club Highlights, 2005). Following study of the accident statistics report spanning 7-year (2005–2011) was published in 2012 (The Swedish Club Highlights, 2012). A comparison between the two statistical results is shown in Figure 1. The cost of the two periods was shown in Table 1 and Table 2 respectively. From the Figure 1, it can be seen that the percentages of machinery malfunctions still occupy most part of all malfunctions and have an obvi-

ous rise in past 15 years. Meanwhile the cost of mending the propulsion system in the machinery claims rises sharply in the past 15 years. Therefore, the investigations on propulsion system to avoid more damages are very necessary. Vibration is one of the damage factors for propulsion system. It

Figure 1. Comparison of H&M claims by number from 1998 to 2004 and from 2005 to 2011.

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Table 1.

Machinery claims (1998–2004).

Claims type

Number

Total cost (USD)

Avg. cost (USD)

Main engine Steering gear Aux, engine Boilers Propulsion Other Total

232 66 120 65 63 12 558

69,744,597 15,636,563 27,257,436 18,138,065 17,798,483 2,559,295 151,134,439

300,623 236,918 227,145 279,047 282,516 213,275 270,850

Table 2.

Machinery claims (2005–2011).

Claims type

Number

Total cost (USD)

Avg, cost (USD)

Main engine Steering gear Aux, engine Boilers Propulsion Other Total

370 55 185 59 174 139 982

201,536,086 36,319,922 72,167,047 21,028,882 132,587,850 45,626,125 509,265,911

544,692 660,362 390,092 356,422 761,999 328,246 518,601

is significance to do some researches on the shaft vibration characteristics caused by the vibration. Nowadays lots of scholars have do many researches on the propulsion system vibrations. Zhang and Zhao (2014) have series researches on the longitudinal vibration characteristics of shafting system excited by propeller. Zhang (2014) forces on the dynamic modeling on the diesel engine propeller shafting system. Murawski (2004) has done lots of works on the longitudinal vibrations of a propulsion system taking the coupling and boundary conditions into considerations. Tang and Brennan (2013) research the torsional vibration of a marine propulsion system. Figureari (2007) studies the dynamic behavior of marine propulsion based on numerical simulation which is mainly used for engine-propulsion matching. Mihajlovic and Wouw (2007) analyze the interaction between torsional and lateral vibrations in flexible rotor system. Even though lots of researches have been done for the vibrations on the propulsion system, taking the hull deformation into account is rare found in those researches. In the paper, vibration characteristics of propulsion system subjected by the hull deformation will be studied. 2

method to analyze some key components on the large vessels. In general, specialized software is based on the finite element method. Taking the 8530TEU (Hudong Co. Ltd., 2008) as an example (shown in Fig. 2), finite element model is established in this paper. Some parameters of the ship are shown in Table 3. In the FEM model, 3-dimension propulsion system is established and mounted on the ship hull through the bearings. The propulsion system is divided into three intermediate shafts, one stern shaft, a propeller, three intermediate bearings, two stern bearings, shown as Figure 3. 2-node combin14 elements are used for modeling the five bearings. Their stiffness, damping values may differ by several orders of magnitude. Owing to the low rotational speed of the shaft (below 104 rpm) and low initial loading, there are no gyroscopic effects (Murawski, 2005). The propulsion system is consisted of solid 185 elements. There are totally 29960 elements

Figure 2.

The real ship of 8530TEU container ship.

Table 3.

The parameters of the 8530TEU.

Parameters

Value

L.OA L.B.P Breadth mld Depth mld Design draught Scantling draught Deadweight Container loading

335.00 m 320.00 m 42.80 m 24.80 m 13.00 m 14.65 m 101,000 t 8530TEU

MODEL ESTABLISHMENT

As experimental test on real ship is usually difficult to realize, numerical analysis becomes an effective

Figure 3.

Propulsion system with bearings.

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in the model. Though a 3-dimension model is established, this paper focuses on the vibration subjected by the hull deformation in transverse direction. Constraint conditions for the shaft is firstly constraint the dof Ux, Uz, Rotx, Roty. As the shaft is connected to the main engine, the shaft is regarded as a cantilever beam with the lumped mass in a 2-dimension domain. Shaft is divided by bearings into five segments with the lengths L1, L2, ..., L6. Tian (2014) gives the method to analyze the segmented beam problems. For the cantilever beam, the free vibration kinematic equation is shown as follow: Ei I i

∂ 4vi ( y,t ) ∂ 2vi ( y,t ) + ρi = 0, i = 1, 2, ..., 5 4 ∂y ∂t 2

(1)

where EiIi denotes the bending stiffness of each shaft part, ρi is density per unit length of each shaft part. Based on the method of variable separation (Clough & Penzien, 1995), the solution of the equation is the mode function. Propeller is regarded as a lumped mass in the propulsion system. The boundary conditions of the propulsion are shown as follows: vi (yy t )

i ( y ) zi (t )

(2)

where zi (t ) is generalized coordinate and φi ( y ) is

φ1(0 ) 0, φ1(0 ) 0

Table 4.

(3)

Parameter of the propulsion system.

Parameter

Unit

Value

Shaft total length L Shaft diameter D Stiffness of 1# stern bearing Stiffness of 2# stern bearing Stiffness of 3# intermediate bearing Stiffness of 4# intermediate bearing Stiffness of 5# intermediate bearing Mass of propeller Rotate inertia

m m N/m N/m N/m N/m N/m kg kg ⋅ m2

49.275 1.4 9.8 × 108 9.8 × 108 9.8 × 108 9.8 × 108 9.8 × 108 92580 290720

Table 5. The frequencies of the propulsion with bearings. Orders

Frequencies (Hz)

1 2 3 4

1.3775 2.5913 3.3449 3.9315

Figure 4.

The modal shapes of the shaft with bearings.

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EI φ6′′′′(′ L6 )

ω 2φ6 ( L6 )m1

EI φ6′′′( L6 )

ω 2φ ( L6 ) j1

(4)

where m1, j1 is the transverse inertia force and rotate inertia force of the propeller respectively. L is the length of the shaft. At the same time, all dof of the bearings are fixed. For the numerical simulation of the shaft model, the parameter details are shown in Table 4. The frequencies of the shaft are shown in Table 5. Meanwhile the modal shapes are shown in Figure 4. As the ship is floating on the water, it can be regarded as free-free boundary conditions (Dong, 1991) (Xiong & Xing, 2005). The boundary is as follows (Xing & Price, 1996):

φ ′′( ) = 0 φ ′′′′( ) = 0

(5)

φ ′′( ′′ ) = φ ′′′′( ) = 0

(6)

where l is the length of the ship hull. The ship FEM model is consisted of 180624 shell elements. The frequencies of the ship are shown in Table 6. The modal shapes of the ship hull are shown in Figure 5. When the propulsion system mounted on the ship hull through the bearings, inherent characteristic of the ship structures could be changed. Based on the same boundary of the free-free ship beam, frequencies of the coupled system are obtained which is shown in Table 7. From Table 7, it can be seen that the frequencies of the coupled ship system is changed compared with that of ship hull without propulsion. When mounting the propulsion on the stern part of the ships, the mass distribution and stiffness of the ships could be changed obviously. Comparing the 4th order of modal shapes in Figure 6 with that in Figure 5, the displacement amplitude of the ship is changed from stern to head. As a result, the propulsion system has effect on the ship system. On the other hand, some changes in ship hull structure would affect the operation of the propulsion system.

Table 6. The frequencies of the ship hull. Orders

Frequencies (Hz)

1 2 3 4

0.45078 × 10−5 2.7158 4.7163 6.7851

Figure 5.

The modal shapes of the ship hull.

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Table 7. The frequencies of the coupled ship system.

3

Orders

Frequencies (Hz)

1 2 3 4

0.36003 × 10 2.6931 4.6953 6.7954

The propulsion could receive excitations from the ship hull. The ship hull is subjected the wave loads from different directions which lead to the deformations. As the research is on the preliminary analytic stage, some assumptions are done. As hull deformations react on the propulsion system through bearings, the interactions along the axial direction are ignored in the system. This paper focuses on transverse vibration on the 2-dimensional domain. The 8530TEU container is situated in the head wave conditions with no navigational speed while the wave is about 10 meters high and the length of the wave is 319.98 meters. The wave period is 14.32 s. The deformations on ship hull are delivered to the five bearings’ locations where the deformations on the bearings are shown in Figure 7. From Figure 7, it can be seen that the ship hull has the maximum deformation on 4# bearing in the wave condition that the deformations on the after-stern bearing are much lower than any other bearing positions. As a result, the forces on each bearing’s positions could be obtained which is shown as Figure 8.

–5

SHIP EXCITATIONS

Figure 7. The hull deformations in the bearings location.

Figure 6. The modal shapes of the coupled ship system.

Figure 8.

Forces on the bearings position.

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Figure 9.

4

the transverse vibration circumstance. Based on the AYSYS, a propulsion system of the 8530TEU is established and mounted on the ship hull to form a coupled system. In this paper, as the main purpose is studying the vibration of propulsion excited by ship hull deformation, the load of the wave is not presented. However the deformations and forces of the ship hull on the bearings are shown. From the results, it can be seen that the suitable methods of controlling the propulsion vibration to avoid the ship hull deformations’ excitations could improve the operation performance of the propulsion and reduce the malfunctions. Though this study uses a numerical approach to analyze the vibration characteristics of the propulsion system excited by ship hull deformations, an abundance of parameters should be taken into account in the future studies. Based on this study, more circumstances should be taken in the future investigations as follow:

Vibration displacement of the propulsion.

VIBRATION CHARACTERISTICS

Based on ANSYS, dynamic behavior of the propulsion is obtained which are shown in Figure 9. From the Figure, it can be seen that when the frequencies of ship hull deformation are around 0.3 Hz and 1.4 Hz, the displacement of the propulsion on the after-stern bearing (1#) position could have an obvious change which means it reaches the resonant frequencies in the wave case mentioned above. Meanwhile when the frequency of the ship hull deformation is around 1.8 Hz, the displacements of the propulsion on the 3# intermediate bearing and 2# stern bearing positions could reach to the peaks where are much larger than that on any other places. For the propulsion on the 4# bearing location, when the frequencies of the ship hull deformation are around 1.8 Hz and 3.7 Hz, it could have a large displacement where it reach more intense around the 3.7 Hz. The displacements of the propulsion on the 5# bearing have an obvious fluctuation around 4.6 Hz. In a word, the low frequencies are threatened the reliability of the long propulsion system. As the large ship hull is a typical hug thin-wall structure, it is very easy deformed by the wave excitations. Low frequencies are a characteristic for the ship hull. When the frequencies of the ship hull deformation in the wave case conditions are around the frequencies of analysis above, monitoring must be enhanced to avoid the harmful vibrations for the propulsion in different positions exceeding the allowable value which could guarantee the safety of the ship propulsion. 5

CONCLUSIONS

This study researches the dynamic behavior of the propulsion system excited by hull deformation in

1. Adding the interactions between the propeller and water to and the interactions between the main engine and the shaft the system; 2. Adding these excitations of main engine and propeller, the system could be extended to 3-dimensional domain. 3. More circumstances could be taken into consideration. This paper provides one wave condition which still more conditions to state the vibration characteristics clearly. 4. More calculations should be done to find the critical wave conditions which threaten the ship’s safety. ACKNOWLEDGEMENTS This project is sponsored by the grants from the State Key Program of National Natural Sciences Foundation of China (NSFC) (No.581139005), the Fundamental Research Funds for the Central Universities (WUT: 2014-JL-006), (WUT: 2014IV-030) and the Program of Introducing Talents of Discipline to Universities (B08031). REFERENCES Clough, R.W. & Penzien, J. 1995. Dynamics of structures. Computers & Structures, Inc, University Ave. Berkeley, USA. Dong, Y.Q. 1991. The ship wave outside water load and elastic. Tianjin UP. Figureari, M. & Altosole, M. 2007. Dynamic behaviour and stability of marine propulsion systems. Proceedings of the Institution of Mechanical Engineers Part M 221: 187–209. Hudong Zhonghua Shipbuilding Co. Ltd. 2008. 8530TEU container ship. Ship Engineering 4: 91.

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Murawski, L. 2004. Axial vibrations of a propulsion system taking into account the couplings and boundary conditions. Journal of Marine Science and Technology 9: 171–181. Mihajlović, N. & Wouw, N. 2007. Interaction between torsional and lateral vibrations in flexible rotor systems with discontinuous friction. Nonlinear Dynamics 50: 679–699. Murawski, L. 2005. Shaft line alignment analysis taking ship construction flexibility and deformations into consideration. Marine Structures 18: 62–84. The Swedish Club Highlights, 2005. The Swedish Club Highlights, 2012. Tang, B. & Brennan, M.J. 2013. On The Influence of the mode-shapes of a marine propulsion shafting system on the prediction of torsional stresses. Journal of Marine Science and Technology 21(2): 209–214. Tian, Z., Yan, X.-P. & Li Z.X. 2014. Dynamic interaction analysis of a 2D propulsion shaft-ship hull system subjected by sea wave. ASME 2014 33rd International Conference on Ocean, Offshore and Arctic Engineering.

Xiong, Y.P., Xing, J.T. & Price, W.G. 2005. Interactive power flow characteristics of an integrated equipment nonlinear isolator travelling flexible ship excited by sea waves, Journal of sound and vibration 287: 245–276. Xing, J.T., Price, W.G. & Du, Q.H. 1996. Mixed finite element substructure-subdomain methods for the dynamical analysis of coupled fluid-solid interaction problems. Phil. Trans. R. S. Lond 354: 259–295. Yan, X.P. & Li, Z.X. 2013. Study on coupling dynamical theory of interaction of propulsion system and hull of large ships: a review. Journal of Ship Mechanics 17: 439–450. Zhang, G.B., Zhao Y., Li, T.Y. & Zhu, X. 2014. Propeller excitation of longitudinal vibration characteristics of marine propulsion shafting system. Shock and Vibration 19: 100–120. Zhang, Q.L & Duan J.G. 2014. Nonlinear Dynamic Modeling for a Diesel Engine Propeller Shafting Used in Large Marines. Chinese Journal of Mechanical Engineering 27: 937–948.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

A new approach to analyze the underwater vibration of double layer ribbed cylinder Fei Wang Key Laboratory of Ship Vibration and Noise, China Ship Scientific Research Center, Wuxi, P.R. China Key Laboratory of Ship Vibration and Noise, Institute of Vibration and Noise, Naval University of Engineering, Wuhan, P.R. China Faculty of Engineering and the Environment, Fluid-Structure Interaction Research Group, University of Southampton, Boldrewood Innovation Centre, Southampton, UK

Yeping Xiong Faculty of Engineering and the Environment, Fluid-Structure Interaction Research Group, University of Southampton, Boldrewood Innovation Centre, Southampton, UK

Zhenping Weng Key Laboratory of Ship Vibration and Noise, China Ship Scientific Research Center, Wuxi, P.R. China

Lin He Key Laboratory of Ship Vibration and Noise, Institute of Vibration and Noise, Naval University of Engineering, Wuhan, P.R. China

ABSTRACT: Analyzing double layer ribbed cylinder’s underwater vibration is of great importance to the design and evaluation of vessels’ dynamic performance of operation and human comfort level. The two main traditional methods that are applied to solve this work are Flügge’s equations of motion and numerical method, such as FEM. Even though both methods have their own special advantages, complicated model developing process and comparative low accuracy (compared with data from real measurement) are their common drawbacks. In this paper, Neural Network is used to analyze the dynamic character of this double layer ribbed cylinder. Real measurement data are used to construct Neural Network, and after this Neural Network model is well built, comparison studies are processed between the vibration data of some positions of interest on the cylinder acquired from Neural Network and real test, respectively. The results show that Neural Network can be used to analyze the dynamic character of double layer ribbed cylinder and is of high accuracy and timesaving. The next step of this research will be concentrated on using more representative vibration data to develop Neural Network model. 1

INTRODUCTION

Analyzing double layer ribbed cylinder’s underwater vibration is of great importance to the design and evaluation of vessels’ dynamic performance of operation and human comfort level. Flügge (1973) presented a three-dimensional equation of motion of the vibrating cylindrical shell in 1973 firstly. Leissa (1973) then acquired similar results through numerical analysis. Also, many researchers studied the dynamic character of various cylindrical shells, such as Garnet et al. (1962), Forsberg (1964, 1966) and Wah et al. (1968). Merz et al. (2009) studied a structure interacts with the fluid by a thin-walled axisymmetric shell of finite elements based on Reissner-Mindlin theory, where transverse shear

deformation is allowed, whereas the stress component normal to the shell is assumed zero throughout the shell thickness. Caresta et al. (2010) used Flügge’s equations of motion to model the fluidloaded cylindrical shell by applying smeared theory in dynamic modeling which includes stiffeners, and an infinite shell model was used to solve fluidstructure interaction problem. Results showed a good approximation for finite shell. Recently, Qu et al. (2013) analyzed the free and forced vibration of ring-stiffened conical-cylindrical shell combinations using a modified vibratory approach. Even though methods presented above can solve problems of cylinders’ vibration, when equipped with devices and working underwater, the cylinders’ dynamic character will not agree with results

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acquired by above methods. Ma et al. (2014) presented a free and forced vibration analysis of coupled conical-cylindrical shells with arbitrary boundary conditions through a modified FourierRitz method. With this method, a unified solution for the coupled conical-cylindrical shells with classical and non-classical boundary conditions can be directly derived instead of changing either the equations of motion or the expressions of the displacements. Also, the reliability and accuracy of the present method are validated by comparison with FEM results and those from the literature. Li et al. (2014) analyzed the free vibration of joined conical-cylindrical shells. Firstly, the governing equations of vibration of a conical shell, including a cylindrical shell as a special case, are written as a coupled set of first order differential equations by using the transfer matrix of the shell. Then the entire structure matrix is obtained by the product of the transfer matrices of the shells and the point matrix at the joint, and the frequency equation is derived with terms of the elements of the structure matrix under the boundary conditions. The method had been applied to a joined truncated conicalcylindrical shell and an annular plate-cylindrical shell system. Wang et al. (2014) used Neural Network to develop a combined cylinder’s model, and results acquired from simulation showed that Neural Network model had a comparatively accuracy. In this paper, Radial basis networks were used to develop the double layer ribbed cylinder’s model, in which spatial coordinates and actuated frequency were used as input vectors while accelerometers’ output were used as targets. After network models were well built, the outputs of some concerned positions from real testing and network models were compared. Results showed that the network model developed in this way could fully applied to predict complicated structures’ vibration under special working condition while models can hardly be built through other methods.

Figure 1.

Table 1. Detail parameters of double layer ribbed cylinder. Parameter

Value

Cylinder inner radius Cylinder outer radius Cylinder length Cylinder inner shell thickness Cylinder outer shell thickness Rib thickness Cap radius Cap thickness Distance between ribs

0.492 m 0.548 m 2.0 m 0.008 m 0.002 m 0.008 m 0.585 m 0.008 m 0.4 m

Figure 2.

2

EXPERIMENT ON DOUBLE LAYER RIBBED CYLINDER

Compared with the work presented in Wang et al. (2014), in which the dynamic information i.e. acceleration data were acquired from FEM model, in this paper these data were measured from a real double layer ribbed cylinder as is shown in Figure 1. The details of this model’s geometrical dimensions are listed in Table 1, whilst it is made of steel whose density is 7800 kg/m3, Poisson ratio 0.3 and Young’s Module 2.1 × 1011 N/mm2, respectively. Globally, 32 accelerometers were mounted on the shell corresponding to ribs’ positions. The concrete positions can refer to Figure 2. Meanwhile,

Double layer ribbed cylinder.

The scheme of accelerometers’ arrangement.

according to the coordinate system of Figure 2, the coordinates of accelerometers’ position are listed in Table 2. Furthermore, in order to simulate different conditions, the model was actuated by two electromagnetic shakers which were mounted in the cylinder, which were also shown in Figure 2. Figure 3 is the experiment scheme which shows how to measure the vibration of this double layer ribbed cylinder’s shell. As can be seen from the chart, it was suspended in an anechoic tank by four meters away from the water surface. The anechoic tank here is 10 meters deep, and covers an area of 200 square meters.

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Table 2. The coordinate of each accelerometer’s position. No.

Coordinate value

No.

Coordinate value

1# 2# 3# 4# 5# 6# 7# 8# 9# 10# 11# 12# 13# 14# 15# 16#

0, 0.5, 0.4 0.35, 0.35, 0.4 0.5, 0, 0.4 0.35, −0.35, 0.4 0, −0.5, 0.4 −0.35, −0.35, 0.4 −0.5, 0, 0.4 −0.35, 0.35, 0.4 0, 0.5, 0.8 0.35, 0.35, 0.8 0.5, 0, 0.8 0.35, −0.35, 0.8 0, −0.5, 0.8 −0.35, −0.35, 0.8 −0.5, 0, 0.8 −0.35, 0.35, 0.8

17# 18# 19# 20# 21# 22# 23# 24# 25# 26# 27# 28# 29# 30# 31# 32#

0, 0.5, 1.2 0.35, 0.35, 1.2 0.5, 0, 1.2 0.35, −0.35, 1.2 0, −0.5, 1.2 −0.35, −0.35, 1.2 −0.5, 0, 1.2 −0.35, 0.35, 1.2 0, 0.5, 1.6 0.35, 0.35, 1.6 0.5, 0, 1.6 0.35, −0.35, 1.6 0, −0.5, 1.6 −0.35, −0.35, 1.6 −0.5, 0, 1.6 −0.35, 0.35, 1.6

Figure 3.

3

3.9446

The experiment scheme.

MODEL DEVELOPMENT USING NEURAL NETWORK

In this paper Neural Network Fitting Tool GUI was used to construct, train and verify the cylinder’s model. In the first place, the spatial coordinate of each accelerometer’s position on the shell as well as actuated frequency was used as input vector. Accordingly, the acceleration magnitude of this point was used as target, i.e. ⎧x ⎫ ⎪y⎪ ⎪ ⎪ ς = net ⎨ ⎬ ⎪z⎪ ⎪⎩ f ⎪⎭

where x, y and z represent the three coordinates of this point, f is the frequency, ς is the acceleration magnitude and net represents the developed Neural Network model. Specifically, take accelerometer-1 for example, also assumed this model was actuated by a point force of 167.5 Hz, then taking FFT to the measured acceleration data of this point. After that extracting frequency 167.5 Hz and relevant magnitude i.e. −3.9446 from these transformed data. Meanwhile, according to Table 2 the coordinate of this point is [0, 0.5, 0.4], therefore

(1)

⎧ 0 ⎫ ⎪ 0.5 ⎪ ⎪ ⎪ net ⎨ ⎬ ⎪ 0.4 ⎪ ⎪⎩167 ⎪⎭

Considering the Radial basis networks which require more neurons than standard feed-forward back propagation network, but they work best when many training vectors are available. Radial basis networks consist of two layers: a hidden radial basis layer of S1 neurons, and an output linear layer of S2 neurons as shown in Figure 4. The dist box in this figure accepts the input vector p and the input weight matrix IW1,1, and produces a vector having S1 elements. The elements are the distances between the input vector and vectorsi IW1,1 formed from the rows of the input weight matrix. In experiment, shaker1 was adjusted to work in 167.5, 341, 549 and 739 Hz, whilst shaker2 was adjusted to work in 167.5, 345 and 442 Hz, respectively. Therefore, with the aim of predicting specified points’ vibration under specified frequencies, two neural Network models were developed which were Net1# and Net2#. According to equation (1), every point has one target on one frequency, e.g. actuated frequency 167.5 Hz. Consequently, in Net1# 32 points have 32 4 (167.5, 341, 549 and 739 Hz) = 128 input vectors, while in Net2# 32 points have 32 3(167.5, 345 and 442 Hz) = 96 input vectors. On the other hand, to test the accuracy of Net1# and Net2#, four input vectors were extracted randomly from the input vectors respectively as shown in Table 3. Figure 5 and Figure 6 show the outputs of Net1# and targets as well as the outputs of Net2# and targets respectively. As can be seen from the lines, the outputs track the targets very well. However, one thing worth noting is that according to Figure 6 the testing results acquired under actuated frequencies 345 and 442 Hz were almost the same, while this phenomenon didn’t appear in Figure 5. The reason for this is that the two

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Figure 4.

Radial basis networks’ architecture.

Table 3.

The vibration of points that need to be predicted. Net1#

X Y Z f

Figure 5.

−0.5 0 0.4 167.5

Net2# −0.35 −0.35 0.8 341

0 0.5 1.2 549

0 0.5 1.6 739

−0.5 0 0.4 167.5

−0.35 −0.35 0.8 345

0 0.5 1.2 345

0 0.5 1.6 442

Final outputs of Net1# in developing and real testing data.

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frequencies were too close, which should have been avoided in experiment to get more useful data. Then apply these two developed models to predict the four points in Table 3, the results are shown in Figure 7 and Figure 8, which show comparatively high accuracy. 4

Figure 6. Final outputs of Net2# in developing and real testing data.

Figure 7. Comparative results acquired from Net1# and testing data.

Figure 8. Comparative results acquired from Net2# and testing data.

DISCUSSION

As aforementioned we can find that: When further observation is taken on Figure 5 and Figure 6, an apparent fact is that the vibratory character of double layer ribbed cylinder is very complicated even though actuated under single frequency, and this demonstrates two main aspects. Firstly, different points have different vibratory character, which reflects in Figure 5 and Figure 6 are they have variant acceleration magnitudes. Secondly, same points’ vibratory character varies dramatically from frequency to frequency. Besides, in developing neural network, one critical thing should be borne in mind is cylinder’s structural character, especially for complicated structures e.g. double layer ribbed cylinder. Furthermore, in this paper, to simulate different conditions two shakers were installed in the cylinder, which caused the structure’s dynamic behavior much more complicated. That is why some points’ results were worse than others when 32 points were predicted respectively (31 points as input vectors while one points as target) as is shown in Figure 9. Take Point-26 and Point-28 for example, their predicting results and testing results have the most significant difference. Nevertheless, if we considering their positions on the shell as well as shaker’s positions in the cylinder, it can be found that they almost located in the same area.

Figure 9.

Predicting results of 32 points.

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Consequently, there are three points worth noting in applying neural network to develop double layer ribbed cylinder’s model: First and foremost, theoretically, more input vectors lead to more precise model in neural network. This reflects more accelerators in experiment. However, from the standpoint of engineering, accelerators in increasing numbers will cause experiment hard to carry out. As a result, there is tradeoff between model precision and experiment’s complexity. Second, as for points such as Point-26 and Point-28, they have unique structural properties that could hardly be represented by other points. Therefore, in developing network model they should be contained in input vectors, which means they should be measured in experiment as well. Finally, it is crucial to choose an appropriate neural network function. As is well known different functions have different applied domains. In this paper, Radial basis network was chosen to develop double layer ribbed cylinder’s model. Besides, the weighted input SPREAD also has significant effects on model. The definition of SPREAD is if a neuron’s weight vector is a distance of spread from the input vector, then its weighted input is SPREAD. And the larger SPREAD is, the smoother the function approximation. Too large a SPREAD means a lot of neurons are required to fit a fast-changing function. Too small a SPREAD means many neurons are required to fit a smooth function, and the network might not generalize well.

5

CONCLUSIONS

Double layer ribbed cylinder is widely used in aerospace, orbital vehicles and naval vessels. It is of great importance to analyze its dynamic character. In this paper, Radial basis network was applied to develop its vibratory model. Dynamic responses at the same points acquired from experiment and network model respectively were compared. Results showed that neural network model allows

obtaining an accurate prediction of double layer ribbed cylinder, although the network models developed are only valid for the conditions that input vectors corresponding for. This would lay a solid foundation for predicting and controlling the vibration of double layer ribbed cylinder. REFERENCES Caresta, M., Kessissoglou, N.J. 2010. Acoustic signature of a submarine hull under harmonic excitation. Applied Acoustics, 71(01): 17–31. Flügge, W., 1973. Stresses in Shells. Berlin: SpringerVerlag, 204–259. Forsberg, K., 1964. Influence of boundary conditions on the modal characteristics of thin cylindrical shells. American Institute of Aeronautics and Astronautics Journal, 2(12): 2150–2157. Forsberg, K., 1966. A review of analytical methods used to determine the modal characteristics of cylindrical shells. NASA CR-613. Garnet, H., Goldberg, M.A., 1962. Free vibration of ring-stiffened shells. Grumann Aircraft Engrg. Cor. Res. Dedt. Mech, 21: 156–106. Leissa, A.W., 1973. Vibration of Shells. NASA SP-288: 1–184. Li, T.Y., Xiong, L., Zhu, X., Xiong, Y-P., Zhang, G.J. 2014. Free vibration of joined conical-cylindrical shells. Journal of Sound and Vibration, 84: 255–262. Ma, X., Jin, G., Xiong, Y., Liu, Z. 2014. Free and forced vibration analysis of coupled conical-cylindrical shells with arbitrary boundary conditions. International Journal of Mechanical Sciences, 88: 122–137. Merz, S., Kinns, R., Kessissoglou, N. 2009. Structural and acoustic responses of a submarine hull due to propeller forces. Journal of Sound and Vibration, 325: 266–286. Qu YG, Chen Y, Long XH, Hua HX, Meng G., 2013. A modified variational approach for vibration analysis of ring-stiffened conical-cylindrical shell combinations. Eur J Mech A Solids, 37: 200–215. Wah, T., Hu, W.C.L.,1968. Vibration analysis of stiffened cylinders including inter-ring motion. JASA, 43(5): 1005–1016. Wang F., Xiong, Y., Weng, Z. 2014, Neural Network Modeling of Submarine Shell. Springer. Vibration Engineering and Technology of Machinery, 23: 1055–1064.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Impact resistance assessment of shipboard equipment considering spectrum dip effect Yongjia Yang, Min Yu & Xiaobin Li School of Transportation, Wuhan University of Technology, China

Linhan Feng Naval Academy of Armament, Beijing, China

ABSTRACT: Due to the interaction between structural inertia force and impact input, spectrum dip effect exists in the impact environment of shipboard equipment. This phenomenon should not be ignored, because it can bring errors to impact resistance assessment results. In this paper, mass-spring system model was adopted to analyze the mechanism and influential factors of spectrum dip effect. Considering the fluid-structure coupling, the improved Taylor plate theory was employed to build an impact environment model of shipboard equipment. The spectrum dip effect occurs at the natural frequency of installed equipment, and spectrum dip amplitude becomes more obvious when equipment mass increasing. Compared to the BV reduction formula, the decline in spectrum value calculated by DDAM (Dynamic Design Analysis Method) is closer to the simulation value when equipment mass changes. The mounting frequency of equipment has little effect on the amplitude of spectrum dip. The model based on improved Taylor plate theory and mass-spring system can be used to assess the impact resistance performance of the shipboard equipment effectively when spectrum dip effect and fluid-structure coupling are both taken into consideration. 1

INTRODUCTION

Shipboard equipment is an essential subsystem of ships. However, naval battles show that shipboard equipment is a weak link to ship vitality because of its poor impact resistance performance. Hence, study on the impact resistance performance c of shipboard equipment undoubtedly has great significance in improving the vitality and fighting capacity of ships. Spectral analysis method becomes a main method of impact environment analysis because of its simplicity and intuition. However, the “spectrum dip” effect resulted from the interaction of shipboard equipment and basic structure changes spectrum value to a certain extent. Concept of “Spectrum dip” was first proposed by the US Navy in DDAM (Dynamic Design Analysis Method) (DeRuntz 1989, Edward 1995). Since the impact assessment determines the Research and Development (R&D) standard of equipment’s impact resistance performance, naval academy has established correction regulations on “spectrum dip” effect. Related to military secrets, public researches made on the “spectrum dip” effect mainly focus on impact theory and response spectrum analysis by adopting Single Degree of Freedom (SDOF) system (Edward 1998). The law

of influential factors of spectrum dip effect has received limited investigation and the simplified model system have not taken the fluid-structure interaction into account as well. The organization of this paper proceeds as follows: in the next section the author explains the mechanism of spectrum dip effect. The effect laws of spectrum dip effect is discussed in section III. Section IV establishes a ship-equipment integrated impact environment predictive model considering the spectrum dip effect and fluid-structure coupling effect based on the optimized Taylor Plate theory. The concluding remarks are presented in Section V. 2

MECHANISM OF SPECTRUM DIP EFFECT

Based on the concept of response spectrum, the peak displacement of construction is calculated by superposition of a series of static peak vibration from the simplification of continuous dynamic mechanics analysis in seismology. But an obvious error was found in calculating shock spectrum of large buildings under free field, the mass of building changes the movement spectrum of ground

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(Xie 2005). Similarly, this phenomenon also exists in shipboard equipment shock environment analysis. Ignoring water environment, the mounting structure of multiple deck ship can be simplified to mass-spring system model, as shown in Figure 1. Taking four mass-spring system for instance to analyse the mechanism of “spectrum dip” effect, M0 = M1 = M2 = 1463 kg, M3 = 723 kg, K0 = K1 = K3 = 3 × 105 N/m, K2 = 5 × 105 N/m, M0 is exerted with speed pulse V = 2 m/s. Figure 2 illustrates the impact response spectrum of M0. When system’s DOF is one, no equipment installed, the model has a single velocity spectrum peak at its natural frequency 2.28 Hz. After equipment installed, the peak number changes to 4 due to the DOF increasing and peaks drift aside. Due to the dynamic interaction between the upper mass and foundation, impact spectrum value drops sharply at the mounting frequency (f1 = 1.27 Hz, f2 = 3.24 Hz, f3 = 4.45 Hz) of every mass piece and

Figure 2. system.

Impact spectrum of M1 in four mass-spring

makes the actual impact environment of equipment damp significantly. This phenomenon is regard as the “spectrum dip” effect. “Excessive assessment” error of shipboard equipment may occur if spectrum dip effect is ignored, and high impact resistance standard leads to the increase of design difficulty and R&D costs. O’Hara (1956, 1962) utilized medium impact tester experiment to investigate the spectrum dip effect and found the measured impact velocity spectrum at mounting frequency of the structure was located at the minimum envelope spectrum by changing the counterweight. The minimum envelope of the spectrum corresponds to the spectrum dip value and is only 1/10 of maximum envelope spectrum value that does not consider the impact dip effect. From the point of current research results, the cause of spectrum dip effect is the interaction between the inertial force of shipboard equipment response to impact and the input excitation on basic structure. Taking the SDOF system for example, its dynamic forced state is shown in Figure 3. The relative displacement is defined as y = x − z, the impact dynamic response equation of SDOF system subjected to stiffness force k ( z x ) and damping force c( z x ) from basis is given by y + 2ξ

 + ω n2 y ny

z(t )

(1)

Considering the non-contact underwater explosion shock is transient and ignoring the damping effects on the peak of the structural response, the solution of equation (1) is given by

Figure 1.

y(t ) = −

Four mass-spring system model.

1 t z( )s ) in (t − τ )dτ ω ∫0

(2)

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Figure 3. model.

Figure 4.

Simplified model of shipboard equipment.

Figure 5. mass.

Impact spectrum of equipment with different

Forced state of single mass-spring system

The inertial force Fg exerted on the SDOF system becomes Fg

t

mω 2 y(t ) = −m mω ∫ z(τ ) in ω (t τ )dτ 0

(3)

From the equation above, we can find that the equipment mass and mounting frequency are two important factors that affect the amplitude of spectrum dip. 3 3.1

INFLUENCE FACTORS OF SPECTRUM DIP EFFECT Mass of equipment

The mounting environment of shipboard equipment can be simplified as Figure 4. According to the study of Cunniff and Collins (1968), 5% of damping to installation basis has no significant influence on spectrum value at the mounting frequency. Thus, the damping coefficient in this model is assumed as zero, M1 = 10 t, K1 = 2.4 N/m. The installation basis is exerted with speed pulse V = 2 m/s. Provided the actual design structure of ship hull is invariant and the equipment mass is less than basis, m1/m2 = λ < 1 is set as the research variable to evaluate the impact response spectrum of various installed equipment. The inherent installation frequency of M2 is set as 2.47 Hz. The calculation result is shown in Figure 5. Obviously, when λ = 0, means no equipment installed, the peak of impact velocity spectrum located at the natural frequency is up to 9.7 m/s.

The number of peaks comes to two after equipment installed. With the increase of λ, velocity spectrum peaks drifts to both sides, first order peak decays and second order rises. The mounting frequency (fm2 = 2.47 Hz) of equipment is located between the two peaks and close to the natural frequency of installation basis. As a result, impact spectrum dips significantly and the velocity spectrum value at the mounting frequency decreases while the mass of M2 increasing. According to equation (3), the inertia force Fg related to the reaction between the equipment and installation basis increases with the mass of equipment. Hence, impact spectrum dip generates with the maximum inertial force and is positively related to the mass of equipment. Since the maximum participation coefficient of dynamic response is corresponding to the natural frequency of equipment, the spectrum value at this

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frequency should be adopted to the design standard of impact resistance assessment.

Aa = 196.2

3.2

Va = 1.52

Mounting frequency

When the mass of equipment is invariant, the mounting frequency of equipment varies with mounting stiffness. Basic parameters are constant, M1 = 10 t, M2 = 5 t, λ = 0.5, K1 = 2.4 × 106 N/m, the mounting stiffness of equipments (K1 − K5) is adjusted to make the corresponding mounting frequency rang between 1–5 Hz. The effect laws of mounting frequency to the impact spectrum is illustrated in Figure 6. As shown in Figure 6, the interval of two peaks becomes greater when the mounting frequency of equipment increasing. Predictably, if the mounting stiffness is infinity or close to zero, the velocity spectrum trend to be same as the non-equipment installed spectral line. Spectrum dip effect occurs at the mounting frequency as well. Within a certain range, the mounting frequency of equipment affect little on the amplitude of spectrum dip, drawing a conclusion consistent with the spectrum design theory of DDAM.

(17.01 + ma )(5.44 + ma ) (2.72 + ma )2

5.44 + ma 2.72 + ma

(4) (5)

where, ma denotes the modal mass (t), means the effective mass participated in the certain order vibration. In BV043/85, the impact response spectrum of equipment is given in the specification of part VS. Especially for equipment mass greater than 5t, the following reduction formula is put forward: Velocity spectrum formula: V ′ ⎛ m′ ⎞ =⎜ ⎟ V ⎝ m⎠

−0.4

(6)

Acceleration spectrum formula: A′ ⎛ m ′ ⎞ =⎜ ⎟ A ⎝ m⎠

−0.537

(7)

With regard to the effect of equipment mass on the impact environment, DDAM and Germany military standard BV043/85 have given reference spectrum design method based on large numbers of experiments. The former norm is on account of modal superposition principle and takes the installation environment of equipment into account. Each order modal mass of the equipment installed in deck installation area contributes to the vertical input acceleration and velocity impact value by

where: m denotes the equipment weight is 5 t, V and A represent the velocity and acceleration spectrum value of 5 t equipment respectively, m′′ , V′′ and A′′ represent the corresponding parameters of testing equipment. Since the impact spectrum value of the equipment is related to the input excitation, numerical value of impact spectrum in Figure 7 and Figure 8 is not for reference. Figure 7 reveals that the reduction tendency of BV formula is greatest, and DDAM is consistent with the numerical simulation. Reduction tendency for acceleration spectrum of simulation results is not so distinct as BV formula and DDAM due to the shipboard equipment is mounted at intermediate frequency.

Figure 6. Impact spectrum of equipment with different mounting frequency.

Figure 7. Velocity spectrum curve of various equipment mass.

3.3

Compared with DDAM and BV formula

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Figure 9. plate.

UNDEX response model of constrained rigid

Figure 8. Acceleration spectrum curve of various equipment mass.

4

4.1

According to Newton’s second law, the movement equilibrium equation of floating mass body is given by

IMPACT ENVIRONMENT PREDICTION MODEL OF SHIPBOARD EQUIPMENT CONSIDERING SPECTRUM DIP EFFECT

mz + ρccz + N

Improved model of Taylor plate theory

The underwater explosion (UNDEX) shock load on ship determines the magnitude of impact spectrum value directly. Taylor utilized the ray theory (1944), Newton’s second law and the wave equation to derive the control equation of floating body subjected to exponential attenuation shock wave in water, based on the step response of ship hull and unconstrained rigid body assumption. Wang Yu (2013) introduced the constrained force including ambient pressure, static water buoyancy and weight of floating body into the basic model and proposed the improved one dimension mass model that floating on water surface, as shown in Figure 9. In Figure 9, Pi, Pr, Ph and Pb represent the incident shock wave pressure, reflected shock wave pressure, hydrostatic pressure and atmospheric pressure respectively. According to the exponential decay expression of UNDEX shock wave, the incident shock wave pressure is given by

Pi ( t, y )

P0e

−

t+

θ

y c

(8)

where, variable y denotes the distance between water points and the floating body. According to the interface continuous condition of floating body and water: z ui − ur and the relation of acoustic fluid medium pressure and velocity: p ρccu, before the local cavitation occurs, the equation above can be written as

ρcz = Pi

Pr

2 P0e

−

t θ

(10)

where, m denotes the mass of floating body; N F + Pb Ph ; y = 0 at the interface of floating body and water. The displacement and velocity of floating body can be obtained from the equation (10) with the initial conditions: z( ) 0 and  ) 0, that read z( z (t ) =

z(t ) =

t − ⎞ p0θ ⎡ 2γ ⎛ 1 ⎛ 2γ N⎞ ⎢ − ⎟ ⎜1 − e θ ⎟ − ⎜ ρ c ⎢γ − 1⎝ γ γ − 1 P ⎝ 0⎠ ⎠ ⎣ t ⎛ −γ ⎞ N t⎤ ⎥( < ) × ⎜1 − e θ ⎟ − ⎝ ⎠ p0 θ ⎥⎦ t 2 P0γ ⎛ −θ ⎜e ρ c( − ) ⎝

e

−γ

t⎞ θ

(11)

t −γ ⎞ N⎛ ⎟ − ⎜1 − e θ ⎟ (t < t ⎠ ρc ⎝ ⎠

)

(12) where, γ ρ cθ /m, t0 is the moment when local cavitation occurs. According to Kennard theory (1947), partial cavitation propagates when the total pressure of water near the floating body is zero. Pi

Pr + Ph = 0

(13)

The reflection of shock wave derived from equation (9) is

(9)

Pr (t 0)

Pi (t, t,0) − ρccz = − P0

t

⎛ γ + 1⎞ − θ e ⎝ γ − 1⎠

t

⎛ 2P γ ⎞ −γ +⎜ 0 −N e θ +N ⎝ γ −1 ⎠

(14)

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Combined with the equation (10), w = (t − y/c)/θ, the equation above can be derived into 2 P0 e 1− γ

⎛ 2P γ ⎞ +⎜ 0 −N e ⎝ γ −1 ⎠

γw

+ F + Pb = 0

(15)

Ignoring N, F, and Pb, namely basic Taylor plate model, when total pressure is 0, the expression of w is given by t−

w=

θ

y c =

1 1 ln 1− γ γ

(16)

The moment and position of location cavitation’s occurrence are t=

θ 1 y ln + 1− γ γ c

(17)

y

tc −

cθ 1 ln 1− γ γ

(18)

Centered this point, cavitation spreads towards and away from the two direction of floating body extension as shown in Figure 10. This law can also be obtained from equation (16). The expansion of the cavitation spreads away from the floating body with the sound velocity c if ignoring the rigid constraints of floating body. Considering the relevant constraints, explicit expression of the w, y, and t can’t be obtained, whereas the relation of cavitation time and position with w are given as t

⎡⎛ γ + 1⎞ −ww ⎛ 2γ θ⎧ N⎞ ⎨ w + ln ⎢⎜ ⎟⎠ e − ⎜⎝ γ − 1 − P ⎟⎠ e 2 ⎩⎪ ⎝ γ − 1 0 ⎣

γw

⎛ F Pa ⎞ ⎤⎪⎫ −⎜ ⎥⎬ ⎝ P0 ⎟⎠ ⎦⎭⎪

cθ ⎧ ⎨w + 2 ⎪⎩

⎡⎛ γ + 1⎞ −ww ⎛ 2γ N⎞ e −⎜ − e ⎢ ⎝ γ − 1 P0 ⎟⎠ ⎣⎝ γ − 1⎠

γw

Spectrum analysis of shipboard equipment subjected to UNDEX loading

Based on the improved Taylor plate theory, numerical simulation of non-contact UNDEX loading can be more accurate by utilizing subsection model displayed in Figure 11. The decks are connected by bulkhead and hull plate. Its stiffness can be approximately calculated by pull rod equation K = EA/L, where A is the available section area, L is the length. The stiffness of Deck 2 and the ship bottom is 1.71 × 1010 N/m, weighting 23.1t. According to Taylor plate theory, the impact environment analysis integration model of ship structure and installed equipment is taken some proper simplifications and assumptions as follows: Only shock wave is load on the ship subsection and internal equipment. The effect of shock wave is regarded as a planar linear wave, and only coupling effect between the vertical plane and the incident shock wave is considered. The research object is ship subsection, coupling effects of shock dynamic response from other sections are ignored temporarily. The structure of hull bottom, each deck and equipment can be equivalent to a series of linear or nonlinear damping spring connected with MultiDegree of Freedom (MDOF) system theoretically, as shown in Figure 12. The stiffness and damping of springs represent the dynamic connection characteristic of each deck, equipment, bulkhead, pedestal and shock absorber. For this system, the inertia force exerted on hull bottom from the movement of the upper deck and equipment is given as Fg

t

∑ mbω b ∫0 z(

) in

b (t

− τ )dτ

(21)

b

(19) y=−

4.2

⎛ F Pa ⎞ ⎤⎫⎪ −⎜ ⎥⎬ ⎝ P0 ⎟⎠ ⎦⎪⎭

According to simulation and experimental results from Rhett Shaw and Mark Gillcrist (2003), spectrum dip effect also exists in the impact environment of the vertical launch system. This conclusion

(20)

Figure 10.

Local cavitation water structure.

Figure 11.

Impact response model of cabin.

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of flow field and the hull structure subjected to UNDEX shock. 5

Figure 12.

CONCLUSIONS

Based on the mass-spring system model, this paper investigated the mechanism of spectrum dip effect and discussed the effect laws of equipment mass and mounting frequency to spectrum dip effect. Impact spectrum value calculated by DDAM method, the formula of BV and the numerical simulation are compared. The prediction model of shipboard equipment’ impact environment subjected to UNDEX is established based on the improved Taylor plate theory. The main conclusions are as follows:

Cabin-equipment integrated model.

Figure 13. Impact environment of equipment in simplified model.

confirms the feasibility of utilizing MDOF system model to investigate the dynamics impact response of large shipboard equipment. Through Fourier integral transform scheme, the acceleration time history response curve of hull bottom is transformed to non-contact UNDEX shock spectrum. Setting this impact spectrum as input load, combined with the simplified model in Figure 12 and parameters of tank structure, the simulation impact environment of equipment installed on Deck 2 is displayed in Figure 13. There is a peak at the natural frequency 18 Hz of the cabin subsection. Velocity spectrum has an obvious spectrum dip phenomenon at 43 Hz contrasted with the design spectrum before equipment’s installation. In the preliminary evaluation stage, subsection model and MDOF system can be combined as a method to forecast the impact environment of shipboard equipment with uncertain structure parameters, considering the coupling interaction

1. Spectrum dip effect of equipment is mainly related to factors including the mass and mounting frequency of the equipment. Spectrum value falls sharply at the mounting frequency of equipment and this value should be adopted to the design standard of the impact resistant assessment. The spectral value dip magnitude increases with the equipment mass, while the mounting frequency of the equipment does little difference to the magnitude of spectrum dip effect. 2. Velocity spectrum calculated by DDAM method is similar to simulation results when equipment mass changes. Since the mounting frequency of shipboard equipment is usually located at intermediate frequency, acceleration spectrum dip is not obvious. 3. Based on the improved Taylor plate theory, the shock environment of shipboard equipment subjected to UNDEX has been obtained by MDOF system model considering the fluid-structure interaction. The spectrum dip laws are consistent with the conclusion of mass-spring system. REFERENCES Cunnniff P.F., Collins R.P. (1968). Structural interaction effects on shock spectra. The Journal of the Acoustical Society of American. 43, 239–244. DeRuntz Jr J.A., Rankin C.C. (1989). The underwater shock analysis code and its applications. SAVIAC Proceeding of the 60th Shock and Vibration Symposium. Virginia, United States. Edward Alexander J. (1995). A nonlinear DDAM procedure. SAVIAC Proceeding of the 66th Shock and Vibration Symposium. Biloxi, Mississippi, United States. Edward Alexander J (1998), Structural Analysis of a Nonlinear System Given a Shock Response Spectrum Imput, SAVIAC Proceeding of the 69th Shock and Vibration Symposium. Biloxi, Mississippi, United States.

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Kennard E.H. (1944). The effect of a pressure wave on a plate or diaphragm. David Taylor Model Basin, March. Kennard E.H. (1947). Explosion load on underwater structures as modified by bulk cavitation. Underwater Explosions Research Field Report 511, Norfolk Naval Shipyard, Portsmouth, VA. O’Hara G.J. (1956). Shock spectra and shock design spectra. Nacal Research Laboratory, NRL report 5368, Nov. O’Hara G.J. (1962). A numerical procedure for shock and Fourier analysis. Nacal Research Laboratory, NRL report 5772, June.

Shaw R., Gillcrist M. (2003). Near-Miss Shock Response Analysis of an Encanistered Vertical Launch Antisubmarine Rocket Carrying a MK 54 Torpedo. Journal of Critical Technologies in Shock and Vibration. Wang Yu, Ji Chen, Du Zhipeng (2003). Integrative dynamic model of the ship hull and equipments subjected to far field underwater explosion. Engineering Mechanics. 30, 390–394. (in Chinese). Xie Xu (2005). Analysis of seismic response and seismic design of bridge structure. China Communications Press. (in Chinese).

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Natural frequencies of eccentric cylindrical shells filled with pressurized fluid G.J. Zhang, T.Y. Li, X. Zhu & L. Xiong School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan, China

Y.P. Xiong Faculty of Engineering and the Environment, University of Southampton, Highfield, Southampton, UK

ABSTRACT: Cylindrical shells are widely used in industry, aerospace, underwater structures and many other fields. While eccentric cylindrical shells are often met in practical applications due to the machining process or some other special requirements. Because of the change of the cross section, the theory and method which were used in the vibration analysis of concentric cylindrical shells cannot be used directly to the eccentric cylindrical shell. Based on the geometry characteristics of the cross section of eccentric cylindrical shell, the eccentric problem is converted into a circumferentially varying thickness problem. Taking hydrostatic pressure of inner fluid into account, the vibration equations are rewritten in the form of a matrix differential equation of the state variables of the shell. By using the index matrix precise integration method of the transfer matrix method, the main parameters which influence the natural frequency characteristics are analyzed, the relationships between these parameters and the natural frequencies are given, and these results are compared to those which do not consider the effect of the hydrostatic pressure. The characteristics between natural frequencies and hydrostatic pressures of the concentric cylindrical shell are also compared with those of eccentric cylindrical shell to observe the differences of the two systems with different hydrostatic pressures. 1

INTRODUCTION

Cylindrical shells are widely used in industry, aerospace, underwater structures and many other fields. Researches on these structures have important significance. Leissa (1973) had made a summary and review of the vibration cases of circular cylindrical shells, including more than 500 publications, analyzing and discussing the linear and nonlinear vibration problems of circular cylindrical shells. For vibro-acoustic coupled systems, Junger (1952) analyzed the structural response of thin shell vibrations in acoustic fluid medium and Lin (1956) had researched the propagation of waves through fluid in a cylindrical shell. Then Fuller & Fahy (1982) investigated the dispersion characteristics of free wave propagation in an infinite fluid-filled cylindrical shell. Scott (1988) made an analysis of the free vibration of an infinite fluid-loaded cylindrical shell. Recently, a wave propagation approach in which a similar beam axial wavenumber used to substituting the axial wavenumber of the shell has been applied to compute the natural frequency of a finite structural-fluid coupled cylindrical shell by Zhang (2002).

The static pre-stress caused by the hydrostatic pressure changes structural stiffness characteristics. The effect of initial hydrostatic pressure fields on the structural and acoustic response of a cylindrical shell which is immersed in water has been analyzed by Keltie (1983). Liu (2010) had studied the effect of hydrostatic pressure on the dispersion characteristic of a cylindrical shell immersed in water and filled with water. These researches mainly focused on concentric cylindrical shells. While eccentric cylindrical shells are often met in practical applications due to the machining process or some other special requirements. Because of the change of the cross section, the theory and method which were used in the vibration analysis of concentric cylindrical shells cannot be used directly to the eccentric cylindrical shell. Because of the special geometry of the cross section, it’s quite difficult to build the vibration equations with analytical method. Therefore, researches on eccentric cylindrical shells are seldom found from published literature. Chang & Chang (1994) had researched the relationships between stress, displacement and the eccentricity of an eccentric cylindrical shell, but the analysis of coupled

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structural-acoustic vibrations of the eccentric cylindrical shell especially with hydrostatic pressure fields is seldom reported. In this paper, based on the geometry characteristic of the cross section of eccentric cylindrical shells, the eccentric problem is converted into a circumferentially varying thickness problem. The vibration equations containing the hydrostatic pressure term are written in a matrix differential equation by using the transfer matrix method, and the fluid-loading term is represented in the form of Bessel functions. The results of the eccentric cylindrical shell obtained by the present method are compared with existing literature, and it is shown that the present approach is sufficient, correct and also reasonably accurate. 2

2

OE =| rw cos θ | 2

(

)

CE + OE − OA

2

= r12

rw2 +

e2 + errw cos θ = r12 4

(2)

Similarly, the equation for inner circle can be derived: rn2 +

e2 − errn cos θ = r22 4

(3)

Equation (2) minus equation (3) yields: n )( w

cos n ) + ecos

(rw + rn ) (r (r (r1 + r2 )(r1 − r2 ) (4)

For convenience, supposing:

The geometry of cross section of the single layer eccentric cylindrical thin shell is shown in Figure 1. Point A is the center of outer circle with radius r1, whereas point B is the center of inner circle with radius r2 . Point O is the center of line AB, AB = e is the eccentricity. C is a point on the outer circle; line OC and inner circle intersect at point D. Defining OC = rw and OD = rn , according to the geometrical relationships, the following equations can be obtained: 2

(1d)

Combining with equations (1a)∼(1d), we can get the equation for outer circle:

(w

GEOMETRY OF THE CROSS SECTION OF THE SINGLE LAYER ECCENTRIC SHELL

CE + OE = rw2

AO = e / 2

h

rw − rn , h0 = r1 r2 ,

(5)

where h represents the thickness of eccentric cylindrical shell, h0 represents the thickness of the concentric cylindrical shell. When eccentricity e ⎜⎢ ⎥⎟ 2.5 ⎝ ⎣ σ y ⎦⎠

2

(1)

3.2 Testing 3 3.1

EXPERIMENTAL METHODS Specimens

The specimens used in order to evaluate the FCG performance of the welded parts, were three point bending specimens, following BS 7448-2, with the weldment in the middle of the specimen (Fig. 12).

As aforementioned the specimens were fatigue precracked up to a0 = 9 mm total initial crack length, thus satisfying the requirement of BS7448-2, which dictates that 0.45 W 2 −1 σ 2 σ π r + cos b ⎪⎩ b e − Considering this phenomenon that the experimental data are closure free at higher load ratio, it is proposed that the term ΔKeff in the constitution relation can be expressed by the above piece-wise function. And Newman’s function (Newman 1984) for fop is modified by introducing a constraint factor α′ as follows,

( )

(

)

⎧max{R, A0 + A1R + A2 R 2 + A3R 3 } 0 ≤ R < 1

fop = ⎨

⎩A0 + A1R

−2 ≤ R < 0

⎧A0 = ( .825 − 0..34α ′ + 0.05α ′ ) [cos((πσ max / ⎪ A1 = ( .415 − 0. ′ ) ⋅ σ max /σ fl ⎪⎪A 1 A A A ; A 2 A + A 1; 2 0 1 3 3 0 1 ⎨σ ⎪ fl (σ Y σ u )/2 ⎪α ′ = 1 / (1 − 2ν ) [1 1 / (1 2ν )] / 2251 0.75952 [1 0.8861 ⋅ ( /( / max /σ Y )2 )3.2251 ] 5 ⎪⎩ 2

fl

)]1/ ′ ;

(2)

⎪⎧K max (ath ) ⋅ ⎡⎣1 − fop ath ) ⎤⎦ R < 0.7 ΔK effth (R ) = ⎨ R ≥ 0.7 ⎩⎪ΔK th (R )

(3)

where,

Figure 1. (a) A schematic representation of fatigue loading history with intermittent dwell time for pressure hull of deep sea manned submersibles; (b) Schematic representations of cyclic loading with dwell time and cyclic loading with overload.

⎧( R )β ⎪ Δ th ( ))/Δ ΔK ΔK th 0 = ⎨( R ) β ⎪( .05 1.4R 4R 0.6R 2 )β ⎩

5≤R 1 mm, where the Mk factors are no more effective; see Hobbacher (1993). Again, the results of the integrations were converted into stress-based values, and the factor FS results from a regression analysis (R2 = 0.99) in which the relative height h/t is the dependent variable; see Eq. (11) being valid to h/t = 10. ⎛ h⎞

− ⎛ h⎞ FS = 0.943 − 7.9 ⋅ 10 −4 ⎜ ⎟ + 0.056e ⎝ t ⎠ ⎝t⎠

4.3

(11)

Factor FW

This factor takes into account the load-carrying grade of the weld. Its effect is included in the investigated variants as the load-carrying grade increases when adding the upper longitudinal attachment. In the structural HSS approach, this effect is considered on the fatigue strength side. The FAT class is decreased from FAT 100 (non-load-carrying fillet welds) to FAT 90 if the weld is full or partial load-carrying; see Hobbacher (2009). The different load-carrying grades of variant 3 and 4 are directly included in the effective notch stress. This means that the structural HSS becomes

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smaller for the same σeff according to Eq. (7). As this slows down the crack propagation resulting in an increased life, a correction factor < 1 is reasonable here. If the same difference between fatigue lives of non-load and load-carrying fillet welds is assumed, FW becomes for the effective notch stress approach: FW = 0.9 4.4

(12)

Factor FB

If the structure is vertically supported so that it is constrained against bending, a constant factor FB is suggested, Eq. (14). It is derived from the comparison of the last two variants in terms of fatigue strength. The factor could be different for other complex structures. FB = 0.96 5

(13)

EXAMPLE OF APPLICATION

The detail shown in Figure 1 is similar to the complex structure. The statistical evaluation of the fatigue tests yields a fatigue strength Δσc = 67 MPa regarding the nominal stress range at 2 ∙ 106 cycles and a survival probability PS = 97.7%; see Kim & Lotsberg (2005) for the regression curve and the standard deviation. The geometry is reported by DNVGL (2014). Storsul et al. (2004) found a Stress Concentration Factor (SCF) of 2.08 when applying the structural HSS approach with a coarse mesh according to the IIW guideline (Hobbacher 2009). Greulich (2011) obtained an SCF of 5.0 with respect to the effective notch stress. The re-analysis of Greulich’s FE model yields a degree of bending δ = 0.411 according to Eq. (4) and a relative height h/t = 7.0, but no additional bending constraint is existent (FB = 1.0). The corresponding correction factors are summarized in Table 3. Additionally, a crack propagation simulation was performed for comparison. An initial crack, ai = ci = 0.15 mm, was placed at the hot-spot, similar as for the complex structure (5). The characteristic fatigue strengths calculated by the stress-based approaches is the FAT class Table 3. Correction factors according to Eq. (10)–(14) for the detail shown in Figure 1. Ft

FS

FW

FB

0.88

0.94

0.90

1.00

Table 4. SCF and assessed nominal characteristic fatigue strength Δσc at 2 ∙ 106 cycles and PS = 97.7%. SCF Experiments Structural HSS approach Corrected acc. to Eq. (8) Eff. notch stress approach Corrected acc. to Eq. (9) Crack propagation simulation

2.08 1.72 5.00 4.50

Δσ50% [MPa] 67 43 52 45 50 55

(90 MPa for σs and 225 MPa for σeff) divided by the SCF. The fatigue assessments are very conservative compared to the experiments; see Table 4. A possible reason could be the relatively high initially applied pre-loading together with the acting mean stress; see Kim & Lotsberg (2005). Generally, the two stress-based approaches achieve similar fatigue strengths, approaching the crack propagation simulation when the correction is applied. 6

CONCLUSIONS AND OUTLOOK

The paper presents crack propagation simulations which were performed for different welded geometries—from a transverse attachment up to a complex structure. The determined fatigue lives were compared assuming equal structural HSS and alternatively effective notch stress in all variants. The complex structures showed 60% to 70% longer fatigue lives. Four effects extending the fatigue life were identified, but they are not fully considered in the fatigue approaches. Hence, correction factors changing the reference stress of the applied approach are introduced, and they are applied to an example. The following conclusion can be drawn: – The longer crack propagation phase observed in experiments of complex structures can be found also in numerical crack propagation simulation; – Four effects slowing down the crack propagation rate in complex structures are identified: the stress gradient over the plate thickness, the apparent plate thickness, the load-carrying grade of the weld and the bending constraint due to the surrounding structure; – The stress gradient along the weld line does not affect the life but the aspect ratio of the semielliptical crack shape; – The effects have been confirmed by crack propagation analyses, but the structural HSS and the effective notch stress approaches do not fully account for all of them;

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– The application to an example closes the gap to the performed crack propagation simulation, whereas differences with respect to the experiments still exist. The correction factors should be further validated by different structural configurations, varying in particular the plate thicknesses and the slope angle of vertical plates. REFERENCES BS 7910:2005. Guide to methods for assessing the acceptability of flaws in metallic structures, British Standards Institution, London. Dijkstra, O., Janssen, G. & Ludolphy, J. 2001. Fatigue tests on large scale knuckle specimens. In W.-C. Cui, Y.-S. Wu and G.-J. Zhou (ed.), Proc. of the 8th Int. Symp. on Practical Design of Ships and Other Floating Structures (PRADS): 1145–1151. Amsterdam: Elsevier. DNVGL. 2014. RP-C203: Fatigue design of offshore steel structures. DNVGL AS, Høvik. Dong, P. 2001. A structural stress definition and numerical implementation for fatigue analysis of welded joints. Int. J. Fatigue 23: 865–876. Fischer, C., Fricke, W., Rizzo, C.M. 2012. N-SIF based fatigue assessment of hopper knuckle details. In E. Rizzuto and C. Guedes Soares (ed.), Substainable Maritime Transportation and Exploitation of Sea Resources: 411–418. London: Taylor & Francis. Fischer, C. & Fricke, W. 2013. Realistic fatigue life prediction of weld toe and weld root failure in loadcarrying cruciform joints by crack propagation analysis. In C. Guedes Soares and J. Romanoff (ed.), Analysis and Design of Marine Structures: 241–248. London: Taylor & Francis. Fricke, W. 2001. Recommended hot-spot analysis procedure for structural details of FPSO’s and ships based on round-robin FE analysis. In J. Chung (ed.), Proc. 11th Int. Offshore and Polar Engineering Conference, 89–96. Cupertino, CA: ISOPE. Fricke, W. 2013. IIW recommendations for the fatigue assessment of welded structures by notch stress analysis. Cambridge: Woodhead Publishing Limited. Gimperlein, D. 1990. Tragverhalten von Rahmenecken mit geknickten Gurten. Schweißen und Schneiden 42 (5): 234–240. Górski, Z. & Kozak, J. 2010. Estimation of development of fatigue damage in a bilge corner of a Ro-Ro ship. In 5th Int. ASRANet Confernce. Self-published. Greulich, M. 2011. Bewertung der Schwingfestigkeit von geschweißten Knicken mittels der effektiven Kerbspannung und der mittleren Formänderungsenergiedichte. Bachelor Thesis, Institut für Konstruktion und Festigkeit von Schiffen, TU Hamburg-Harburg, Hamburg.

Hobbacher, A. 1993. Stress intensity factors of welded joints. Engineering Fracture Mechanics 46 (2): 173–182. Hobbacher, A. 2009. Recommendations for fatigue design of welded joints and components, WRC Bulletin 520. New York: Welding Research Council. Kang, J., Kim, Y. & Heo, J. 2004. Fatigue strength of bent type hopper corner detail in double hull structure. In M.M. Salama, P.K. Gorf, C.F. Mastrangelo and S. Balint, Proc. of OMAE Specialty Symp. on Integrity of Floating Production, Storage & Offloading (FSPO) Systems. Houston: ASME. Kim, W. & Lotsberg, I. 2005. Fatigue Test Data for Welded Connections in Ship-Shaped Structures. Journal of Offshore Mechanics and Arctic Engineering 127: 359–365. Lotsberg, I. & Sigurdsson, G. 2006. Hot spot stress S-N curve for fatigue analysis of plated structures. Journal of Offshore Mechanics and Arctic Engineering 128: 330–336. Mori, M., Matoba, M., Umezaki, K. & Hirose, M. 1969. Application of the program fatigue tests on the members of ship structures (in Japanese). Journal of the Society of Naval Architects of Japan 125: 275–285. Osawa, N. 2005. Fatigue assessment of bilge knuckle joint of VLCC according to JTP/JBP rules. In Y. Sumi(ed.), Comparative studies on the evaluations of buckling/ ultimate strength and fatigue strength based on IACS JTP and JBP rules, 3.1–3.5. The Japan Society of Naval Architects and Ocean Engineers (JASNAOE), self-published. Osawa, N., Yamamoto, N., Fukuoka, T., Sawamura, J., Nagai, H., & Maeda, S. 2010. Development of a Structural Hot Spot Stress Estimation Technique Based on Shell FE Analysis for Ship Structural Details under the Real Load. In Proc. of 11th Int. Symp. on Practical Design of Ships and Other Floating Structures (PRADS). Rio de Janeiro: COPE/UFR. Paris, P. & Erdogan, F. 1963. A critical analysis of crack propagation laws. Journal of Fluid Engineering 85 (4): 528–533. Peterson, R.E. 1974. Stress concentration factors. New York: J. Wiley & Sons. Poutiainen, I. 2006. A modified structural stress method for fatigue assessment of welded structures, Doct Thesis 251, Lappeenranta University of Technology. Radaj, D., Sonsino, C.M. & Fricke, W. 2006. Fatigue Assessment of Welded Joints by Local Approaches. 2nd edition, Cambridge: Woodhead Publishing Limited. Storsul, R., Landet, E. & Lotsberg, I. 2004. Convergence analysis for welded details in ship shaped structures. In M.M. Salama, P.K. Gorf, C.F. Mastrangelo and S. Balint (ed.): Proc. of OMAE Speciality Symposium on Integrity of Floating Production, Storage & Offloading (FPSO) Systems. Houston: ASME. Tada, Hiroshi, Paris, P.C. & Irwin, G.R. (1985): The stress analysis of cracks handbook. 2nd edition, St. Louis, Missouri: Del Research Corporation.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Fatigue assessment of joints at bulb profiles by local approaches C. Fischer & W. Fricke Institute of Ship Structural Design and Analysis, Hamburg University of Technology (TUHH), Hamburg, Germany

C.M. Rizzo Marine Structures Testing Lab, DITEN, Polo Navale, University of Genova, Genova, Italy

ABSTRACT: This paper summarizes the research work carried out in Germany and in Italy about the fatigue strength of joints between bulb plate profiles, typically applied by the shipbuilding industry. Fatigue tests of different joint configurations were carried out and several local approaches were applied. The paper focuses on the fatigue assessment approaches based on the notch stress intensity factor, which assume that fatigue is governed by the singular stress field at notch tips. Two approaches, referring to either the local strain energy density or to the peak stress value, are presented and applied. Additionally, a sensitivity investigation regarding the geometry of the weld seam of butt joints was performed on the basis of strain energy density approach. Comparisons with test data and with assessments of other common approaches offer a critical review about where the pros and the limitations of each approach are. 1

INTRODUCTION

The worldwide shipbuilding industry typically applies bulb profiles—so-called “Holland Profiles” (HP)—to stiffen shell plates. Two aligned HP stiffeners are often connected by a butt joint; see Fig. 1 left. This requires various weld layers for the bulb region. Alternatively, a patch plate welded on the top of the bulb in various configurations avoids the multi-run welding (Fig. 1 right is an example). Such configurations were and are still used in shipbuilding and in ship repair. A local stress concentration at the weld toe appears in both cases and could be critical for crack initiation under cyclic loads. If cracks are not detected during surveys, the strength and the stiffness of the stiffened plate are impaired, and it may collapse in the worst case. Hence, a reliable fatigue assessment is of interest. This can be based on experimental investigations of small—and large scale specimens. Alternatively, there are different approaches proposed in literature in order to assess the fatigue strength, but only a few are currently applied in design practice; see Fricke (2014). On the one hand, there are approaches using either the nominal stress or the structural stress or the elastic notch stress. These approaches are already established in the guideline of classification societies (e.g. IACS, 2012) and of the International Institute of Welding (IIW) (Hobbacher, 2009).

Figure 1. stiffener.

Butt joint (left) and patch plate (right) at HP

On the other hand, the approaches assessing the short and long crack propagation are applied rather seldom due to challenging calculation effort in the Finite-Element (FE) method. The recently introduced notch stress intensity factor (N-SIF) concept is in between these. The elastic stress field at the notch is described by the N-SIF due to a vanishing notch radius. Lazzarin & Zambardi (2001) and Meneghetti (2008) show that the N-SIF can be determined by means of the strain energy density and by the peak stress when using a suitably defined FE mesh. These latter approaches show promise thanks to the limited requirements on the local mesh pattern compared to the effective notch stress approach. Some fatigue assessments have been carried out for the captioned joints at HP stiffener in the past. Rizzo & Fricke (2013) applied different stress based

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fatigue assessment approaches and compared the obtained results. Monti & Rizzo (2013) attempted to simulate the stable crack propagation at a butt joint configuration. Moreover, Fischer et al. (2013) applied the peak stress method to the reinforced specimens. This paper gives a comprehensive overview of the fatigue strength assessment as well as of the comparison with test results when applying different local approaches to the described joint configurations at bulb profiles—the butt joint and the added patch plate. Additional results referring to the strain energy density are presented, completing the survey on different fatigue approaches. Since the local geometry is relatively complex, the definition of the structural stress is rather challenging, and other concepts seem to be more suitable. Finally, all obtained results are compared with the experiments and the applicability of the different approaches is discussed. 2

FATIGUE TESTS

Different joint configurations were tested in the Marine Structures Testing Lab of the University of Genova and by the Institute of Ship Structural Design and Analysis of the Hamburg University of Technology; see (Notaro et al., 2007; Rizzo & Codda, 2009 and Fricke, 1986). The experiments were stopped when a complete failure of the bulb head had been reached (i.e. stiffener web not fractured). Experiments performed in Genova were eight plate panels (2400 × 1600 mm) made of high strength steel with a nominal yield strength σy = 355 MPa. Each panel was stiffened by four HP 120 × 7 profiles having a frame spacing of 400 mm. The tests were carried out under 3-point bending resulting in a stress ratio R = 0 at the bulb head. The bulb profiles were joined either by a butt joint or by a flat patch plate, Fig. 2. In the first case, the bulb head features a V-shaped gap with an angle of 50°. These specimens are named 120BJ50. The patch plate is 240 mm in length, 55 mm in width and 7 mm thick. A fillet weld seam is completely running around the flat patch plate. The specimens are called 120FPP in the following. The specimens tested in Hamburg consist of HP profiles which were welded on a narrow plate strip, Fig. 3. The profiles were sized 160 × 7 mm, and their heads were machined in the region of the welded joint. The resulting gap features an angle of 60°. The patch plate having a thickness t = 12 mm is vertically arranged here. All specimens are made of mild steel and are named 160BJ60 (butt joint) and 160VPP (vertical patch plate). The fatigue tests were performed under tension loading realizing R = −1. The applied stress amplitude was kept constant for all specimens.

Figure 2. Design of specimens tested in Genova (Rizzo & Fricke, 2013).

Figure 3.

Design of specimens tested by Fricke (1986).

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Table 1.

Summary of test results.

Specimen

Characteristic nominal fatigue strength Δσc [MPa]

Stress ratio R

Scatter index TN,10/90

120BJ50 120FPP 160BJ60 160VPP

84 45 70 70

0 0 −1 −1

1:2.39 1:5.88 1:3.12 1:2.37

The test results of each series were statistically evaluated assuming a slope exponent m = 3 for the regression lines. The obtained nominal characteristic fatigue strengths Δσc refer to 2⋅106 cycles to failure and to a survival probability PS = 97.7%, Tab. 1. The corresponding scatter index TN is rather large and describes the ratio between the number of cycles referring to PS = 10% and 90%. Rizzo & Fricke (2013) discussed the possibility to consider a normalized scatter for comparison purposes of the test results. It is worth noting that generally component tests include residual stresses, fabrication defects and/or misalignments; see Fig. 1 showing exemplarily weld seam scattered shape and misalignments. While the larger Italian specimens were built according to usual shipyard practice (but within usual rule fabrication tolerances), the smaller German ones were made on purpose for laboratory tests. The relatively low fatigue strength of the series 120FPP is possibly due to more complex fabrication procedures and geometries involved in this case. 3

ASSESSMENT OF REINFORCED JOINTS WITH STRESS BASED APPROACHES

3.1 Nominal stress approach Different welded details are assigned to different detail classes defining a permissible nominal stress range. The external loads are generally converted into a nominal stress by means of the beam theory. In the case of the patch plates, an appropriate detail class—FAT 56 for 120FPP and FAT 71 for 160VPP)—can be selected in the guideline of the IIW (Hobbacher, 2009), but this is not the case for the butt joint, whose geometry is not included in standard detail categories. In any case, the definition of the reference plate thickness necessary to properly select the detail class is rather difficult as the bulb profile has a rather variable thickness.

ity and by the notch effect; see Hobbacher (2009). The structural discontinuity causes a linear stress increase and is covered by the so-called structural stress, which is estimated by means of a linear extrapolation according to the IIW guideline (Hobbacher, 2009). The location of the corresponding reference points for extrapolations depends on the plate thickness and on the type of weld. When excluding the stress increase due to misalignments, the structural stress is per definition equal to the nominal stress in the case of butt joints. Hence, the structural stress approach is only applied to the specimens 120FPP and 160VPP. However, the application is still ill-posed as a suitable reference plate thickness cannot be identified at the bulb head. Consequently, the reference points cannot be easily defined. Moreover, the use of a coarse mesh or of a refined one could result in different structural stresses and, hence, in deviating assessments. Nevertheless, Notaro et al. (2007) considered different mesh patterns to obtain the extrapolated structural stress at the weld toe and concluded that a weld toe of type “a” is better fitting test data. Fig. 4 shows exemplarily the FE model of specimen 120FPP, having a relatively fine mesh. The stress distribution is not symmetric along the weld toe line, and it is considered by different extrapolation path. The reference points at 0.4⋅t and 1.0⋅t are directly nodes. The target stress concentration factor SCF, assumed as FAT 90 here, is the ratio of the FAT class of the structural stress approach to the experimentally determined characteristic fatigue strength. The FAT class refers to 2⋅106 cycles and PS = 97.7%, too. Table 2 shows the obtained SCF of the tests and of various FE analyses. Depending on the extrapolation technique and on the selected path, the numerical assessment scatters and shows lower SCFs for the 120FPP

3.2 Structural stress approach It is commonly assumed that the stress increases at a weld toe is caused by the structural discontinu-

Figure 4. FE Model of the reinforced joints being used for the structural stress approach.

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Table 2. Target SCF and calculated SCF referring to structural stresses and effective notch stress. Structural stress

Eff. notch stress

Specimen

Target SCF

Calculated SCF

Target SCF

Calculated SCF

120FPP 160VPP

2.0 1.86

1.53–1.86 1.80–1.88

5.0 4.18

4.61 4.50

joint. Notaro et al. (2007) found a deviation up to −23.5%; i.e. the assessment is on the non-conservative side. The SCF would be slightly higher if misalignments or FAT 100, which is used for a non-load-carrying fillet welds, are considered. FAT 100 is assumed for the fatigue strength of the 160VPP joint. The experimental results were corrected by a factor f(R) = 1.3 according to the IIW guideline (Hobbacher, 2009) in order to account for the stress ratio realized in the tests and for possibly existing residual stress of small scale specimens. The FE models show a good agreement. The asymmetry of the bulb profile contributes to increase the stress concentration and, probably, the principal stress should be used instead of the directional stress. If the effects of misalignments, weld defects and residual stresses are considered, the experimental data are in better agreement with the numerical analyses. 3.3

Effective notch stress approach

Recalling Neuber’s paper (Neuber, 1968) about the micro-structural support effect, the elastic notch stress being effective for fatigue can be determined by means of a fictitiously increased notch radius. According to Radaj’s worst case assumption, the weld toe at a welded joint is rounded by a radius rref = 1 mm; see Fricke (2013). The highest principal stress on that radius is considered as the effective notch stress, and it accounts directly for the nonlinear notch effect. The approach is questionable if the height of the weld bead is low, i.e. the resulting notch effect is rather weak. Pedersen et al. (2010) discussed the effect of different geometries of butt joints and the correction factor recommended by the guideline of the IIW (Hobbacher, 2009). The FE mesh used for the reinforced specimens is refined according to Fricke’s (2013) requirements, leading to the distribution of the 1st principal stress as shown in Fig. 5 for the specimen 120FPP. A similar stress field is obtained for the case of the 160VPP joint. The corresponding S-N curve features FAT 225, resulting in a target SCF KS = 5.0 for the 120FPP joint and KS = 3.2 for the 160VPP joint. The SCF achieved by FE analyses are non-

Figure 5. Distribution of the 1st principal stress being used for the effective notch stress approach.

conservative by 8% (120FPP) and conservative by 8% (160VPP), respectively; see Table 2. Rizzo & Fricke (2013) found that the local stress concentrations are dependent on the mesh type and size. According to Fricke (2013), at least four elements should be placed on the fillet radius, i.e. the element size is ≈0.25 mm. Such dimension should be matched along the weld toe contour girth as well as in the surrounding elements. Additionally, higher order elements may provide more accurate results. The proper mesh refinement of this relatively complex geometry is rather time consuming and sharp corners are unavoidable in practice. Hence, the obtained results need to be carefully reviewed. 4

N-SIF BASED APPROACHES

The N-SIF based approaches assume that fatigue is governed by the singular stress field at notch tips, a vanishing notch radius is considered as a worst case assumption; see Lazzarin & Tovo (1998). The resulting stress singularity in the elastic theory can be described by the mode I loading N-SIF K1 using a cylindrical coordinate system (Fig. 6) and extending the fracture mechanics theory. Gross and Mendelson (1972) defined the N-SIF under mode I loading as follows: K1

2π li ⎡⎣σ θθ r1− λ1 ⎤⎦ r→0

(1)

where σθθ is the acting stress along the bi-sector, r the distance from the coordinate origin and λ1 the eigenvalue. The latter represents the strength of the singularity and depends on the opening angle 2α of the notch. For crack-like notches is λ1 = 0.5 and

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Figure 6. Weld like geometry with usual reference system for N-SIF based approaches (Meneghetti 2008)

0.326 for 2α = 135°, being typical for the weld toe of a fillet weld. Livieri & Lazzarin (2005) showed that the N-SIF can describe the fatigue strength of welded joints failing from the weld toe as well as from the weld root. The scale effect is fully covered, and different loading modes can be superposed. However, the N-SIF depends on 2α and requires highly refined FE meshes for a direct determination. Both, the strain energy density approach (SED) and the Peak Stress Method (PSM) overcome these drawbacks. Further information of the N-SIF based approaches are given by Radaj & Vormwald (2013) and by Rizzo (2011). 4.1

Peak Stress Method

The PSM is a simplified, FE-based technique to readily estimate the N-SIFs of any sharply shaped notch. Due to its computational simplicity, the PSM appears rather useful for fatigue assessment in the everyday design practice of the shipbuilding industry. It takes advantage of both the simplicity of a point-wise approach and the robustness of N-SIF approach. Nisitani and Teranishi (2004) discovered that the ratio between the SIF of geometrically similar cracks is constant if the element size and type of the used mesh are similar, too. Meneghetti & Lazzarin (2011) discovered that this is also valid for the elastic peak stress at the notch root considering the same FE mesh. Hence, the peak stress depends on the type and size of the elements adopted in the discretization. They found a relation between the numerically determined N-SIF KFE* and the exact value K1 for mode I loading, Eq. 2. Here, σpeak is the 1st principal stress at the notch and d the element size: * K FE =

K1 σ ppeak ⋅ d (

λ1 )

≅ 1.38

(2)

However, Eq. 2 is only valid for elements with linear shape function (PLANE42) of the software ANSYS®, a size of d = 1 mm and a defined mesh pattern; see Meneghetti & Lazzarin (2011). Later on, Meneghetti et al. (2014) and Fischer et al. (2013) applied successfully the method to 3D models. All analyzed cases are summarized in a uniform S-N curve having a slope exponent m = 3. The characteristic fatigue strength is 156 MPa together with a scatter index Tσ = 1.90 (defined as the ratio of the fatigue strengths referring to Ps = 2.3% and 97.7% at 2⋅106 cycles). The possibility to apply different element types and different software environment is still to be checked. 4.2 Application of PSM to reinforced joints The butt joint configurations are not considered since the nominal geometries of the weld shape were not available. The results of a variation study should be similar to those achieved by means of the SED approach presented in the following. Figure 7 shows the solid mesh adopted at weld toe which was valid for the evaluation of the peak stress at the reinforced joints. The highest value of the 1st principal stress along the weld toe was taken, and a weighting factor fw = 1.064 was applied as suggested in Meneghetti & Lazzarin (2011). The specimens 120FPP feature an SCF 3.08 and the 160VPP an SCF 3.39, being again respectively non-conservative and conservative as assessed by the other approaches; see Table 3. Figure 8 compares the experimental data expressed as peak stress values with the design scatter band proposed by Meneghetti & Lazzarin (2011) for steel. All data refer to R = 0. A good agreement is found.

Figure 7. Detail of the FE model used for the PSM (120FPP joint).

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Table 3. Target SCF and calculated SCF referring to peak stress and SED. Peak stress

SED

Specimen

Target SCF

Calculated SCF

Target SCF

Calculated SCF

120FPP 160VPP

3.47 2.90

3.08 3.39

3.60 3.05

3.15 3.42

The radius R0 = 0.28 mm has been derived for welds at steel using Beltrami’s failure criterion and the high cycle fatigue strength referring to the N-SIF; see Livieri & Lazzarin (2005). Berto & Lazzarin (2009) re-analyzed various fatigue tests to determine a uniform S-N curve including both failures from the weld toe and the weld root. The evaluation is based on a fixed radius R0. The slope exponent (m = 1.5) is half of the stress based approaches and the characteristic fatigue strength (PS = 97.7% and 2⋅106 cycles) is ΔW = 0.058 Nmm/ mm3. It is also possible to convert the SED W into an equivalent stress σeq, Eq. (3), where, E is the Young’s modulus and ν the Poisson ratio. Additionally, the SED can be determined in a larger control radius R* > R0 and converted a posteriori; see Lazzarin et al. (2008). This offers a coarser FE mesh.

σ eq

Figure 8. Comparison of experimental results of reinforced joints with S-N curve proposed for the PSM.

4.3

Strain energy density approach

The use of energy-based approaches with respect to fatigue is relatively old; see Sih (1974) or the literature review of Berto and Lazzarin (2009). The latter reminisces about the work of Beltrami, dated back 1885. Nevertheless, energy parameters are not currently used in common industrial practice even if such methods are believed becoming more and more popular. The SED approach proposed by Lazzarin & Zambardi (2001) is based on the strain energy averaged over a relatively small cylindrical volume of material. It follows in principle Neuber’s idea of micro-structural support. Lazzarin & Zambardi (2001) demonstrated that the SED is directly related to the N-SIF. Therefore, it is a parameter describing the fatigue strength, too. The volume is thought as a material property, and it degenerates into a circular sector in case of two-dimensional problems.

(

2W E 1 ν 2

)

(3)

4.3.1 Results of reinforced joints The SED approach was applied to both the reinforced joints and to the butt joint. The FE models consist of solid elements with quadratic shape function. Five prismatic elements having a depth of 0.28 mm were arranged in the control volume, Fig. 9. The achieved SCF using σeq are 3.15 (120FPP) and 3.42 (160VPP); see Table 3 comparing such values with the corresponding targets. Figure 10 shows the comparison of experimental results for the reinforced joints with the scatter band proposed by Berto & Lazzarin (2009). Again, the experimental data of 160VPP are corrected

Figure 9. Detail of the FE model used for the SED approach.

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Figure 11.

Half FE model of the butt joint.

Figure 10. Comparison of experimental results of reinforced joints with S-N curve proposed for the SED approach.

to account for the deviating stress ratio and are reported for R = 0 here. A rather good agreement is noted, too. It is evident that PSM and SED are based on the same basic concept by comparing Fig. 8 and Fig. 9. 4.3.2 Assessment of butt joint by SED The butt joint is rather interesting as the geometry of the weld seam is varying at the bulb head and causes different notch shapes and orientations of the notch to the loading direction. A parametric FE model was built up for the butt joint at the HP120 × 7 stiffener. Since the analysis focus on a very local geometry, the HP160 × 7 stiffener should behave in a rather similar way. The gap opening angle φ of the butt weld governs the direction of the weld line with respect to the loading, Fig. 11. The weld seam height h governs the opening angle 2α of the weld toe since it increases linearly towards the side of the bulb. A weld reinforcement h = 3 mm and a gap opening angle φ = 50° are assumed as the reference weld geometry according to the original detail drawings. The variations consider four reinforcement heights and four gap opening angles as follows: − h = 1 mm to 4 mm with an increment of 1 mm − φ = 30° to 60° with an increment of 10°. The modeled weld geometry is a nominal one while the actual is typically rather scattered. The control volume at the weld toe has a constant

Figure 12. Distribution of the SED along the weld line for the reference weld geometry.

Figure 13. est SED.

Influence of the weld geometry on the high-

radius of 0.28 mm and consists of five elements, similarly as shown in Fig. 9. The volume, however, varies along the weld line due to the changing opening angle 2α. The smaller 2α is, the higher the stress concentration and the strain energy are.

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Figure 12 shows the distribution of the SED along the weld line for the reference weld geometry. The nominal stress is 100 MPa. The SED ranges from W = 0.032 Nmm/mm3 to 0.053 Nmm/mm3, being a ratio of 1.66. The highest value occurs at bulb rounding where the smallest flank angle occurs. The assessed location of possible failure point is in agreement with the reported experimental evidence. Figure 13 summarizes the results of the variation study, showing the highest SED value. Again, the nominal stress is 100 MPa. The results appear reasonable. When h is small, large opening angles at the weld toe occur. Consequently, the SED level is relatively low. If the weld gap angle φ is small, the increasing weld reinforcement results in sharper opening angles 2α. Hence, the SED rises more for φ = 30° than for φ = 60°. In case of h = 1 mm, the notch effect is weak; therefore the SED is marginally influenced by the gap angle. On the contrary, in case of h = 4 mm, the SED decreases more for increasing gap opening angles φ as the weld profile gets smoother. Finally, it can be noted that the test results, expressed in terms of the SED, would fall within the proposed scatter band. The fatigue strength of butt joints at plates has been extensively measured in the past. Gurney (1979) pointed out that the fatigue strength increased by about 25% if the opening angle is increases by 10°. A similar influence can be found in the variation study. The SED increases in average by about 56%, being 25% if converted to equivalent stress, when reinforcement is h = 4 mm instead of 1 mm. 5

COMPARISON OF THE APPROACHES

The results of the approaches, applied to the reinforced joints, are compared with the experiments on the basis of the nominal characteristic fatigue strength Δσc at R = 0, Table 4. The assessment of the series 120FPP is slightly non-conservative for all approaches because the test results feature a high scatter. For the other configurations, conservative results are achieved. Moreover, the assessment using the structural stress and the effective notch stress approach is nearly similar, whereas the N-SIF based approaches Table 4. Assessed characteristic fatigue strength Δσc [MPa] of the reinforced joints. Detail

Structural Tests stress

Eff. Notch stress PSM

SED

120FPP 160VPP

45 54

49 50

51 47

48–59 53–55

51 46

yield more conservative results for the series 160VPP. 6

CONCLUSION AND OUTLOOK

After a brief description of the specimens and of the relevant test results, the fundamentals and the assumptions of the various applied approaches used for fatigue assessment are summarized in this paper. The numerical models are generally challenging to create due to the rather complex bulb shape. The fatigue strength of the reinforced joints is assessed similarly by all applied approaches. However, it is non-conservative for series 120FPP. Moreover, the SED approach covers the dependency of the fatigue strength on the notch opening angle of butt joints sufficiently. Finally, the selected welded details provide an overview of the application of the approaches to a typical structural detail and offer a critical review about the pros and the limitations of each approach. Following conclusions can be drawn: − The large scatter of the specimens 120FPP yields low fatigue strength considering PS = 97.7%; − A detail class is not defined for butt joints at bulb profiles, and the selection of a reference thickness ratio is undetermined for the patch plates; − There is no structural stress at butt joints existing if misalignments are excluded; − The definition of a proper plate thickness is challenging in the structural stress approach as well, and the fatigue assessment depends on the extrapolation technique; − The effective notch stress approach yields a conservative assessment if recommendations on meshing—especially in the surrounding regions—are followed; − The application of the PSM is smart due to its computational simplicity, but the approach is for the time being limited to the software ANSYS®; − The determination of the SED is theoretically independent of the FE software, and the fatigue assessment reaches similar or even better accuracy; − The modeling effort of these two approaches is not larger than for the common ones. However, the two N-SIF based approaches are still not widely applied, and the corresponding S-N curves have been calibrated by means of data from literature. Possibly existing misalignments have not been identified during the evaluation and, hence, are included in the S-N curves. Moreover, effects of stress ratio and of complex loadings are still to be fully validated by

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performing and analyzing dedicated and already available data. ACKNOWLEDGEMENTS This work has been carried out in the frame of a cooperation exchange project funded by the Alexander von Humboldt Stiftung, Germany. REFERENCES Berto, F. & Lazzarin, P. 2009. A review of the volumebased strain energy density approach applied to V-notches and welded structures. Theor. Applied Fract. Mech. 52, 83–194. Fischer, C. Fricke, W. Meneghetti, G. & Rizzo, C.M. 2013. Fatigue strength assessment of HP stiffener joints with fillet-welded attachments using the peak stress method. In: Comp. Methods in Marine Eng. V, MARINE 2013 (eds.: B. Brinkmann and P. Wriggers), Barcelona: CIMNE, 660–669. Fricke W. 1986. Untersuchungen zur Schwingfestigkeit von Längsspantstößen aus HP-Profilen. Schiff & Hafen. 2/1986, 50–54. Fricke, W. 2013b. IIW recommendations for the fatigue assessment of welded structures by notch stress analysis, Oxford: Woodhead Publishing Limited. Fricke, W. 2014. Recent developments and future challenges in fatigue strength assessment of welded joints. Proc.s of the Institution of Mechanical Engineers, Part C: J. Mech. Eng. Sc. DOI: 10.1177/0954406214550015. Gross, R. & Mendelson, A. 1972. Plane Elastostatic Analysis of V-Notched Plates. Int. J. Fract. Mech. 8, 267–272. Gurney, T.R. 1979. Fatigue of welded joints. 2nd edition. Cambridge (UK): Cambridge University Press. Hobbacher A. 2009. Recommendations for Fatigue Design of Welded Joints and Components. WRC Bulletin 520. New York (USA): Welding Research Council. IACS. 2012. Common Structural Rules for Bulk Carriers. International Association of Classification Societies (IACS). self-published: London. Lazzarin, P. & Tovo, R. 1998. A notch stress intensity factor approach to the stress analysis of welds. Fatigue and Fract. Eng. Mat. and Struct. 21, 1089–1103. Lazzarin, P. & Zambardi, R. 2001. A finite-volumeenergy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int. J. Fract. 112, 275–298. Lazzarin, P., Berto, F., Gomwz, F.J. & Zappalorto, M. 2008. Some advantages derived from the use of the strain energy density over a control volume in fatigue strength assessment of welded joints. Int. J. Fatigue 30, 1345–1357.

Livieri, P. & Lazzarin, P. 2005. Fatigue strength of steel and alminium welded joints based on generalised stress intensity factors and local strain energy values. Int. J. Fract. 133, 247–276. Meneghetti, G. 2008. The peak stress method applied to fatigue assessments of steel and aluminium filletwelded joints subjected to mode I loading. Fatigue and Fract. Eng. Mat. and Struct. 31, 346–369. Meneghetti, G. & Lazzarin, P. 2011. The peak stress method for fatigue strength assessments of welded joints with weld toe or weld root failures. Welding in the World 55(7/8), 22–29. Meneghetti, G., Guzzella, C. & Atzori, B. 2014. The peak stress method combined with 3D finite element models for fatigue assessment of toe and root cracking in steel welded joints subjected to axial or bending loading. Fatigue and Fract. Eng. Mat. and Struct. 37 (7), 722–739. Monti M. & Rizzo C.M. 2013. Studio numerico della propagazione di un difetto in un tipico giunto navale saldato di geometria complessa. GNS7 Giornate Nazionali di Saldatura, Istituto Italiano della Saldatura: Genova, Italy. Neuber, H. 1968. Über die Berücksichtigung der Spannungskonzentration bei Festigkeitsberechnungen. Konstruktion 20 (7), 245–251. Nisitani, H. & Teranishi, T. 2004. KI of a circumferential crack emanating from an ellipsoidal cavity obtained by the crack tip stress method in FEM. Eng. Fract. Mech. 71, 579–585. Notaro, G. Rizzo, C.M. Casuscelli, F. & Codda, M. 2007. An application of the hot spot stress approach to a complex structural detail. In: Maritime Industry, Ocean Engineering and Coastal Resource. Proc.s 12th Int. Congr. Int. Marit. Ass. Mediterranean IMAM 2007 (eds.: C. Guedes Soares and P. Kolev). London: Taylor & Francis. Pedersen, M.M. Mouritsen, O.Ø. Hansen, M.R. Andersen, J.G. & Wenderby, J. 2010. Re-analysis of fatigue data of welded joints using the notch stress approach. Int. J. Fatigue 32 (10), 1620–1626. Radaj, D. & Vormwald, M. 2013. Advanced methods of fatigue assessment. Heidelberg: Springer-Verlag. Rizzo, C.M. & Codda, M. 2009. Application of the ‘structural stress’ approach to a welded joint with complex geometry. Welding Int. 23(12), 904–915. Rizzo, C.M. (2011): Application of advanced notch stress approaches to assess fatigue strength of ship structural detail: literature review. Schriftenreihe Schiffbau 655. Hamburg: Schriftenreihe Schiffbau. Rizzo C.M. & Fricke W. 2013. Fatigue assessment of bulb stiffener joints according to local approaches. Ship and Offshore Structures 8(1), 73–83. Sih, G.C. 1974. Strain-energy-density factor applied to mixed mode crack problems. Int. J. Fracture 10(3), 305–321.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Fatigue life improvement of laser-welded web-core steel sandwich panels using filling materials D. Frank As2con-Alveus Ltd., Rijeka, Croatia School of Engineering, Aalto University, Espoo, Finland

J. Romanoff & H. Remes School of Engineering, Aalto University, Espoo, Finland

ABSTRACT: Steel sandwich panels are being investigated for several decades because of their favourable strength-to-weight and stiffness-to-weight ratios. However, their application in shipbuilding is still limited since their response under fatigue loading is insufficiently explored. Recent literature demonstrated that the J-integral is convenient fatigue strength parameter for assessment of web-core steel sandwich panels. This paper uses the J-integral concept in the assessment of web-core panels filled with polyurethane (PUR) and Divinycell® foams of different densities. It is well known that adding a filling material increases flexural stiffness of the panel. The results obtained here show how low-density foam already significantly increases the fatigue life of a panel subjected to lateral fatigue loading. The paper outlines the possibility of multiplying panel’s fatigue life several times, while negligibly increasing its mass. 1

INTRODUCTION

Civil engineering and shipbuilding industry are seeking lightweight and space-saving structural solutions such as sandwich panels that offer high strength-to-weight ratios (Bright and Smith 2004, Roland and Reinert 2000, Kortenoeven et al. 2008, Romanoff et al. 2011). A core can be homogenous or made of various unidirectional shapes that are periodically distributed across the panel; Figure 1. Such panels are usually made of steel only and are assembled by laser stake-welding. The voids between the core shapes can be filled with lowstrength materials. In this case, bending and especially shear stiffness of the panel are additionally increased (Kolsters and Zenkert 2002, Romanoff et al. 2009). One of the all-steel sandwich panels is the web-core type. It is unique because the assembling utilizes a laser stake-welded T-joint instead of a lap joint as in the case of the panels with other core types shown in Figure 1. The T-joint connects a vertical core plate called the web plate to the horizontal face plates (Roland and Reinert 2000). The breadthwise penetration of the laser beam determines the thickness of the weld, which is less than half of the thickness of the web plate. Consequently, the joint has two crack-like notches on each side of the weld and it is considerably less stiff than an equivalent fillet-welded T-joint (Romanoff et al. 2007a). This stiffness needs to be accurately considered in the panel assessment (Romanoff and

Figure 1. Sandwich panels and the definition of a laser stake-welded T-joint.

Varsta 2007, Romanoff et al. 2007b). However, in design of the panels the fatigue resistance of the joint and the panel itself is insufficiently explored. For that reason, fatigue tests of T-joints and webcore panels were made (Socha et al. 1998, Frank et al. 2013a, Boronski and Szala 2006a, Boronski and Szala 2006b, Sandwich Consortium 2002, Kozak 2007). The common observation in these experiments is that the joints often fail by a fatigue crack propagating through the weld. The failure in the weld of the panel joint is caused by local bending of the plates in the vicinity of the weld, as shown in Figure 2. This local bending occurs due to the fact the core is periodic and the shear force distribution

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causes the warping and the secondary shear-induced bending stresses in the face and web plates (Romanoff and Varsta 2007). As a result, the critical joint in the panel has one notch in a tensile stress state, whereas the other is in a compressive state. Because of that the nominal stress approach yields large scatter of results (Kozak 2007). Therefore, a local approach to the fatigue strength assessment is necessary to account for stress and strains at the tensile notch tip where the fatigue crack initiates. In general, the requirement for sandwich panels is that they withstand more than 100.000 cycles and thus the high-cycle fatigue regime is of primary interest. The investigation by Frank et al. (2011) made a systematic comparison of local stress and energy-based approaches for laser stake-welded T-joints to determine their applicability for this type of joint in the high-cycle fatigue regime. It was determined that fatigue strength can be expressed in a form of the strain energy release rate, i.e. the difference between the work of external forces and the change in the internal energy of the body (Griffith 1921, Irwin 1957). Rice (1968) devised the analytical expression called the J-integral that evaluates the strain energy release rate over a contour path, Γ, surrounding the notch tip; see Figure 3. The J-integral formulation can be applied whether the material model is linear or nonlinear-

elastic. Therefore, the linear-elastic hypothesis was used to evaluate the J-integral-based fatigue strength of T-joints in web-core sandwich panels under lateral loading (Frank et al. 2013b, 2013c). It was observed that the slope value of the fatigue resistance curve, m, largely differs from the usual 3 for welded joints and reaches over 8 in the case of the web-core panel’s joints (Frank et al. 2013b). However, it was also observed that because of the low-density (105 kg/m3) polyurethane filling (PUR) contained in panel voids, the shear stiffness of the panel increases and the resulting bending moment in the joint decreases significantly. As a result, the fatigue life of the filled panel is prolonged by about 50 times in the high cycle regime when the empty and the filled panel are subjected to the same range of the external lateral force (Frank et al. 2013b). This study aims to investigate the impact of different filling materials to the fatigue life of the web-core steel sandwich panels. The investigation considers the loading and boundary conditions imposed to the empty and filled panels in the experimental setups (Frank et al. 2013b). In this paper, five different densities of PUR and also five densities of PVC-based Divinycell® foams are considered. Firstly, the regression of the fatigue lives from the tests and Finite Element (FE) analysis data by Frank et al. (2013b) is given in a form of the J-integral-based fatigue resistance curve. Then, the obtained fatigue resistance curve is used for the assessment of fatigue lives of panels filled with PUR and Divinycell® foams. 2

FATIGUE EXPERIMENTS

This study utilizes the fatigue lives of the experiments performed on two series of panels. The series, denoted “EMPTY” and “PUR”, were bent by a force applied to the middle of the panel spans in both x- and z-directions (Frank et al. 2013b); Figure 4. Figure 2. Weld failure resulting from the bending of the joint. Tension and compression in the plates are distinguished by + and − signs, respectively.

Figure 3. Contour path Γ for evaluation of the J-integral at the tensile notch tip.

Figure 4. Description of fatigue tests. The distance between the face plates is 40 mm, whereas the distance between the web plates is 120 mm. Load ratio R = 0.

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Both series of panels had thicknesses of 2.86 and 3.97 mm for the face and web plates, respectively. The web plates were made of DIN S235 JR structural steel with a yield strength of 235 MPa and an ultimate strength of 355 MPa. The face plates were made of DIN S355 J2G2 steel with the corresponding values being 355 MPa and 400 MPa. The only difference between the series is that the specimens from the PUR series had panel voids filled with PUR foam with the density of 105 kg/m3. The tests were force-controlled. The cylinder-induced force was transmitted to the panel by a disc-shaped contact plate with a diameter of 40 mm; Figure 4. The applied force range and the achieved fatigue lives are given in Table 1. The panel fatigue life, Nf , was taken as the number of load cycles until the final separation of the face and web plates at one of the panel joints. In all the panels from both series, the separation occurred at one of the four joints denoted as “C” in Figure 4, i.e. the web plates that are next to the loaded web plate. The diagram in Figure 5 and Table 1 show the range of the applied force, ΔF, with respect to the panel fatigue life for both series. It can be noticed from the diagram that the fatigue strength of the PUR series is significantly higher than in the case of the empty series although their steel structures are identical. It can be noticed that the fatigue lives of the PUR specimens are 50 times longer than the lives of the EMPTY specimens at the same level of force range; see the mean curves plotted in the Figure 5. Table 1.

Fatigue test results.

Specimen

ΔF [N]

Nf

√ΔJ [MPa0.5 * mm0.5]

Empty 1 Empty 2 Empty 3 Empty 4 Empty 5 Empty 6 Empty 7 Empty 8 Empty 9 Empty 10 Empty 11 Empty 12 Empty 13 Empty 14 PUR 1 PUR 2 PUR 3 PUR 4 PUR 5

2500 3250 3375 3500 3750 4000 4250 4418 4500 4750 4995 5000 5250 5500 5014 6877 7459 7857 8853

10,000,000* 7,000,000* 2,400,000 2,850,000 1,100,000 880,000 508,000 341,384 287,500 409,000 60,296 153,000 143,000 23,000 2,000,000* 567,168 401,448 203,376 32,644

0.268 0.349 0.363 0.377 0.405 0.431 0.459 0.477 0.484 0.513 0.538 0.539 0.567 0.593 0.310 0.426 0.462 0.487 0.549

*Runout test, i.e. no failure occurred.

Figure 5. Fatigue resistance curves for panel series based on the force range as the fatigue strength parameter. Ps stands for the probability of survival.

Obviously, when the fatigue strength is expressed in a form of the force range, a fatigue life can vary based on the mechanical properties of the filling material. This means that if the type or the density of the filling material is changed, none of the fatigue resistance curves from Figure 5 is able to estimate the fatigue life of the panel. Therefore, the local assessment of the critical joints is made to determine the stress levels in the joint and around the notch tips using FE analysis (Frank et al. 2013b). Previous studies showed that the deformations of the critical joint are mainly two-dimensional (2D) in the plane orthogonal to the weld line, i.e. the x-y plane in Figure 4 (Frank et al. 2013b, 2013c). Therefore, 2D stress investigation of the critical joint is possible using the plane strain assumption. However, since the panel response determines the response of the joint, a two-stage FE analysis is necessary. The two-stage analysis means that the local FE assessment requires the displacements from the three-dimensional (3D) analysis of the panel to be used as the boundary conditions in the local 2D model of the joint. The details of the 3D and 2D FE analyses are given in Appendix. The analysis by Frank et al. (2013b) demonstrated that all four critical joints from Figure 4 have the identical J-integral values and thus, only one of these joints needs to be evaluated. 3

FATIGUE LIFE ASSESSMENT OF FILLED PANELS

Figure 6 summarises the range of the J-integral, √ΔJ, for specimens form Empty and PUR series with their corresponding fatigue life from the tests. The diagram shows that the specimens from the both panel series can be fitted into a narrow scatterband if √ΔJ is used as the fatigue strength parameter. The fatigue resistance curve is a power function defined as:

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Nf

(

ΔJ

)

m

=C

(1)

where m is the slope of the fatigue resistance curve and the symbol C represents the material constant. Note that the fatigue curve is a straight line when plotted in double-logarithmic scale and thus m represents the slope value of the line; see Figure 6 for the definition of m. Worth noting is that the square root of the J-Integral range, √ΔJ, is utilized in order to obtain the slope of the fatigue resistance curve, such that it is directly comparable with the stress-based local approaches instead of the local energy as in the case of the J-Integral (Radaj et al. 2006). The usual slope for the welded steel joints is m = 3, whereas the critical joints in web-core panels subjected to the lateral loading obtained about 9; Figure 6. This means that at the higher load levels laser stake-welded T-joints

Figure 6. Fatigue resistance curves for panel series based on the √ΔJ as the fatigue strength parameter. Table 2.

have shorter fatigue life than other steel joints. The mean curve from Figure 6 is the one that has the probability of survival, Ps, equal to 50%. This curve has the following parameters: m = 9.1 and C = 294 MPa4.55 mm4.55. The mean curve is further used in the study to estimate the fatigue life of sandwich panels filled with the materials listed in Table 2. As stated, five different densities of in-situ PUR and also five densities of Divinycell® foams are considered. The loading and boundary conditions are assumed to be identical as in the experiments (Frank et al. 2013b) as described in the previous section. The simulations used the force range 6 kN for all panels. The FE assessment is also identical to the one from Frank et al. (2013b) and it is described in Appendix. Young’s modulus for PUR is taken from Goods et al. (1997), whereas Poisson’s ratio from Gibson and Ashby (1997). The data for Divinycell® foams is taken from The Diab Group (2012). The fatigue lives are assessed using the J-integral range and Equation 1. The diagram in Figure 7 shows the estimated fatigue life as a function of the mass of the filling material. The horizontal axis additionally shows the percentage of the increase in panel’s total mass with respect to the empty panel, which has 33 kg. Obviously, Divinycell® foam is more efficient in extending the fatigue life, i.e. with the same added mass, the Divinycell® foam provides a longer fatigue life. The reason why the Divinycell® foam is more efficient is because it has higher Young’s modulus than PUR foam when the same density is considered; Table 2. Consequently, the shear stiffness of the

Computed fatigue life for ΔF = 6 kN. Filling

Critical joint

Case

Density [kg/m3]

Young’s modulus [MPa]

Poisson’s ratio

Mass [kg]*

√ΔJ [MPa0.5 mm0.5]

Fatigue life, Nf [cycles]

EMPTY PUR15 PUR60 PUR100 PUR120 PUR140 H35 H45 H60 H80 H100

0 15 60 100 120 140 38 48 60 80 100

0 1 19 25 37 55 44.5 52.5 72.5 92.5 132.5

0 0.33 0.33 0.33 0.33 0.33 0.4 0.4 0.4 0.4 0.4

0.0 0.3 1.2 1.9 2.3 2.7 0.7 0.9 1,2 1.5 1.9

0.636 0.626 0.399 0.359 0.303 0.249 0.283 0.261 0.221 0.194 0.159

18,026 20,913 1,268,334 3,274,644 15,475,030 90,490,102 28,775,058 59,730,156 270,390,869 888,523,982 5,434,562,360

*All panels have the area of 0.48 m2.

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4

Figure 7. Fatigue life improvement with respect to the additional mass of filled panels.

CONCLUSION

This paper aimed to investigate the impact of polyurethane and Divinycell® fillings to the fatigue life of the web-core steel sandwich panels. The analysis employed two-stage finite element analysis in which the displacements from the global panel assessment were used as the boundary conditions in the local analysis of the fatigue-critical joint. Then, the regression formula based on the J-integral as the fatigue strength parameter is used to evaluate the fatigue life of the critical joint. The results obtained here show how low-density Divinycell® filling of about 38 kg/m3 can significantly increase the fatigue life of a panel by more than 500 times, while increasing its mass only about 2%. The future work could explore the fatigue life improvement when other filling materials are used, such as balsa or polystyrene. Then, the cost assessment should be made in order to determine which of the filling materials is the most favourable regarding its cost and benefit. ACKNOWLEDGEMENTS

Figure 8. Fatigue life improvement with respect to the Young’s modulus of filling material.

panel increases more in the Divinycell® case and the local bending of the critical joint’s plates decreases. The increased stiffness of the panel due to the filling material is a known fact confirmed by Kolsters and Zenkert (2002) and also Romanoff et al. (2009). When the estimated fatigue life is plotted as a function of the Young’s modulus in Figure 8, it becomes obvious that the extension of the fatigue life is related to the modulus of the filling material, i.e. the shear stiffness of the sandwich panel. It is worth noting that all considered Divinycell® foams achieved fatigue lives over 10 million cycles for the chosen load level, what is beyond the scope of the experimental results from Frank et al. (2013b). Therefore it is unknown whether fatigue failure of the panel would occur at the critical joint or in the face plate as reported by Kozak (2007) for filled panels. However, it can be expected that the Divinycell® H35 foam would increase the panel’s fatigue life to at least 10 million cycles for the considered load level of 6 kN, while increasing the panel’s mass only about 2%. Considering the fact that the empty panel’s fatigue life is about 18.000 cycles, Divinycell® H35 filling would increase the panel’s fatigue life more than 500 times. In such case, the unfavourable slope of the fatigue resistance curve would be completely suppressed.

This study was funded by the Finnish Academy of Science project: “Fatigue of Steel Sandwich Panels”. The financial support is received from “Merenkulunsäätiö”—the Finnish Maritime Foundation. CSC—IT Centre for Science provided the ABAQUS and ANSYS licences necessary for the calculations. All the support is gratefully appreciated. APPENDIX The material models were assumed to be linearelastic. The steel was modelled using the Young’s modulus E = 206,000 MPa and the Poisson’s ratio ν = 0.3. The geometrical nonlinearity was employed in all FE models. Figure 9 presents a detailed view of the ANSYS FE model that utilises the shell elements. The web and face plates were modelled using 4-node shell elements (SHELL 181). The top and bottom face plate elements had the thickness offset in the y-direction by tf /2 and −tf /2 respectively, while the web plate elements had no offset; see Figure 9 (Detail A). The same detail shows the nodes connecting the face and the web plate elements. Instead of one node shared among all the adjacent elements, there are two coincident nodes. One node belongs to face plate elements, whereas the other to the web plate elements. Two coincident nodes are connected to each other using one rigid element for each degree of freedom, except the rotation about the z-axis. Therefore, the coincident nodes are

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Figure 9.

Definition of 3D FE models.

additionally connected by one rotational spring element (COMBIN14) with a single degree of freedom about the z-axis and a unit rotational stiffness 53.5 kN. The rotational stiffness was determined experimentally by Romanoff et al. (2007a) for this type of joint with the identical web and face plate thicknesses to the panels from this study. The filling material was modelled using 8-node solid elements (SOLID185). Poisson’s ratio and Young’s modulus for filling materials are given in Table 2. In all panel models, the default element edge size was 10 mm. The loading and boundary conditions are shown in Figure 9. The rolls that support the panel are modelled as absolutely rigid and fixed in all the degrees of freedom. The contact elements (CONTA178) were placed between the roll and the panel nodes, thus allowing all displacements in the panel nodes except translation in the negative y-direction. Previous studies (Frank et al. 2013b, 2013c) showed that the deformations of the critical joint are mainly two-dimensional (2D) in the plane orthogonal to the weld line. Therefore, 2D stress investigation of the critical joint is possible using the plane strain assumption. However, since the panel response determines the response of the joint, the local FE assessment requires the displacements from the panel analysis to be used as the boundary conditions in the local 2D model of the joint. The 2D FE model of the joint is shown in Figure 10. The modelling was done using the 8-node 2D plane strain elements (CPE8). The joint model is a representation of a panel cross-section between the face plate and the web plate nodes that are closest to the critical weld; see Figure 9, Detail A. The weld is considered to have the statistical mean values of notch depths equal to 1.28 mm as determined by Frank et al. (2013a). The displacement controlled 2D analysis is made in ABAQUS. The nodal translations from the panel nodes are applied to the joint model nodes denoted by the same letters M, N and O in the panel as well as in the joint model; see Figure 10. The rotations

Figure 10.

Definition of 2D FE model.

from the panel nodes are applied by rotation of the boundary cross-sections in the 2D model where the joint nodes M, N and O were used as centres of rotation and the rotating cross-section was assumed to remain plane during the rotation. The contact option is introduced between the free surfaces of the notches; Figure 10, Detail A. The contact definition is necessary in order to capture the accurate response of the joint that largely depends on the contact effect on the compressive side of the weld (Frank et al. 2013b). The J-integral at the notch tips is obtained using the “Contour Integral” option available in ABAQUS. The J-integral is evaluated at the 6th contour; Figure 10, Detail A. REFERENCES Boronski, D.; Szala, J. 2006a. Fatigue Life Tests of Steel Laser-Welded Sandwich Structures, Polish Maritime Research, Special Issue 2006/S1, pp. 27–30. Boronski, D; Szala, J. 2006b. Tests of Local Strains in Steel Laser-Welded Sandwich Structure, Polish Maritime Research, Special Issue 2006/S1, pp. 31–36. Bright, S.R.; Smith, J.W. 2004. Fatigue Performance of Laser-Welded Steel Bridge Decks, The Structural Engineer, 82, 21, pp. 31–39. Diab Group, 2012. Divinycell® H—Technical Data, http://www.atlcomposites.com.au. Frank, D.; Remes, H.; Romanoff, J. 2011. Fatigue assessment of laser stake-welded T-joints, International Journal of Fatigue, 33, pp. 102–114. Frank, D.; Remes, H.; Romanoff, J. 2013a. “J-integralbased approach to fatigue assessment of laser stakewelded T-joints, International Journal of Fatigue, 47, pp. 340–350. Frank, D.; Romanoff, J.; Remes, H. 2013b. Fatigue strength assessment of laser stake-welded web-core steel sandwich panels, Fatigue & Fracture of Engineering Materials & Structures, 36, 8, pp. 724–737.

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Frank, D.; Remes, H.; Romanoff, J. 2013c. On the slope of the fatigue resistance curve for laser stakewelded T-joints, Fatigue & Fracture of Engineering Materials & Structures, 36, 12, pp. 1336–1351. Gibson, L.J.; Ashby, M.F. 1997. Cellular Solids— Structure and properties, Second ed., Cambridge University Press, Cambridge, UK. Goods, S.H.; Neuschwanger, C.L.; Henderson, C; Skala, D.M. 1997. Mechanical properties and energy absorption characteristics of a polyurethane foam. Sandia National Laboratories, Albuquerque, NM, USA. Griffith, A.A. 1921. The Phenomena of Rupture and Flow in Solids, Philosophical Transactions of the Royal Society of London, A 221, pp. 163–198. Irwin, G., 1957. Analysis of Stresses and Strains near the End of a Crack Traversing a Plate, Journal of Applied Mechanics, 24, pp. 361–364. Kolsters, H.; Zenkert, D. 2002. Numerical and Experimental Validation of a Stiffness Model for LaserWelded Sandwich Panels With Vertical Webs and a Low-Density Core, In: Hans Kolsters, Licentiate Thesis—Paper B, Kunliga Tekniska Högskolan, Stockholm, Sweden. Kortenoeven, J.; Boon, B.; de Bruijn, A. 2008. Application of sandwich panels in design and building of dredging ships, Journal of Ship Production, 24, pp. 125–134. Kozak, J. 2007. Forecasting of Fatigue Life of Laser Welded Joints, Zagadnienia Eksploatacji Maszyn, 149, 1, pp. 85–94. Radaj, D.; Sonsino, C.M; Fricke, W. 2006. Fatigue Assessment of Welded Joints by Local Approaches, 2nd ed., Woodhead Publishing, Cambridge, UK. Rice, J.R. 1968. A Path Independent Integral and the Approximate Analysis of Strain Concentration by

Notches and Cracks, Journal of Applied Mechanics, 35, pp. 379–386. Roland, F.; Reinert, T. 2000. Laser Welded Sandwich Panels for the Shipbuilding Industry, Lightweight Construction—Latest Developments, February 24–25, London, pp. 1–12. Romanoff, J.; Varsta, P. 2007. Bending Response of WebCore Sandwich Plates, Composite Structures, 81, 2, pp. 292–302. Romanoff, J.; Remes, H.; Socha, G.; Jutila, M.; Varsta, P. 2007a. The Stiffness of Laser Stake Welded T-joints in Web-Core Sandwich Structures, Thin-Walled Structures, 45, 4, pp. 453–462. Romanoff, J.; Varsta, P.; Remes, H. 2007b. Laser-Welded Web-Core Sandwich Plates under Patch-Loading, Marine Structures, 20, 1, pp. 25–48. Romanoff, J.; Laakso, A.; Varsta, P. 2009. Improving the Shear Properties of Web-Core Sandwich Structures Using Filling Material, Analysis and Design of Marine Structures, Ch. 14, pp. 133–138, Taylor & Francis Group, London, UK. Romanoff, J.; Naar, H.; Varsta, P. 2011. Interaction Between Web-Core Sandwich Deck and Hull Girder of Passenger Ship, Marine Systems & Ocean Technology, 6, 1, pp. 39–45. Sandwich Consortium, 2002. Balance User Group Report No. 3—Version 3, Helsinki University of Technology, Espoo, Finland. Socha, G.; Koli, K.; Kujala, P. 1998. Mechanical Tests and Metallurgical Investigation on Weld Samples, Report AWCS—Project No. BE 96-3932—Task A4, Helsinki University of Technology, Espoo, Finland.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

A full-scale fatigue test on longitudinal-through-bulkhead detail by using equivalent angle bar Q. Yi, J. Yue & Y. Liu Departments of Naval Architecture, Ocean and Structural Engineering, School of Transportation, Wuhan University of Technology, Wuhan, China

W. Tang Laboratory Center, School of Transportation, Wuhan University of Technology, Wuhan, China

Z. He NERCMS, Wuhan University, Wuhan, China

ABSTRACT: Fatigue experiment on full-scale details is significant for ship structural fatigue life evaluation. However, the flat bulb for full-scale test model is usually not available at laboratory, especially at the ship’s design stage. This paper proposes to use an equivalent angle bar with uniform thickness which is cold-formed from flat plates to take place the flat bulb in the full-scale fatigue test model. Firstly, Finite Element (FE) analysis is performed on the model with such equivalent angle bar to verify that the Hotspot Stress (HSS) distribution around its weld joint is similar as the real ship detail. For comparison, a non-uniform angle bar suggested by some classification rules is also investigated for its HSS distribution. Then, a bilge Longitudinal-through-bulkhead at a river-to-sea ship structure detail is chosen as the test model. The fatigue test on such full-scale model under biaxial load is carried out. And the HSS distribution and fatigue crack propagation are demonstrated. At last, the fatigue life of this ship structure detail with flat bulbs is calculated based on classification rules. By comparison of the rule-based result and the fatigue test result, the proposed test model fabricated from the equivalent angle bar can give good prediction of the fatigue life of structure details. 1

INTRODUCTION

The connection of Longitudinal-through-bulkhead at mid-ship, one of the most easily fatigued ship details, draws much attention in ship structure strength assessment. And a full-scale fatigue test on such structural detail is common method to estimate the ship’s fatigue life. (Fricke et al. 2012; Erny et al. 2012; Kim et al. 2010; Lotsberg & Landet 2005) In general, longitudinals in ship structure are usually flat bulb section. However, the flat bulb in ship is hardly available in laboratory due to its mass production, especially at the design stage. This paper would propose an equivalent angle bar instead of flat bulb in the full-scale test model for fatigue test. Based on Common Structural Rules for Bulk Carriers and Oil Tankers-Harmonized (CSR-H) (IACS 2014), an equivalent angle is recommended to idealize the common d bulb in structure strength assessment, on the premise that their section area and inertial moment keep accordance and their web height and thickness remain same. However,

the above rule-based angle bar has different thickness in web and face, which is also difficult to process in laboratory. Therefore, an equivalent angle bar with uniform thickness, which could be coldformed from flat plates, is designed to substitute for the flat bulb in the full-scale fatigue test in this paper. As a consequence, the full-scale test model may vary with the real ship structural detail, and the varied longitudinal in the test model will affect the Hotspot Stress (HSS) distributions in ship structure detail, since the HSS is sensitive to the connection and weld joint types, with reference to Yue et al 2012; Osawa et al. 2011; Fricke & Kahl 2005. Thus the HSS measurement result and the fatigue life evaluation may be influenced by the structure variation of the proposed equivalent angle bar. On this consideration, Finite Element (FE) analysis is performed on such equivalent angle bar to verify that the HSS distribution around weld joint is similar as the real ship detail. A full-scale fatigue test on ship structural detail with equivalent angle bar instead of flat bulb is

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carried out for validation. A bilge longitudinal-gothrough-bulkhead in a river-to-sea ship is chosen as the test model. All three kinds of loads: axial, vertical and biaxial load were applied to the structure detail in order to investigate on the HSS. By the comparison of the rule-based result and the fatigue test result, the proposed test model fabricated from the equivalent angle bar can provide good prediction of the fatigue life of structure details.

pure vertical load and bi-axial load were applied on this structure detail, to simulate longitudinal bending, transverse deformation and overall deformation respectively. Moreover, the full-scale fatigue test was carried out to obtain HSS and S-N curves were so as to make fatigue strength assessment on the ship detail. Besides, an angle bar was made at laboratory and used to replace the flat bulb in the connection of longitudinal-go-through bulkhead. 3

2

DESIGN OF EQUIVALENT ANGLE BAR

DESCRIPTION OF STRUCTURE DETAIL 3.1

In this paper, an 18000DWT river-to-sea ship structure was initially designed, see Figure 1. The ship’s draft is limited by shallow waterway, and its breadth-depth ratio (B/D) is up to about 2.9. Thus the wave fluctuation near the ship bilge is remarkable due to the small draft. And the large ship breadth may lead to great inertial load variation during ship rolling (Dong, 2014). With combination of longitudinal strength, the structure connections at ship bilge have complicated load condition and have tendency to fatigue. On the other hand, the ship’s principle dimensions cannot satisfy the requirements of Regulations for the Construction Classification of Sea-going Steel Ships (CCS, 2012), and there is no suitable guidance for its structural design. So the structure designed according to classification rules leads to conservative structure and underestimated fatigue strength. Therefore, the connection of bottom longitudinal and bulkhead at mid-ship bilge was investigated for its fatigue property by both FE analysis and full-scale test. The longitudinal is interrupted by the watertight bulkhead as Figure 1 shows. With consideration of the complicated stress condition, pure axial load,

Figure 1. The longitudinal-through-bulkhead 18000DWT river-to-sea ship cross section.

Rule-based equivalent angle bar

In ship structural strength analysis, common stiffeners, such as beam, longitudinals, and other primary supporting members are idealized for the convenience of calculation. According to the idealization rules in CSR-H (IACS 2014), the properties of bulb profile sections are usually to be determined by direct calculations. Where direct calculation of properties is not possible, a bulb section may be taken equivalent to a built-up section. The net dimensions of the equivalent built-up section are to be obtained from the following formulae. h′ +2 9.2 h′ ⎛ ⎞ b1 = α t ′ + −2 ⎝ ⎠ 6.7 h′ tf1 = −2 9.2 h1

tw1

h′ −

(1)

t′

where h′ and t′ are net height and thickness of bulb section in mm; hw1 are tw1 are height and thickness of the web of the equivalent angle bar, in mm; bf1 and tf1 are breadth and thickness of the face of the equivalent angle bar, in mm; α = 1.0 if h′ > 120 mm. See Figure 2.

of Figure 2.

Section profile of equivalent angle bar.

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3.2

Equivalent angle bar with uniform thickness

As the above build-up section of the equivalent angle bar, the thickness of its web and face are different, and it is difficult to make in laboratory. So in our study, an equivalent angle bar with uniform thickness is designed, and both the inertial moment and section area are kept constant with bulb section. Besides, the total height of the designed angle also keeps same. Then the net dimensions of the designed equivalent built-up section can be obtained from the following formulae. ⎧h2 t2 + (b2 t2 ) ⋅ t2 A′ ⎪ 3 ⎪(b2 t2 )t2 + t2 b2 − t2 ) ⎪ 12 2 ⎪ h2 t2 ⎤ ⎡ h t + t ( )( ) ( ) ⎪ 2 2 2 2 2 2 ⎢ ⎥ 2 2 ⎪ × h − t2 − ⎢2 ⎥ ⎨ 2 h2 t2 + ( 2 2 ) ⋅ t2 ⎣ ⎦ ⎪ 2 h2 t2 ⎡ ⎤ ⎪ h t + t b − t h − ) 3 2 2 2 2 2 2 ⎢ ⎥ ⎪ t2 h2 2 2 − h2 = I ′ + h2t2 ⎢ ⎥ ⎪ + 12 h t + ( b t ) ⋅ t 2⎦ 2 2 2 2 2 ⎣ ⎪ ⎪⎩h2 = h′

Figure 3. analysis.

Locations of hotspot considered in FE

(2) where h′ is the net height a bulb section; A′, I′: section area and inertial moment of a bulb section; h2, b2, t2: height of web, breadth of face and plate thickness of the designed angle with uniform thickness, which also illustrated in Figure 2. Thus, the thickness t2 and breadth b2 of the designed section can be solved. It should be noted that the designed angel section has same area but different centroid compared with the bulb section. 3.3

FE model

The connection of longitudinal-through-bulkhead at mid-ship bilge (see Fig. 1) is considered for the HSS analysis. A FE model for this structure detail was setup as Figure 3. The length is the distance between side transverses, and the width is about the longitudinal distance. The height of the model is over the double bottom distance, while the inner bottom is neglected in depth direction. Locations near the weld joints which connect bracket and longitudinal at both sides of the bulkhead were regarded as hotspots HS1 (connected with bulkhead stiffener) and HS2 (connected with bulkhead), see Figure 3. For the longitudinal, three section profiles were analyzed for parallel comparison. They are flat bulb, angle with different thickness (refer to CSR-H rules) and angle with uniform thickness as Figure 2 shows. 6-node hexahedron elements were

Figure 4.

Mesh of different longitudinals.

adopted to mesh the whole model. Figure 4 shows the mesh of three kinds of longitudinals. 3.4

Load cases

The boundary conditions on FE model refer to the requirements of calculation method specified in Guidelines for Direct Strength Analysis of Double Side Skin Bulk Carriers (CCS, 2005), as the Table 1 illustrates. With consideration of the force and deformation features of the bilge Longitudinal-throughbulkhead connection discussed in Section 2, both uniaxial loading and lateral load was acted on the FE model. The load case is defined in Table 2. The load along the longitudinals is applied through displacement control and the load perpendicular to the longitudinals is applied by point force. The nominal stress is defined as the first principal stress on the longitudinal face 100 mm from the weld, and also given in Table 2. 3.5

HSS from FE analysis

From FE analysis, the maximum principal stress σ1max distributions along the longitudinal around the weld toe at HS1 and HS2 were obtained for

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Table 1.

Boundary conditions in FE analysis.

Location Liner displacement Angular displacement

δx δy δz θx θy θz

Aft ship

Fore ship

Constrained Constrained Constrained – – –

– Constrained Constrained – – –

Table 2.

Load cases in FE analysis.

Loads cases

Displacement (mm)

Force (kN)

Nominal stress (MPa)

LC 1 LC 2 LC 3

(0.5, 0, 0) – (0.5, 0, 0)

– (0, 0, 10) (0, 0, 10)

43.28 8.57 45.62

three kinds of longitudinals, see Figure 5. From the comparison, it can be found that, under all load cases, the distribution of σ1max near the weld toe is not sensitive to the varied section profile of the longitudinal itself. Furthermore, based on the extrapolation method suggested by Guidelines for Fatigue Strength of Ship Structure (CCS 2007), the HSS was calculated from the σ1max distributions obtained from FE analysis and listed in Table 3. Based on the above FE analysis result, both stress distribution and HSS at weld toes of longitudinal-through-bulkhead seem to be not sensitive to the varied longitudinals with different section profiles. Therefore it is reasonable to use an equivalent angle bar to substitute for the flat bulb, so long as the section area and inertia moment keep same. 3.6

HSS validation for cold-formed equivalent angle bar

Figure 5. The distributions of σ1max near weld toe under different load cases. Table 3. section.

Error

HS1

At laboratory, the proposed equivalent angle bar with uniform thickness can be easily cold-formed from steel plate. So to use such kind of angle bar as ship longitudinals is convenient for fatigue test on ship detail. However, the cold-formed angle bar unavoidably has a round angle due to the cold frame processing, on which the bracket is attached, see Figure 6. Then FE analysis was carried out on the angle bar with different angle radius r for all three load cases. The HSS was calculated and listed in Figure 7. As presented in Figure 7, the HSS is minimum if the angle radius r = 0, which means the angle bar has a right angle instead of a round angle. With

HSS at weld toe of longitudinals with varied

HS2

LC1 LC2 LC3 LC1 LC2 LC3

Bulb HSS/MPa

CSR-H angle

Proposed angle

97.65 27.48 116.06 86.94 12.47 98.82

4.51% 1.32% 1.94% 5.18% 3.26% 1.05%

2.59% 1.83% 0.06% 3.81% 3.40% 3.81%

angle radius r rising, the HSS slightly decreases and changes to increase when the angle radius becomes larger than plate thickness. Considering the processing ability of laboratory, it is suggested that the round angle of the cold-formed angle bar should be limited as small as possible. Generally, the influence of round angle on the HSS can be neglected.

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Table 4. model.

The geometry and material parameters of test

Member

Dimensions (mm)

σyeild (MPa)

1 2 3 4 5 6

10 L260 × 100 × 10 12 FB150 × 12 12 × 330 × 330 10 × 580 × 580

235 352 235 235 235 235

Bottom plate Bottom longitudinal Bulkhead plate Bulkhead stiffener Fore bracket Aft bracket

Figure 6. Round angle of the equivalent angle bar due to cold-frame processing.

Figure 7. Normalized HSS varied with different angle radius r of the equivalent angle bar.

Above all, the equivalent angle bar proposed in this paper not only has similar mechanical property but also keeps the similar stress distribution and HSS with the flat bulb. Therefore, it is possible to adopt this kind of equivalent angle bar to make full-scale model of the ship detail of longitudinalgo-through bulkhead. 4 4.1

FULL-SCALE FATIGUE TEST Test model

A Full-scale model was fabricated according to the ship structural detail given by Figure 1. The geometry and material parameters of all structural members are listed in Table 4. The overall size of the test model is about 3000 mm in length, 650 mm in width and 1500 mm in height. In this model, the original bilge longitudinal at design stage is flat bulb HP260 × 10. Based on the equivalent principle introduced in section 3.2, as well as the laboratory processing ability, an equivalent angle bar L260 × 100 × 10 was taken as the

Figure 8. The full-scale test model of longitudinal-gothrough bulkhead.

bilge longitudinal. The radius of round angle of this angle bar is 15 mm which would cause little error discussed in section 4.4. Figure 8 shows pictures of longitudinal-through-bulkhead test model. 4.2 Test setup Since such structural detail in designed river-to-sea ship may undergo fluctuant loads in both longitudinal and transversal directions, the bi-axial load was applied on the test model, as Figure 9 shows. The longitudinal member was fixed in one end and acted by axial load with MTS actuator at the other end. A vertical load was acted to the bulkhead by another MTS actuator. Generally, the hotspot in such structural detail is located at the weld toe of the intersection of the bracket and longitudinal on weld, and measurement strain gages were arranged near the locations HS1 and HS2 (see Fig. 3). Each hotspot has three measurement points for axial stress measurement in order to obtain HSS by extrapolation. Besides, some stress measurements are also carried out on the weld joint and other possible hotspots in

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Figure 10.

Figure 9.

Setup of the full-scale fatigue test.

Table 5.

order to ensure the measurement reliability during the fatigue test. Measurement points are shown in Figure 10. In order to eliminate the influence of welding residual stress, specimen was pre-loaded before fatigue test. All three kinds of loads were tested: axial load, vertical load and biaxial load. The loading plan is listed in Table 5. Among of all 17 load cases, both static and dynamic loads (average and amplitude are also listed in the table) were applied on the test model, stress distributions were obtained. 4.3

Test result of HSS

The measured axial stress around the hotspot is plotted in the Figure 11. For comparison, the principal stress distributions near the hotspot were calculated by FE analysis and also plotted. It should be noted that the FE analysis was based on the flat bulb. From the comparison, the stress distributions on the proposed equivalent angle bar L260 × 100 × 10 are similar with the flat bulb HP260 × 10. So using the equivalent angle bar to replace the flat bulb in fatigue test is reasonable. Moreover, HSSs under three different kinds of load conditions were calculated based on the measured stress distribution. To investigate the influence of biaxial load on the HSS, a biaxial influential factor C is defined as the equation below. C=

σb σ a + σv

HSS measurement points on the test model.

The test loading plan.

No.

Load case

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Axial

Lateral

Biaxial

LC11 LC12 LC13 LC21 LC22 LC23 LC24 LC25 LC26 LC27 LC28 LC31 LC32 LC33 LC34 LC35 LC36

Axial load (kN)

Vertical (kN)

120 65 ± 55 60 ± 60 – – – – – – – – 120 65 ± 55 65 ± 55 60 ± 60 65 ± 55 65 ± 55

– – – −40 −25 ± 15 −25 ± 25 −25 ± 20 −20 ± 15 −30 ± 15 25 ± 15 ±15 −25 ± 15 −40 −25 ± 15 −25 ± 25 25 ± 15 ±15

(3)

where σb is HSS under biaxial load; σa is HSS under axial load; σv is HSS under vertical load.

Figure 11. Comparison of stress distributions near HS1 between test measurement and FE analysis.

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Table 6 lists the biaxial influential factor C at HS1. Except for LC31 and LC32 which are combinations of static and dynamic loads, biaxial fatigue loads have a tendency to increase the HSS value about 10∼20%. Therefore, it is suggested that, for the structural details such as bilge longitudinal-through-bulkhead connection, HSS from both load components can simply be superimposed. 4.4

Results of fatigue test

The fatigue test was carried out under varied axial compressive load. Based on the measured data, the S-N curve HS1 was fitted (C = 3.103 × 1012) by using of the least square method, in which the inverse slope m was kept to invariant (m = 3), as Figure 12 shows. It should be pointed out that, compared with the S-N curve E provided by classification rules (CCS, 2012), the experiment result tends to be conservative since it was obtained from only one specimen. The fatigue crack initiated under the load with amplitude of 120 kN and average of 70 kN, and with the HSS of about 400 MPa at HS1. Since the fatigue crack initiation was observed, the relation between crack length and load cycles was recorded in Figure 13. Figure 14 demonstrates the crack propagation process. The crack initiated at weld toe at first,

Figure 13.

HSS measurement points on the test model.

Figure 14.

Fatigue crack propagation at HS1.

Table 6. Biaxial influential factor C at HS1 under combined axial and vertical dynamic loads. LC31 LC32 LC33 LC34 LC35 LC36

120 65 ± 55 65 ± 55 60 ± 60 65 ± 55 65 ± 55

−25 ± 15 −40 −25 ± 15 −25 ± 25 25 ± 15 15

0.942 1.631 1.113 1.018 1.060 1.196

then propagated on both the face and web of the angle bar synchronously until it went through the thickness of the angle bar. Above all, the full-scale model with the equivalent angle bar instead of flat bulb can fulfill the fatigue test on structural detail. From the fatigue test, both HSS and S-N curve can be well obtained for further fatigue life prediction. 5

Figure 12.

S-N curves fitted from varied fatigue loads.

CONCLUSIONS

This paper proposed an equivalent angle bar with uniform thickness, which can be easily cold-formed at laboratory, to substitute for flat bar in the fullscale fatigue test on structural detail. FE analysis was carried out to prove that the stress distribution and HSS were not influenced by the varied section profile of the angle bar. Then, a bilge Longitudinal-through-bulkhead at a river-to-sea ship structure detail is chosen as the fatigue test model. Fatigue test on such equivalent angle bar also demonstrated that both HSS and S-N curve can be well obtained with comparison to the FE analysis

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result and rule-based result. The main findings of this study can be summarized as follows:

REFERENCES

• The bilge longitudinal-go-through bulkhead was under both longitudinal and transversal loads, and the application of the proposed equivalent angle bar into such structural detail was suitable for various load conditions, all of which have similar stress distribution. • The proposed equivalent angle bar with uniform thickness is a reasonable substitute for flat bar in laboratorial fatigue test. While it is suggested that the round angle radius r of the equivalent angle bar should keep close to the plate thickness t as possible. • HSS on the longitudinal-go-through bulkhead connection under biaxial load should not be simply combined from separate HSS under axial load. Otherwise, the HSS may be underestimated based on the test results. • Near the weld toe of the bracket attached to the equivalent angle bar, the fatigue crack initiated, and then propagated both on the face and web of the angle bar. Both the fatigue parameter and phenomenon can be obtained from the full-scale test model.

CCS. 2005. Guidelines for Direct Strength Analysis of Double Side Skin Bulk Carriers. Beijing: CCS. CCS. 2007. Guidelines for fatigue strength of ship structure. Shanghai: CCS. CCS. 2012. Regulations for the construction and classification of sea-going steel ships. Beijing: CCS. Dong, Y., Yue, J., Wu, W. & Zang, W. 2014. Simplified direct calculation of torsion-bending strength of a broad and flat river-sea transportation ship. Ship Engineering 36(1): 6–9. Erny, C., Thevenet, D., Cognard, J.Y. & Körner, M. 2012. Fatigue life prediction of welded ship details. Marine Structures 25(1): 13–32. Fricke, W. & Kahl, A. 2005. Comparison of different structural stress approaches for fatigue assessment of welded ship structures. Marine Structures 18(7–8): 473–488. Fricke, W., Anatole von Lilienfeld-Toal & Paetzold, H. 2012. Fatigue strength investigations of welded details of stiffened plate structures in steel ships. International Journal of Fatigue 34(1): 17–26. IACS. 2013. Common structural rules for bulk carriers and oil tankers-Harmonized. Kim, M.H., Kang, S.W., Kim, J.H., Kim, K.S., Kang J.K. & Heo, J.H. 2010. An experimental study on the fatigue strength assessment of longi-web connections in ship structures using structural stress. International Journal of Fatigue 32(2): 318–329. Lotsberg, I. & Landet, E. 2005. Fatigue capacity of side longitudinals in floating structures. Marine Structures 18(1): 25–42. Osawa, N., Yamamoto, N., Fukuoka, T., Sawamura, J., Nagai, H. & Maeda, S. 2011. Study on the preciseness of hot spot stress of web-stiffened cruciform welded joints derived from shell finite element analyses. Marine Structures 24(3): 207–238. Yue, J., Dong, Y., Zhou, H. 2012. Fatigue analysis on a multi-planer tubular KK joints based on scalded model test. Proc. of the 22ed Inter. Offshore and Polar Engineering Conference, Rhodes, 17–23 June, 2012: P174–179.

Furthermore, it should be noted that, because the equivalent angle bar was made from steel plate by cold-framing, the strain-hardening effect cannot be avoided, which may influence the fatigue property of structural details. ACKNOWLEDGEMENTS This work was supported by Chinese 111 Project (B08031), ESI Plan of Wuhan University of Technology and WUT-UoS Joint Research Center of High Performance Ship. The author would like to express sincere appreciation to Prof. Carlos Guedes Soares for his valuable comments and suggestions on this paper.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Springing effect on the fatigue life of an 8000TEU container ship P.-K. Liao Research Department, CR Classification Society, Kobe, Japan

Y.-J. Lee & H.-J. Lin Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan

S.-C. Tsai & H.-L. Chien Department of Design, CSBC Corporation, Taiwan

B.C. Chang & G.M. Luo Department of Naval Architecture and Ocean Engineering, National Kaohsiung Marine University, Kaohsiung, Taiwan

ABSTRACT: This study investigated the springing effect on the fatigue life of an 8000TEU container ship by performing hydro-elastic analyses. The hydro-structure computations consist in evaluating the hydrodynamic loads and transferring them directly to the finite element model of the ship structure. The quasistatic and dynamic responses of the structure can then be evaluated and the fatigue life assessed through spectral analysis. This study evaluated the fatigue life at two hot spot locations and considered several wave environments at two ship speeds. In addition, this study evaluated the effect on the fatigue of various natural modes of hull girder vibration. The results showed that because of the springing effect, the ship fatigue life decreased by 16% to 46% depending on the hot spots locations and the operating conditions. Therefore, the effect of the springing phenomenon on the fatigue life is non-negligible in large container vessels design and should be carefully evaluated. 1

INTRODUCTION

The springing is the phenomenon of global ship structure resonance that is induced by wave loads. The springing response can considerably reduce the fatigue life of ship. On one hand, because of the large openings on the strength deck, container ship structures are naturally soft leading to low frequency vibration modes. The great length and the extensive use of high tensile steel reduce also the structure rigidity and thus the global structure natural frequency. On the other hand, the high service speed of container ship can increase the encounter frequency of incoming waves. As a result, ship hull girder vibrations can occur for wave periods that correspond to more probable sea state, increasing thus the springing effect on the fatigue. Several researchers have performed hydro-elastic analyses to quantify the springing effect contribution to the fatigue life. Yang (2011) established that the springing effect contributes by approximately 8% to the fatigue life of a 203,000 DWT ore carrier. Wu (2013) evaluated that the springing effect represents about 14% to the fatigue life of an 8000TEU container ship under North Atlantic

wave environment. Furthermore, Koo et al (2013) determined that the springing effect increases by 24% to 64% the fatigue damage of an 18,000 TEU container ship. This study presents a quantitative evaluation of the springing effect on the fatigue life of an 8000TEU container ship under various operating conditions. In particular, this study performs hydro-elastic computations by using HOMER software developed by Bureau Veritas (BV). The calculations include direct hydrodynamic analysis conducted using the potential flow theory software HYDROSTAR (BV), and Finite Element Analysis (FEA) by NX NASTRAN. The obtained response amplitude factors (RAO) of hot spot stress are then used to evaluate the fatigue life by spectral analysis. This article consists of three sections. The first section presents the spectral fatigue assessment derived from the hydro-elastic computations. The second section compares the springing effect on the fatigue life under the considered operating conditions including various wave environments and ship speeds. The third section evaluates the contribution of each natural vibration mode to the springing.

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2

SPECTRAL FATIGUE ASSESSMENT

2.1

Hydro-elastic analysis

The rigid body model, also called quasistatic method, considers the six rigid body modes of motion. This method is widely used in seakeeping analysis. For vessels with sufficient structure rigidity, the rigid body model produces accurate results within a reasonable computation time. However, the hydro-elastic model includes additional modes of deformation chosen from a series of dry structural natural modes. After determining hydrodynamic coefficients and excitations, the modal amplitude is obtained by solving the following equation. {

2

([ ] [ A]) − iω (([ B ] [ b ]) + ([ k ] [C ])}{ξ } {F DI }

(1)

where [m] = structural mass, [A] = hydrodynamic added mass; [B] = hydrodynamic damping; [b] = structural damping; [k] = structural stiffness; [C] = hydrostatic restoring; {ξ} = modal amplitudes; and {FDI} = modal hydrodynamic excitation. This study performs hydro-elastic analyses using HOMER (BV). Figure 1 shows the flowchart of HOMER. Using the global Finite Element (FE) model of the ship including the mass of the containers, the HmFEM module produces the ship mass properties and conducts the modal analysis required to obtain the natural modes and their corresponding frequency. Using the hydro model of the ship, the HmSWB module then calculates the hydrostatic coefficients required to balance the models by adjusting the draft and trim. The HmHST module generates hydro models for each elastic mode shape obtained by the HmFEM module. The HmHST module then solves the radiation and diffraction problems to compute the pressures acting over the hull, and then produces the hydrodynamic coefficients of ship. The HmMCN

Figure 1.

Flowchart of HOMER.

module solves the motion equations, as shown in Equation (1). The HmFEA module then enables the transfer of the hydrodynamic pressure to the structural model through the integration model that is generated from the outer shell of the structural FE model. This step is simply executed thanks to the source method employed by HYDROSTAR that provides a continuous representation of the potential through the wetted part of hull structural meshes. HOMER can then recalculate the hydrodynamic coefficients by integrating the pressure over the hull structural mesh. Finally, the motion equations are solved with the new coefficients. The HmFEA module then conducts the FEA of the global ship structure, and the nodal displacement of every node is extracted. This study used separate local FE models to evaluate the stress at the considered hot spots. In the local model, the hot spot areas were modeled according to the very fine mesh rules requirements of BV (2008b). The separate model extent was bounded by primary supporting structures. The nodal displacements obtained from the global model FE analyses were then applied to the corresponding boundary nodes on the local model. This module thus conducts the FEA of the very fine mesh models. The HmRAO module can then generate the stress RAO file of the elements surrounding the hot spots. Finally, spectral analyses were performed to determine the fatigue life by using the RAO of hot spot stress accordingly to BV (2008a). The springing effect on the fatigue can be obtained by subtracting the fatigue life evaluated by the quasistatic analysis from that produced by the hydro-elastic analysis. 2.2 Case study This study considers an 8000TEU container ship constructed by CSBC Corporation, Taiwan. Table 1 lists the principal particulars of the ship. Figure 2 shows the ship coarse mesh FE model and the examined hot spot locations. Mass elements are also added to the structure model, thus including the weight of equipments and containers based on the loading manual full load condition. The RBE3 elements of NX NASTRAN ensure an adequate load distribution from the mass elements to the affected structural components nodes. Figure 3 shows the very fine mesh models used to analyze the stress at the hot spots. Hot Spot I is the radiused edge of the hatch corner at coaming top located amidship, and it is expected to be mainly sensitive to the vertical bending mode. The Hot Spot II is the toe of the coaming end bracket

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at the upper deck, and it is expected to be mainly sensitive to the torsion mode. The seakeeping analysis was performed for wave headings ranging from 0° to 345° with 15° interval and for wave frequencies ranging from 0.1 rad/s to 2.0 rad/s with 0.05 rad/s interval. In addition, two ship average speeds were considered separately: – 60% of the ships service speed as suggested by BV (2008). – 80% of the ships service speed as determined by Koo et al (2013) based on full scale measurements collected on an 8000TEU container ship.

– Case I: The ship operates in the North Atlantic environment (Table 2, IACS (2011)) during its entire lifetime – Case II: The ship operates in the Worldwide environment (Table 3, BV (2012)) during its entire lifetime – Case III: The ship operates on the Asia-Europe route (see Table 4, Koo (2013)) during its entire lifetime – Case IV: The ship operates on the Asia-West America route (see Table 5, Koo (2013)) during its entire lifetime

This study then evaluated the fatigue life by spectral analysis using the Pierson-Moskowitz spectrum and cos n spreading function with n = 2 to model the sea state and extract the short term response. The fatigue life was thus assessed for five cases of wave environment, the distribution of sea states are as shown in Figures 4 to 7:

Table 1. Principal particulars of the 8000 TEU container ship. Length Breadth Draft Service speed (Vs)

317 m 46 m 14 m 24.5 knt

Figure 2.

Ship global FE model.

Figure 3.

Structural details fine mesh models.

Figure 4.

North Atlantic wave scatter diagram.

Figure 5.

Worldwide wave scatter diagram.

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Table 2.

Figure 6.

Asia-Europe route scatter diagram.

Figure 7.

Asia-West America route scatter diagram.

Natural frequency and type of wet modes.

Table 3. modes.

– Case V: The ship operates on the Asia-Europe route for half of its lifetime and on the Asia-West America route for the other half of its lifetime. During the hydro-elastic seakeeping analysis, eight elastic modes were considered. Table 2 presents the wet mode frequencies (i.e. natural modes that consider the added mass of sea water) and types of mode obtained by FEA. The structural damping values used for elastic modes are as shown in Table 3, ranging from 1% to 7% of critical damping.

Structural damping for elastic

Elastic mode

Structural damping used

I II III IV V VI VII VIII

1% 2% 3% 5% 5.5% 6% 6.5% 7%

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3 3.1

ANALYSIS RESULTS Stress RAO

Figures 8 and 9 show the stress RAO at Hot Spot I for 60% and 80% of the service speed respectively. Figures 10 and 11 show the stress RAO at Hot Spot II for 60% and 80% of the service speed, respectively. In Figures 8 to 11, it can be observed that the RAOs amplitude increases with the speed, leading to higher level of long term stress range distribution that would decrease the fatigue life. It can also be observed that, in the dynamic part, the RAO peaks frequency decreases as the ship speed increases

because the encounter wave frequency increase for those head sea and quartering sea headings (head sea = 180 deg). Tables 4 and 5 present for each hot spot, the peak RAO’s wave frequency in the dynamic part. The corresponding structural wet mode number (see Tab. 2) is also marked with parentheses beside the encounter frequency value. For Hot Spot I, Table 4 shows that the encountered

Figure 11. speed.

Figure 8. speed.

Stress RAO at Hot Spot I for 60% of service

Table 4.

Stress RAO at Hot Spot II for 80% of service

Peak condition of stress RAO at Hot Spot I. 60%Vs

80%Vs

Wave Encounter Wave Encounter Heading frequency frequency frequency frequency (degree) (rad/s) (rad/s) (rad/s) (rad/s)

Figure 9. speed.

120 135 150 165 180

1.75 1.6 1.5 1.45 1.45

2.93 (II) 3.00 (II) 3.002 (II) 3.016 (II) 3.071 (II)

1.65 1.45 1.35 1.3 1.3

3.049 (II) 2.978 (II) 2.972 (II) 2.978 (II) 3.037 (II)

Table 5.

Peak condition of stress RAO at Hot Spot II.

Stress RAO at Hot Spot I for 80% of service

60%Vs

80%Vs

Wave Encounter Wave Encounter Heading frequency frequency frequency frequency (degree) (rad/s) (rad/s) (rad/s) (rad/s) 120

150 165

1.65 1.75 1.5 1.6 1.5 1.45

2.67 (I) 2.93 (II) 2.73 (I) 3.00 (II) 3.00 (II) 3.02 (II)

180

1.45

3.07 (II)

135

Figure 10. speed.

Stress RAO at Hot Spot II for 60% of service

1.5 1.65 1.35

2.66 (I) 3.05 (II) 2.68 (I)

1.35 1.3 1.9 1.3

2.97 (II) 2.98 (II) 5.48 (IV) 3.04 (II)

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wave frequency of all peaks is approximately equal to 3.0 rad/s, which is consistent with the natural frequency of the 2-node vertical bending (i.e. Mode II). For Hot Spot II, Table 4 shows that the RAO has more than one peak in the dynamic part. However, the encountered wave frequency of most of the peaks is consistent with the structural wet Mode II (see Table 2). Mode I and IV can also occur. In addition, Tables 4 and 5 show that, for a ship operating at 80% of the service speed, the springing can occur for a wave frequency of approximately 1.3 rad/s that corresponds to a wave period of 4.5s. Based on the scatter diagrams (see Figures 4 to 7) sea states distribution, it can thus be anticipated that the springing would occur more often for higher speeds, because short waves (i.e. Tz < 7s) sea states become more probable with longer waves. 3.2

Figure 12.

Fatigue life at Hot Spot I.

Figure 13.

Fatigue life at Hot Spot II.

Spectral fatigue analysis

Figures 12 and 13 show the fatigue life assessed by spectral analysis for the quasistatic and hydro-elastic approaches. It can be observed that the fatigue life varies significantly according to the operating conditions. First, it can be observed that the predicted fatigue life assuming operations in the North Atlantic wave environment (i.e. Case I) are approximately three to five times lower than for the other environments examined in this study. It appears also that the speed and the hydro-elastic structure response consideration have a nonnegligible effect on the fatigue life assessment. Table 6 presents the ship speed effect on the fatigue life that is calculated using Equation (2). It appears that, for a speed increase of 33% (i.e from 60% to 80%Vs), the fatigue life decreases by 12% to 62%. It is thus verified that the fatigue life decreases at higher speeds as anticipated in section 3.1, based on the observed amplitude of the RAOs. It can also be observed that the speed effect on the fatigue is similar for every considered case of wave environment. However, the ship speed effect is two to three times higher for the hydro-elastic approach than for the quasistatic. The springing evaluated by the hydro-elastic analysis is thus very sensitive to the ship speed. Speed Effect (%) =

L60%Vs L L60%Vs

%Vs

(2)

where L60%Vs = the fatigue life in years, predicted for an average speed corresponding to 60% of the ship service speed and L80%Vs = fatigue life in year predicted for an average speed corresponding to 80% of the ship service speed. Table 7 presents the springing effect on the fatigue life that is calculated using Equation (3).

Table 6.

Ship averaged speed effect on fatigue life. Case I Case II Case III Case IV Case V

Hot Spot I Quasistatic 12% 12% Hydro-elastic 24% 28%

10% 33%

11% 29%

10% 32%

Hot Spot II Quasistatic 49% 56% Hydro-elastic 55% 61%

57% 62%

56% 61%

57% 62%

Table 7.

Springing effect on fatigue life. Case I

Case II

Case III

Case IV

Case V

60%Vs 80%Vs

Hot Spot I 16% 21% 28% 35%

27% 46%

23% 39%

26% 43%

60%Vs 80%Vs

Hot Spot II 17% 22% 26% 30%

29% 37%

24% 32%

27% 35%

For a speed increase of 33% (i.e from 60% to 80%Vs), the springing effect contribution to the fatigue increases by approximately 75% and 30% at the Hot Spots 1 and 2 respectively. Therefore, it is verified that the springing occurs more often

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at higher speed because the sea states corresponding to higher wave periods are more probable, as anticipated in section 3.1 based on the observation of the RAO peaks frequency in the dynamic part. Springing i Effect (%) =

LQS

LHE

(3)

LQS

where LQS = the fatigue life in years, as predicted by the quasistatic approach; and LHE = the fatigue life in years considering the hydro-elastic effect. Furthermore, in Table 7 it can be observed that the evaluated springing effect of a ship operating in the North Atlantic is slightly lower than that of a ship in other wave environments. 4

Figure 14. Contribution of each mode to the springing effect for 60% of service speed and Worldwide environment.

EFFECT OF NATURAL MODES

This study investigates the influence on the fatigue of each natural mode. Several hydro-elastic analyses are thus performed considering: – – – – –

Mode I only, Modes I to II, Modes I to III, Modes I to IV, and Modes I to V.

The contribution to the springing effect of each individual mode can be computed by Equation (4). SE (Mode X ) S SE (Mode M s I to X ) − SE(Modes SE (Modes I to X

)

Figure 15. Contribution of each mode to the springing effect for 80% of service speed and Worldwide environment.

(4)

where SE = springing effect (%). Figures 14 and 15 show the contribution of the individual modes to the springing for the Worldwide wave scatter diagram. The contribution of natural modes to the springing effect depends on the location of the considered hot spot. The springing effect at Hot Spot I is mainly caused by the elastic Mode II (vertical bending mode), whereas the springing effect at Hot Spot II is mainly caused by elastic Mode I (torsion mode) and, less considerably, by elastic Mode II. For the one-node torsion mode, the loads are higher at location away from node of torsion, which is near to midship. Therefore, the Hot Spot II is very sensitive to that mode. For the vertical bending mode, the loads are higher amidship. Therefore, the Hot Spot I is more sensitive to the vertical bending mode and Hot Spot II is less affected. Table 8 and 9 present the contribution of the individual modes to the springing effect for 60% and 80% of service speed for all the wave environments considered. The results confirm that

Table 8. Individual modes contribution to the springing for 60% of service speed. Modes

North Atlantic Worldwide Asia-Europe Asia-West America North Atlantic Worldwide Asia-Europe Asia-West America

I

II

III

IV

V

VI–VII

Hot Spot I 0% 14% 0% 18% 0% 24% 0% 20%

1% 1% 1% 1%

1% 1% 1% 1%

0% 0% 0% 0%

1% 1% 1% 1%

Hot Spot II 9% 3% 12% 5% 17% 7% 13% 5%

2% 3% 12% 2%

1% 1% -8% 2%

1% 1% 0% 1%

0% 0% 0% 0%

the individual mode contribution to the springing effect is highly dependent to the hot spot location. In addition, the contribution of individual modes does not deviate considerably between the wave environments.

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Table 9. Individual modes contribution to the springing effect for 80% of service speed. Modes

North Atlantic Worldwide Asia-Europe Asia-West America North Atlantic Worldwide Asia-Europe Asia-West America

5

I

II

III

IV

V

VI–VII

Hot Spot I 0% 25% 0% 32% 0% 41% 0% 35%

0% 0% 0% 0%

2% 2% 2% 2%

0% 0% 0% 0%

1% 1% 1% 1%

Hot Spot II 18% 4% 21% 5% 25% 8% 22% 6%

−1% −1% −2% −1%

6% 7% 9% 7%

0% 0% −1% −1%

−1% −1% −2% −1%

CONCLUSIONS

This study has evaluated the fatigue life of an 8000 TEU container vessel by using spectral fatigue analysis. In particular, hydro-elastic analyses were performed to assess the springing effect on the fatigue life of two hot spots. This study compared the springing effect under various operating conditions, including two ship speeds and five wave environments. The evaluated springing effect contribution to the fatigue life is comprised between 16% and 46% for the radiused edge of the hatch corner at the coaming top located amidship and between 17% and 37% for the toe of the coaming end (bracket) at the upper deck. The springing effect varies with the operating conditions and the hot spot location. First, this study has shown that higher ship speeds generate a greater springing effect contribution to the fatigue life due to the rise of the RAOs amplitude, and due to the decrease of the wave RAO peaks frequency in the dynamic part of the RAOs. The RAO amplitude rise generates higher long term stress range distribution at the considered hot spots. The drop of RAO peaks frequency in the dynamic part leads to springing phenomena occurring at higher wave periods for which the sea states are more probable. This study has then evaluated that the North Atlantic wave environment generates less springing effect than others, such as the Worldwide wave environment. However, because the springing can occur for sea states with low probability (i.e corresponding to Tz < 4.5s), the influence of this wave

environment on the springing effect contribution to the fatigue life cannot be accurately evaluated; the scatter diagram uncertainties may be high at those sea states. Besides, although the North Atlantic wave environment might generate less springing, the total fatigue life including the quasistatic ship response remains much shorter than that evaluated assuming other environments. Finally, this study has demonstrated that the hot spot location in the ship determines directly which individual elastic modes are involved in the springing effect contribution to the fatigue life. The results have revealed that for the hot spots located amidship, the springing are highly sensitive to the vertical bending mode, corresponding to the second lowest natural frequency mode. The hot spot located on the fore part of the ship are highly sensitive to the torsional bending mode which is the first lowest natural frequency mode. This study assessed the springing effect due to linear wave loads. However, the springing response may be influenced by the interaction between longcrested wave systems. For the sake of accuracy, a time domain approach should thus be employed to evaluate the influence of those critical nonlinear waves on the springing response. ACKNOWLEDGEMENTS This research is supported by CSBC Corporation, Taiwan and Ministry of Science and Technology. REFERENCES BV 2008a. Spectral Fatigue Analysis Methodology for Ships and Offshore Units, Guidance Note NI 539. BV 2008b. Guidelines for Structural Analysis of Container Ships, Guidance Note NI 532. BV 2012. Whipping and Springing Assessment, Rule Note NR 583. IACS 2001. Standard Wave Data. IACS Rec. No. 34. Koo, J.B., Kim, B.J., Jang, K.B., Suh, Y.S. & Bigot, F. 2013. Fatigue Assessment of the 18,000TEU Container Vessel Considering the Effect of Springing, Proceeding of ISOPE2013, Alaska, USA. Wu, Z.W. 2013. Fatigue Strength Analysis of Container Ship by Hydroelastic Dynamics. Master Thesis, National Taiwan University, Department of Engineering Science and Ocean Engineering, Taipei, Taiwan. Yang, S.H., Lee Y.J., Tasai, S.C., Hsu, S.H. & Malenica, S. 2011. Hydro-elastic effect on spectral fatigue damage assessment of bulk carriers, Asian-Pacific Technical Exchange and Advisory Meeting on Marine Structures (TEAM), Korea.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

A fracture mechanics based approach for the analysis of crack growth at weld joints of ship structures Benqiang Lou, Shengming Zhang, Jimmy Tong, Sai Wong & Fai Cheng Global Technology Centre, Lloyd’s Register EMEA, Southampton, UK

Spyros Hirdaris Technology Centre—Korea, Lloyd’s Register Asia, South Korea

ABSTRACT: Existing spectral fatigue analysis techniques used for the assessment of fatigue life of ship structures are based on the S-N curve approach. Such methods do not distinguish between the crack initiation and propagation stages and therefore they do not account for the influence of the size and growth of crack defects on local structural details. The application of fracture mechanics concepts could offer a meaningful alternative as they allow for the estimation of the residual fatigue life once cracks are identified. As a first step toward the pragmatic use of such methods this paper introduces a semi-analytical approach which leads to the derivation of analytical Stress Intensity Factors (SIF) by combining hybrid fracture mechanics theory with FEA. The potential of the method is demonstrated by studying the crack propagation effects on the fatigue life of typical joint idealization. 1

INTRODUCTION

During 1980’s and early 1990’s crack and fracture induced failures were common in way of the joint connections of longitudinal stiffeners and transverse frames of large oil tankers and bulk carriers constructed of higher tensile steel. To resolve this problem Lloyd’s Register introduced spectral Fatigue Design Assessment procedures (ShipRight FDA), that today are integral part of the Societies’ Classification assurance framework as stipulated in the Rules for Ships (Lloyd’s Register, 2013a, b). Existing spectral fatigue assessment methods assess the overall fatigue damage (or fatigue life) using the S-N curve approach. Accordingly, they do not distinguish between the crack initiation and propagation stages and they cannot accommodate for the development or facilitation of a time efficient approach whereby the size and growth of crack defects on local structural details is controlled in a way that allows for safe, yet efficient ship repairs. The practical application of fracture mechanics concepts could offer a meaningful alternative as they allow for the estimation of the prediction of fatigue life once cracks are identified. Traditional fracture mechanics theories (e.g. ParisErdogan equation) are “phenomenological”, i.e. they are established in a semi-empirical fashion based on results from fatigue or fracture tests on key structural details. Modern techniques with

potential application on real crack defects found in ship structures introduce empirical correction factors. Those, may be derived by various methods such as empirical solutions, finite element analysis, the green function method and last but not least hybrid methods where the influence of Stress Intensity Factors (SIF) is decomposed in the form of empirical estimates that are able to correct the

Figure 1.

Typical crack propagation defect.

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influence of structural details in different phases of the crack propagation (Xu and Bea, 1997). It is believed that hybrid methods, depending on the configuration and complexity of crack defects, have good potential to assess the effects of fracture by realistic idealization of the load shedding effects and are therefore integral part of relevant engineering standards (e.g. BS7910, 2005). The work presented in this paper is based on the hybrid method of Lou et al. (2012 and 2014). Accordingly, the publication introduces an empirical formula for the idealisation of cracks in way of cruciform joints found in typical structural details of ships. With the aim to demonstrate the significance of suitable modelling of geometric discontinuities, emphasis is attributed to the derivation of analytical SIF by combining fundamental fracture mechanics and FEA.

Equation (2) defines K as the characteristic parameter for the stress intensity around the crack tip. The fatigue life due to crack propagation that emerges in way of stress concentrations is expressed using Paris-Erdogan Law (1963) as follows:

2

N =∫

2.1

where: σ peak is the (engineering) peak (or maximum) stress and σ 0 is the nominal stress. The driving force for a crack to grow is the SIF that embodies both the stress, the crack size and describes the crack tip stress field. By direct perturbation of the geometry correction function (F) into the SIF of a specimen that is subject to a crack size of length (a) is defined as:

af

a0

THEORY General

Based on Euler’s Elastica (Love, 1844), the twodimensional stress distribution of a homogeneous structure that is subject to linear loading depends on the body geometry. Stress raises occur in way of joint connections, the theoretical stress at the zero radius connection ends is infinity. However, for engineering application the peak stress even at zero radius is not infinitive, and it is also depend on the methods (e.g. FEA, model test) to obtain the stress values. In application, local abrupt amplifications of stresses will be defined as a peak critical value as illustrated in Figure 2. The ratio of the peak stress and the nominal stress in the net section leads to the commonly used definition of the stress concentration factor Kt, given: Kt = σ peak σ 0

F ⋅σ0 ⋅ πa

K

(3)

where: a0 = initial crack length; af = initial crack length; C, m are empirical factors depending to the material properties. According to Hobbacher (2009) for typical steel grades C = 1.65 ×10−12 and m = 3.0 (units in MPa and m). For numerical calculation purposes, it will apply equation 3 into its discretised form that considers the influence of (assumed) small crack steps as follows:

∑ ΔNi

Ni

(4)

where: ΔNi =

δ

(1)

i

2.2

Figure 2. Concentrated stress demonstration of peak stress increase at connection.

da C [ K ]m

(2)

1 δ ai C δ K im i

i −1

δ a1 = a1 ΔN1 = 0

Hybrid method

Commonly used methods for the estimation of SIF are based on empirical solutions, numerical solutions (e.g. FEA), and the Green Function Method and Hybrid methods. Empirical methods are based on experimental data and observations (e.g. Newman and Raju, 1981). FEA solutions are provided by the energy release rate method, stress or displacement approach and the linear spring method. The Green Function Method assumes that SIF due to complex stress fields can be expressed by integrating the contribution of unit stress amplitudes along the crack (Xu and Bea, 1997). Accordingly, the solution is achieved first

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by assuming that the stress distribution is a nonsingular elasticity problem that may be solved by FEA and then the SIF is estimated by the so-called influence function that depends on the geometric properties of the cracked body. The hybrid method is essentially a Green Function Method with superposition that makes use of available solutions for two and three-dimensional crack problems. Based on this approach for a through thickness crack defect that is common in ship structural details; the geometry correction function (F) can be expressed as the bi-product of the geometry correction factor Fa for longitudinal stiffener (here is a flat bar plate) & correction fact fg for web plate connection. Accordingly, the influence of correction factors on SIF is expressed as: K

Fa ⋅ fg

0

a

1.122 0.231(a/b ) . (a/b )2 −21.71(a/b )3 + 30.382(a/b )4

(6)

Typical crack locations in ship structures.

K FEEA Fa σ 0 π a

(7)

Stresses are extremely steep in way of the joint connections. We suppose that when the crack size is very small or approaching zero, the flange correction factor (fg) is approaching the stress concentration factor (Kt). That is:

(5)

Practical experience suggests that cracks in ship structures usually occur in way of the connections of the longitudinal stiffeners with the transverse frames of side shells; bottom and deck structures (see Figure 3). The presence of a flange in such structural details is important as it influences the distribution of stress flows across the main plate. The introduction of correction functions such as the one outlined in equation

Figure 3.

fg =

Kt

In equation 5, Fa denotes the propagation of crack sizes across the critical dimensions of a structural detail. According to Lou et al. (2014), for typical joint structures such as the ones studied in this paper the following formulae by Gross (1964) may be considered appropriate provided that it is suitably modified depending on the type of the specimen under consideration: Fa

(7) implies the need to idealise the stress influence in way of critical locations. Therefore, in the method presented here flange correction factor (fg) is introduced as an additional item that may be evaluated from FEA (ANSYS, 2005) according to equation:

3

σ max σ0

i

a→0

fg

(8)

PRACTICAL APPLICATIONS

The elementary stress formulas used in the design of structural members are based on key structural locations having similar sections. The presence of flat bar in such structures is important. This is because the supporting attachment (e.g. bracket) helps to increase the strength in way of joint connections and influences the distribution of stress flows across the main part of the structure. Fracture mechanics based fatigue growth analysis can be carried out in the following steps: • Step 1: Select key geometries of typical ship structure joints and identify the critical crack locations for those in accordance with the specimen libraries or databases (e.g. Lloyd’s Register FDA, 2013a); • Step 2: Assume formation and propagation of single edge cracks and select a suitable geometry correction function; • Step 3: Determine an equivalent SIF by combining the correction factors for the general plate geometry and the joint connections. At first instance FEA may be used to derive these correction factors and empirical values can be calibrated by curve fitting; • Step 4: Calculate the fatigue life under a nominal tensile stress conditions by using equations 3 or 4. Case studies have been carried out. All models are subject to cyclic tensile stress of 50 MPa and the initial crack size has been restricted to 5 mm. For all idealisations it is assumed that crack propagation is single sided, it initiates in way of structural connections and develops along the direction of the stiffeners.

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3.1

Stiffener plate with flange bar (model I)

In practical cases most plate components are connected with a supporting attachment (e.g. flat bar or bracket) that helps to increase strength in way of joint connections (see Figure 4). Flat bar width (W) is an important geometry factor that may affect fg and for this reason the parametric study presented here considered the influence flat bar widths that vary from 150 mm to 300 mm in steps of 50 mm. In the recent work by Lou et al. (2014) empirically derived fg solutions have been compared against the numerical results. It was shown that the traditional formula by Gross (1964) has limits on its applications and validity. With this in mind this paper introduces a geometry function formula expressed by the following polynomial function: 1

fg

3

⎛ a ⎞2 ⎛ a⎞ ⎛ a ⎞2 ⎛ a⎞ C0 + C1 ⎜ ⎟ + C2 ⎜ ⎟ + C3 ⎜ ⎟ + C4 ⎜ ⎟ ⎝ D⎠ ⎝ D⎠ ⎝ D⎠ ⎝ D⎠

2

(9) In equation 9 ‘D’ presents the height of the specimen. For small cracks where a < 0.094W/D + 0.032 (m), the coefficients C0 to C4 are given below functions: C0 = 1.0 + 4.0

W ⎛W ⎞ − 1.70 ⎜ ⎟ ⎝ D⎠ D

C1 = −13.119 −18 18.982 C2 = 137.98 − 92.893

W ⎛W ⎞ + 10.255 ⎜ ⎟ ⎝ D⎠ D

Figure 4.

Based on the above equations fg values reduce with increasing crack size, while SIFs are considered to increase slowly. On the other hand for large cracks where a > 0.094W/D + 0.032 (m), the approach assumes that the influence of flat bar effects are not considered significant and hence fg = 1.0. From the comparison between this formulae and FEA results (see Figure 5), it is demonstrated that absolute differences are within 1%. Thus the calculated SIF values using the semi-analytical solution may be considered satisfactory. When the crack size is very small or approaching to zero, the correction factor (fg) is approaching the stress concentration factor (Kt). Thus we obtain:

W ⎛W ⎞ + 42.197 ⎜ ⎟ ⎝ D⎠ D

W ⎛W ⎞ C3 = −501.1 + 714.59 − 383.38 ⎜ ⎟ ⎝ D⎠ D C4 = 622.82 − 1143.9

Figure 5. Relative SIF derivations between formula and FE solutions.

W ⎛W ⎞ + 657.5 ⎜ ⎟ ⎝ D⎠ D

Longitudinal with ‘buckling stiffener’.

Kt = 1.0 + 4.0

W ⎛W ⎞ − 1.70 ⎝ D⎠ D

(10)

This is obtained at crack size of 5 mm; this formula reflects the stress concentration with the sharp crack. The fatigue life for specimens incorporating different flat bar widths (W) was also evaluated. Figure 6 demonstrates that the results derived by FEA and the new semi-empirical method agree well for all different cases. For all models after the crack length reaches 100 mm, the crack propagation grows fast. Table 1 presents a summary of the fatigue life of the different specimens at crack length of 100 mm. Assuming average wave period of 8 seconds and tensile stress of 50 MPa the results show that the fatigue life is reduced from 94.3 days for the flat bar width of 150 mm to 70.3 days for the flat bar of 300 mm. This demonstrates the importance of the effect of structural details on fatigue life.

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Figure 7.

Longitudinal with attached shell plate.

Figure 6. Fatigue life comparisons models 1–4, FEA vs semi-empirical formula.

Table 1.

Fatigue life for different models (1–4).

Model no.

1#

2#

3#

4#

W (mm) Fatigue life (days)

150 94.3

200 83.7

250 76.1

300 70.3

3.2

Stiffener plate with attached shell plate (model II)

Figure 7 shows a typical longitudinal stiffener attached with shell plate. This was used to investigate the effect of the attached shell plate on SIF and fatigue life. The effect of the attached shell plate was simulated by the so-called equivalent plate concept; i.e. by un-folding the attached shell plate into the same plane of the longitudinal. With the help of FEA simulations, the equivalent plate width (b1) can be determined from: b1 = (1 + ϑ) b0

(11)

where:

ϑ = 0.125 ×

2 /t1

(12)

Figure 8. Geometry correction function comparisons between FEA, proposed formula and the basic formula of gross (1964).

and 20 mm, and the stiffener plate thickness (t1) was kept constant at 10 mm. Figure 8 presents a comparison between FEA and Equation (13) for t2 = 15 mm and the results from a specimen without an attached plate as represented by Gross (1964). The crack analysis on stiffener plate can be based on:

Therefore Equation 6 for the equivalent plate becomes: Fa

1.122 0.231(a/b1 a/b1 )2 3 −21.71(a/b1 ) + 30.382(a/b1 )4

(13)

Three different thicknesses of attached plate were investigated where t2 is 10 mm, 15 mm,

K

Fa (

) ⋅ fg

0

a

(14)

Figure 9 presents a summary of comparison of SIF between the semi empirical method and FEA. Good agreement is achieved over a range of typical crack lengths and plate thicknesses for t2 = 15 mm and 20 mm. For thinner plates (e.g. t2 = 10 mm) the numerical deviation is slightly higher (aprox. 4%).

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main advantage of the proposed method is that it is time efficient and hence useful for practical applications. 3.3

Figure 9. SIF comparisons between FE and semianalytical solution.

Stiffener plate with flat bar & attached shell plate (model III)

In this application example, plate thickness ratios between various width of the flat bar, and the attached plate on the longitudinal are considered for fatigue life analysis (see Figure 11). The K prediction formula will be modified by the two correction formulae of fg and Fa ( b1 ) as presented in previous sections. The calculated fatigue life for a crack length of 100 mm under stress of 50 MPa is presented in Table 3. It demonstrated that the new method can estimate the remaining fatigue life quickly after initial crack identified. The later suggests that subject to further testing and validation of this method, it is

Figure 10. Life comparisons between FEA and semiempirical formula, model 6 (t2 = 15 mm). Table 2.

Fatigue life comparison (formula vs. FEA).

Model no.

5#

6#

7#

t1 (mm) t2 (mm) 1+ϑ b1 (mm) Fatigue life by formula (days) Fatigue life by FEA (days) Difference %

10 10 1.125 562.5 169.9

10 15 1.15 576.5 170.1

10 20 1.18 588.5 170.3

172.7

173.0

173.2

1.61%

1.66%

1.66%

Figure 11. Illustrations of the equivalent model transformation from real components (intersection of flat bar & attached shell plate).

Table 3.

Fatigue life predictions determined by the proposed method and FEA results also compare well (see Figure 10). Table 2 shows a comparison of fatigue life predictions for crack lengths of the order of 100 mm and stress of 50 MPa. The

Formula applications on typical joints.

Model no.

8#

9#

10#

W (mm) t1 (mm) t2 (mm) 1+ϑ Fatigue life (days)

200 10 24 1.19 85.4

250 10 18 1.17 77.5

300 10 20 1.18 71.8

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under development may be useful for maintenance and repair plans. 4

CONCLUSIONS

This paper describes a fracture mechanics based methodology that may be used for the practical fatigue assessment of ship structures. The method is demonstrated by studying the crack propagation effects on the fatigue life of typical joint idealisations. Conclusions can be summarised as follows: 1. Suitable idealisation of the structural geometry plays a role for the evaluation of SIF. Modelling of the geometric discontinuities is critical. This is because they introduce singularities in the stress distributions and influence stress concentrations in way of critical structural geometries. 2. The method developed can predict the SIF by suitable combination of an empirical solution with FEA in way of the longitudinal stiffener end connections. 3. It is indicated that the main advantage of the proposed method is its time efficiency and good accuracy. In this sense, subject to further testing and validation, the method presented could be implemented for crack management procedures. 4. Future research will concentrate on more systematic numerical studies and experimental verification for various types of practical structural joints. ACKNOWLEDGEMENTS The views expressed in this paper are those of the authors and are not necessarily those of Lloyd’s Register. Dr. Benqiang Lou would like to acknowledge Professor Nigel Barltrop for his guidance, during his PhD studies at the Department of NAME, University of Strathclyde (UK). Lloyd’s Register and variants of it are trading names of Lloyd’s Register Group Limited, its subsidiaries and affiliates. Lloyd’s Register EMEA (Reg. no. 29592R) is an Industrial and Provident Society registered in England and Wales. Registered office: 71 Fenchurch Street, London, EC3M 4BS, UK. A member of the Lloyd’s Register group. Lloyd’s Register Group Limited, its affiliates and subsidiaries and their respective officers, employees or agents are, individually and collectively, referred to in this clause as the ‘Lloyd’s Register’. Lloyd’s Register assumes no responsibility and shall not be liable to any person for any loss, damage or expense caused by

reliance on the information or advice in this document or howsoever provided, unless that person has signed a contract with the relevant Lloyd’s Register entity for the provision of this information or advice and in that case any responsibility or liability is exclusively on the terms and conditions set out in that contract. REFERENCES ANSYS, 2005, Modelling and Meshing Guide, for ANSYS Release 10.0., ANSYS Inc. Canonsburg, PA, USA. BS7908, Code of practice for Fatigue design and assessment of steel structures, London, British Standards Institution, 1993a. BS7910, Guide to methods for assessing the acceptability of flaws in metallic structures, London, British Standards Institution, 2005. Gross, B., Srawley, J.E. and Brown, W.F., 1964, Stressintensity factors for a single-edge-notch tension specimen by boundary collocation of a stress function, NASA Technical Report TN D-2395. Hobbacher, A., 2009, Recommendations for fatigue design of welded joints and components, International Institute of Welding, WRC Bulletin 520, Welding Research Council, New York. Lloyd’s Register, 2013a, ShipRight FDA Fatigue Design Assessments Procedures, London, UK. Lloyd’s Register, 2013b, Rules and regulations for the classification of ships, London, UK. Lou, B.Q., 2012, A Geometric Methodology of Fatigue SCF and Fracture SIF Assessment, PhD thesis, University of Strathclyde, UK. Lou, B.Q., Zhang, S.M., Tong, J., Wong, S., and Hirdaris, S., 2014, Analysis methods for crack growth at the connections of longitudinal stiffeners and transverse frames of ship structures, ICTWS2014, ICTWS2014-120135, Busan. Love AEH. The Mathematical Theory of Elasticity. Dover: New York, 1844. Newman, J.C., and Raju, I.C., 1981 An empirical stress intensity factor equation for the surface crack. Engineering Fracture Mechanics, 51:185–192. Paris, P., Erdogan. F., 1963, A critical analysis of crack propagation laws, Journal of Basic Engineering, Transactions of the American Society of Mechanical Engineers, 85:528–534. Pilkey, W.D., 1997, Peterson’s Stress Concentration Factors 2nd Edition, John Wiley & Sons, Inc., New York, USA, ISBN 0-471-53849-3. Xu, T., and Bea, R., 1997, Load shedding of fatigue fracture in ship structures, Marine Structures, 10:49–80. Xu, L., Lou, B.Q., and Barltrop, N., 2013, Considerations on the fatigue assessment methods of floatingstructure details, Proceedings of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment, 227(3):284–294. ISSN 1475-0902.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

A study on the effect of hull girder vibration on the fatigue strength Masayoshi Oka & Toshio Niwa National Maritime Research Institute, Japan

Ken Takagi The University of Tokyo, Tokyo, Japan

ABSTRACT: The effects of superposed elastic vibrations on linear wave loads in the fatigue life estimation were researched. Using created time history of vertical bending stress, numerical simulations for fatigue damage of a post-panamax container ship has been conducted. Random stress time history is created using a storm-model, and a high frequency stress is proposed taking the effects of the hull girder vibration into account. Fatigue crack propagation which simulates the nonlinear retardation phenomena and Miner’s Law were applied. In a period of the maximum storm during ship’s life in the North Atlantic show that the fatigue crack length has a similar trend as the damage factor calculated by Miner’s Law. The difference of fatigue damage between RAW and Low Frequency waveform (LF) is remarkable, e.g. 50%–100%. The difference between RAW and envelope waveform (ENV) is relatively small, e.g. 0.1–2.0%. 1

INTRODUCTION

With the larger size of ships, the natural period of hull girder vibration is close to the encounter wave period. As a result, whipping and springing is easily to occur. In the structural design, high frequency stress caused by hull girder vibration has been assumed to be smaller than the quasi-static stress caused by ship motion. Thus, the effect of whipping and springing is considered to have been covered by safety factor. However, the range and incidence of vibration stress assumed to become higher as ship size is bigger. It is important to clarify the effect of the hull girder vibration on ship strength in order to ensure the future structural safety. Authors (Oka et al., 2011) researched the fatigue strength against a hull girder vertical bending stress of post-panamax container ship. The stress was obtained by on-board monitoring. Two types of stress waveform was prepared in order to confirm the effect of high frequency stress such as whipping. The one is RAW waveform, and the other is waveform of Low-Frequency component (LF). LF was obtained by removing the High Frequency component (HF) from RAW by low-pass filter. Fatigue damage factor (D) was calculated by rain-flow method and Miner’s law. The difference of D of RAW and D of LF are twice. However, in actual situation, the amount of fracture on large ships has not increased. The gap occurs between the estimation and actual situation. Two causes are considered as the reason for the gap occurs. The one is concerning with a low

pass filter. And, the other is the accuracy of the Miner’s law. LF has been used as a basis for the evaluation. But it has been a problem that low-pass filter decreases the peak value of stress range. Therefore, the peak value of the stress range is lower than the RAW, especially when the slamming occurs. It leads to an underestimation for D of LF. Therefore, it may be better compared with RAW and the envelope waveform (ENV) in order to compare with the superimposed effect of HF without reducing the peak value of the stress range. In this paper, the fatigue strength was examined using the three types of stress waveform RAW, LF and ENV. Fukasawa (Fuwasawa et al., 2013) calculated the long-term D using two types of stresses. One is RAW obtained by the nonlinear strip method as elastic hull. And the other is corresponding to LF which calculated as rigid hull. The difference of D of RAW and LF were significant approximately twice. And he pointed out it is necessary to simulate the nonlinear retardation phenomena of crack propagation. Miner’s law can’t take into account the retardation of crack growth according to the load transition. Therefore, the fatigue crack propagation analysis should be applied to simulate the retardation phenomena. Goto (Goto et al., 2013) and Sumi (Sumi et al., 2013) performed the fatigue test and fatigue clack propagation analysis using RAW, LF and ENV in order to confirm the tendency of crack growth under superimposed stress waveform. It was indicated that significant difference between

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the amount of crack growth of RAW and LF, and the difference between RAW and ENV was not significant so much. However, they did not use random loads, it is necessary to simulate the crack propagation under random loading conditions. In this study, random stress time histories were generated. And the fatigue crack propagation analyses were performed using random stress. A numerical program was developed to simulate the retardation phenomena. The program is in accordance with the cohesive force model that takes into account the plastic behavior of the crack tip. The fatigue strength based on crack growth was compared with the D of Miner’s law. Furthermore, it was examined the fatigue strength under the three types of stress waveform RAW, LF and ENV. 2

ACTUAL STRESS ON LARGE SHIPS

Figure 1.

Longitudinal bending stress of the large ship have the multi-peak spectrum composed with high frequency components due to longitudinal bending vibration and a low frequency component in the vicinity of wave encounter frequency. In the analysis of the actual ship’s stress, low frequency component is often referred as wave component (e.g. Okada et al., 2006, Heffelund et al., 2011). 3 3.1

EVALUATION METHOD FOR FATIGUE STRENGTH

D Ni

case of applying the loading pattern and/or structure style beyond the scope of the fatigue test. On the other hand, fatigue crack propagation shown in equation (2) can be evaluated the effect of the loading pattern and the damage condition becomes clear. The authors thought that the evaluation utilized by the crack propagation analysis is necessary in addition to Miner’s law, in order to prevent inexperienced fracture on new type ship or new structure style. da m =C( K) dN

Miner’s law and crack propagation analysis

Figure 1 shows flows of fatigue crack propagation analysis and Miner’s law. Miner’s law is adopted current fatigue strength evaluation for ship structure. In Miner’s law, long term distribution of stress range fit the S-N curve to estimate a fatigue damage factor (D). Rain flow method is generally applied in order to obtain the frequency of stress range from the stress waveform. The Miner’s law is shown in equation (1).

∑ (n j

Nj

)

(1)

K SM j

where: D : Damage factor Sj : Stress range on rank j nj : Number of cycles at Sj Nj : Number of cycles to failure at Sj K, M : Constants of S-N curve. Miner’s law can’t express the damage condition. In addition, S-N curve is obtained by the fatigue test. Therefore, it is necessary to pay attention in

Flows of each fatigue strength evaluation.

(2)

where: a : Clack length N : Number of load cycle ΔK : Range of stress intensity factor C, m : Constants of fatigue crack propagation law. 3.2 Crack propagation analysis program The numerical simulation program is necessary to follow the plastic behavior of the crack tip depending on the variation at every load cycle. In this study, we have developed a program in accordance with the methodology of Goto (Goto et al., 2013) and Sumi (Sumi et al., 2011). This program has a function extracting the effective load cycle for crack growth. Moreover, the fatigue crack propagation analysis based on cohesive force model focuses on the plastic behavior of the crack tip (Toyosada & Niwa, 2001) was adopted in order to simulate the retardation phenomena. Figure 2 shows a comparison of numerical simulations and fatigue tests. Fatigue tests under

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p(

La ) L

=

⎛ σ LLa 2 ⎞ σ La exp p ⎜− 2 ⎟ R2 ⎝ 2R ⎠

∞

∫−−∞ S (

R2

(5)

)dω

(6)

where: σLa : Stress amplitude of LF R : Standard deviation. 4.3 High frequency component waveform (HF)

Figure 2. An example of verification of analysis program.

superimposed stress waveform were reported by Kitamura (Kitamura et al., 2012). The material constants C and m was using the value indicated by Goto (Goto et al., 2013). Fine estimation is indicated. 4

EVALUATION METHOD FOR FATIGUE STRENGTH

4.1 Object of evaluation The vertical bending stress acting on the deck structure of the post-Panamax container ship was evaluated. Analyzed period was assumed to the one storm (about 5 days), and created a storm model (Tomita et al., 1995). 4.2 Low frequency component waveform (LF) The stress of LF in the short-term sea state was created utilizing superimposition the wave spectrum such as P-M type or JONSWAP type and the stress RAO. The stress RAO is determined by the direct loads and structure analysis utilizing 3D FE model. The wave form was obtained by inverse Fourier transform of RAO.

σ L (t) S( )

∞ iω t

∫0

( )2

2S (ω )dω

(3)

Sw ( )

(4)

The stress of HF was expressed the initial amplitude caused by the bow slamming (σslam) followed damping vibrations, expressed in equation (7). When the bow slamming occurs, compression stress acts on the deck structure.

σ H (t)

⎛ δt ⎞ σ sllam exp ⎜ − cos (ω ⎝ T2 n ⎟⎠

+ ε)

(7)

where: σH : Stress of HF σslam : Initial stress amplitude caused by the bow slamming T2n : Natural period of 2-node hull-girder vibration(s) δ : Logarithmic damping coefficient ε : Phase of whipping. 4.4 Initial stress amplitude (σslam) The value of the σslam was decided in accordance with the following assumptions. 1. Maximum value in one storm The maximum stress during one storm occurs under the worst sea state that is the maximum significant wave height (Hmax). It was decided that the maximum stress is not over the yield stress of the material. The maximum stress is obtained by adding three kind of stress component. First one is the maximum expected value of σslam. Second one is the maximum expected value of LF stress in worst sea state. Third one is the mean stress according to the static load. The maximum of σslam in one storm is expressed in equation (8) to summarize.

where: σL(t) : Stress of LF S(ω) : Spectrum of response G(ω) : Stress RAO Sw(ω) : Wave spectrum. Generally, the frequency of the stress amplitude follows a Rayleigh distribution expressed in equation (5).

σ slam _ max

σY − σ mean

where: MAX X1/ N [

L ]i max

N σY σmean

MAX

/ N [σ L ]i max

(8)

: The maximum expected value of LF stress in Hmax : Number of encounter wave in short sea state : Yield stress of the material : Mean stress.

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2. Short term maximum expected value σslam was decided in relation to the ship motion. In accordance with momentum theory, excitation force due to slamming is proportional to the square of the relative velocity between the hull and the wave. And the relative velocity is proportional to the wave height. Therefore, the short-term maximum expected value of σslam can be assumed to be proportional to the square of the significant wave height (H). The short term maximum expected value of σslam is expressed by the equation (9) to summarize. 2

MAX X1 / NSi [

slam ]i

where: MAX X1/ NSi [

sslam ]i

(9)

sslam )

RH

=

⎛ σ sslam 2 ⎞ σ slam exp ⎜− 2⎟ RH 2 ⎝ 2RH ⎠

M MAX X1 / NSi [

]

sslam i

2 ln Ni +

⎧⎪ d 2f V 2 ⎫⎪ exp ⎨− 2 − cr2 ⎬ ⎪⎩ 2Rzr 2Rvr ⎪⎭

(11)

where: Qsl : Probability of occurrence of slamming Rzr : Standard deviation of the relative distance between hull and wave df : Height of target position Rvr : Standard deviation of the relative velocity between hull and wave Vcr : Rvr corresponding to limitation of slamming occurrence. 4.6

⎛ Hi ⎞ σ sslam _ max ⎝ H max ⎠

: The maximum expected value of σslam in i-th short term sea state Hi : Significant wave height of i-th short term sea state Hmax : Maximum significant wave height NSi : Number of occurrence of whipping in i-th short term sea state (= Pi × Ni) Pi : Probability of occurrence of whipping in i-th short term sea state Ni : Number of encounter wave in i-th short term sea state. 3. Probability distribution in short term sea state The probability distribution of σslam in short term sea state was assumed Rayleigh distribution following the Jiao (Jiao & Moan, 1990). The Rayleigh distribution expressed by the equation (10). p(

Qsl ( H ,T )

Waveform of random stress

The waveform of random stress (subscript RAW) is expressed by sum of LF stress (subscript L) and HF stress (subscript H).

σ RAW (t) = σ L (t) + σ H (t)

(12)

The effect of whipping on the fatigue strength is investigated by comparing with the analysis results of following waveforms. − Superimposed waveform (RAW) − Low frequency component waveform (LF) − Envelope waveform (ENV) An example of the stress waveforms are shown in Figure 3. The peak stress of LF is lower than RAW because of low pass filter. With regard to ENV, peak stress is equal to RAW. In addition, the number of mean cross of ENV is same as LF.

(10)

γ 2 ln Ni

where: P(.) : Short term probability distribution of σslam RH : Standard deviation of σslam γ = 0.5772 (Euler’s constant). 4.5

Probability of whipping occurrence

Probability of whipping occurrence is equal to the probability of slamming occurrence. It can be determined from the relative velocity between the hull and wave in accordance with Ochi’s (Ochi, 1967) equation (11).

Figure 3. analysis.

Examples of time histories for comparative

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5

curves were applied. One is for the base material (B-curve) and the other one was for the butt welding (D-curve) in accordance with the CSR-T (IACS, 2012). The constant parameter of S-N curves are shown in below. These S-N curves have Haibach correction in load level 107.

FATIGUE STRENGTH ANALYSIS

5.1

Analysis condition

1. Environmental condition The maximum storm in the North Atlantic during 25 years was assumed as the environmental condition. Number and sequence of H in the storm were decided in accordance with the model proposed by Kawabe (Kawabe et al., 2003). The mean wave period of short-term sea was decided according to ITTC equation (13) (Price, 1974). T

3.86 H

(13)

where: T : Mean wave period (s) H : Significant wave height (m). Profile of the storm is shown in Table 1. The storm composed by 109 short-term seas, and the period of the short-term sea was set at 1.2 h. Wave spectrum of short-term sea was adopted a P-M type of ISSC1964 spectrum. The number of frequency component to generate the time history was set to 20. Stress RAO was obtained by the strip method and whole ship FE analysis (Oka et al., 2011). The operational condition was set to 3/4 of service speed and head sea. Moreover, 100 MPa static tensile stress was acted considering with the hogging condition of container ship. The natural period and the logarithmic decay rate of hull girder vibration were assumed to be 1.3 s and 0.1. 2. Probability of whipping occurrence Qsl was calculated as Vcr = 4.27 m/s and df = 8 m in accordance with Ochi (Ochi, 1967). The lower limit σslam was provided, therefore whipping does not occur when less than 10 MPa of σslam. 5.2

Result

1. Miner’s law D was calculated by means of the rain-flow method and Miner’s law. Two kinds of S-N Table 1.

K = 1.013E15, M = 4.0 (Sj > 100.2 MPa) B-curve K = 1.020E19, M = 6.0 (Sj  100.2 MPa) B-curve K = 1.519E12, M = 3.0 ( Sj > 53.4 MPa) D-curve K = 4.239E15, M = 5.0 (Sj  53.4 MPa) D-curve The frequency distributions of stress range by rain-flow method are shown in Figure 4. LF and ENV is close to the Rayleigh distribution. RAW is close to the shape of the exponential distribution. The cumulative D is shown in Figure 5. Firstly, comparing with the RAW and ENV, the difference in D is not significant 2–5%. In Figure 4, histogram of ENV become the large part of histogram of RAW. It can be said that the vibration stress, which account for much of RAW, is not so effective for D. Secondly, comparing with the RAW and LF, the difference in D is about twice. The significant effect is confirmed. 2. Fatigue crack propagation analysis Fatigue crack propagation analysis using the random stress time histories were performed. The analysis conditions are shown in below. – – – – – –

E = 206 GPa, Poisson’s ratio 0.3 Yield stress of the material 460 MPa Initial Crack (a0 = 5 mm, 20 mm, 80 mm) Uniform stress of-infinite plate Plastic constraint factor λ = 1.21 (plane strain) Material constant (Toyosada, 2001): C = 4.505e-11, m = 2.692 (SI unit)

The results of is shown in Figure 6 (The curve of RAW and ENV is over lapped in case of a0 = 80 mm). With regard to the difference of the initial crack size, the order between the RAW, LF and ENV unchanged. Therefore, the influence of the initial crack size is considered negligible in this study.

Composition of short-term significant wave height in the maximum storm.

i

1

2

3

4

5

6

7

8

9

10

11

H T Ns

5.5 8.5 10

6.5 9.5 10

7.5 9.5 9

8.5 10.5 9

9.5 10.5 8

10.5 11.5 3

11.5 12.5 2

12.5 12.5 1

13.5 13.5 1

14.5 13.5 1

15.5 13.5 1

i

12

13

14

15

16

17

18

19

20

21

Total

H T Ns

14.5 13.5 1

13.5 13.5 1

12.5 12.5 1

11.5 12.5 2

10.5 11.5 3

9.5 10.5 8

8.5 10.5 9

7.5 9.5 9

6.5 9.5 10

5.5 8.5 10

109

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Figure 5. law.

Cumulative fatigue damage based on Miner’s

Figure 6.

Simulated fatigue crack propagations.

Figure 4. Example of the stress range histogram analyzed by Rain-Flow method. (H = 15.5 m).

The fatigue crack growth can’t directly compare with D because Miner’s law can’t express the damage condition, discussed in 3.1. Therefore, the relative relationship between RAW, LF and ENV is discussed here. The crack growth curves on RAW, LF and ENV close to the trend of D. It is considered the effects of whipping in single storm condition may be evaluated in Miner’s rule. However, the trend of D shown in Figure 5 is the shape of the point symmetry on 66 hour, but crack growth is not. It caused

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by the retardation of crack growth. It is necessary to examine in multiple storm condition and confirm the effect of the retardation of crack growth during long term. Moreover, it is necessary to perform considering with hot-spot stress although the nominal stress was used in this study. These are to be further study. With regard to the effect of whipping, the difference of crack growth between RAW and ENV is 0.1 to 2.0%. Similar results indicated by Goto (Goto et al., 2013) and Sumi (Sumi et al., 2013) were obtained. Therefore, it is considered that simple analysis under random loading condition is possible. The reduction of analysis time will be possible due to reduce the number of cycles by means of ENV waveform. On the other hands, the difference between RAW and LF appear 1.5 times. It is discussed with respect to the gap between the actual situations that described in introduction. Even if utilizing the fatigue crack propagation analysis simulating the retardation phenomena, the difference between LF and RAW is still significant. On the other hand, ENV is close to RAW. It was not possible to clear the gap in this study. The reason for this is considered the effects of multiple storm and hot spot stress. Moreover it is considered the influence of maneuvering that is not considered explicitly in the design. In future, to clarify these are considered important. 6

CONCLUSIONS

Regarding to the fatigue strength of deck structure in container ship, it is performed the fatigue crack propagation analysis taking account of retardation phenomena under random loading conditions. The relationship between three type of stress waveform RAW, LF and ENV were evaluated. It is concluded that, 1. Fatigue crack propagation analysis and Miner’s law indicates same tendency in maximum storm at the North Atlantic during 25 years. Therefore, it is considered that the effect of whipping in one storm period can be evaluated in Miner’s law. 2. The differences in the fatigue strength between RAW and LF is about 100% in Miner’s law, and about 50% in the fatigue crack propagation analysis. The difference between LF and RAW was still significant. And, the difference between ENV and RAW was found to be from 0.1 to 2.0%. It is considered that simple analyses under random loading conditions are possible by means of ENV waveform. 3. In this study, it was not possible to clear the gap between the actual situations. The major reason is considered the effects of multiple storm and hot spot stress. Moreover it is considered the

influence of maneuvering that is not considered explicitly in the design. In future, to clarify these are considered important. ACKNOWLEDGEMENTS This study was supported by a Grant-in Aid for Scientific Research of the Japan society for Promotion of Science (No. 24560995). REFERENCES Fukasawa, T. & Mukai, K. 2013. On the effect of hullgirder vibration upon fatigue strength of a Post-Panamax container ship disaggregated by short-term sea state, PRADS 2013, pp. 479–485. Goto, K. & Matsuda, K. 2013. Numerical Simulation of Fatigue Crack Propagation under Variable Amplitude Loading Containing Two Different Frequency Components, Journal of the Japan society of naval architects and ocean engineers, vol. 17, pp. 75–81. (in Japanese). Heggelund, S., Gaute S. & Byong-Ki C. 2011. Full scale measurement of fatigue and extreme loading including whipping on an 8600TEU post panama container vessel in the Asia to Europe trade, Proceedings of 30th International Conference on Ocean, Offshore and Arctic Engineering (OMAE2011), Rotterdam, the Netherlands. IACS (International Association classification society)/ Common structural rules for double hull oil tankers, Rules July 2012. Jiao, G. & Moan, T. 1990. Probabilistic analysis of fatigue due to Gaussian load processes, Probabilistic Engineering Mechanics, vol.5, pp. 76–83. Kawabe, H., Oka, S. & Oka, M. 2003. The study of storm loading simulation model for fatigue strength assessment of ship structural members -1st report new storm loading simulation model which is consistent with a wave frequency table-, Journal of the society of naval architects of Japan, vol193, pp. 39–47. Kitamura, O, Sugimura, T., Nakayama, S. & Hirota, K. 2012. A Study of the Results of Fatigue Crack Propagation Tests under Combination Loads of High & Low Cycles consideration to the Effect of Whipping on the Fatigue Strength, Conference proceedings of the Japan society of naval architects and ocean engineers, vol.14 pp. 13–16. Ochi, Michel K. 1967. Ships slamming-hydrodynamic impact between waves and ship bottom forward, Symposium on Fluid-Solid Interaction, ASME, pp. 223–240. Oka, M., Takami, T., Ogawa, Y. & Takagi, K. 2011. A study of design loads for fatigue strength utilizing direct calculation under real operational conditions, Advances in Marine structures (MARSTRUCT2011), pp. 317–323. Oka, M., Ogawa, Y. & Takagi, K. 2011. A fatigue design for large container ship taking long-term environmental condition into account, Proceedings of 30th International Conference on Ocean, Offshore and Arctic Engineering (OMAE2011), Rotterdam, the Netherlands.

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Okada, T., Takeda, Y. & Maeda, T. 2006. On board measurement of stresses and deflections of a Post-Panamax containership and its feed-back to rational design. Marine Structures, No.19 Elsevier Ltd: 141–172. Sumi, Y., Takasaki, S. & Hayakawa, G. 2013. Effects of Superimposed Stress of Small Amplitude to Fatigue Crack Propagation, Conference proceedings of the Japan society of naval architects and ocean engineers, vol16, pp. 519–522. Toyosada, M. & Niwa, T. 2001. Fatigue Life Assessment for Steel Structures, Kyoritsu Shuppan Co., Ltd. on Dec. 2001.

Tomita, Y., Hashimoto, K., Nagamoto, R., Kawabe, H. & Fukuoka, T. 1995. Stochastic characteristic of long term distribution of wave-induced loading an simulation method for fatigue strength analysis of ship structural member (3rd report), Journal of the society of naval architects of Japan, vol.177, pp. 381–390 (in Japanese). W.G. Price & R.E.D. Bishop 1964. Probabilistic Theory of Ship Dynamics, Chapman and Hall Ltd, 1974, pp. 161.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Fatigue strength of welded extra high-strength and thin steel plates H. Remes, M. Peltonen, T. Seppänen, A. Kukkonen, S. Liinalampi, I. Lillemäe, P. Lehto, H. Hänninen & J. Romanoff School of Engineering, Aalto University, Espoo, Finland

S. Nummela Ruukki Metals Oy, Finland

ABSTRACT: Weight reduction of structures is possible using thinner plates, i.e. with thickness between 3 and 4 mm, and high-strength steels. However, current design rules do not support their utilization in ship structures. This paper investigates the fatigue strength of welded Extra High-Strength (EHS) and thin plate structures. Fatigue tests were carried out for butt welded specimens and the results were analysed using the structural stress approach, which considerers the actual geometry of the specimens. The results show that the present nominal stress design S-N curve FAT80 with the slope value of three is applicable to thin welded EHS plates, but it is somewhat conservative especially at high cycle regime. The fatigue test results in terms of structural hot-spot stress are significantly above the design S-N curve FAT100. For the welded thin EHS plates, the slope value of five shows better agreement when the structural hot-spot stress approach is applied. 1

INTRODUCTION

To improve the energy efficiency and load carrying capacity, new light-weight solutions are required. The weight reduction of current ship structures is possible using higher strength steel and thinner plates, i.e. with thickness between 3 and 4 mm. However, the utilization of advanced solutions is limited due to lack of knowledge about the fatigue behaviour of these materials. Fatigue strength of thin plate ship structures has been recently investigated by several researchers. In thin plates, the main challenge is distortions caused by the fabrication process (Lillemäe et al. 2012, Eggert et al. 2012, Fricke et al. 2013). Misalignments, particularly the angular misalignment, are harmful since they cause additional secondary bending stress at the weld fusion line and, thus, reduce the fatigue strength. In addition, thin plate structures are sensitive to welding-induced flaws such as undercut (Remes and Fricke, 2014). Previous investigations show also that by using low heat-input welding and welldefined welding parameters, these challenges can be avoided and similar fatigue strength properties can be obtained as those of thick plates both in terms of nominal and structural stress (Fricke et al. 2013, Remes and Fricke 2014, von Selle, 2013). However, these investigations were limited to steels with the yield strength of less

than 390 MPa. Thus, they did not consider Extra High-strength Steels (EHS). Fatigue strength of welded EHS joints are investigated, e.g., by Herzog and Stein (2008) and Costa et al. (2010). The yield strength of the base material was 690 and 670 MPa, respectively. While Costa et al. (2010) reported considerable increase in the fatigue strength compared to IIW recommendations (Hobbacher 2009), very small improvement was seen in the results by Herzog and Stein (2008). High variation in fatigue strength of EHS joints is also visible in FATHOMS (2010). In these studies the analysis was mainly limited to nominal stress approach and, thus, deeper understanding of the influence of misalignments, weld geometry and material yield strength was not obtained. At present, large variation in the results and limited amount of available test data is preventing the utilization of higher fatigue strength of EHS welded joints. The objective of this paper is to study the fatigue strength of welded high-strength and thin plate structures. The experimental and numerical investigations are focused on the EHS joints, which were welded by three different arc welding methods. A special emphasis is given to the influence of misalignments and weld geometry on fatigue strength. The fatigue test results were analysed by nominal and structural stress approaches. In the latter analysis, the actual geometry of the specimen was taken into account.

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2

FATIGUE EXPERIMENTS

2.1

Fatigue test specimens and program

The fatigue test specimens were water-cut from the Metal Active Gas (MAG), Plasma Arc (PAW) and Submerged Arc Welded (SAW) extra high-strength steel plates with 4 mm plate thickness t. The length of the specimens was 252 mm and the width varied from 18 mm in the middle, where the weld was positioned, to 52 mm at the ends, according to the hourglass shape with the radius of 159 mm as shown in Figure 1. The specimen edges were grinded and two holes were drilled at both ends for clamping. The yield strength of the base material is 900 MPa for the welded specimens. The chemical composition of the base material is given in Table 1. The fatigue tests consisted of 28 butt welded specimens. All specimens were processed using one-sided mechanical welding. For submerged arc welding the composite backing rod was used on the root side, while metal active gas and plasma welding was performed without backing rod. Out of 28 specimens six were plasma arc welded, ten were submerged arc welded and twelve were metal active gas welded as shown in Table 2.The different welding methods led to different weld geometries and heat-affected zones, see Figures 2–4. Welding

Figure 1.

Material

C

Si

Mn

P

S

Ti

Optim 900 QC

0.1

0.25

1.15

0.020

0.010

0.070

Fatigue test program.

Series Joint description symbol welding method PAW SAW MAG

Figure 2.

Macro-section of the PAW specimen.

Figure 3.

Macro-section of the SAW specimen.

Figure 4.

Macro-section of the MAG specimen.

Fatigue test specimen.

Table 1. Chemical composition of the base material, maximum content in wt. % (ladle analysis).

Table 2.

parameters are shown in Table 3. In plasma arc welding, the power supply used was LTG 400 and control panel was PW 3000. The welding system for submerged arc welding was ESAB A6 with power supply ESAB LAE 1250. In metal active gas welding, the welding system was Kemppi FastMig Pulse 450 and power supply was FastMig MXF

t Number [mm] of tests

Plasma arc welded butt joint 4 Submerged arc welded butt joint 4 Metal active gas welded butt joint 4

6 10 12

Table 3. Parameters of plasma, Submerged Arc (SAW) and Metal Active Gas (MAG) welding. Welding method Welding parameter Voltage U (V) Current I (A) Welding speed v (m/min) Welding wire Arc energy; U⋅I/v (kJ/mm)

Plasma

SAW

MAG

26.4 214 0.32

28.5 501 1.00

29 330 1.00

Coreweld 89 1.06

OK Tubrod 15.27S 0.86

Coreweld 89 0.57

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Table 4. Test series

Tensile test results for welded joints. Yield strength Tensile strength Elongation to Rm [MPa] fracture A [%] ReH [MPa]

PAW 647 SAW 622 MAG 729

807 821 873

8.8 8.5 6

Figure 7. Test arrangement with the servo hydraulic test machine and the special rotating clamping device designed for testing the specimens with high angular misalignment. Figure 5. Example of 2D cross-section of the SAW specimen.

Figure 6.

Axial e and angular α misalignments.

65. Mechanical properties of the welded specimens from the tensile tests are shown in Table 4. For all welded joints the strength mismatch can be observed as the yield strength is lower than the nominal yield strength of the base material, i.e. 900 MPa. 2.2

Geometry measurements

The geometry measurements of the specimens were performed using laser measuring system with the measuring spot size of 30 × 70 μm. The geometry points were recorded with two laser sensors simultaneously on both top and bottom surfaces of the specimen in order to obtain a full 3D geometry. From this data the whole shape of the specimen was captured including the weld geometry and the distortions of the plates as shown in Figure 5. The geometry data was measured for all test specimens and it was utilized later to define the misalignments (Figure 6) and to create finite element model for structural hot-spot stress analysis. 2.3

were attached onto the middle of the specimens on the top and bottom surfaces close to weld fusion line. During the tests the applied force and cycles to failure were recorded. For fixing the specimens, special clamping device was used as shown in Figure 7. The device allows the rotation of the specimen ends, and thus, avoids bending stresses caused by misalignments during the clamping. After clamping the rotation movement is restricted by locking the rotating wheels. In fatigue tests, a load frequency of 5 Hz and load ratio of R = 0.1 were used. At the beginning of each test 3 cycles were first performed with a frequency of 1/60 Hz in order to have reliable strain measurement for the validation of numerical simulation. The definition of cycles to failure was the occurrence of the final fracture. If fatigue failure was not observed before 2 million load cycles, the test was considered as a Run-out.

Fatigue tests

Fatigue tests were done using servo hydraulic MTS 810 machine. Strain gages with a gage length of 5 mm were used for strain measurement. They

3

STRUCTURAL STRESS ANALYSIS

The structural hot-spot stress for the crack initiation location was calculated for each test specimen using the Finite Element (FE) method and plane stress elements with linear shape functions. A 2D plane stress element model was created based on the measured geometries similar to Lillemäe et al. (2012). Abaqus version 6.13 was used including geometrical non-linearity, i.e. the load was applied step by step and geometry was updated for each step according to specimen straightening. The analysis used linear elastic material model with a Young’s modulus of 207 GPa and a Poisson’s ratio of 0.3. The element size at the weld area was less than 0.2 mm and elsewhere about 0.5 mm. The FE simulations were validated with the measured strains 5‒8 mm from the weld fusion line and about ±5% accuracy was observed.

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Figure 8. FE-model of butt joint and trough-thickness linearization for hot-spot structural stress.

The structural stress was defined using throughthickness linearization approach similar to the previous study for thin plates (Remes and Fricke, 2014). This approach neglects the nonlinear stress peak induced by the weld notch, but it considers the secondary bending stress at the location of the crack initiation caused by misalignments as shown in Figure 8. The structural stress σs, composed of membrane σm and bending stress σb, is calculated from normal stress distribution σxx:

σS = σm + σb t

σS

1 c 6 σ xx z dz + tc ∫0 tc

(1a) tc

∫ σ xx ( z ) (tc

− z ) dz d , (1b)

0

where tc is the thickness of the critical crosssection. 4 4.1

Figure 9. The minimum, maximum and average value of axial misalignment for PAW, SAW and MAG joint.

RESULTS Joint geometry

The summary of the measured axial and angular misalignments is presented in Figure 9 and 10. The minimum, maximum and average values are determined for all test series and compared with the limit values from ISO 5817 (2003). As shown in Figure 9 the axial misalignment is close to zero for all joints and, thus, well below the both ISO limit values: e/t = 0.15 and e/t = 0.1. The angular misalignment, given in Figure 10, varies between the joints. The highest values and variation of angular misalignment were observed for PAW joints, which also have the highest welding energy, see Table 3. In this case, some values are above the limit value of α = 2 deg. The smallest angular misalignment was observed in SAW joints, where the weld width is almost equal for both top and bottom surfaces as seen in Figure 3. For SAW joints, all angular misalignment values are within the limit of α = 1 deg. For MAG the maximum value is slightly smaller

Figure 10. The minimum, maximum and average value of angular misalignment for PAW, SAW and MAG joint.

than the limit value of α = 2 deg. For all joints, the mean value of angular misalignment is negative, which means that the highest stresses occur at weld root side; see Figure 8. 4.2

Fatigue strength

In the fatigue tests, the crack initiated at the weld fusion line of all specimens. However, the side of the failure location varied between joints. For PAW and MAG joints the fatigue crack was initiated only from the root side while for SAW joints the crack initiated from both toe and root sides. The failure location, fatigue life Nf, nominal stress Δσnom and structural hot-spot stress range Δσs, are summarised in Table 5. The nominal stress range was calculated based on the applied load range ΔF, specimen width at the failure location w and the plate thickness t: Δσ nom =

ΔF . w t

(2)

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Table 5.

Fatigue test results.

Specimen

Nf

Failure

Δσnom

Δσs

km1

PAW-41 PAW-42 PAW-43 PAW-44 PAW-45 PAW-46 MAG-26 MAG-27 MAG-28 MAG-29 MAG-30 MAG-31 MAG-32 MAG-33 MAG-34 MAG-35 MAG-36 MAG-37 SAW-8 SAW-9 SAW-10 SAW-11 SAW-12 SAW-13 SAW-14 SAW-15 SAW-16 SAW-17

689 266 122 109 333 571 201 331 3 000 000 373 119 487 891 384 407 329 513 3 000 000 252 466 3 000 000 354 181 165 495 572 407 117 149 496 124 464 658 48 284 1 101 449 526 967 698 534 99 549 122 989 747 222 349 101 66 950 1 451 573

Root Root Root Root –2 Root Root Root Root –2 Root –2 Root Root Root Root Root Root Toe Toe Root Toe Root Toe Toe Toe 3 Toe Root

358 380 266 265 182 200 198 185 195 147 199 127 184 307 214 300 205 235 474 262 287 251 408 403 288 271 405 252

397 459 371 437 363 414 271 265 281 236 329 211 278 345 265 380 254 279 492 264 367 245 543 350 257 344 441 308

1.11 1.21 1.39 1.65 1.99 2.07 1.37 1.43 1.44 1.61 1.66 1.67 1.51 1.12 1.24 1.27 1.24 1.19 1.04 1.01 1.28 0.98 1.33 0.87 0.89 1.27 1.09 1.22

Figure 11. Fatigue test results in terms of nominal stresses. The characteristic fatigue strength Δσ97,7% (−), Run-out—tests ( ) and design S-N curve (Hobbacher 2009) are also presented.

1. Value was calculated for crack initiation location, thus km < 1 is possible. 2. Run-out—test. 3. Failure on root side was also observed.

The ratio between the structural hot-spot and nominal stress range defines the stress magnification factor km for the crack initiation site and, thus, illustrates the severity of misalignments. The km factors had significant variation corresponding to the high variation of angular misalignments. The highest km factors are observed for PAW specimens, but also for other joints the maximum value of km factor is higher than km = 1.30, which is applied for butt joints in IIW recommendations (Hobbacher 2009). Fatigue test results in terms of nominal stress ranges are presented in Figure 11. A high variation in test results is observed especially for PAW joint. The scatter index Tσ for all tests results is 1:1.92. This is higher than commonly observed (1:1.41) for thick plates (Radaj 2009), but similar to the previous results for normal strength thin plates (Fricke et al. 2014). Using the slope of S-N curve m = 3, which gives reasonable fit for the test data, the fatigue strength at the survival probability of 97.7% Δσ97,7% is 91 MPa. This is only slightly

Figure 12. Structural stress results for PAW joint. The characteristic fatigue strength Δσs_97.7% (−), Run-out—tests ( ) and design S-N curve FAT100 are also presented.

higher than FAT80 given by IIW recommendations (Hobbacher 2009). However, the FAT80 design curve is conservative at high cycle regime in comparison to the test data. The highest fatigue strength is observed for SAW joint with very small misalignments. For this series, also the slope value of 5 agrees better with the test data. 4.3

Fatigue strength in terms of structural stress

The structural hot-spot stress approach was applied to investigate the influence of the misalignments

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Table 6. Fatigue strength values (structural hot-spot stress), the “best” and “worst” weld profiles for the studied joints.

Figure 13.

Figure 14.

Structural stress results for SAW joint.

Structural stress results for MAG joint.

and weld geometry on fatigue strength. The structural stress results are plotted in Figures 12–14 for PAW, SAW and MAG joints, respectively. Table 6 presents the characteristic fatigue strength values and the weld profile variation, i.e. the “best” and “worst” weld shapes. In the case of structural stress analysis, the scatter of the test data is reduced in comparison to the nominal stress analysis. Using the fitted slope of S-N curves, the scatter range index Tσ was less than 1:1.13 for PAW and MAG joints. For these joints, weld geometry did not vary in welding direction and it is almost equal for all specimens. For SAW joints the variation of weld notch quality is higher and undercut was observed on weld toe side for

some specimens. Consequently, the scatter range index was higher, i.e. Tσ = 1:1.49 for SAW joints. Similar to the scatter of test results, the weld geometry affects the fatigue strength and the slope value. PAW joints with a smooth geometry had the highest characteristic fatigue strength and slope value. For SAW joints the fatigue strength was lower than that of PAW joints due to the top side undercut. MAG joints had a high flank angle, which reduced the fatigue strength. In addition, the slope values of SAW and MAG are smaller than that of PAW. The characteristic fatigue strength for EHS joints was above 150 MPa, and significantly higher than that of the design curve FAT100 given by IIW recommendations (Hobbacher 2009). In addition, the slope value m = 5 agrees better for the test data than the commonly used slope value m = 3. Similar finding about the slope value was observed in the previous studies for normal strength thin plates (Lillemäe et al. 2012, Fricke et al. 2014). However, the fatigue test data for thin and normal strength plates were not significantly above of the design curve FAT100 differing from the results for EHS joints, see Table 6 and Figures 12–14. 5

DISCUSSION AND CONCLUSIONS

Fatigue tests were performed for thin and extra high-strength butt joints welded by plasma arc, submerged arc and metal active gas welding methods. Welding distortions and misalignments were measured for all specimens. Furthermore, the nominal and structural hot-spot stress analyses were carried out. The angular misalignments considerably affect the fatigue strength when nominal stress approach was applied. This is visible in large scatter of the

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results. The fatigue strength is above the fatigue class FAT80 used in design codes although the stress magnification factor km was much higher than considered, e.g., in the IIW recommendations (Hobbacher 2009). Thus, the results indicate that the S-N curve FAT80 with the slope value of m = 3 is applicable for thin welded EHS plates, but it gives somewhat conservative estimation at high cycle regime. Based on the structural hot-spot stress, which includes the effects of angular misalignment, all results are significantly above the relevant fatigue class FAT100. The fatigue strength in terms of the structural stress was the highest for PAW joints with a smooth weld geometry and, thus, the results indicate that fatigue strength of EHS plates is strongly dependent on the weld quality. This finding is different from the previous studies for thin normal strength joints, where fatigue class FAT100 shows good agreement with the test data (Fricke et al. 2014). Similar to the previous studies for thin plates (Lillemäe et al. 2012, Fricke et al. 2014), the present results showed that the slope value of m = 5 gives better agreement when the structural hot-spot stress approach is applied. The present study was limited to the nominal and structural hot-spot stress analyses. Therefore, further investigations are required on the influence of the weld notch shape using the local approaches such as the notch stress approach. Furthermore, the testing of large-scale EHS structures is recommended to consider all influencing factors such as different loading types and residual stress effects. ACKNOWLEDGEMENTS The research has received funding from TEKES, the Technology Agency of Finland. The work is part of the Light and Breakthrough Steel Application project within the scope of the Finnish Metals and Engineering Competence Centre. The financial support is gratefully appreciated.

Eggert, L.; Fricke, W.; Paetzhold, H. 2012. Fatigue strength of thin-plated block joints with typical shipbuilding imperfections. Weld World 56(11–12): 119–128. FATHOMS, 2010. Fatigue behaviour of high-strength steel-welded joints in offshore and marine systems (FATHOMS). European Commission, Research Fund for Coal and Steel. Fricke, W.; Remes, H.; Feltz, O.; Lillemäe, I.; Tchuindjang, D.; Reinert, T.; Nevierov, A.; Sichermann, W.; Brinkmann, M.; Kontkanen, T.; Bohlmann, B.; Molter, L. 2013. Fatigue strength of laser-welded thin-plate ship structures based on nominal and structural hot-spot stress approach. Ships and Offshore Structures, DOI: 10.1080/17445302.2013.850208 Herzog, D.; Stein, J. 2008 Induction assisted welding technologies in steel utilisation—INDUCEWELD; RFSR-CT-2005–00040; Res. Prog. Fund for Coal and Steel. Hobbacher, A. 2009. Recommendations for Fatigue Design of Welded Joints and Components, IIW doc.1823–07, Welding Research Council Bulletin 520, New York. ISO 5817. 2003. Welding—fusion-welded joints in steel, nickel, titanium and their alloys (beam welding excluded)—quality levels for imperfections. Geneva: International Standardisation Organisation. Lillemäe, I.; Lammi, H.; Molter, L.; Remes, H. 2012. Fatigue strength of welded butt joints in thin and slender specimens. Int J Fatigue 44: 98–106. Radaj, D.; Sonsino, C.M.; Fricke, W. 2006. Fatigue assessment of welded joints by local approaches. Woodhead Publ, Cambridge. Remes, H.; Fricke, W. 2014. Influencing factors on fatigue strength of welded thin plates based on structural stress assessment. Weld World 58: 915–923. Sonsino, C.M.; Bruder, T.; Baumgartner, J. 2010. S-N lines for welded thin joints—suggested slopes and FAT values for applying the notch stress concept with various reference radii. Weld World. 54(11/12): 375–392. von Selle, H.; Peschmann, J.; Eylmann, S. 2013. Implementation of fatigue properties of laser welds into classification rules. Proceedings of 4th International Conference on Marine Structures MARSTRUCT2013, Espoo Finland, pp. 273–280.

REFERENCES Costa, J.D.M.; Ferreira, J.A.M.; Abreu, L.P.M. 2010. Fatigue behaviour of butt welded joints in a high strength steel. Procedia Eng 2: 697–705.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Investigation of weld root fatigue of single-sided welded T-joints W. Sundermeyer Overdick GmbH and Co. KG, Hamburg, Germany

W. Fricke & H. Paetzold Hamburg University of Technology (TUHH), Hamburg, Germany

ABSTRACT: Several welded joints in ship and offshore structures can be performed only from one side, e.g. tubular connections in offshore structures. Full penetration is usually aimed at, but sometimes partial penetration occurs, either unintended or intended in order to reduce the welding effort. Such welds are assessed with the lowest fatigue class in the nominal stress approach. However, weld root failures are rarely known although fabrication is frequently performed under unfavorable conditions regarding welding position and gap width. Therefore, fatigue tests were performed with T-joints subjected to bending to investigate weld root fatigue and possibly optimize the welds. 25 mm thick plates were joined with singlesided welds having full and partial penetration (HV and HY-welds), partly performed without and with gap. The results are assessed using the nominal, structural hot-spot and effective notch stress approaches with the aim to rationally consider the actual weld throat thickness of partial penetration welds. 1

INTRODUCTION

Welded ship and offshore structures are prone to fatigue which is not only due to the cyclic loading, but also to local geometric and material effects of the welded joints. In addition to the weld toes, also the weld roots, partly showing non-fused root faces, may be the origin of fatigue cracks. Weld roots are particularly critical if welding can be performed only from one side. This situation occurs typically at tubular joints which are especially used in offshore structures, partly with large wall thickness. Owing to the cyclic loading, full penetration welds have usually to be performed. In normal cases, the structural hot-spot stress is larger at the weld toes than weld roots in tubular joints. Therefore, weld root failure is usually not observed in fatigue tests of tubular structures with full-penetration welds (e.g. Kuhlmann et al. 2015). On the other hand, the structural hotspot stress at the weld root can reach 80% of that at the weld toe (Baerheim, 1996) so that this should also be assessed. One major problem is the weld quality due to unfavourable welding conditions occurring around the circumference of the tubular connection requiring certain chamfering and gap widths. Marshall et al. (2013) address this problem and propose partial weld penetration performed under better conditions and with higher quality. In the past, many investigations have been performed on single-sided welds at tubulars, several of them concerning butt joints of pipelines, e.g. by

Maddox & Razmjoo (1998) and Haagensen et al. (2003). In addition to the quality of the weld root, also the effect of misalignment has to be observed (Lotsberg, 2009). Fricke et al. (2006) investigated fillet-welded ends of Rectangular Hollow Sections (RHS) subjected to axial and bending loads. Cracks initiated from the weld root. Zenner and Grzesiuk (2004) performed tests on single-sided aluminium welds at cruciform joints subjected to axial loading. One S-N curve could be derived from the test results of both full and partial penetration welds if secondary bending was included in the stress range. Several investigations in the recent past have been devoted to the assessment of weld root fatigue. An overview of suitable approaches and several application examples is given in the Recommendations of the International Institute of Welding (IIW), see Fricke (2013). Mostly applied is today the nominal stress approach using the nominal stress in the weld throat area, which is implemented in several rules and guidelines (e.g. DNV 2012). Due to the uncertainties regarding the quality and penetration at the weld root, a very low fatigue class of FAT36 for steel is usually assumed. If primary and secondary bending is included, a higher class such as FAT71 applies (Hobbacher 2009). Alternatively the structural stress approach may be applied, e.g. using the linearized stress distribution in the weld throat. If axial and bending stresses are mainly acting in the weld leg plane without significant shear stress, a simplified approach using

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FAT80 has been proposed by Fricke (2006). In case of significant shear, the structural stress should be analyzed in different sections in the weld throat (Turlier et al. 2014). Also the effective notch stress approach and the crack propagation approach (Radaj et al. 2006; Hobbacher 2009) are well-suited for the assessment of weld root fatigue, the first using a fictitious notch rounding with a reference radius of rref = 1 mm and a fatigue class FAT225 for steel. Further details with respect to weld root fatigue have been given by Fricke (2013). In view of the development of the numerical methods and computer hardware there is a need for refined fatigue assessments also for weld roots of single-sided welds, e.g. to increase the safety or to optimize the fabrication. As not many fatigue test data of thick-walled joints are available, additional tests of T-joints with single-sided welds have recently been performed and are described in this paper. The 25 mm thick T-joints have been subjected to bending such that root failure is expected. This type of loading is possible even in tubular joints due to tensile stresses acting in the weld root caused by ovalization of the tubes. Full and partial penetration welds are considered and the gap width has been varied. Finally, the nominal, structural and effective notch stress approaches are applied in order to validate the approaches and to draw conclusions for practical application. 2 2.1

FATIGUE TESTS Overall specimen geometry and loading

The material used was higher-tensile steel S355J2 according to EN 10025-2 with minimum yield strength of 355 MPa. 2.2 Types and fabrication of welded joints Different types of welded joints were tested, see Figure 2. The first two were partial penetration welds (c > 0 mm), one with nominal gap of 0 mm (denoted HY) and the other with nominal gap of 5 mm (HY-G). The third was a full penetration weld (c = 0 mm) with 5 mm gap, denoted HV. The fourth was also a full penetration weld with a small backlay welding (HV-B), used for comparison purposes. The welding was performed by a shipyard under practical conditions with the MAG process (process 135 acc. to EN ISO 4065:2000) in PC position after pre-heating. Although 3–4 auxiliary plates were used it turned out that it was difficult to keep the desired gap width. The gap width g and also the average length of non-fused root faces c recorded from the specimens varied considerably between the specimens, see Table 1. In one plate, the desired full-penetration weld (HV) finally turned out to be a partial-penetration weld so that it was included in the HY series. Two specimens of HV series were excluded because of lack of fusion found in the weld root. The macro-sections in Figure 3 give an impression of the realized weld profiles. The number of weld passes was between 11–12 for the partial penetration welds and 21–22 for the full penetration welds. The weld surface was partly irregular which will be further discussed in connection with the numerical analysis.

A right-angled T-joint was chosen for the fatigue tests with plates of nominally 25 mm thickness (actual: 24.5 mm), see Figure 1. The specimens, having a width of 125 mm, were cut from 1300 mm wide welded plates. The load was introduced perpendicular to the end of the branch plate so that mainly bending and some shear were acting in the welded joint. Figure 2. Investigated weld types and geometry parameters. Table 1. Geometry specimens

Figure 1.

Geometry of the investigated specimens.

parameters

realized

in

the

Weld type

No. of c or d specim. mm

b mm

g mm

a mm

HY HY-G HV HV-B

17 18 8 10

15–19 14–18 21–25 16–21

0 0.1–5.4 5 5

22–29 27–32 33–35 30–32

5.1–12 1.9–6.3 0 6.5–9.0

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Figure 5. Typical fracture surface of a partial penetration weld.

Figure 3.

ratio R = Fmin/Fmax = 0 such that pulsating tensile stresses are acting on the weld root side. Fatigue cracks initiated from this side in all specimens. The tests were stopped after the cracks had penetrated most of the weld area so that specimen fracture can be assumed as failure criterion. Fig. 5 shows a typical fracture surface of a partial penetration weld from a similar viewpoint as in Figure 4, with the non-fused root face and some slag inclusion at the bottom, followed by the surface of stable crack growth and the final unstable fracture surface. The average depth of the non-fused root face was measured at each HY specimen.

Macro-sections of the different weld types.

3

FATIGUE ASSESSMENT USING THE NOMINAL STRESS APPROACH

3.1 Basis of the assessment Usually the nominal stress is based on the throat area. For the partial penetration welds (HY), the individually measured depth of the non-fused root faces was used to determine the throat thickness a as minimum distance to the actual weld surface as shown in the left part of Figure 6. For the full penetration welds (HV), the throat thickness was measured from the theoretical root position, i.e. the intersection point of the plate surfaces, whereas the upper weld toe was taken for the HV-B-welds as illustrated in the right part of Figure 6. The nominal bending stress can be calculated using the applied bending moment M and the section modulus W (see Fig. 1)

σn = Figure 4. Fatigue test set-up (seen from weld root side).

2.3

Performance of fatigue tests

The clamping of the base plate was realized along the edges of the base plate as shown in Figure 4. Strain gages were applied to check that the clamping is effective and the bending stresses are acting as planned. The tests were performed with force

M F e = W b ⋅ a2 6

(1)

where b is the width of the specimen. The section modulus simply considers the area with minimum throat thickness a rotated into the horizontal (weld leg) plane. The fatigue test results are not corrected to consider high stress ratios (e.g. R = 0.5 instead of R = 0) because high residual stresses are assumed in the relatively thick specimens.

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3.2

Fatigue test results

The S-N diagram showing the fatigue test results of the partial penetration welds without and with gap is displayed in Figure 7. Also plotted are the curves for 50% and 97.7% probability of survival Ps, the latter representing the mean curve minus two standard deviations. A slope exponent m = 3 was assumed, being typical for welded joints. A regression analysis would yield almost the same slope exponent. The influence of the gap width is rather small. The characteristic fatigue strengths for Ps = 97.7% at two million cycles, i.e. 60.0 and 57.9 MPa, are well above the mentioned fatigue class of FAT36,

Figure 6.

Determination of the throat thickness a.

Table 2. Mean and characteristic fatigue strengths at N = 2⋅106, their difference indicating twice the standard deviation. Weld type

Δσn,50 MPa

Δσn,97.7 MPa

Δσs,50 MPa

Δσs,97.7 MPa

Δσe,50 MPa

Δσe,97.7 MPa

HY HY-G HV HV-B

73.0 76.7 99.4 107.3

57.9 60.0 81.2 84.5

88.4 105.7 201.1 223.8

68.3 80.6 164.1 183.0

275.1 279.5 378.9 343.8

211.6 213.1 307.9 271.8

but below FAT71 given in the IIW recommendations (Hobbacher 2009). It should be noted that no allowance for possible deviations in the throat thickness is considered here. The fatigue test results for the full penetration welds are summarized in Figure 8. Again, a slope exponent m = 3 has been assumed for the statistical evaluation. The characteristic fatigue strength of 81.2 MPa is close to FAT90 given in rules (e.g. DNV 2012), full penetration provided. The left columns in Table 2 summarize the obtained mean and characteristic fatigue strengths Δσ50 and Δσ97.7 at two million cycles. 4

Figure 7. Fatigue test results for partial penetration welds based on nominal bending stress range Δσn in throat thickness a.

Figure 8. Fatigue test results for full penetration welds based on nominal bending stress range Δσn in throat thickness a.

FATIGUE ASSESSMENT USING THE STRUCTURAL STRESS APPROACH

As mentioned in the beginning, the structural stress approach can be applied to weld root failure by linearizing the applied stress in the weld throat. According to the proposal by Fricke (2006), the linearized axial and bending stress can be assessed in the weld leg area as long as the shear stress is small which is the case here. To achieve conservative results, the throat thickness should be chosen instead of the leg length, however not more than the plate thickness. This corresponds in principle to the procedure described above for the nominal stress approach, i.e. eq. (1), with the additional limitation to the plate thickness. If this is considered, the fatigue assessment yields the fatigue strengths listed in the fourth and fifth column of Tab. 2, i. e. higher values than for the nominal stress approach. It should be noted that the proposed fatigue class FAT80 is not reached by the partial penetration welds (HY), whereas the full penetration welds exceed far this value. Furthermore, the approach by Turlier et al. (2014) has been applied, using the linearized stresses in different sections through the weld throat. Similar values were found as for the above mentioned approach, reaching FAT80 for partial penetration welds only if the weld throat thickness is again limited to the plate thickness.

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5 5.1

FATIGUE ASSESSMENT USING THE EFFECTIVE NOTCH STRESS APPROACH Computation of the effective notch stress

The effective notch stress requires the computation of the linear-elastic stress in the notch root rounded with a reference radius rref = 1 mm. For this purpose 2D finite element models of the specimens have been created, Figure 9. The clamping in the 2D model differs somewhat from the real specimen, however the relevant bending stress in the branch is acting as measured in the specimens (max. deviation 5%). Plane strain elements with quadratic shape function have been used. The actual weld shape was approximated by a bi-linear line. The mesh has been refined at the notch according to the IIW recommendations (Fricke 2013). Figure 10 shows the typical keyhole notch chosen for partial penetration welds with non-fused root faces. The vertex point of the circle is located at the end of the slit as measured from the specimens. The modelling of the partial penetration welds with gap follows the rule that the rounding of the notch—in this case of the two notches due to the gap—should keep the given corner point. The resulting mesh is shown in Figure 11. The rule results in a slightly increased depth. The full penetration welds were simply modelled with a fillet radius. Figure 12 shows a typical model. The models of the HV-B welds include the backlay weld. Figure 13 illustrates the computed stress distribution in a model of the HY-G series. The largest notch stress occurs at the lower notch in case of

Figure 10. welds.

Finite element mesh for partial penetration

Figure 11. (HY-G).

Finite element mesh for the welds with gap

Figure 12. welds.

Finite element mesh for the full penetration

shorter non-fused root faces, whereas the upper notch shows the largest stress in case of long root faces. 5.2 Figure 9. 2D finite element model of the T-joint with load and clamping.

Fatigue test results based on effective notch stress

Figure 14 displays the fatigue test results for the partial penetration welds based on computed notch

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stress ranges at two million cycles are also listed in Table 2. 6

Figure 13. Distribution of the max. principal stress in an idealized partial penetration weld with gap.

Figure 14. Fatigue test results for partial penetration welds based on effective notch stress range Δσe.

Figure 15. Fatigue test results for full penetration welds based on effective notch stress range Δσe.

stress ranges. The specimens with gap (HY-G) show again almost the same fatigue life as those with small gap (HY) when the same effective notch stress is acting. The characteristic fatigue strengths for both series are slightly below FAT225 given in the IIW recommendations (Hobbacher 2009). The results for the full penetration welds (HV and HV-B) based on effective notch stress range are summarized in Figure 15. These results are far above the FAT225 line. The mean and characteristic

DISCUSSION OF THE RESULTS

The evaluation of the fatigue test results for the partial penetration welds, based on the nominal bending stress in the actual weld throat thickness a, yields characteristic fatigue strength of well above FAT36, however FAT71 defined for such welds including secondary bending is not reached. Also the fatigue strengths according to the structural stress approach, using the linearized stress in the weld throat, and the effective notch stress approach do not reach the recommended classes FAT80 and FAT225, respectively. The reason for these surprising results is not fully clear. The weld root contained blow-holes and partly relatively sharp ends of the slit (incomplete root penetration), probably caused by 45° bevelling. This could have resulted in high crack propagation rate right from the beginning of the tests. In several specimens, the crack surface showed different zones with very smooth and rougher surface indicating different fracture behaviour. Further metallographic investigations have not yet been performed, but are planned. In general, the welding of the specimens revealed some problems of achieving the desired partial penetration. Obviously, large variations of the non-fused weld root faces occur, which might even be larger when tubular joints are welded. This would mean that additional safety has to be considered during the design of such welded joints to cover uncertainties. In the nominal stress approach, the relatively low class FAT36 might already cover these uncertainties. However in local approaches, where a refined stress analysis is performed and a relatively high fatigue class is applied, an additional incomplete root penetration might have to be considered if the actual penetration cannot be verified or weld roots with blow-holes and lack of fusion are to be expected. 7

CONCLUSIONS

Fatigue tests were performed with relatively thickplated T-joints with partial penetration welds (HY) as well as full penetration welds (HV) for comparison, a part of them having a small backlay weld on the weld root side. The results were assessed using the nominal and structural stress in the actual weld throat area as well as the effective notch stress at the weld root.

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The following conclusions can be drawn from the results: • The fatigue strength of the partial penetration welds based on nominal stress is far above the relevant fatigue class FAT36. The full penetration welds without and with backlay weld showed a fatigue class of about FAT80. • The influence of the gap width, which was varied for the partial penetration welds, on the fatigue life was negligible. • The application of the structural stress approach showed that the partial penetration welds did not reach the target fatigue class FAT80. Also in the effective notch stress approach, the class FAT225 was not reached, in contrast to the full penetration welds. A reason could be the rather poor quality of the weld root, to which the 45° bevelling of the joints might have contributed. • The fabrication of the specimens has shown that it was very difficult to keep the desired partial penetration of the HY welds. Therefore, an additional incomplete root penetration should be considered particularly in the local approaches if the actual penetration cannot be verified or poor quality of the weld root has to be expected.

ACKNOWLEDGEMENTS The authors appreciate the support of OVERDICK GmbH & Co. KG with regard to fabrication of specimens and general support of the presented tests and studies. REFERENCES Baerheim, M. 1996. Stress concentrations in tubular joints welded from one side. Proc. 6th Int. Ocean and Polar Engineering Conference, ISOPE-I-96-251, Los Angeles.

DNV (2012). Recommended Practice DNV-RP-C203: Fatigue Design of Offshore Steel Structures. Det Norske Veritas, Høvik. Fricke, W. 2006. Weld root fatigue assessment of filletwelded structures based on structural stresses. Proc. Int. Conf. on Offshore Mech. & Arctic Engng (OMAE2006-92207), New York: ASME. Fricke, W. 2013. IIW Guideline for the assessment of weld root fatigue. Welding in the World 57 (6), 753–791. Fricke, W., Kahl, A. & Paetzold, H. 2006. Fatigue Assessment of Root Cracking of Fillet Welds subject to Throat Bending using the Structural Stress Approach. Welding in the World, 50 (7/8), 64–74. Haagensen, P., Maddox, S.J. & Macdonald, K.A. 2003. Guidance for fatigue design and assessment of pipeline girth welds. Proc. of 22nd Int. Conf. on Offshore Mechanics and Arctic Engng., OMAE2003-37496, New York, ASME. Hobbacher, A. (2009). Recommendations for Fatigue Design of Welded Joints and Components. IIW doc.1823-07, Welding Research Council Bulletin 520, New York. Kuhlmann, U., Bucak, Ö., Mangerig, I., Kranz, B., Euler, M., Hubmann, M., Fischl, A., Hess, A., Herrmann, J. & Zschech, R. 2015. Fatigue-resistant trusses of circular hollow sections with thick-walled chords (in German). Submitted for publication in Stahlbau. Lotsberg, I. 2009. Stress concentration factors at welds in pipelines and tanks subjected to internal pressure and axial force. Marine Structures 21, 138–159. Maddox, S.J. & Razmjoo, G.R. 1998. Fatigue Performance of Large Girth Welded Steel Tubes. Proc. 17th Int. Conf. Offshore Mechanics and Arctic Engng. (OMAE’98), New York, ASME. Marshall, P., Qian, X. Nguyen, C.T. & Yuthdani, P. 2013. Welder-optimized CJP-equivalent welds for tubular connections. Welding in the World 57(4), 569–579. Turlier, D., Klein, P. & Berard, F. 2014. FEA shell element model for enhanced structural stress analysis of seam welds. Welding in the World 58 (4), 511–528. Zenner, H. & Grzesiuk, J. 2004. Influence of the weld preparation and weld execution on the fatigue strength of high-quality aluminium structures. Welding and Cutting 2004 (3), 220–223.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Fatigue data of High-Frequency Mechanical Impact (HFMI) improved welded joints subjected to overloads H.C. Yildirim Department of Applied Mechanics, Aalto University, Finland

G. Marquis Department of Applied Mechanics, Aalto University, Finland Department of Aeronautical and Vehicle Engineering, KTH—Royal Institute of Technology, Stockholm, Sweden

ABSTRACT: In the past decade, High Frequency Mechanical Impact (HFMI) has significantly developed as a reliable, effective and user-friendly method for post-weld fatigue strength improvement technique for welded steel structures. The evaluation and features of developing guideline within the International Institute of Welding (IIW) for the design of structures improved using HFMI is briefly discussed. So far, the extra fatigue strength benefit for HFMI-treated high strength steels has been mostly shown for constant amplitude loading. This paper reports 68 available HFMI-improved welds subjected to overloads or pre-fatigue loads at various loading conditions prior to fatigue testing. These loading conditions are often seen for marine structures. 1

NOMENCLATURE

fy FAT

Kn m1 Nf R S ΔS t ρ σN 2

Yield strength (MPa) IIW fatigue class, i.e., the stress range in MPa corresponding to 95% survival probability at 2 ⋅ 106 cycles to failure (a discrete variable with 10–15% increase in stress between steps) Notch stress concentration factor Slope of the S-N line for stress cycles above the knee point Cycles to failure Stress ratio (σmin/σmax) Nominal stress (MPa) Nominal stress range (MPa) Plate thickness of the specimen (mm) Radius (mm) Standard deviation in Log (Nf)

INTRODUCTION

Recommendations for fatigue design of HFMItreated welded steel structures have been proposed by Marquis et al. (2013) according to a pool of the collected experimental data of axially-loaded specimens obtained mainly at constant amplitude loading for longitudinal, cruciform and butt welds. The fatigue data has been extracted in an overview study focusing on the S-N slope identification by Yildirim & Marquis (2012a). Further, the data has been analysed by Yildirim and Marquis

(2012b) with an assessment procedure considering the influence of material yield strength on fatigue strength of HFMI-improved welds identification. Evaluations for local approaches have also been performed in separate studies (Yildirim et al. 2013) (Yildirim & Marquis 2014). FAT classes (The International Institute of WeldingIIW fatigue class, i.e. the nominal or effective notch stress range in mega pascals corresponding to 95% survival probability at 2 × 106 cycles to failure) have been proposed based on Nominal Stress (NS), Structural Hot Spot Stress (SHSS) or Effective Notch Stress (ENS) methods by using stress analysis procedures. Those design procedures have been defined by the IIW Commission XIII-Fatigue of Welded Components and Structures. FAT values have been suggested for each type of weld detail in the case of NS based assessment. On the other hand, two sets of characteristic S-N curves according to non-load carrying and load-carrying HFMI-improved welded joints have been defined in terms of SHSS whereas only one set of S-N curves has been given in the case of ENS method. Besides, special requirements for low-stress-concentration welded details have been suggested. Cautions on high R-ratio loading conditions and variable amplitude loading have also been suggested in that recommendation. The design recommendations include one fatigue class increase in strength (about 12.5%) for every 200 MPa increase in static yield strength, and they have been shown to be conservative with

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respect to all available fatigue data. This stepwise increase due to the HFMI treatment is shown in Figure 1 with reference to the same as-welded detail and improved weld detail by the IIW recommendations. With these suggestions, a four (4) fatigue class increase in strength has been defined for joints fabricated from 235 < fy ≤ 355 MPa steel with respect to the nominal fatigue class in the as-welded condition. One additional fatigue class increase in fatigue strength for every 200 MPa increase in static yield strength is proposed. Namely, this stepwise increase has been extended up to an eight (8) fatigue class improvement for fy > 950 MPa. All the fatigue classes are defined for N = 2 × 106 cycles and assume an S-N slope of m1 = 5 for HFMI-treated welds. So far we have mainly focused on the degree of improvement in fatigue strength after the HFMI treatment of welds subjected to constant amplitude loading. Studies on HFMI material characterization have also been done by other scholars. These studies include investigations on highly coldworked region (Revilla-Gomez et al. 2013), hardness dependency on steel strength (Weich 2011, Weich et al. 2009), grain elongation (Le Quilliec et al. 2011) and the microstructure after fatigue loading (Abdullah et al. 2012). Recently, Mikkola & Marquis (2014) have shown that HFMI treatment increases the fatigue life of the steel under low applied strains and decreases it under high applied strains. However, less attention has been paid to the fatigue strength assessment of HFMI-treated welded joints subjected to pre-fatigue loading, especially under variable amplitude loading. Therefore the aim of this paper has been to present the available HFMI improved test data obtained at various loading conditions including overloads prior to cyclic loading. This is a necessary step for the development of HFMI guideline since the induced beneficial compressive stresses by HFMI might be diminished due to those overloads, or pre-fatigue

Figure 1. Proposed maximum increases in the number of FAT classes as a function of fy from Yıldırım and Marquis (2012b).

loads etc. Thus proposed design curves should be checked for those afore mentioned data too. 3 3.1

METHODS Available data in the literature

The authors were able to identify five publications containing fatigue data for welded details improved by HFMI treatment and subjected to overloads prior to fatigue loading. Table 1 shows the material yield strength, weld detail type and HFMI treatment method. Stress ratio and loading condition for each data set is separately given on the graphs. In the table, many of the references considered in this study provide fatigue data only as points on a graph. When numerical values were not provided, they were extracted from the S-N plots using open source software. This was not considered to introduce significant errors in the results or conclusion. Maddox et al. (2011) have worked on as-welded and HFMI-treated longitudinal non-loading joints manufactured from 30 mm steel sheets with a steel grade of 390 MPa. Basically, two different loading conditions have been considered with the application of the improvement method. Firstly, prefatigue loading has been performed on as-welded specimens for 10% of the expected life, and further samples have been HFMI-treated either at an applied stress of 120 MPa or zero applied stress. In each case, subsequent fatigue testing has been carried out under axial tension-tension loading with a constant minimum stress of 120 MPa. The resulted applied stress ratios have ranged 0.5 to 0.69. Deguchi et al. (2012) have studied longitudinal non-load carrying specimens in order to confirm the benefits of the improvement techniques for ship structures. They have considered two load histories in conjunction with the treatment method; HFMI treatment has been done before and after a scenario of ship launching. For the former case, specimens have been pre-fatigue loaded at 100 MPa, then they have been treated and tested at a mean stress of σm = 100 MPa at R = −1. For the latter one, specimens have been treated and then tested at a mean stress of σm = 100 MPa at R = −1. Okawa et al. (2011) and Ummenhofer et al. (2011) have reported fatigue test results on HFMItreated transverse non-load carrying specimens and butt joints, respectively, in order to investigate both tensile and compressible pre-fatigue loads effect. In the study of Okawa et al. (2011), tensile pre-fatigue load has been applied up to 90% of the fy and compressive pre-fatigue load has been applied up to 60% of the fy. Ummenhofer et al. (2011), on the other hand, have applied the overloads reaching just below the fy of the material. Recently, Polezhayeva et al. (2014) have investigated the influence of compressive fatigue loads

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Table 1.

Extracted experimental fatigue data for HFMI-treated welds subjected to overloads.

Ref.

fy [MPa]

t [mm]

HFMI method

Weld detail

Kn

(Maddox et al. 2011) (Deguchi et al. 2012) (Okawa et al. 2011) (Ummenhofer et al. 2011) (Polezhayeva et al. 2014)

390 355 392 422 560

30 16 20 16 20

UIT UP UIT UIT/HiFiT UIT

Longitudinal Longitudinal Transverse Butt Transverse

– 3.75 2.33 2.11 –

Table 2. The previously-proposed FAT classes in the NS system for HFMI-treated joints as a function of fy (Yıldırım and Marquis 2012b). fy (MPa) All fy fy ≤ 355 355 < fy 235 < fy ≤ 355 355 < fy ≤ 550 550 < fy ≤ 750 750 < fy ≤ 950 950 < fy

Longitudinal welds

Transverse welds

Butt welds

As-welded, m1 = 3 Hobbacher (2009) 71 80 90 Improved by hammer peening, m1 = 3 Haagensen & Maddox (2013) 90 100 112 100 112 125 Improved by HFMI, m1 = 5 112 125* 140* 125 140 160 140 160 180 160 180* – 180 – –

fy (MPa) All fy fy ≤ 355 355 < fy

*no data available.

on fatigue strength of HFMI-improved transverse non-load carrying specimens with the fy of 560– 570 MPa. Both constant and variable amplitude loading with changing mean stresses have been studied. Maximum tensile and compressive stresses have varied from 200 MPa to −270 MPa for constant amplitude whereas they have varied from 400 MPa to −200 MPa for variable amplitude scheme. 3.2

Table 3. Existing IIW FAT classes for SHSS and ENS approaches for as-welded and improved joints and the proposed FAT classes for HFMI-treated joints as a function of fy.

HFMI design curves

As mentioned before, FAT values have been previously suggested for each type of weld detail in the case of nominal stress based assessment method (Marquis et al. 2013). These values are given as a function of fy in Table 2. Design proposals for the structural hot spot stress method have been defined separately for load and non-load carrying fillet specimens. On the other hand, only one set of FAT values has been given for the effective notch stress method. These are presented in Table 3 with respect to the existing IIW design recommendations for as-welded and improved welds. In contrast to needle or hammer peened welded joints where the maximum FAT improvement is three (3) by Haagensen & Maddox (2013), HFMI

235 < fy ≤ 355 355 < fy ≤ 550 550 < fy ≤ 750 750 < fy ≤ 950 950 < fy

Structural Hot-Spot Stress (SHSS) method

Effective Notch Stress (ENS) method

Load carrying fillet welds

For all welds

Non-load carrying fillet welds

As-welded, m = 3 Hobbacher (2009) 90 100 225a Improved by hammer peening, m = 3 Haagensen & Maddox (2013) 112 125 –b 125 140 –b Improved by HFMI, m1 = 5 140 160 320 160 180 360 180 200 400 200 225 450 225 250 500

a some studies suggest that FAT 200 is a better fit for the experimental data. b no proposals were ever developed.

Table 4. Minimum reductions in the number of FAT classes in fatigue strength improvement for HFMI treated welded joints as presented in Figure 1 based on R ratio. R ratio

Minimum FAT class reduction

R ≤ 0.15 0.15 < R ≤ 0.28 0.28 < R ≤ 0.4 0.4 < R ≤ 0.52 0.52 < R

No reduction due to stress ratio One FAT class reduction Two FAT classes reduction Three FAT classes reduction No data available. The degree of improvement must be confirmed by testing

treated welds can have up to eight (8) FAT classes of improvement depending on the material strength, welded joint geometry, etc. It is therefore easier to express the stress ratio influence as a penalty with respect to the maximum increase in the number

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of FAT classes as a function of fy as presented in Figure 1. These are given in Table 4. 4

RESULTS AND DISCUSSIONS

4.1 Available test data and HFMI design curve comparison Figures 2–7 show the available fatigue data presented in Table 1. These figures give also the previously-proposed HFMI design curves from Tables 1, 2 and 3 depending on the specimen geometry and the assessment method as defined by the IIW. In Figure 2, fatigue test results for longitudinal non-load carrying welds with 355 < fy ≤ 550 MPa are presented. Fatigue strength value for this type of specimen is defined as FAT 63 in as-welded case and FAT 112 in HFMI-treated conditions by Hobbacher (2009) and Marquis et al. (2013), respectively. It is worth to remind that all FAT proposals for HFMI are based on failures initiated from the weld toe and designated to serve for both constant and variable amplitude loadings. In the study of Maddox et al. (2010), failure started from

Figure 4. Fatigue data of pre-loaded and tested HFMI welds for transverse non-load carrying welds specimens with 355 < fy ≤ 550 MPa.

Figure 5. Fatigue data of transverse non-load carrying attachments with 550 < fy ≤ 750 MPa tested at different levels of mean stresses at both CA and VA loading.

Figure 2. Fatigue data of pre-loaded and tested HFMI welds for longitudinal non load carrying welds with 355 < fy ≤ 550 MPa.

Figure 6. Fatigue data of pre-loaded and tested HFMI welds with 355 < fy ≤ 550 MPa in the effective notch stress method.

Figure 3. Fatigue data of tensile and compressive preloaded and tested HFMI welds for transverse non-load carrying welds with 355 < fy ≤ 550 MPa.

the weld root in all cases and the stress ratio varied from 0.5 to 0.69. Table 4 does not define any penalty value for stress ratios higher than R = 0.52. Nevertheless, a three FAT class reduction was led to FAT 71 from FAT 112 for R = 0.1, which is conservative to the fatigue data. In the case of Deguchi

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Figure 7. Residual stress distributions before and after the fatigue loading (Yıldırım & Marquis 2013).

et al. (2010), on the other hand, specimens were subjected to constant amplitude fatigue loading at a stress ratio of R = −1 after or before pre-fatigue loadings. FAT 112 is shown without any reduction due to the stress ratio. One important result is clear from the fact that pre-fatigue loading after the HFMI treatment allows less applied stress as compared to the treatment after pre-fatigue loading. In Figure 3, Okawa et al. (2012) considered the effect of both tensile and compressive pre-fatigue loads by using transverse non-load carrying welds with 355 < fy ≤ 550 MPa. For this type of yield strength, Figure 1 and Table 2 allow maximum of five FAT class increase in fatigue strength after HFMI, which result in FAT 140 in the conservative side for both loads. In Figure 4, test results of Ummenhofer et al. (2011) are shown with the expected improvement due to HFMI treatment. This corresponds to FAT 160 without any stress ratio correction. Treated butt joints were tested two conditions at tensile and compressive pre-fatigue loading that is about the yield strength of the base material. Nonetheless, all the data points, which are failed from weld toe, are well above the previously-proposed design curve. A comprehensive study by Polezhayeva et al. (2014) has been recently completed for fatigue loadings that are often encountered in marine structures. Basically, numerous loading cases were considered by using constant and variable amplitude loadings with changing compressive and tensile mean stresses. Details of loading conditions are shown in Figure 5 and they can also be found in the original paper. In this study, variable amplitude test results are represented by equivalent stress ranges to make a comparison with respect to the proposed HFMI curves. In contrast to other studies, as-welded test results are also shown because of the significance of the loading type. Based on Figure 1 and Table 2 maximum allowable improvement for this type of specimen is six. This results in FAT 160 without any penalty due to stress ratio. It is important to note that all weld toe failures of HFMI-treated test results are either on or

above the previously-proposed design curve even the loadings are different. The root failures in the figure are shown with a bracket around the data points. In Figure 6, fatigue test results of specimens with 355 < fy ≤ 550 MPa failed at the weld toe are shown in the effective notch stress method for comparison to previously-proposed design curve. Stress concentration values for the assessment method are evaluated as given in the IIW guideline by Fricke (2012) and presented in Table 1. First impression from the figure is that all the notch stress values are above the previously proposed design curve FAT 360 for this type of steel grade. The scatter, on the other hand, is larger because the evaluated fatigue data includes a variety of stress ratios, e.g. overloads both in tension and compression. 4.2 Residual stress distributions Significant variations in the measured surface residual stress distributions at the fatigue-critical locations before and after testing of HFMI-treated longitudinal non-load carrying welds are observed in the study of Yildirim & Marquis (2013), see Figure 7. The loading was variable amplitude loading at a stress ratio of R = −1. Residual stress measurements were carried out non-destructively using X-ray diffraction method. Measurements before the test were performed at the bottom of the HFMI groove at the end of the stiffener. For the tested specimens, measurements were taken at the unbroken side of the stiffener. Measured stresses were perpendicular to the weld toe, i.e., longitudinal with respect to the axis of the specimen. The dashed lines in Figure 7 represent the 10% and 90% scatter limits of the measured data. It can be clearly seen that the surface residual stresses in compression state were all diminished after fatigue testing as expected. After test measurements were done on broken specimens, the change in residual stress state is also expected even after the very beginning of fatigue testing depending on the overloads, which may result in shorter fatigue life. However, there is no clear evidence for this behaviour therefore this has to be studied in detailed in the future. 5

CONCLUSIONS

Experimental fatigue data collected from the literature for HFMI-treated welded joints have been presented. The samples were subjected to numerous loading types including overloads and pre-fatigue loads prior to fatigue testing. The treatments in all cases have been performed on the critical regions of the components. In total, 68 data points considered in this study are composed of varying yield strength (355 ≤ fy ≤ 560 MPa) and loading conditions (0.0 ≤ R ≤ −1) either at constant

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or variable amplitude loading. According the findings following conclusions can be done: • The previously-proposed fatigue classes which represent the fatigue strength of HFMIimproved joints at 2 × 106 cycles and assume an S-N slope of m1 = 5 have shown a good agreement (except two points) with the available data for HFMI-treated welds subjected to overloads prior to fatigue loading. • For one set of data, penalty factors due to stress ratio was applied, and found to be consistent with the FAT values. • It has been shown in two cases that application of tensile or compressive pre-fatigue loads have almost similar effect on fatigue strength. • By using three sets of data for material yield strength of 355 < fy ≤ 550 MPa, previously-proposed FAT value in the effective notch stress method have been verified with the considered fatigue data. • Preliminary results show that HFMI treatment method has a huge potential for application in marine structures. In the future, more investigations of fatigue data at R > 0.1 and variable amplitude testing with overloads are needed. More fatigue tests on all types of specimens with yield strength greater than 550 MPa should be encouraged. Studies on large-scale structures improved by HFMI treatment method might be studied extensively. ACKNOWLEDGEMENTS Support for this work has been partially provided by the Finnish Cultural Foundation and research programme of the Finnish Metals and Engineering Competence Cluster (FIMECC) BSA project. REFERENCES Abdullah, A., Malaki, M., Eskandari A. 2012. Strength enhancement of the welded structures by ultrasonic peening. Materials and Design 38:7–18. Deguchi T., Mouri M., Hara J., Kano D., Shimoda T., Inamura F., Fukuoka T., Koshio K. 2012. Fatigue strength improvement for ship structures by Ultrasonic Peening, J. Mar. Sci. Technol., 7(3), 360–369. Fricke, W., IIW Recommendations for the Fatigue Assessment of Welded Structures by Notch Stress Analysis, Woodhead Publishing Ltd., Cambridge, 2012. Haagensen, P.J., Maddox, S.J. 2013. IIW Recommendations on methods for improving the fatigue lives of welded joints. Woodhead Publishing Ltd., Cambridge. International Institute of Welding, Paris. Hobbacher, A. 2009. IIW Recommendations for Fatigue Design of Welded Joints and Components. WRC Bulletin 520, The Welding Research Council, New York.

Le Quilliec, G., Lieurade, H.-P., Bousseau, M., DrissiHabti, M., Inglebert, G., Macquet, P., Jubin, L. 2011. Fatigue behaviour of welded joints treated by high frequency hammer peening: Part I, Experimental study. International Institute of Welding, Paris, IIW document XIII-2394-11. Maddox, S.J., Doré, M.J., Smith, S.D. 2011. A case study of the use of ultrasonic peening for upgrading a welded steel structure, Welding in the World, 55(9–10):56–67. Marquis, G.B., Mikkola, E., Yildirim, H.C., Barsoum, Z. 2013. Fatigue strength improvement of steel structures by HFMI: proposed fatigue assessment guidelines. Welding in the World 57(6): 803–822. Mikkola, E., Marquis, G.B. 2014. Material characterization of high frequency mechanical treated high strength steel. International Institute of Welding, Paris, IIW Document XIII-2528-14. Okawa, T., Shimanuki, H., Funatsu, Y., Nose, T., Sumi, Y. 2011. Effect of preload and stress ratio on fatigue strength of welded joints improved by ultrasonic impact treatment. International Institute of Welding, Paris, IIW Document XIII-2377-11. Polezhayeva H., Howarth D., Kumar M., Ahmad B., Fitzpatrick M.E. 2014. The effect of compressive fatigue loads on fatigue strength of non-load carrying specimens subjected to ultrasonic impact treatment, International Institute of Welding, Paris, IIW Document XIII-2530-14. Revilla-Gomez, C., Buffiere, J.-Y., Verdu, C., Peyrac, C., Daflon, L., Lefebvre, F. 2013. Assessment of the surface hardening effects from hammer peening on high strength steel. Fatigue Design 2013, International Conference Proceedings, Procedia Engineering, 66:150–160. Ummenhofer T., Herion S., Hrabowsky J., Rack S., Weich I., Telljohann G., 2011. REFRESH—Extension of the fatigue life of existing and new welded steel structures (Lebensdauerverlängerung bestehender und neuer geschweißterStahlkonstruktionen). FOSTA Research Association for Steel Applications (Forschungsvereinigung Stahlanwendung e.V.) P 702, Verlag, Düsseldorf. Weich, I. 2011. Edge layer condition and fatigue strength of welds improved by mechanical post-weld treatment. Welding in the World, 55(1/2):3–12. Weich, I., Ummenhofer, T., Nitschke-Pagel, Th., Dilger, K., Eslami, H. 2009. Fatigue behaviour of welded highstrength steels after high frequency mechanical post-weld treatments. Welding in the World, 53(11–12):322–332. Yildirim, H.C., Marquis, G.B. 2012a. Overview of fatigue data for high frequency mechanical impact treated welded joints. Welding in the World 56(7/8):82–96. Yildirim, H.C., Marquis, G.B. 2012b. Fatigue strength improvement factors for high strength steel welded joints treated by high frequency mechanical impact. International Journal of Fatigue 44:168–176. Yildirim, H.C., Marquis, G.B., Barsoum, Z. 2013. Fatigue assessment of High Frequency Mechanical Impact (HFMI)-improved fillet welds by local approaches. International Journal of Fatigue 52:57–67. Yildirim, H.C., Marquis, G. 2013. A round robin study of high frequency mechanical impact (HFMI)-treated welded joints subjected to variable amplitude loading, Welding in the World, 3, pp. 437–447. Yildirim, H.C., Marquis, G.B. 2014. Fatigue design of axially-loaded high frequency mechanical impact treated welds by the effective notch stress method. Materials and Design, 58(0), 543–550, 2014.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Application of wave models to fatigue assessment of offshore floating structures T. Zou, M.L. Kaminski & X.L. Jiang Delft University of Technology, Delft, The Netherlands

ABSTRACT: Offshore floating structure design requires long-term sea states data in order to make fatigue assessments. However, measured wave data may be limited at the designated sea area. The development of measurement and hindcast technology has enhanced long-term atmospheric data availability in most sea areas. This paper couples a wave model-SWAN to the atmospheric model of the ERA-interim project. Sable field wind data from ERA-interim project is combined to produce six-hourly sea states which permits fatigue assessments for FPSO-Glas Dowr. Validating this methodology requires comparing this fatigue assessment results to the fatigue damages which are calculated based on the sea states from the wave model of ERA-interim and buoy-measurement. The result indicates that this methodology has merits, but still requires further improvement. 1

INTRODUCTION

Offshore floating structures designs estimate longterm fatigue damage consumption using wave climate data, as represented by the distribution of significant wave height (Hs) and zero crossing period (Tz) (DNV, 2010). These sea states data is obtained through past measurements and is assumed to represent the wave environment the offshore structure is expected to encounter. A spatially or temporally limitation of these measurement is that site-measured wave data is not always available in oil fields or the amount of wave data is not enough for the fatigue assessment. In addition, more and more researchers have realized that the effect of climate change on sea states should not be neglected and both average sea states and extreme sea states cannot be considered stationary. Young et al. (2011) analyzed a 23-year database of global wind speed and wave height, and estimated a linear trend for both extreme events and mean conditions. Others developed a variety of different joint distributions based on the empirical relationship among wave variables like wave height or wave period (Bitener-Gregersen & Haver, 1991; Repko et al. 2004). Nonetheless, these researches all need improvement for making fatigue assessments as they only focus on wave height change. In addition, wave height, period and direction are all the key elements for fatigue assessment. Change of wave period and wave direction should also be identified. The development of measurement (especially satellite observation) and hindcast technology has

made global atmospheric data available for the past 50 years. Related reanalysis projects include ERA-interim which addresses the global atmosphere covering the data-rich period since 1979 and continuing in real time. The atmospheric model is coupled to an ocean-wave model (WAM), which can output wave data at the nodes of its 1.0° × 1.0° latitude/longitude grid. By analyzing the wind data variation, wave models can simulate the changing trend of wave height, wave period and wave directions. Challenges remain for this project. First, the ERA-interim resolution is too coarse. Within these 1.0° × 1.0° latitude/longitude “boxes”, all the wave parameters are assumed to be uniform. The area of this “box” is obviously too big for fatigue assessment. Offshore floating structures are designed to work in one particular location for certain years. In order to make an accurate fatigue assessment, a higher resolution is required. Second, ERA-interim still requires validation. Caires and Sterl (2003) validated the reanalysis result of ERA-40 (ERA-interim is an updated project based on ERA-40) against buoy and satellite measurements over 17 sea areas along the coast of North America. The results show that in terms of significant wave height the satellite and buoy observations compare well with each other, and that the ERA-40 values of Hs slightly overestimate low ( 6 m) in the buoy-measured data indicates that EAR-interim reanalysis model does not simulate extreme events. Only 10 percent of the sea states are plotted in Figure 2b in order to simplify the plot, but all the sea states are used to make the contours. The yellow regions indicate the limit of wave steepness (H/L > 0.07). All the sea states falling in these regions are omitted in this paper. As a reanalysis project, the ERA data should not be expected to agree with buoy-measured data 100 percent. Possible reasons for the differences are: the resolution of ERA-interim is 1.0° × 1.0° latitude/longitude and an interpolation is used, whereas a buoy only measures one spatial point; different wave partitioning methods and human errors can both reduce the degree of agreement. The comparisons indicate that the reliability of ERA-interim’s sea states data is acceptable. 4

Figure 2. Comparison of wave data from ERA-interim and MARIN.

FATIGUE CALCULATION

In ultimate strength analysis, the extreme loading condition always corresponds to extreme sea states, whereas the other relatively moderate sea states are not so important (Bitner-Gregersen & Skjong 2011). However, in fatigue damage calculation, each sea state should be taken into account. This paper utilizes SWAN wave model to simulate the sea states for floating structure fatigue assessment. wind data input is from ERA-interim. In order to validate this methodology, the monthly fatigue damage was calculated with three different datasets at the same hotspot of a FPSO-Glas Dowr, which was working in Sable field from July 2007 to October 2008. These three datasets of sixhourly sea states include data from ERA-interim wave model, buoy-measured data and SWANsimulated data. Part of these data are shown in Table 1 as an example. These three datasets are all for the same spot of Sable field (35.25°S/21°E) and cover the same period (from July, 2007 to October, 2008). The Floating Production Storage and Offloading system (FPSO) is a vessel used for offshore oil and gas exploitation. There are two types of FPSOs in terms of mooring system: spread-moored and turret-moored. A spread-moored FPSO is positioned by several mooring legs which can maintain the vessel in a fixed orientation. Turret-moored

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Table 1.

Example of sea states from buoy-measurement. Windsea

Swell

Date

Hs [m]

Tz [s]

Direction [º]

Hs [m]

Tz [s]

Direction [º]

07070102* 07070108 07070114 07070120 07070202 07070208 07070214 07070220 07070302 … 08103112

0** 0 0 0 0 0 0 0 0.5 … 0

5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 2.8 … 5.0

191 191 196 200 208 200 214 269 132 … 205

3.8 3.5 3.6 3.4 2.9 3.4 2.4 2.2 2.1 … 2.3

7.3 7.5 8.0 8.3 8.2 8.3 8.2 7.3 8.7 … 7.1

205 206 205 211 216 227 216 263 271 … 228

*The format of date is: yy/mm/dd/hh. **Wave heights (0.5 m) is displayed as 0 m.

FPSOs (like Glas Dowr) are designed to allow FPSOs to rotate depending on the direction and strength of waves, currents and winds. The details of Glas Dowr and the coordinates of the hotspot are listed in Table 2. All the fatigue calculations are carried out by the fatigue damage calculation program-Bluefat. The software is suitable for fatigue calculations in the deck, side shell and bottom structure (Aalberts et al. 2010). The calculation process of Bluefat, which requires the short term statistical input, in this case six-hourly sea states for both wind waves and swell separately, is shown in Figure 3. The related functionalities are listed in Table 3. According to the Bluefat calculation shown in Figure 4, the monthly fatigue damage with those three datasets have the same seasonal trend. Monthly fatigue damage is high during winter and low during summer. SWAN and ERA fatigue lines fit well with each other. However, the average fatigue of SWAN line is higher than buoy line. They are only close to each other at those “peak” months (such as Nov. 2007 and Sep. 2008). Comparison with the ERA-interim wave dataset reveal no SWAN dataset accuracy advantages. However, calculating fatigue damage just by inputting wind data is obviously more promising. First, the wind measurement technology has been developed much further than wave measurement. Second, during the design of floating structures, the expected sea states should be estimated; however, because of climate change, past sea states are less accurate predictors of future sea states. Thus, the wave change trend must be predicted. Predicting wave trends still requires more development, whereas more research has been done in

Table 2. The details of Glas Dowr and the coordinates of the hotspot.

Glas Dowr

Hotspot

Figure 3.

Table 3.

Length (m)

Breadth (m)

Mean draughts (m)

232

42

12.99

X (m)

Y (m)

Z (m)

112.85

18.3

21.31

The calculation process of Bluefat.

The functionalities of Bluefat.

Functionality

Bluefat

Non-linear roll damping Input RAO database Radiation/diffraction Load combination

Linearization 2-D diffraction (seaway) Empirical method 3D Standard deviation (DNV method) Spectral Rayleigh Method DNV Method DNV

Calculation method Short-term distribution Tank pressure method Intermittent wetting method

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the effect of climate change on floating structures’ fatigue consumption during their service life. It should be noted that the methodology still requires further improvement before it can guide any engineering activity.

ACKNOWLEDGEMENTS The authors acknowledge the financial support from Chinese Scholarship Council (CSC).

REFERENCES Figure 4. The calculation result with three datasets (ERA-interim wave model, buoy-measured data and SWAN-simulated data).

the field of wind change, rather than wave change (Zou et al. 2013). As a result, by predicting the wind change trend, the sea states that floating structures may encounter can be estimated with a wave model, such as SWAN. Such a methodology still requires improvement, as the calculated fatigue damage doesn’t fit the damage calculated by the buoy measured data very well. One reason may be that the wind data in Sable field is only the result of interpolation. That indicates directly-measured wind data may improve the accuracy of fatigue calculation. In addition, the SWAN dataset and the buoymeasured dataset used different wave partitioning methods. This may also result in the fatigue calculation difference. 5

CONCLUSIONS

This paper has presented a methodology, in which offshore floating structures fatigue damage can be calculated without directly inputting wave data. This methodology first simulates the sea states in Sable field with SWAN wave model by inputting the wind data from ERA-interim project. Then, the fatigue assessment for FPSO-Glas Dowr is made. In order to validate this methodology, this calculation is compared with the fatigue damages which are calculated based on the sea states from ERAinterim and buoy-measurement. The comparison indicates that this methodology has its own advantages: firstly, during the process of fatigue damage, the sea state data is required. However, with the help of wave models, if the wave data is limited, the wind data can take its place. Second, by predicting the wind change trend, designers can estimate

Aalberts P., Cammen J. & Kaminski M.L., 2010. The Monitas system for the Glas Dowr FPSO, Offshore Technology Conference (OTC 2010), Houston, USA. Berrisford P., Dee D., Poli P., Brugge R., Fielding K., Fuentes M., Kallberg P., Kobayashi S., Uppala K. & Simmons A. 2011. The ERA-interim Archive, European Centre for Medium Range Weather Forecasts, United Kingdom. Bitner-Gregersen E.M. & Skjong R. 2011. Potential impact on climate change on tanker design, Proc. 30th International Conference on Ocean, Offshore and Arctic Engineering (OMAE 2011), Rotterdam, the Netherlands. Bitner-Gregersen E.M. & Haver S., 1991, Joint environmental model for reliability calculations, Proceedings of the First International Offshore and Polar Engineering Conference (ISOPE 1991), Edinburgh, UK. Caires S. & Sterl A. 2003. Validation of ocean wind and wave data using triple collocation, Journal of Geophysical Research, v.108, issue c3, 403–427. Dee D.P., Uppala S.M., Simmons A.J., Berrisford P., Poli P., Kobayashi S., Andrae U., Balmaseda M.A., Balsamo G., Bauer P., Bechtold P., Beljaars A.C.M., van de Berg L., Bidlot J., Bormann N., Delsol C., Dragani R., Fuentes M., Geer A.J., Haimberger L., Healy S.B., Hersbach H., Hólm E.V., Isaksen L., Kållberg P., Köhler M., Matricardi M., McNally A.P., Monge-Sanz B.M., Morcrette J.J., Park B.K., Peubey C., de Rosnay P., Tavolato C., Thépaut J.N. & Vitart F. 2011. The ERA-Interim reanalysis: configuration and performance of the data assimilation system. Q.J.R. Meteorological Society, 137, 553–597, doi: 10.1002/ qj.828. DNV. Fatigue Assessment of Ship Structures, Det Norske Veritas, 2010, Classification Note 30.7. Durrant T., Diana J., Greenslade M. & Simmonds I. Validation of Jason-1 and Envisat Remotely Sensed Wave Heights, Journal of Atmospheric Oceanic Technology, v.26, 123–134. Hanson J., Lubben A., Aalberts P. & Kaminski M.L., 2001. Wave Measurements for the Monitas System, Offshore Technology Conference (OTC 2010), Houston, USA. Hanson J. & Phillips O.M., Automated Analysis of Ocean Surface Directional Wave Spectra, Journal of Oceanic Technology, v.18, 277–293.

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Holthuijsen L. H. 2007. Waves in oceanic and coastal waters, New York, Cambridge University Press, 305–361. Repko A., Van Gelder P.H.A.J.M., Voortman H.G. & Vrijling, J.K., 2004. Bivariate description of offshore wave conditions with physics-based extreme value statistics, Journal of Applied Ocean Research, v.26, 162–170. Semedo A., Sušelj K., Rutgersson A. & Sterl A., 2011. A global view on the wind sea and swell climate an variability from ERA-4, Journal of Climate, v.24, 1461–1479.

Young I.R., Zieger S. & Babanin A.V., 2011. Global trends in wind speed and wave height, Journal of Science, v.332, 451–455. Zou T., Jiang X.L. & Kaminski M.L., 2014, Possible Solutions for Climate Change Impact on Fatigue Assessment of Floating Structures, The International Society of Offshore and Polar Engineering (ISOPE 2014), Pusan, South Korea.

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Structural analysis

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

A simplified FE model for the non-linear analysis of container stacks subject to inertial loads due to ship motions E. Brocco, L. Moro & M. Biot University of Trieste, Trieste, Italy

ABSTRACT: On board ships, cargo-related failure conditions are strongly dependent on cargo and lashing systems. This study presents a procedure for achieving an effective FE structural model of a container stack for carrying out the numerical simulations with the aim to define a numerical model that accurately simulates the actual deformation of a stack, making it as simple as possible in order to decrease the computational cost of the simulations. Using this model, a series of numerical analyses can be carried out for a huge amount of loading conditions obtained by non-linear numerical simulations of ship motions. 1

INTRODUCTION

During the last decades, the number of containers transported has grown exponentially. Following the need of a greater capacity of container transportation, the dimension of the ship along with the number of containers carried has also increased, thus increasing the forces acting on the containers selves, which has increased the occurrences of damage or container loss overboard. Loss of containers overboard translates into economic losses, environmental hazard especially when carrying dangerous goods and potential hazard for every ship traveling on the route because containers can float for long time when lost and a ship can easily hit them and be damaged without any evidence of impact. Having a thorough insight in the forces acting between the containers of the stack and the lashing equipment is still difficult due to the large number of variables to be taken into consideration. Variables as payload weight, inertial load induced by the sea, wind load, nonlinearities due to the lashing equipment and contact with other stacks. In order to investigate the container loss phenomenon, to verify if containers and lashing equipment standards can still be considered up to date and to improve the efficiency of container lashing systems a better understanding of container stacks behaviour is necessary. Classification societies produce regulation for the design and approval of the Cargo Securing Manual, which provides general loading conditions and lashing equipment arrangement. It uses simplified calculation methods based on the assumption that containers behave like bidimensional bodies. In order to evaluate the

maximum forces acting on the containers and the lashing equipment, the following forces are taken into consideration: gravitational loads, wind loads and inertial loads due to ship motion. Last developments in regulation have been the introduction of route specific container stowage where ship motion is seen to depend from the route followed (GL 2013, ABS 2010). Recently DNV (DNV 2011) has proposed a direct calculation method based on beam analysis (which is at the basis of the NAUTICUS Container Securing software), where container stacks are modelled as two independent bi-dimensional beam models to take into account for the different racking stiffness of the closed and open side of the containers. In the early 2000s studies focused on the prediction of maximum forces acting on the containers (Nakamura et al. 2001), using container stack linear numerical models and long term acceleration distribution. Others investigated the influence of parametric roll on the forces acting on the containers carried on deck (Frances et al. 2003) where the forces due to parametric roll were evaluated using the method proposed by the classification society and the accelerations were both calculated and derived from test model. Recently, research focused on the influence of some major variables, as the gaps due to the twist locks and the contact between adjacent container stacks, in the prediction and control of the dynamic forces acting on the containers. These studies used experimental scaled models of container stacks and a simplified numerical model created and calibrated on the experimental one (Kirkayak et al. 2010, Aguiar de Souza 2010, Aguiar De Souza et al. 2011). Finite element model of a stack has been used also to evaluate the forces acting on

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containers aiming at creating a test procedure for fully automatic twist locks (Wolf et al. 2011). Besides this recent development, a deeper knowledge of container stacks is necessary, especially when considering the nonlinearities due to the securing equipment and load due to ship motions. In this paper, the procedure for achieving an effective FE structural model of a container stack for carrying out the numerical simulations is presented. The aim of such a procedure is to define a numerical model that accurately simulates the actual deformation of a stack, making it as simple as possible in order to decrease the computational cost of the simulations. Using this model, a series of quasistatic numerical analyses can be carried out for a huge amount of loading conditions obtained by non-linear numerical simulations of ship motions. Figure 1 shows a procedure for the structural assessment of a container stack under the action of inertial load, evaluated using extensive ship motion simulations. It is structured as follows: 1. Creating a fine mesh FE model of the container 2. Calculating its inertial and mechanical properties 3. Creating and calibrating a simplified FE container model 4. Generating a stack of containers 5. Calculating the inertial loads acting on the stack 6. Applying the loads on the stack 7. Evaluating the forces acting on the containers of the stack. In this paper, the attention is focused on generating a simplified container stack model and identifying the most effective approach to solve such nonlinear structural problem. It first presents the fine mesh FE model of a container used as benchmark for calculating torsional stiffness and transversal and longitudinal racking stiffness. Then, it defines a coarse mesh FE model of a container stack, which is made of few 1D elements tuned so

that the coarse mesh model has the same torsional and racking stiffness as the fine mesh model. In the stack coarse mesh model, the non-linear behaviour of the lashing system that secures the container stack to the ship main deck is properly modelled. The implemented procedure, along with exhaustive ship motion simulations and cargo plan data, could be useful to have an insight on the forces arising on the containers in order to verify if current standards can still be considered up to date. The procedure could also be used to calculate the securing manual basing on a direct approach both from the hydrodynamic and the structural point of view. All models and structural analyses have been carried out using MSC Patran/Nastran suite. 2

Containers loaded on board of a ship are stacked and connected one to the other and to the ship structure by means of twist locks. When loaded on deck, they are secured using lashing bars that reduce transversal forces acting on the lower containers in the stack. Containers stacked on deck are most subject to transversal acceleration due to the lack of any lateral restraint (as for example the cell guides used for containers loaded in the holds). The forces acting on containers on board of a ship are: gravitational loads, inertial loads and when considering stacks carried on deck, also wind forces, green water, impact with adjacent stacks and ship structure relative motion. Wind loads have to be considered only for the outer stack on the deck while effect of green water on deck and other loads are generally considered of minor importance. Gravitational load and inertial load strictly depend on the payload carried within the containers of the stack. In this study, the attention is focused on the global effect of the cargo related forces while local loads on the container structure are neglected. A thorough study on local loads on the container structure would require a more complex model and detailed information on the cargo– container structure interaction (for instance the good weight, their dimension and disposition and how they are fastened in the container). A more complex model would imply a higher computational effort and more detailed information on the goods carried which is usually not available going besides the aim of this work. 2.1

Figure 1.

Scheme of the proposed procedure.

THE CONTAINER STACK FINITE ELEMENT MODEL

The container model

The first step was the creation of a detailed FE model of a container used to calculate the

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mechanical quantities necessary to set up a simplified model. A 20 feet general cargo container has been modelled. The detailed model, based on a real container design, has been created using only low order quadrilateral and triangular plate (2D) elements, with respectively four or three nodes. The beams composing the main supporting structure of the container have been explicitly modelled. They, along with the container shell plating, have been modelled on the mid thickness while the corner castings have been modelled on their external sides to reduce the approximation of the meshed structure. The corner castings have been modelled with continuous faces. Aim of this work is to investigate the global structure of the container, therefore disregarding the discontinuities induced by the holes has been considered acceptable. Non-structural components as wooden floor and the door assembly have not been modelled. The bottom panels have been considered non-structural because they are made of wood and they are simply screwed on the transversal supports. The doors, when closed, create a gap of about 18 mm with the rear side frame and are surrounded by rubber seals to make the assembly water tight. In order to take into account the influence of the doors on the transversal stiffness of the open side of the container, a nonlinear problem with big displacement and contact boundary condition must be solved. At a first modelling stage, considering small deformations, door assembly influence to the transversal stiffness of the rear side can be neglected without introducing a significant approximation. The container model has been meshed mainly using quadrilateral elements with a minimum use of triangular elements. The edge element size has been kept as uniform as possible with an average length of 10 mm. The material adopted in the definition of the element properties is steel. Figure 2 and Figure 3 show the geometry of the model and a meshing detail.

Figure 2.

Container finite element model.

Figure 3. Particular of the bottom rear corner casting and cross members.

2.2

Strategy for container stiffness evaluation

Aim of the simplified model is to reduce the computational effort in calculating forces acting on containers when acceleration derived from extensive ship motion simulation are applied to the stack. In order to have a good approximation of the behaviour of the stack, the mechanical and inertial properties of the coarse mesh model have to be as similar as possible as those of the detailed model. In the stack, only corner castings transmit forces between containers. Therefore, in order to be less calculation expensive, eight node, at least, are necessary to define the coarse mesh model. Following a rigorous definition of the stiffness of a structure, and taking into account that in order to be less calculation expensive, the simplified model has to be made only by eight nodes, a complete knowledge of the stiffness of the model could be obtained by knowing a very big stiffness matrix. Even when the geometrical symmetry with respect to the mid plane is considered, the number of parameters to be calculated still remains very large. In any case, after calculating the stiffness on the detailed model it’s necessary to set the coarse mesh parameter by parameter in order to fulfil the stiffness properties. It is clear that the whole operation is hard because of the high number of parameters to be taken into account. On the other hand, by considering that the aim of this work is not to perform an optimization process, in order to reach good results in a short time, few parameters defining the global behaviour of the container basing on just the macroscopic effects can be used. The constraints and the applied forces as defined in the standards, left the container structure free to deform also in parts other than those directly involved by the application of the force. Consequently, the effect of the load is shared by many elements, so making any setting process very difficult. In other words, this excessive freedom

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in deformation is not suitable in an equivalence process, because it involves tuning operations on a large number of structural elements at the same time. The guidelines stated in the standards are just a starting point for the investigation, and a more precise identification of the equivalence parameters is needed. Qualitative considerations on the main forces acting on the containers lead to the identification of the main global parameters summarizing the behaviour of the container structure. In the analysis, the superposition principle is used and the forces, acting on the four upper corner castings of the containers, are applied separately for each direction while the container is considered constrained at the base. The effects of the applied forces are evaluated in terms of single edge relative displacement, where relative displacement of an edge of the container with respect to the opposite edge is considered, and in terms of face relative displacement, where relative displacement of one face with respect to the opposite face of the container is considered. This is why longitudinal and transversal racking and torsional stiffness both at close and open end are the parameters used to characterize the container structure behaviour and to set the coarse mesh model. To have a better description of the racking phenomenon, the longitudinal and transversal racking have been calculated with reference to two conditions: forces applied on the upper side and forces applied on the lower side. This because of the difference of the stiffness in the structure of the bottom and the roof of the container. 2.3

Techniques for FE calculation of container stiffness

Aim of the coarse mesh finite element model of the container is to allow extensive non-linear structural calculations and therefore the model has to be as light as possible from the computational point of view. The key factor for the most simplified FE model of a container working in a stack is the set of the 8 nodes corresponding to the corner castings of the container, placed in the geometric centre of the corner castings and connected by beams simulating the main structure of the container. This kind of modelling derives from the considerations that the force transmission between containers in a stack takes place through the twist locks that connect the corner fittings of the containers facing each other’s, being the twist locks the only devices by which containers are connected to each other. Moreover, former studies discussing how containers interact in a stack (Kirkayak et al. 2010, Aguiar de Souza 2010, Aguiar De Souza et al. 2011) and

the recommendations of the DNV register given in the “Direct calculation using beam model” guide (DNV 2011) lead to use this kind of modelling. Considering the approach outlined above, in the stiffness evaluation of the detailed model, forces and constraints have to be applied in the geometric centre of each corner casting. An additional node has been placed in the geometric centre of each corner casting and it has been connected to the nodes belonging to the hole zone of the corresponding corner casting by means of rigid connectors. Forces and constraints on the detailed model have been applied to the 8 additional nodes. The stiffness parameters have been derived from the detailed model and then the same procedure used on the detailed model has been applied on the coarse mesh model in order to set the model. The procedure used to evaluate the stiffness parameters consists of allowing only the deformation of the elements under analysis in the direction along which the forces are acting and constraining all other deformations both in the other direction and for the other elements. For instance, in the case of transversal racking stiffness evaluated for the upper close end two equal forces acting in the transversal direction are applied to the nodes located at the centre of the two upper corner castings of the close end. In these same nodes, the deformation has been allowed only in the transversal direction, constraining the translation in the longitudinal and vertical direction. Translational constraints in all the three orthogonal directions have been applied to the central nodes of all the remaining corner castings. All the eight corner castings nodes have been left free to rotate around the three axes. This procedure allows the identification of only the transversal racking stiffness associated with the upper edge at close side end of the container. The same procedure is used to calculate the remaining transversal stiffness. When calculating longitudinal and torsional stiffness a similar procedure has been adopted. In particular, when longitudinal racking stiffness is investigated, only the upper or lower face of the container is left free to translate in longitudinal direction while, when the torsional stiffness is evaluated the closed or the open end are left free to rotate maintaining the faces on the YZ plane. Figures 4 to 6 show the loading condition used to perform the stiffness evaluation. 2.4

The coarse mesh model for extensive calculations

The aim of the coarse mesh model is to allow agile extensive calculations on the different load time histories derived from the ship motion analyses. In order to perform thousands of calculations, the computational effort to solve the problem for one

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Figure 4. Constraints and load application for transverse stiffness calculation.

Figure 5. Constraints and load application for longitudinal stiffness calculation.

Figure 6. Constraints and load application for torsional stiffness calculation.

time instant of the motion history (i.e., one step) has indeed to be as lower as possible. Therefore, it is clear that the finite element model used for the calculation has to be as simple as possible in order to reduce the computational time. The need of a very simple model, as already mentioned, leads to a schematization that consists of frame basing on a set of only 8 main nodes to represent the corner castings of the container, which are the points through which the container transmits and receives the forces from and to the other containers when placed in a stack. The eight nodes are connected by BEAM elements, which respond to axial load, bending moment and shear (MSC Nastran 2013), representing the twelve beams supporting the container plating. Even if such a model should be adequate to simulate the behaviour of the real container structure, other elements need to be added to the model in order to reach an accurate tuning on the detailed model stiffness parameters. So, ROD elements (which only respond to axial load), positioned on the diagonals of the container faces, have been used to add additional stiffness to the coarse mesh model. Mass elements could be added to simulate the cargo mass. The expected results of such a simplified model are the displacements and the forces mutually exerted by the container through the corner castings. According to a multi-level approach, such displacements and forces could be used to perform a punctual stress analysis on the detailed model, by applying them on the additional nodes of the detailed model, or to directly perform a failure assessment basing on the limit forces derived from the standards. The main frame of the container structure has been modelled taking the sectional area of the main beam of the container with a quote of associated plate equal to the 50% of the plating connected to that beam. Even though this percentage is extremely high, the stiffness target values cannot be reached so other stiffening element with only axial behaviour has been added. The final results of the setting are shown in Table 1. As regards the transversal and longitudinal racking stiffness, which may be considered the main stiffness parameters, a good grade of approximation has been reached, while some differences remain between the benchmark value and the obtained value of the torsional stiffness. These parameters can be considered of lower importance and such rough approximation can be considered acceptable. In Figure 7 the final configuration of the coarse mesh model is shown. Moreover, by this approach the real column buckling capability of the corner posts cannot be simulated. This means that the results of the FE

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Table 1.

Container stiffness.

Model

Fine mesh

Transverse stiffness (MN/m) Closed top 89.5 Open top 22.2 Closed bottom 73.1 Open bottom 0.59

Coarse mesh

89.3 21.3 72.2 0.60

Differences %

0 −4 −1 2

Longitudinal stiffness (MN/m) Top 220.8 224.3 Bottom 220.8 224.5

2 2

Torsional stiffness (MN m/rad) Closed 56.1 Open 11.5

−23 −11

43.0 10.2

Figure 7. Complete coarse mesh FE model of the container.

analysis on the container stack can be considered accurate till the buckling phenomenon triggers. As a consequence, the buckling checking of the corner posts and the ultimate capability of the container structure can only be evaluated in the post processing analysis. In the tuning of the inertial properties, the mass of the empty container has been associated to the beam elements (using a proper material density), while the rod elements, with only axial behaviour, have been modelled without mass. In order to decrease the error on the inertial moment a concentrated mass has been added to the coarse model with a value of 10% of the weight of the empty container maintaining the mass of the coarse mesh model the same of the fine mesh model. 2.5

that in this stage the wind loads are neglected since they affect only the stacks in the outer rows. The stack under investigation is secured on a hatch cover using base sockets, twist locks, lashing rod and turnbuckles. In the actual loading condition no lashing bridge is considered so lashing of the container stack is symmetric considering the close and open end. As usual on board container ships, the lashing rods with turnbuckles are tightened when the ship is leaving the port and they are checked and tightened again during sailing. The lashing rods are usually not pre-tensioned. All the containers tiers are fastened to each other by means of twist locks that are considered locked when in close position. Each component of the securing system is characterized by a non-linear behaviour. Each lashing rod reacts to a tensile force, whose line of action is the axis of the rod, while it is ineffective when a compression load is applied. Two different types of fittings can be identified: the short lashing rod represents the lashing equipment used to secure the base of the second container tier to the deck’s structures, the long lashing rod connects the base of the third tier of containers to the deck’s structures. For these two types of equipment, the estimated stiffness curves are showed in Figure 8. The twist locks connect the container facing each other and are characterized by a strong nonlinear behaviour. In the vertical direction when traction force is acting, after a first deformation where the reaction of the twist lock is null, the twist lock starts to react with its proper stiffness. When a compressive force is acting, the twist lock behaves like a rigid body because the footprint area is very high. This behaviour is showed in Figure 8.

The lashing equipment

The equipment used to secure the container stack depends on the position on which the same is loaded. In this first work the stack is assumed to be located on ship deck in an inner row, this means

Figure 8.

Lashing equipment stiffness curves.

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Along the transversal and longitudinal directions, because of the constructive and working tolerances, there is a gap that has to be filled before the twist lock start to react. To simplify the problem, first, the nonlinearities related to longitudinal and transversal translation have been neglected because of their lower importance if compared to those associated with vertical translation. When considering rotation allowed by the twist lock, some uncertainties are present, in fact, because of the gaps a little rotation around all of the three axes is possible. Taking advice from the guidelines reported in DNV classification notes 32.2 “Container securing” (DNV, 2011) at the section “Direct Calculation using Beam Analysis” the behaviour of the twist lock against the rotation is simplified by using a hinge that allows the relative rotation of the two containers. This representation is valid under the assumption of small rotations and will be used also in this study, so introducing a small and acceptable approximation in the FE model. 2.6

Techniques for the simulation of a container stack

Concerning the nonlinearities due to lashing equipment, two different simplified model types have been tested. The first approach uses fully nonlinear elements (CBUSH data entry of MSC Nastran software) to simulate both the lashing bar and twist locks. This element allows the definition of a nonlinear stiffness law in all the three orthogonal directions and for the three associated rotations. In the case of twist locks a stiffness curve has been defined for the vertical direction, while for the two other directions and for the three rotations the value of the stiffness has been set very high to simulate a rigid connection. The rotations of the two facing corner castings have been allowed with the insertion of a spherical hinge using an RJOINT element. When modelling the lashing bar, a proper stiffness law has been assigned to the direction corresponding to the longitudinal axis of the element while along the two other perpendicular directions the stiffness has been set to zero; since the lashing bar is effective only against axial stress, also the rotational stiffness has been set to zero. Using such elements makes convergence difficult when solving the nonlinear problem with a lot of time spent to reach a solution. Moreover, the simulations of different load cases required different tunes of the solver parameters in order to reach a solution. This is why a more suitable approach to the analysis of extensive simulation was found, using different element types. In the second approach, some simplifications have been introduced especially in the simulation

of twist lock gaps. In this model, lashing bar and turnbuckles have been modelled using ROD element associated with a nonlinear definition of the material Young Modulus in order to simulate the ineffectiveness of this lashing element against compression. The twist locks have been modelled using beam elements with high sectional area in order to simulate a rigid connection. In this case, nonlinearities due to twist lock gaps have been neglected considering that compressive forces drive the most dangerous situation and in this case, the effect of the gap does not affect the transmission of the forces between the containers. As for the behaviour of the stack against transversal load, the two models have been tested under the same external conditions, in the specific case a transversal acceleration of 2 g has been adopted. Figure 9 shows that the maximum deformation of the stack with linear twist locks is about 15% lower if compared with the completely nonlinear twist locks simulation. Also in this second model, hinge connections have been adopted to free the rotations between the two facing corner castings. This second model has the advantage of being very simple and time-efficient, beyond this, the model is very stable and the same tune of the convergence parameter allows the solution of all the tested load cases. The simulations can be performed using a time domain approach or a quasi-static approach. Aim of this study is not to set a procedure to investigate the dynamic behaviour of the stack but to set one in order to assess the stack failure condition and the stresses arising on the containers and the lashing equipment. Time domain simulations involve high computational effort because of the

Figure 9. Deformation, in meters, of the container stack modeled with nonlinear twist lock (leftside) and rigid twist locks (rightside) under the action of a 2 g transversal acceleration. In both the models deformation are scaled of a 10 factor.

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very short time step. In this first part of the work the dynamic effect of the impact of the stacked container driven by twist lock gaps and the impact with near container stacks is neglected and the attention is focused mainly on the effect of the inertial loads, which are carried out from extensive nonlinear ship motion simulations. In order to increase the computational efficiency of the procedure a quasi-static approach has been adopted in this way a limited number of time steps need to be solved decreasing consistently the time to analyse a complete ship motion simulation. Using this approach the accelerations carried out from the ship motion simulations can be analysed using a time step of 0.5 second and, for each time step, the structural problem is solved defining a static load case. It is clear that using this approach, in the resolution of the structural problem, each time step is independent from the others. The time simulations that will be analysed using this procedure are supposed to be enough to describe the ship behaviour in each sea state that could be carried out, for example, from the scatter diagram, for different ship velocities and for different loading condition. Each ship motion simulation is supposed to be 1.5 hours long. The solver used is an implicit nonlinear solver in which also centrifugal terms are taken into account. Inertial loads are introduced, defining, for each time instant, the proper translational accelerations, rotational accelerations and the rotational velocities. This kinematic field has to be defined in a proper reference system that is located in the geometrical centre of the first tier’s container bottom. The containers cargo has been modelled placing 8 lumped masses in the related 8 nodes representing the container corner castings. As prescribed by classification societies, the values of the lumped masses has been set to keep the transversal and longitudinal centre of gravity of the cargo in the geometric centre of the container. Moreover, the vertical coordinate of the centre of gravity of the masses has been placed in way to distribute the 55% of the mass effect on the container’s bottom and the remaining part on the top. The model with rigid twist locks can be used to solve the complete time histories coming from nonlinear ship motion simulations while, once the worst time instants have been identified, the completely nonlinear model can be used to increase the accuracy of the solution. Once the worst load cases have been analysed the deformations evaluated on each container can be applied to the fine mesh model to investigate the stresses arising on the container structure. The forces evaluated on each corner casting and on the lashing equipment can also be compared with

the working or breaking load of the lashing equipment and of the containers to have an easy parameter to assess the container stack failure. 3

CONCLUSIONS

In this study a procedure for the structural assessment of a container stack has been set up. A great effort has been put on facing the nonlinear problem deriving from the interaction between the components of the system, which are the elements used to secure the containers to the ship and to each other’s. A detailed finite element model has been created and it has been used, in a first stage, to define the main quantities needed to create a simple and accurate coarse model, suitable to perform extensive calculations. This model can be used, when a load history is applied, to perform the failure assessment via the force or stress evaluation. The container coarse mesh model fits quite well the detailed model, especially when the main stiffness parameters (transverse and longitudinal racking stiffness) are taken into account, while, as for the torsional stiffness, differences in the two models may be accepted as this does not compromise the accuracy of the analysis. Two models of container stacks have been defined, one more accurate and one more simplified. The more simplified one has been chosen to perform the extensive calculations because it allows to come to the solution of each load case in a fully automatic way, while the more accurate needs a specific setting for each time step to be solved. Even if the contribution of the nonlinearities of the twist locks have been neglected, the model still approximates very closely the behaviour of a container stack loaded on the deck of a container carrier vessel. A more precise analysis can be done by using the more accurate model in an intermediate step between the extensive calculation and the failure check. Basing on the model here defined for a container stack, a simulation can be carried out on the response of the system to the loads deriving by ship motions. ACKNOWLEDGEMENTS This study is part of the project Direct intact stability assessment and operational guidance to the master in the framework of International Maritime Organization Second Generation Intact Stability Criteria: development of tools and procedures for safety assessment with particular attention to cargo securing. The project has been funded by the

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University of Trieste—“Finanziamento per Ricercatori di Ateneo (Announcement FRA2011)” what is gratefully acknowledge.

REFERENCES Aguiar de Souza, V., Study on the Dynamic Response of Container Stack Using Non-Linear Finite Element Analysis, The University of Tokyo, PhD Thesis, 2010. Aguiar de Souza, V., Kirkayak, L., Suzuki, K., Ando, H., Sueoka, H., 2011. Eperimental and numerical analysis of container stack dynamics using a scaled model. Ocean Engineering, vol. 39: 24–42, 2012. American Bureau of Shipping, 2010. Guide for certification of container securing systems—November 2010 (Updated September 20114), Houston, American Bureau of Shipping. Det Norske Veritas, 2011. Classification notes No. 32.2— Container securing—July 2011, Det Norske Veritas AS.

France W., Levadou M., Treakle T.W., Paulling J.R., Michel R.K., Moore C., 2003. An investigation of head-sea parametric rolling and its influence on container lashing systems. Marine Technology, vol. 40 (no 1): 1–19. Germanischer Lloyd, 2013. Rules for Classification and Construction, I—Ship Technology, Part 1—Seagoing Ships, Chapter 20—Stowage and Lashing of Containers. Hamburg, Germanischer Lloyd SE. Kirkayak, L., Aguiar de Souza, V., Suzuki, K. Ando, H., Sueoka, H., 2011. On the vibrational characteristics of a two-tier scaled container stack. Journal of Marine Science and Technology, vol. 16: 354–365. MSC Nastran 2013. Quick Reference Guide, Santa Ana, CA, MSC Software Corporation. Nakamura, T., Ota, S., Nakajima, Y., 2001. Evaluation of expected maximum values of forces acting on containers and lashing rods on container ship. Journal of Marine Science and Technology, vol. 6: 3–12. Wolf, V., Darieand, I., Rathje, H., 2011. Rule development for container stowage on deck. Advances in Marine Structures, pp. 715–722, CRC press.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Structural Health Monitoring of marine structures by using inverse Finite Element Method A. Kefal & E. Oterkus Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow, UK

ABSTRACT: A new state-of-the art methodology named as inverse Finite Element Method (iFEM) is adopted to solve the inverse problem of real-time reconstruction of full-field structural displacements, strains, and stresses. iFEM has shown to be precise, robust, and fast enough to reconstruct the three dimensional displacement field of structures in real-time by utilizing surface strain measurements obtained from strain sensors embedded on the structure. The numerical implementation of the iFEM methodology is done by considering four-node inverse quadrilateral shell element. Two demonstration cases are presented including a quadrilateral plate subjected to bending force and a stiffened plate under bending loading. Finally, the effect of sensor locations, number of sensors and the discretization of the geometry are examined on solution accuracy. 1

INTRODUCTION

Well-maintained structures are more durable. Increase in durability decreases the direct economic losses (repair, maintenance, reconstruction) and also helps to avoid losses for users that may suffer due to a structural malfunction. Furthermore, new materials, new construction technologies and new structural systems are increasingly being used in marine industry; therefore it is necessary to increase knowledge about their on-site structural condition and integrity. Structural Health Monitoring (SHM) certainly provides satisfactory answers to these requests. Reconstruction of the full-field structural deformations, strains and stresses is a key component of SHM by utilizing the strain data obtained from a network of on-board strain sensors located at various sites of a structure (Tessler & Spangler 2005). A regularization term which ensures a certain degree of smoothness to solve this inverse problem was introduced by Tikhonov & Arsenin (1977) and most of the today’s inverse algorithms uses Tikhonov’s regularization (refer to Liu & Lin 1996, Maniatty & Zabaras 1994, Maniatty et al. 1989, Schnur & Zabaras 1990 and references therein). However, most of these inverse methods are not generally appropriate for use in onboard SHM procedures because many of these didn’t take into consideration the complexity of boundary conditions and structural topology. Moreover, they mostly require adequately precise loading and/or material information, although

it shouldn’t be the case for a powerful SHM algorithm. A new state-of-the art methodology named as inverse Finite Element Method (iFEM), which certainly satisfies the necessities of SHM procedure, was developed by Tessler & Spangler (2003, 2005). iFEM algorithm reconstructs the structural deformations from experimentally measured strains based on the minimization of a weightedleast-square functional. Unlike other inverse methods, iFEM methodology possesses a general applicability to complex structures subjected to complicated boundary conditions in real-time (Tessler & Spangler 2005). iFEM framework is precise, powerful and sufficiently fast for real-time applications of any type of static and dynamic loadings, as well as a wide range of elastic materials since only strain-displacement relationship is used in the formulation (Gherlone et al. 2012, 2014). The domain of the structural model can be discretized by using beam, frame, or plate and shell inverse finite elements in order to perform SHM based on iFEM algorithm. In order to monitor truss, beam, and frame structures in real-time, Cerracchio et al. (2010) and Gherlone et al. (2011, 2012, 2014) developed a computationally efficient inverse-frame finite element based on kinematic assumptions of Timoshenko shear-deformation theory. Their numerical and experimental examination of the inverse-frame finite element indicated the superiority of iFEM approach for shape-sensing of three-dimensional frame structures that are subjected to static or/and damped harmonic

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excitations. Tessler and Spangler (2004) used firstorder shear deformation theory to develop a threenode inverse shell element (iMIN3) for analyzing arbitrary plate and shell structures. The precision of iMIN3 element was demonstrated by using experimentally measured strain data by Quach et al. (2005) and Vazquez et al. (2005). Moreover, Tessler et al. (2012) recently improved iMIN3 element formulation to reconstruct deformed shape of plate and shell structures undergoing large displacements. Regarding the application of iFEM analysis on engineering structures, it has been limited to the SHM of aerospace vehicles (Tessler et al. 2011, Gherlone et al. 2013). The main and novel aim of this study is to demonstrate the applicability of iFEM to SHM of marine structures for the first time in the literature. The presented iFEM formulation is based on the minimization of a weighted-least-square functional that uses Mindlin’s first-order plate theory. The numerical implementation of the iFEM methodology is done by developing a four-node inverse shell element (iQS4) including hierarchical drilling rotation degree of freedoms. Various validation and demonstration cases are presented including a quadrilateral plate subjected to bending force and the fundamental problem of a stiffened plate under bending loading which represents the portion of the side of a typical longitudinally and transversely framed tanker. Experimentally measured strains are represented by strain results obtained from a high-fidelity solution using an inhouse finite element code. Several types of discretization strategies are examined and comparisons of the reconstructed iFEM and direct FEM displacement solutions are provided. By exploiting the weighting constants in the least-square functional of iFEM, it is confirmed that a relatively accurate deformed structural shape can be reconstructed in the absence of in-situ strain data. Finally, the effect of sensor locations, number of sensors and the discretization of the geometry on solution accuracy is observed.

2 2.1

xyz is oriented with the reference to the element as shown in Figure 1 to formulate the element stiffness properties. By using global XYZ coordinates of the element nodes, transformation matrix of nodal degrees-offreedom of an element from the local to global coordinate system can be established for assembling the elements. As we assume a flat surface for the element, the transformation procedure is straightforward and the details of how to generate the transformation matrix can be found in Bathe (2006). According to the local xy base plane projected view of the iQS4 element and in terms of usual isoparametric coordinates s and t as shown in Figure 2, the mapping function of the element can be expressed as following: 4

x ( s, t )

x = ∑ Ni xi

(1a)

i =1 4

y( s , t )

y = ∑ Ni yi

(1b)

i =1

where Ni are the standard bilinear shape functions and xi, yi are corner coordinates. The degrees-of-freedoms of x and y translations ui and vi together with drilling rotation θzi, positive

Figure 1. (a) Quadrilateral inverse shell element, showing global and local coordinate systems. (b) Nodal degrees of freedom in the local coordinate system xyz.

INVERSE SHELL FINITE ELEMENT FORMULATION Inverse quadrilateral shell element

A four-node inverse quadrilateral shell element, labeled as iQS4, having six displacement degreesof-freedom per node as shown in Figure 1 is developed in order to represent the formulation of inverse finite element method. The first step is to define convenient coordinate systems to guarantee the geometric uniqueness of the assembled inverse finite element structure. A local coordinate system

Figure 2. (a) Quadrilateral inverse shell element projected onto its local xy base plane. (b) Isoparametric coordinates of parent element.

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counter clockwise, at each corner can be used to express the resulting membrane displacement field u and v given as: 4

4

i =1

i =1

4

4

i =1

i =1

u( x, y ) u = ∑ Ni ui + ∑ Liθ zi

(2a)

v( x, y ) v = ∑ Nivi + ∑ Miθ zi

(2b)

where Li and Mi are the shape functions consistent with ones proposed by Cook (1994) that defines hierarchical drilling rotation degree of freedom. Equation 3a, 3b, and 3c express that the bending displacement field of the element w, θx, and θy are defined by the degrees-of-freedoms of z translation wi and positive counter clockwise rotations around x and y axis, θxi and θyi. Isoparametric shape functions Li and Mi used to formulate drilling rotation can be utilized to describe the flexural capability of the iQS4 element. Hence, mathematical foundation of bending action of iQS4 element becomes identical to the MIN4 (Mindlin-type, four-nodes) provided by Tessler & Hughes (1983). w ( x, y )

θx (

4

4

i =1

i =1

w = ∑ Ni wi − ∑ Liθ xi xi

4

∑ Miθ yyi

(3a)

i =1

4

) θ x = ∑ Niθ xi

(3b)

i =1

θy(

4

) θ y = ∑ Niθ yi

(3c)

i =1

The kinematic relations of the element are prescribed according to the assumptions taken in first order, shear-deformation theory. Using the Equation 1 for membrane action and the Equation 2 for bending action, the three components of the displacement vector of any material point within the element can be described as: ux ( x, y, z ) = ux

u + zθ y

(4a)

uy ( x, y, z ) = uy

v − zθ x

(4b)

uz ( x, y, z ) = uz

w

(4c)

where ux and uy are the average positive in-plane displacements and uz is the displacement across the shell thickness. For the sake of implementation purposes, nodal displacement vector of the element can be stated in compact form by Equation 5. e

⎡ u1e ⎣

u e2

u3e

T

u e4 ⎤⎦

u ie = ⎡⎣ui

vi

T

wi θ xi θ yi θ ziz ⎤⎦ (i = , 2, 3, 4 ) (5b)

After taking the relevant derivatives of the three components of the displacement vector given in Equation 4 and utilizing the nodal displacement vector of element ue, the strain–displacement relations of linear elasticity theory can be written in compact vector forms as given in Equation 6. It is important to mention that εzz has no role in the internal work due to the plane stress assumption σzz = 0. ⎧ε xx ⎫ ⎪ ⎪ ⎨ ε yy ⎬ ≡ e( u e ) + zk( u e ) B m u e + zB k u e ⎪γ ⎪ ⎩ xy ⎭

(6a)

⎧γ xz ⎫ ⎨ ⎬ ≡ g( u e ) = B s u e ⎩γ yz ⎭

(6b)

In equation 6, the membrane strains associated with the stretching of the middle surface are e(ue), therefore Bm matrix stands for the derivatives of the shape functions associated with the membrane behavior. Accordingly, the bending curvatures are k(ue), and the transverse shear strains are g(ue) so that Bk and Bs matrices are the corresponding derivatives of shape functions used to define bending behavior of the element. 2.2 The reference plane strains and curvatures computed from in-situ strain sensors Discrete in-situ strain measures obtained from the embedded sensors are crucial according to the iFEM formulation. Conventional strain rosettes or embedded optical-fiber networks such as Fibre Bragg Grating (FBG) sensors are promising technology to collect large amount of on-board strain data. In order to compute the reference plane strains and curvatures, the necessary orientation of the in-situ strain rosettes on iQS4 elements’ surface is illustrated in Figure 3. According to Tessler & Spangler (2005), at n discrete locations (xi = xi, yi, ±t) (i = 1, …, n) where the surface strains are measured, the reference plane strains eiε, and curvatures kiε that corresponds to the membrane strains e(ue) and bending curvatures k(ue) given in Equation 6 can respectively be determined from the measured surface strains as follows:

(5a)

− ⎫ ⎧ε +xx + ε xx 1 ⎪ ⎪ e εi = ⎨ ε +yy + ε −yy ⎬ 2⎪ + − ⎪ ⎩γ xy + γ xy ⎭i

(i 1, n )

(7a)

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2

Φ e ( e ) we e( e ) e ε + wk k(( e ) − k ε 2 + wg g( g( u e ) − g ε e( u ) − e ε

Figure 3. Discrete surface strains measured at location xi from strain rosettes instrumented on top and bottom of iQS4 elements.

− ⎫ ⎧ε +xx ε xx 1 ⎪ ⎪ k εi = ⎨ ε +yy ε −yy ⎬ 2t ⎪ + − ⎪ ⎩γ xy γ xy ⎭i

(i 1, n )

(7b)

where the measured surface strains are denoted as (εxx+, εyy+, εxy+)i and (εxx−, εyy−, εxy−)i, superscripts ‘+’ and ‘−’ represent the top and bottom surface locations, and the following notation is used (•)i = (•)x = xi. Although, the experimentally measured surface strains can be used to compute in-situ membrane strains eiε and bending curvatures kiε, they cannot be directly used to calculate in-situ transverse shear strains giε. A smoothing procedure, developed by Tessler et al. (1998, 1999), called the Smoothing Element Analysis, enables the first-order derivatives of kiε to be used in computing the transverse shear strains giε. However, in the deformation of thin shells, contributions of giε are much smaller compared to the bending curvatures kiε. Since the marine structures such as ship and offshore structures can be modelled by using thin shells, the giε contributions can be safely omitted in the following formulations. 2.3

Weighted least-squares functional

Accounting for the membrane, bending and transverse shear deformations of the individual element, the inverse finite element method reconstructs the deformed shape by minimizing an element functional, namely a weighted least-squares functional Φe(ue) given in Equation 8a, with respect to the unknown displacement degrees-of-freedom (Tessler & Spangler 2005). The squared norms expressed in Equation 8a can be written in the form of the normalized Euclidean norms as given in Equations from 8b to 8d where Ae represents the area of the element.

2

k( u ) − k

2

g( u ) − g ε

2

n 1 e ( u )i (e(u ∑ ∫∫ n Ae i =1

e i ) dxdy 2

n ( )2 k(u ( u )i (k(u ∑ ∫∫ n Ae i =1 n 1 g ( u )i (g(u ∑ ∫∫ n Ae i =1

2

k i ) dxdy 2

g i ) dxdy 2

(8a) (8b) (8c) (8d)

The weighting constants we, wk, wg in Equation 8a are positive valued and stand for individual section strains. They control the complete coherence between theoretical strain components and their experimentally measured values. The weighting constants are we = wk = wg = 1 for the squared norms given in Equations from 8b to 8d since in-situ strains eiε, kiε, and giε are assumed to be determined. On the other hand, for any case of missing in-situ strain component, the corresponding weighting constants can be selected as a small positive constant α = 10−4. Equation 9a restates the squared norm presented in Equation 8b for an element that has undetermined in-situ strain component of eiε. Accordingly, Equation 8c can be rewritten as Equation 9b for an element that has missing the in-situ strain component of bending curvatures kiε. In addition, Equation 8d can be updated by Equation 9c for the lack of the transverse shear strains giε. e( u ) = ∫∫ e(u ( u )2 dxdy

(9a)

k( u ) = ( )2 ∫∫ k k(( u )2 dxd dy

(9b)

g( u ) = ∫∫ g(u ( u )2 dxdy

(9c)

2

Ae

2

Ae

2

Ae

By virtue of these assumptions, all strain compatibility relations are explicitly satisfied so that Equation 8a can be minimized with respect to nodal displacement vector as shown in Equation 10. After the minimization, the resultant equation is the element matrix equation keue = fe where ke is element stiffness matrix, fe is element right-hand-side vector that is a function of the measured strain values, and ue is the nodal displacement vector of element. ∂Φ e ( e ) = keue − f e = 0 ∂u e

(10)

Once the element (local) matrix equations are established, the element contributions to the global linear equation system of the discretized structure can be performed as follows:

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nel

K

∑ (T )

T

k Te

(11a)

T

fe

(11b)

ue

(11c)

e =1

nel

F

∑ (T ) e =1

nel

U

∑ (T )

T

e =1

KU = F

(11d)

where Te is the transformation matrix of nodal degrees-of-freedom of an element from the local to global coordinate system, K is global stiffness matrix (symmetric and positive definite matrix), U is global nodal displacement vector, F is the global right-hand-side vector (function of the measured strain values), and the script nel stands for total number of inverse elements. The global stiffness matrix K includes the rigid body motion mode of the discretized structure; therefore it is a singular matrix. By prescribing problem-specific displacement boundary conditions, the resulting system of equations can be reduced from Equation 11d to the Equation 12a where KR is a positive definite matrix (always nonsingular), and thus it is invertible. Solving the Equation 12a in order to obtain the global displacement degrees-of-freedoms of all nodes UR, is very fast as represented in Equation 12b because the matrix KR needs to be reversed only once because it remains unchanged for a given distribution of strain sensors and is independent of the measured strain values. K R U R = FR

(12a)

−1 R R

(12b)

UR

K F

However, the right-hand-side vector FR is dependent on the discrete surface strain data obtained from in-situ strain sensors, and hence it needs to be updated during any deformation cycle. Finally, the matrix–vector multiplication KR−1FR gives rise to the unknown degrees-of-freedoms vector UR, which provides the deformed structural shape at any real-time. 3 3.1

concentrated loading in negative z direction as depicted in Figure 4. In the beginning, a linear static direct finite element analysis of the plate is performed based on a high-fidelity mesh consisting of 625 quadrilateral shell elements and possessing 10206 degrees-offreedoms by using an in-house finite element code. The resulting direct FEM deflection and rotation are used as a source for the simulated sensor-strain. In other words, the ‘experimental’ strain measurements used in the following iFEM case analysis are obtained by means of the direct FEM solution. Two SHM case studies of the plate are performed based on iFEM methodology by using different number of strain rosettes and their altered orientation. The strain rosettes can only be placed on one of the bounding surfaces (top or bottom surface) because the material properties of the plate are symmetric with respect to the mid-plane and the resulting deformations are due to bending only so that the strain distribution should be antisymmetric with respect to the mid-plane. In the first example, the quadrilateral plate is uniformly discretized by 18 iQS4 elements possessing 168 degrees-of-freedoms (Figure 5). 18 strain rosettes are placed on each element’s top surface at

Figure 4. Quadrilateral plate and its boundary condition.

NUMERICAL RESULTS Quadrilateral plate

A quadrilateral plate, whose corner coordinates (in meters) are shown in the Figure 4, is considered to be analyzed. The plate has a uniform thickness of 15 mm and it is made of steel having the elastic modulus of 210 GPa and Poissons ratio of 0.3. Left edge of the plate is clamped and the upper right corner is subjected to a static transverse

Figure 5. Approximate locations of 18 strain rosettes on quadrilateral plate discretized by 18 iQS4 elements.

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centroids as shown in Figure 5. The total displacement and rotation results obtained from iFEM analysis are respectively shown together with the direct FEM results in Figures 6–7. According to the distributions, the error of the maximum displacement and rotation obtained from iFEM solution is less than 0.2% with respect to the direct FEM results. Moreover, the real-time monitoring of the plate is performed based on the same mesh, but twelve strain rosettes used in the first case study are removed as shown in Figure 8 in order to assess the precision of iFEM formulation when there are missing in-situ strain measurements. After removal of the strain rosettes, SHM of the quadrilateral plate is conducted by using the strain data obtained from 6 strain rosettes only. For an element that does not have any sensor strains, its weighting coefficients are set to 10−4. The total displacement

Figure 6. (a) iFEM total displacement distribution of quadrilateral plate (18 iQS4 elements—18 strain rosettes). (b) Direct FEM total displacement distribution of quadrilateral plate.

and rotation results obtained from iFEM analysis are respectively compared with direct FEM results as depicted in Figures 9–10. The iFEM-reconstructed displacement and rotation fields almost identically match the reference displacement and rotation fields so that this accuracy confirms the robustness of iFEM framework even if there are missing in-situ strain measurements. 3.2

Longitudinally and transversely stiffened plate

Performing SHM of a longitudinally and transversely stiffened plate is crucial since ship structures are generally consisted of various stiffened plates. Inverse quadrilateral shell elements are clearly appropriate to model these types of structures in order to conduct a precise real-time monitoring. A stiffened square plate that represents the portion of the side of a typical longitudinally and transversely framed tanker is considered to be solved. The plate’s edge length and uniform thickness are 3 m and 15 mm respectively. Each stiffener has a height of 150 mm and uniform thickness of 15 mm. The selected material’s elastic modulus is 210 GPa with the Poisson’s ratio of 0.3. Right, left, upper, and bottom edges of the plate including each stiffener’s end edges are clamped. A static uniform transverse pressure of 40 kPa is subjected to the bottom surface of the plate. Applied boundary conditions and isometric view of the stiffened plate are illustrated in Figure 11. Initially, a direct FEM convergence study of the stiffened plate using meshes of quadrilateral shell elements is carried out in order to establish an accurate numerical solution. The most refined mesh consisted of 5400 square and uniformly

Figure 7. (a) iFEM total rotation distribution of quadrilateral plate (18 iQS4 elements—18 strain rosettes). (b) Direct FEM total rotation distribution of quadrilateral plate. Figure 9. (a) iFEM total displacement distribution of quadrilateral plate (18 iQS4 elements—6 strain rosettes). (b) Direct FEM total displacement distribution of quadrilateral plate.

Figure 8. Approximate locations of 6 strain rosettes on quadrilateral plate discretized by 18 iQS4 elements.

Figure 10. (a) iFEM total rotation distribution of quadrilateral plate (18 iQS4 elements—6 strain rosettes). (b) Direct FEM total rotation distribution of quadrilateral plate.

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Figure 11. Longitudinally and transversely stiffened plate and its applied boundary conditions.

distributed elements, possessing 36966 degrees-offreedoms. The results of displacement and rotation fields obtained from this convergence study are considered as a reference source to generate the ‘experimental’ strain measurements (in-situ strain data) used in iFEM analyses. Two real-time monitoring scenarios of the stiffened plate are respectively executed by using two different iFEM meshes. Although the material properties of the plate and stiffeners are symmetric with respect to the mid-plane, the resulting deformations exhibits both stretching and bending actions due to the complexity of the structure. Hence, the strain rosettes have to be placed on both the top and bottom surfaces of structure for this problem. Firstly, 96 iQS4 elements (36 of them on the plate, 60 of them on the stiffeners) having 1062 degreesof-freedoms are used to generate the iFEM mesh of the stiffened plate as shown in Figure 12. Strain rosettes are positioned at center of each iQS4 element attached to the plate while each iQS4 element on each stiffener has strain rosettes placed near to reverse side of the plate as shown in Figure 12. The distribution of reconstructed total displacement and rotation are respectively compared with those obtained from direct FEM analysis as depicted in Figures 13–14. According to the comparisons, the error of the maximum displacement produced by the iFEM solution is approximately 6% with respect FEM maximum displacement and the results are graphically agreed quite well. Even though the variation of rotation results seems slightly dissimilar for several locations, the maximum rotation found by iFEM analysis is less than 2% in error compared to the FEM maximum rotation. Moreover, the locations of maximum and minimum rotations on plate found in iFEM and FEM analyses are in good agreement. In the second iFEM analysis, the plate is uniformly discretized by using 504 iQS4 elements as illustrated in Figure 15 where the approximate

Figure 12. Approximate locations of 96 × 2 strain rosettes on stiffened plate discretized by 96 iQS4 elements.

Figure 13. (a) iFEM total displacement distribution of stiffened plate (96 iQS4 elements—96 × 2 strain rosettes). (b) Direct FEM total displacement distribution of stiffened plate.

Figure 14. (a) iFEM total rotation distribution of stiffened plate (96 iQS4 elements—96 × 2 strain rosettes). (b) Direct FEM total rotation distribution of stiffened plate.

Figure 15. Approximate locations of 504 × 2 strain rosettes on stiffened plate discretized by 504 iQS4 elements.

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even if a large amount of strain-sensor data is not available. According to results, it has been agreed that iFEM methodology is promising technology for performing an accurate real-time monitoring of marine structures. ACKNOWLEDGEMENTS Figure 16. (a) iFEM total displacement distribution of stiffened plate (504 iQS4 elements—504 × 2 strain rosettes). (b) Direct FEM total displacement distribution of stiffened plate.

The authors would like to acknowledge FP7 project INCASS for financial support. REFERENCES

Figure 17. (a) iFEM total rotation distribution of stiffened plate (504 iQS4 elements—504 × 2 strain rosettes). (b) Direct FEM total rotation distribution of stiffened plate.

locations of the in-situ strain rosettes are included in detail as well. The total number of strain rosettes count as 504 × 2, since each iQS4 element is instrumented with a strain rosette on the top and bottom surfaces. The distribution of reconstructed total displacement and rotation presented in Figures 16–17 confirms the high precision of iFEM framework when a finer iFEM mesh including more strain rosettes is used. 4

CONCLUSIONS

A revised formulation of the inverse Finite Element Method (iFEM) is presented. The presented iFEM methodology is applicable to perform shape-sensing analyses of plate and shell structures by using the strain data obtained from randomly distributed sensors on the structure. A four-node inverse shell element (iQS4) including hierarchical drilling rotation degree of freedoms is formulated to perform numerical simulations. Application of iFEM to SHM of marine structures is established by using various types of low- and high-fidelity discretization strategies of presented problems. The effect of sensor locations, number of sensors and the discretization of the geometry on solution accuracy are pondered. The numerical results have confirmed that it is still possible to reconstruct sufficiently accurate deformed structural shapes,

Bathe, K.J. 2006. Finite element procedures. Klaus-Jurgen Bathe. Cerracchio, P., Gherlone, M., Mattone, M., Di Sciuva, M., & Tessler, A. 2010. Shape sensing of three-dimensional frame structures using the inverse finite element method. In: Proceedings of 5th European Workshop on Structural Health Monitoring, Sorrento, Italy. Cook, R.D. 1994. Four-node ‘flat’shell element: drilling degrees of freedom, membrane-bending coupling, warped geometry, and behavior. Computers & structures, 50(4), 549–555. Gherlone, M., Cerracchio, P., Mattone, M., Di Sciuva, M., & Tessler, A. 2011. Beam shape sensing using inverse finite element method: theory and experimental validation. In: Proceeding of 8th International Workshop on Structural Health Monitoring, Stanford, CA. Gherlone, M., Cerracchio, P., Mattone, M., Di Sciuva, M., & Tessler, A. 2012. Shape sensing of 3D frame structures using an inverse Finite Element Method. International Journal of Solids and Structures, 49(22), 3100–3112. Gherlone, M., Cerracchio, P., Mattone, M., Di Sciuva, M., & Tessler, A. 2014. An inverse finite element method for beam shape sensing: theoretical framework and experimental validation. Smart Materials and Structures, 23(4), 045027. Gherlone, M., Di Sciuva, M., Cerracchio, P., Mattone, M.C., & Tessler, A. 2013. The inverse Finite Element Method for shape sensing of aerospace structures. In: Proceedings of 22nd Conference of Italian Association of Aeronautics and Astronautics, Naples, Italy. Liu, P.L., & Lin, H.T. 1996. Direct identification of nonuniform beams using static strains. International journal of solids and structures, 33(19), 2775–2787. Maniatty, A., Zabaras, N.J., & Stelson, K. 1989. Finite element analysis of some inverse elasticity problems. Journal of engineering mechanics, 115(6), 1303–1317. Maniatty, A.M., & Zabaras, N.J. 1994. Investigation of regularization parameters and error estimating in inverse elasticity problems. International journal for numerical methods in engineering, 37(6), 1039–1052. Quach, C.C., Vazquez, S.L., Tessler, A., Moore, J.P., Cooper, E.G., & Spangler, J.L. 2005. Structural anomaly detection using fiber optic sensors and inverse finite element method. In: Proceedings of AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, California. Schnur, D.S., & Zabaras, N.J. 1990. Finite element solution of two-dimensional inverse elastic problems using

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spatial smoothing. International Journal for Numerical Methods in Engineering, 30(1), 57–75. Tessler, A., & Hughes, T.J. 1983. An improved treatment of transverse shear in the Mindlin-type four-node quadrilateral element. Computer methods in applied mechanics and engineering, 39(3), 311–335. Tessler, A., & Spangler, J.L. 2003. A variational principal for reconstruction of elastic deformation of shear deformable plates and shells, NASA TM-2003212445. Tessler, A., & Spangler, J.L. 2004. Inverse FEM for fullfield reconstruction of elastic deformations in shear deformable plates and shells. In: Proceedings of 2nd European Workshop on Structural Health Monitoring, Munich, Germany. Tessler, A., & Spangler, J.L. 2005. A least-squares variational method for full-field reconstruction of elastic deformations in shear-deformable plates and shells. Computer methods in applied mechanics and engineering, 194(2), 327–339. Tessler, A., Riggs, H.R., & Dambach, M. 1999. A novel four-node quadrilateral smoothing element for stress enhancement and error estimation. International journal for numerical methods in engineering, 44(10), 1527–1541.

Tessler, A., Riggs, H.R., Freese, C.E., & Cook, G.M. 1998. An improved variational method for finite element stress recovery and a posteriori error estimation. Computer methods in applied mechanics and engineering, 155(1), 15–30. Tessler, A., Spangler, J.L., Gherlone, M., Mattone, M., & Di Sciuva, M. 2011. Real-Time characterization of aerospace structures using onboard strain measurement technologies and inverse finite element method. In: Proceedings of the 8th International Workshop on Structural Health Monitoring, Stanford, CA. Tessler, A., Spangler, J.L., Gherlone M., Mattone M., & Di Sciuva, M. 2012. Deformed shape and stress reconstruction in plate and shell structures undergoing large displacements: application of inverse finite element method using fiber-bragg-grating strains. In: Proceedings of 10th World Congress on Computational Mechanics, Sao Paulo, Brazil. Tikhonov, A.N., & Arsenin, V.Y. 1977. Solutions of illposed problems. Winston, Washington, DC. Vazquez, S.L., Tessler, A., Quach, C.C., Cooper, E.G., Parks, J., & Spangler J.L. 2005. Structural health monitoring using high-density fiber optic strain sensor and inverse finite element methods, NASA TM-2005-213761.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Method for estimating soft clay type seabed embedment by pipeline movement D.K. Kim, M.S. Liew & S.Y. Yu Universiti Teknologi PETRONAS, Tronoh, Perak, Malaysia

K.S. Park POSCO Center, Seoul, Republic of Korea

H.S. Choi Pohang University of Science and Technology, Pohang, Republic of Korea

ABSTRACT: The advanced procedure has been proposed to estimate higher accuracy of embedment of pipes that are installed on soft clay seabed. Numerical simulation has been adopted to investigate dynamic seabed embedment and two steps, i.e., static and dynamic analysis, were performed. In total, four empirical curves were developed to estimate dynamic embedment including dynamic phenomena, i.e., behavior of vessel, environmental condition, and behavior of nonlinear soil. The obtained results were compared with existing methods such as design code or guideline to examine the difference of seabed embedment for existing and advance methods. Once this process was performed for each case, a diagram for estimating seabed embedment was established. The applicability of the proposed method was verified through applied examples. This method will be useful in predicting seabed embedment on soft clay, and the structural behaviors of installed subsea pipelines can be changed by the obtained seabed embedment in association with on-bottom stability, free span, and many others. 1

INTRODUCTION

It is well recognized that the deep water offshore development is getting deeper and deeper in association with high demand for natural resources, especially hydrocarbon, due to rapid economic growth in many countries. In regard to deep-water environment, clay seabed is widely distributed in ocean and is closely related with deep-water and ultra-deep water environment. The soft characteristic of clayey soil not only leads to immediate settlement, but also time dependent embedment known as extended settlement. Its characteristics are important variables for the structures installed on seabed and especially for offshore pipelines as the characteristics affect structural design and its capacity such as on-bottom stability, free-span length, and many others. This means that seabed behavior should be estimated and applied to subsea structural design if there is a possibility of seabed embedment that depends on its prospective characteristics such as undrained shear strength, submerged weight of soil, Poisson’s ratio, and friction coefficient. Particularly, this embedment phenomenon might be attributed by dynamic behaviors due to vessel

motion, environmental condition, soil properties, and several other reasons. However, these dynamic behaviors of seabed are not carefully considered in the present design code and guidelines regarding subsea installation of various structures such as pipelines, flowlines, umbilicals, subsea systems and many others. In other words, static embedment of seabed is still assumed and applied to the design. In order to overcome the gap between the reality and current technology, a number of research regarding pipe-soil interaction have been performed extensively by several experts since the end of the 20th century in terms of numerical finite element simulation for soil’s nonlinearity on the softening and large deformation of soil (Merifield et al., 2008; 2009; White et al., 2010), verification of numerical results by geotechnical tests (Dingle et al., 2008; Zhou et al., 2008), and development of advanced soil models considering cyclic loading (Aubeny et al., 2006; Randolph and Quiggin, 2009). For example, the amount of seabed embedment is obtained by simple calculation from submerged weight and reaction force, which is specified in the current design code. Recently, Su et al. (2013) investigated onbottom stability of pipeline on soft clay seabed.

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Many researches are now continuously performing studies worldwide. In addition, the Dynamic Embedment Factor (DEF) concept, which can be defined as dynamic embedment divided by static embedment, has been proposed. However, at the moment, the resolution procedure for calculation of DEF does not clearly exist. In this regards, advanced procedures are proposed for the estimation of pipe embedment on soft clay seabed. The developed procedure will be useful for the calculation of seabed embedment in the early design stages (Pre-Front-End-EngineeringDesign, Pre-FEED), and the procedure can be applied to check the design of on-bottom stability, free span, and others. 2

ESTIMATION METHOD FOR PIPE EMBEDMENT

The procedures for estimating pipe embedment are shown in Figure 1, including general method (i.e., rule-based approach) and proposed method based on DEF. In Figure 1, the left-hand side

Figure 1.

represents the general method and the opposite side shows the proposed method. Once the embedment is obtained by the proposed method, it can be compared with embedment results calculated by existing design code (i.e., general method). If the embedment by proposed method is less than the results from the general method, dynamic installation analysis needs to be repeated. The details of two approaches are covered in subsequent sub-sections, and the following method has been proven by applied examples in Section 3. 2.1

General method

In the early stage of subsea pipeline Front-EndEngineering-Design (FEED), structural safety and reliability during design life should be checked and confirmed by engineers. Among good examples of design checklist in FEED for pipeline structure are on-bottom stability and free span length. Both design check ists are closely related with the amount of seabed embedment. Generally, in order to estimate seabed embedment, design code or rule-based calculation (called general method) is performed.

Proposed procedure for the estimation of pipe embedment on soft clay seabed.

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The basic concept of general approach is minimizing the lift force of pipeline, which means seabed embedment can be calculated based on the assumption of maximum pipe forces to seabed direction. The applied forces on the pipe can be classified as submerged weight, lift force, and others. This method can simply be applied to the pipeline FEED by using empirical formula, which assumes zero lift force in order to maximize subsea embedment. It also considers that different types of submerged weight are assumed depending on internal conditions of pipe such as void, operation, and system test. In addition, the pipe condition is assumed as a shape that rests on the seabed along the pipeline length. In case of real installation condition, pipe structure will be installed with catenary shape. However, in general procedure, catenary shape is not considered and the shape only rests on the seabed condition applied to the design of pipeline. Briefly, the common steps of general method are presented below, and these steps could be equally applied to the proposed method. – Definition of initial design input – Definition of pipe shape depending on its condition – Definition of soil model – Identification of reaction forces – Estimation of pipe embedment. For the first step regarding calculation of subsea pipeline embedment, initial design input should be given such as pipe, soil, and internal fluid data. Once the initial design inputs are identified, pipe embedment can be calculated based on simple empirical formula developed by several test results including soil effect. However, empirical formula can only consider one type of pipe shape, which is the pipe that rests on seabed condition. This means that seabed reaction force is comprised of two parameters, which are submerged weight of pipe and its lift force. In addition, this empirical formula is barely suitable for clay type seabed. If the pipe is in system test condition, i.e., water filled condition, or operation condition, i.e., produced fluid filled condition, the vertical force of pipe is considered as the pipe submerged weight with internal fluid weight. The maximum pipe embedment is produced when the maximum pipe submerged force is applied to the seabed among various pipe conditions. Therefore, general approach requires only two design inputs, i.e., basic pipe and soil data. In addition, it cannot consider any type of dynamic effect that is normally applied to the installation of pipeline due to dynamic installation conditions by complex environmental loads. In this regard, advanced procedure for estimating seabed embedment is proposed based on dynamic embedment effect in the present study.

2.2

Proposed method

In case of the proposed concept that considers DEF as shown in right part of Figure 1, it has more complex procedure, including four types of empirical curves illustrated in the most right side colored box in Figure 1. The core parameters related to the proposed method are presented in Eqs. (1) to (4). Wefff . =

Ws D Su

(1)

where, Weff. = effective pipe weight, Ws = pipe submerged weight, and D = pipe outer diameter. TLF =

RS Ws

(2)

where, TLF = touchdown lay factor, RS = static reaction force, and Ws = pipe submerged weight. DLF =

RD RS

(3)

where, DLF = dynamic lay factor, RD = dynamic reaction force, and RS = static reaction force. DEF =

ED ES

(4)

where, DEF = dynamic embedment factor, ED = dynamic embedment, and ES = static embedment. Four empirical formulas were developed using Eqs. (1) to (4). The DLF versus DEF curve, which is derived in the last step, made possible the easy estimation of dynamic embedment. This procedure is proposed based on the relation between pipe embedment and seabed reaction force, including its static and dynamic phenomena. In general, pipe condition can be expressed as catenary shape when it is under installation. It is certain that the catenary shape of pipe can also be changed depending on its installation method such as S-lay, J-lay, Reel-lay and others. In addition, various linear or nonlinear seabed soil models might be considered, such as rigid seabed, rigid plastic, elasto-plastic model based constitutive equation based models, and many others. Critical parameters, i.e., flexural stiffness (EI) and horizontal top tension (Th) vary with the types of seabed soil model, which change vertical force distribution of pipe at Touchdown Zone (TDZ). For this reason, nonlinear soil model is normally applied to the analysis of subsea pipeline installation with the effect of seabed embedment. Pipe embedment in static condition including laying

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configuration effect and Touchdown Lay Factor (TLF) can be calculated through the static installation analysis. Once the static installation analysis is completed, two empirical formulas can be obtained as follows: First step – Effective pipe weight (Weff.) versus static embedment by pipe diameter (ES/D) curve Second step – Effective pipe weight (Weff.) versus Touchdown Lay Factor (TLF = RS/Ws) curve. The effective pipe weight (Weff.) parameter is commonly adopted in both curves shown in the first and second steps, and it is composed of three variables of submerged pipe weight (Ws), pipe diameter (D), and shear strength of seabed soil (Su). The curve obtained from first step, known as Weff. versus ES/D curve, is compared with dynamic embedment results and used for the determination of final embedment at the final step in Figure 2. In case of the second curve, known as Weff. versus TLF curve, TLF is used as an indicator for the increased reaction force at touchdown point due to the stress concentration effects during the process

Figure 2. line axis.

Wave and current angles with regard to pipe-

of pipe installation. Although this TLF values normally lie between 2.0 and 3.0 according to Palmar (2008), additional parameters should also be considered with regard to laying conditions, i.e., lay tension, departure angle of pipe, bending stiffness and many others in order to obtain accurate range of TLF. It is found that the well-fitted empirical formulas in the first and second steps could be obtained by static installation analysis. In the present study, OrcaFlex numerical simulation code (Orcina, 2013) was applied to both of static and dynamic analysis. Other similar types of simulation codes may also be applied to this analysis. Once the static installation analysis has been completed through the numerical simulation, dynamic installation analysis is performed, and two more curves can be obtained as follows: Third step – Touchdown Lay Factor (TLF = RS/Ws) versus Dynamic Lay Factor (DLF = RD/RS) curve Final step – Dynamic Lay Factor (DLF = RD/RS) versus Dynamic Embedment Factor (DEF = ED/ES) curve. Dynamic Lay Factor (DLF) and Dynamic Embedment Factor (DEF) can be derived by dynamic installation analysis. DLF means the ratio between dynamic reaction force and static reaction force. The obtained TLF values by static analysis can be used as an input data for the development of the third curve, namely TLF versus DLF curve. This static result is connected by dynamic results through the TLF versus DLF curve. Finally, DEF can be estimated by DLF versus DEF curve, which can be derived in the final step. The amount of embedment from dynamic installation effect can be obtained by multiplying the estimated DEF by static embedment (ES), which is obtained in the first step. The comparison of seabed embedment between general and proposed method was performed once the dynamic installation embedment has been decided by the proposed method. In the process of comparison, iteration is required when the pipe embedment by the proposed method is smaller than the obtained embedment by the general method. In this case, the number of cycle should be increased for lateral movement of pipeline at touchdown point. Additional dynamic installation analysis could be performed based on the increased number of cycle of pipeline. Finally, the pipe embedment was determined when the obtained embedment by dynamic installation analysis satisfies the guideline of comparison. This means that the obtained embedment by the proposed method is bigger than the result by the general method. The applicability of the proposed method is investigated by applied example in the next section.

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3

APPLIED EXAMPLE

In the present section, verification work was performed to check the applicability of the proposed method. The general approach has been previously studied by Yu (2014), and only the proposed method part has been covered in this study. The comparison results are briefly illustrated in the last part. 3.1

Applied design input

Three seeds in irregular sea states are summarized in Table 1, and the direction of environmental loads, including wave and current to pipe axis laid on seabed is shown in Figure 2. For the realistic environmental condition, irregular sea states were used instead of regular wave sea states. Also, the applied direction effect reduces the environmental load to pipeline on the seabed. DEF is sensitive regarding environmental data. This approach is recommended because the DEF curve is drawn through dynamic reaction force under random irregular sea states. However, an in-depth study on pipe embedment by various environmental conditions for more general DEF approach is required. 3.2

Results

The dynamic pipe embedment under random sea states with three seeds is shown in Figure 3.

Table 1.

Environmental data.

Irregular sea state

Seed 1

Seed 2

Seed 3

Wave height, Hs (m) Peak period, Tp (S) Current velocity (m/s)

2.6 8.7 0.83

4.7 11 0.33

4.4 10.7 0.35

Figure 3. Dynamic embedment by lateral displacement at TDZ under irregular sea states.

Figure 4. Pipe embedment based on the proposed method under irregular sea states.

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The analysis was performed in the soft clay ranges of shear strength from 0.5 kPa to 25 kPa. It was confirmed that the increase of shear strength at each sea state condition decrease the dynamic embedment. Furthermore, as the lateral displacement at TDZ increased, the pipe embedment increased at each shear strength value. The lateral displacement of the pipe at TDZ has a close relation with pipe embedment. As the lateral displacement due to pipe and vessel motion by environmental load gets larger, cyclic effects related to pipe embedment at TDZ shows a greater increase. Pipe embedment was calculated based on the proposed method and the DEF approach. Dynamic embedment was estimated through four steps. First, from static analysis, Figure 4(a) and (b) were derived, and the static pipe embedment of the ranges from almost 0.0 to 0.3 D and the TLF value from 1.2 to 3.0 by effective pipe weight respectively were drawn. Through dynamic analysis, the results of DLF by TLF in Figure 4(c) and DEF by DLF in Figure 4(d) were illustrated. As the relation between DLF and TLF had a nonlinear trend with TLF of 2.0, the results before TLF of 2.0 are recommended. Furthermore, the value from this result was used as an input for the final DEF curve. Finally, the dynamic embedment at shear strength of 2.1 kPa from the DEF curve by using the proposed approach was approximately 0.43 D, and it shows reasonable result in comparison with field data of 0.56 D (Yu, 2014). In case of the general approach, 0.21 D of pipe embedment was obtained by empirical formula based on Verlay and Lund’s approach (DNV 2010). Although this approach was analysed with limited cases, this procedure could be helpful for dynamic embedment. Based on the obtained results from this study, the proposed method can be applied to estimate realistic embedment of subsea pipeline on soft clay and it will be useful to understand behavior of seabed embedment in association with offshore installation. 4

CONCLUDING REMARKS

Dynamic embedment factor, which is the ratio of dynamic embedment divided by static embedment, is the representative of non-dimensioned factor by considering dynamic effects for pipe embedment. However, there are limit on a wide range of applications, depending on dynamic condition. In this paper, an advanced procedure for the estimation of dynamic embedment based on dynamic embedment factor was suggested. Examples with dynamic conditions of irregular sea states variation were performed to understand the approach with dynamic embedment factor.

DEF is a function of vessel rate, lay tension, pipeline configuration and environmental loading during installation. It means that DEF curve has impacts on various parameters under dynamic installation condition. However, only the procedure for the estimation of dynamic pipe embedment is focused in this study because DEF is a good indicator for dynamic embedment. From the examples based on the proposed method, many uncertainties for dynamic embedment can be reduced, and the understanding on the estimation of dynamic embedment, which is not known prior to installation in field, can be improved. ACKNOWLEDGEMENTS The authors would like to acknowledge for the support by Graduate School of Engineering Mastership (GEM), Pohang University of Science and Technology (POSTECH), POSCO, PETRONAS and Offshore Engineering Center UTP (OECU). The OECU is an emerging research consultancy centre in Malaysia and a sub-institution of Deepwater Technology Mission Oriented Research (MOR), Universiti Teknologi PETRONAS. Since its inception, OECU has been active in the support of PETRONAS Carigali SKG 11 in offshore ventures and researches. REFERENCES Aubeny, C.P., Biscontin, G. & Zhang, J. 2006. Seafloor Interaction with Steel Catenary Risers. Final Project Report (Number: 510, Task order: 35988), Texas A&M University, Houston, Texas, USA. Dingle, H.R.C., White, D.J. & Gaudin, C. 2008. Mechanisms of pipe embedment and lateral breakout in soft clay. Canadian Geotechnical Journal 45(5): 636–652. DNV 2010. On-bottom Stability Design of Submarine Pipelines (DNV-RP-F109). Det Norske Veritas, Oslo, Norway. Merifield, R.S., White, D.J. & Randolph, M.F. 2008. The ultimate undrained resistance of partially embedded pipelines. Geotechnique 58(6): 461–420. Merifield, R.S., White, D.J. & Randolph, M.F. 2009. Effect of surface heave on response of partially embedded pipelines on clay. Journal of Geotechnical and Geoenvironmental Engineering 135(6): 819–829. OrcaFlex 2013. OrcaFlex manual version 9.6C. Orcina Ltd., Daltongate, Ulverston, Cumbria. UK (www. orcina.com). Palmer, A. 2008. Touchdown indentation of the seabed. Applied Ocean Research 30(3): 235–238. Randolph, M.F. & Quiggin, P. 2009. Non-linear hysteretic seabed model for catenary pipeline contact. Proceedings of the 28th International Conference on Offshore Mechanics and Arctic Engineering (OMAE 2009), 31 May–5 June, Honolulu, USA (OMAE2009-79259).

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White, D.J., Gaudin, C., Boylan, N. & Zhou, H. 2010. Interpretation of T-bar penetrometer tests as shallow embedment and in very soft clay. Canadian Geotechnical Journal 47(2): 218–229. Yu, S.Y. 2014. On-bottom stability of offshore pipeline considering dynamic embedment on soft clay. Ph.D. dissertation, Pusan National University, Busan, Republic of Korea. Yu, S.Y., Choi, H.S., Lee, S.K., Do, C.H. & Kim, D.K. 2013. An optimum design of on-bottom stability of offshore pipelines on soft clay. International Journal of Naval Architecture and Ocean Engineering 5(4): 598–613.

Zhou., H., White, D.J & Randolph, M.F. 2008. Physical and numerical simulation of shallow penetration of a cylindrical object into soft clay. International Conference on GeoCongress 2008: Characterization, Monitoring, and Modeling of GeoSystems, 9–12 March, New Orleans, Louisiana, USA (http://dx.doi.org/10.1061/40972(311)14).

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Assumptions and reality: Stress states in uniaxial tension test M. Kõrgesaar Department of Applied Mechanics/Marine Technology, Aalto University School of Engineering, Aalto, Finland

ABSTRACT: This paper contributes to the physical understanding of stress states in different element sizes during uniaxial tension. For this purpose, the results of uniaxial tension tests with the dog-bone sheet specimens are re-analysed to determine the stress state in the material during deformation and the equivalent plastic strain to failure. Failure strains determined this way exhibit the strong size effect, whereby the failure strain decreases for the increasing element size. It is commonly assumed that this failure strain corresponds to uniaxial tension because it is determined with the uniaxial tension test. The aim of this study is to determine the limiting element size for which the condition of uniaxial tension does not hold. The results are especially important for engineers who predict ductile fracture with equivalent plastic failure strain criterion scaled with respect to the element size. 1

INTRODUCTION

Current engineering solutions rely heavily on accurate finite element simulations especially when designing structures to withstand accidental events, for instance, see the review article by Samuelides (2012) and the recent proceedings of the International Conference on Collision and Grounding of Ships and Offshore Structures (Amdahl et al. 2013). Reliable simulations involving impact, crush and crashworthiness require the knowledge of true stress–strain behaviour of the material until fracture initiation. Furthermore, because of the computational reasons, such simulations are carried out with the shell elements utilizing plane stress assumption. Often the material relation is determined with the standard uniaxial tension test. Because of the

necking in tensile test the material relation and the failure strain are element size dependent. For example, in the tensile specimen in Figure 1(a), strains localize in the middle of the specimen in both width and thickness direction as there is no restraint for this localization. Mapping a finite element, or equivalently, some sort of averaging unit, into the middle of the specimen, as shown in Figure 1(a), indicates that average strain in the small element is much higher than in the large element. Such mesh size effects deduced from the tensile test are commonly given in the form of Barba’s law (Yamada et al. 2005, Alsos et al. 2009) as shown in Figure 1(b). Barba’s law is often used in the ship collision and grounding simulations to give the point of shell element removal. Since Barba’s law is determined with the uniaxial

Figure 1. a) Necking in tensile specimen. b) Barba’s law derived from uniaxial tension test for steel S235 JR EN10025. Data from (Alsos et al. 2009).

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tension test the assumption is that it represent the failure strain in case of uniaxial tension. This stems from the fact that traditional approaches to determine Barba’s law focus on the strain state on the specimen surface, but neglect the strain and stress state in the middle of the specimen (Simonsen & Lauridsen 2000, Ehlers & Varsta 2009, Hogström et al. 2009). This issue is also highlighted by Choung (2009). On the other hand, detailed investigations of tensile specimen failure clearly show that the stress state in the middle of the specimen considerably deviates from the uniaxial tension prior to the fracture initiation (Dunand & Mohr 2010, Choung et al. 2012). This fact suggests that there is a certain length scale for which Barba’s law becomes inapplicable—it does not anymore represent the failure strain in uniaxial tension. It is the objective of this paper to determine this limiting length scale. For this purpose, an averaging unit approach introduced by Kõrgesaar et al. (2014) is used to determine the element size dependent failure strain. Differently from the traditional approaches, the stress and strain state inside the specimen is considered. Through analyses of different averaging units and stress state inside those, range of element sizes are established for which the uniaxial tension condition is satisfied. Furthermore, these analyses also show the range of element sizes for which the plane stress condition is satisfied; in practice, this means that plane stress shell elements could be employed. 2

APPROACH

2.1

Description

In this study Barba’s law is defined using the “Averaging Unit” (AU) concept presented by (Kõrgesaar et al. 2014). The basic idea is described in Figure 2. First, a fine solid element model is created and simulated; in this case we simulate a tensile experiment. In the post-processing stage averaging unit is introduced into the model. The size of the averaging unit is defined with the length to thickness ratio, L/t. This is not fixed and can be adjusted according to the needs. The local response averaged over the averaging unit is denoted as the macroscopic response. The macroscopic behaviour of the averaging unit at time instant i is characterized with the volume averaged field variables, such as equivalent stress σ Vi , equivalent plastic strain ε iV and stress triaxiality ηVi evaluated throughout the deformation process:

σ Vi =

∫V σ i dVi ∫V dVi

(1)

Figure 2. Averaging unit mapped to tensile specimen. Specimen dimensions used in the current study are shown; 1/8th of the specimen is modelled.

ε iV =

∫V ε i dVi ∫V dVi

(2)

ηVi =

∫V ηi dVi ∫V dVi

(3)

where σ i ε i , and ηi are the equivalent stress, equivalent plastic strain and stress triaxiality of a single small element at time instant i, respectively; dV is the volume of an element and V is the total volume of the averaging unit; see Figure 2. Stress triaxility η describes the stress state in the material and is defined as the ratio of the hydrostatic stress to von Mises stress, η σ h / σ . Similarly to eqs. (1) to (3), all the stress and strain components can be volume averaged to obtain the complete macroscopic response of the averaging unit. In the following, superscript V denotes the macroscopic response in the averaging unit in contrast to the local response in the single small solid element. Additionally, for the conciseness, the subscript i will be omitted in the following. It is pointed out that the macroscopic response is given until the point of fracture initiation in the averaging unit. The point of fracture initiation corresponds to the failure of the first material point, represented by a single small solid element in the middle of the specimen. Furthermore, it is well known that the stress triaxiality can change considerably during the deformation. In order to plot fracture strain as a function of stress triaxiality for each averaging unit, time averaged stress triaxiality ηVav at the point of fracture initiation is introduced

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Only 1/8th of the specimen was modelled provided by the symmetry of specimen in all three directions. There are six elements through the half-thickness. The number of through-thickness elements is defined on the basis of the convergence analysis carried out. Mesh refinement showed that engineering stress did not change more than 1%. The experimental length scale of 0.88 mm defines the in-plane element length in the model. 3.3 Figure 3.

ηVav =

1 ε Vf

Averaging units.

ε Vf

∫0

ηV ε V

(4)

where ε Vf is the macroscopic equivalent plastic strain at the point of fracture initiation, i.e. failure strain, and ηV is the macroscopic stress triaxiality. 2.2

4

Averaging units

FE simulations

Numerical simulations were carried out with the commercial code ABAQUS/Explicit v6.11-2 (ABAQUS 2011). All the simulations were displacement controlled. The simulations were stopped once the first small solid element was deleted. The total step time was chosen to be long enough, 0.25 sec, to ensure a quasi-static analysis of the simulations, i.e., the ratio of the kinematic energy to the total energy is less than 5%. RESULTS

By employing the described approach the point of fracture initiation can be determined for different averaging units. Furthermore, the approach allows analysing stress state in the averaging unit. Six different averaging units are employed corresponding to approximate L/t ratio of 10, 6, 4, 2, 1, and 0.5, see Figure 3. The largest L/t ratio corresponds to elements used in large-scale ship collision or grounding analyses (Naar et al. 2002), while the smallest corresponds to elements used in smallscale specimen analysis (Li et al. 2010).

Stress triaxiality contours before and after fracture initiation shown in Figure 4 display the strong variations in stress states. The fracture initiates at the centre of the specimen, where the stress triaxiality is close to 0.577 characterising plane strain condition. Plane strain state is commonly associated with the lowest fracture ductility, which also explains the observed fracture location.

3

First, the stress triaxiality in the averaging units was analysed. Figure 5 shows the evolution of the

3.1

FINITE ELEMENT ANALYSIS

4.1 Equivalent plastic strain and stress triaxiality evolution

Specimen and material

The specimen analysed was previously tensile tested by Ehlers & Varsta (2009) using an optical strain measurement system. The specimen was made from Norske Veritas grade A (NVA) RAEX S275 laser steel by Ruukki. The Young’s modulus is 206 GPa, the Poisson’s ratio is 0.3 and the measured yield stress is 349 MPa. For 0.88-mm element length the failure strain, measured from the specimen surface, is ε f = 1.018. True stress-strain relation of the material is given in (Ehlers & Varsta 2009). This is used as the input to finite element simulations. The main dimensions of the specimen are shown in Figure 2. 3.2 FE modelling The finite element model is built from reduced integrated solid elements (C3D8R in ABAQUS).

Figure 4. FE simulation results. Stress triaxiality A) before and B) after fracture initiation.

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as discussed in Section 2. In order to plot fracture strains as a function of stress triaxiality, time averaged stress triaxiality is calculated with eq. (4). The fracture strain is plotted as a function of time averaged stress triaxiality in Figure 6. The figure clearly shows that the limiting size of the AU where uniaxial tension is applicable is L/t = 4.3. This finding is again applicable for both, full 3D response and plane stress case. In Figure 7 failure strains are plotted as a function of the averaging unit size. The figure can also be interpreted as a Barba’s law since the averaging unit can be interpreted as the finite element with a certain size. On the basis of Figure 6, failure strains plotted in Barba’s law represent uniaxial stress state for elements with L/t >∼4. For this reason it becomes meaningless to plot failure strains Figure 5.

Loading paths for the AU.

equivalent plastic strain ε V in the AU as a function of the stress triaxiality ηV in the AU. The black lines correspond to full 3D response as calculated with eq. (3). The grey lines are obtained by neglecting the out-of-plane stress component, σ33, in calculation of stress triaxiality, so that these lines are consistent with the plane stress condition, denoted with subscript *:

σ V*

(

1⎡ V σ 11 − σ V22 2⎣

) (σ ) (σ ) 2

V 2 22

V 2 11

( )

V 2⎤ + 6 ⋅ σ 12 ⎥⎦

(5)

σ Vh*

σ V σ V22 = 11 3

(6)

ηV

σ Vh / σ V*

(7)

Figure 6. Failure strain plotted as a function of time averaged stress triaxiality.

where σ V* is the plane stress equivalent von Mises stress in the AU, σ Vh* is the plane stress hydrostatic stress and ηV* is the plane stress triaxiality in the AU. Figure 5 demonstrates how the stress state locally in the middle of the specimen considerably deviates from the uniaxial tension, irrespective whether full 3D response or plane stress is considered. Locally the stress state advances towards plane strain state ( V / 3 ) as was already indicated in Figure 4. This bifurcation from the uniaxial path occurs at the point of diffuse necking (ε V = n, where n = 0.25 is the strain hardening exponent) at which point the stress state becomes three-dimensional. For larger AU’s this is not as obvious meaning that uniaxial tension remains a valid assumption. 4.2

Fracture strain and stress triaxiality

It is clearly evidenced in Figure 5 that the stress triaxiality is not constant during the deformation

Figure 7. Failure strain as a function of the element size.

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Figure 8. Equivalent plastic strain plotted as a function of normalized out-of-plane stress component σ33.

in Barba’s law for L/t < 4, since failure strain is known to be a strong function of stress triaxiality. In other words, for the range L/t < 4 it becomes unclear weather failure strain changes purely because of the mesh size effect, or because of the stress triaxiality. Furthermore, it is pointed out that Figure 7 applies for full 3D response as well as for assumed plane stress condition. There is only a small difference between 3D and plane stress response in Figure 6, thus in Figure 7 only failure strains corresponding to 3D solution are plotted. 4.3

Plane stress assumption

The averaging unit approach also allowed us to determine the limiting element size for which the plane stress is a valid assumption. Figure 8 shows the evolution of the out-of-plane stress component during the analysis. Thinning of the material is strongest in the middle of the specimen, thus the compressive stress is highest for the smallest unit. Therefore, for elements with length-to-thickness ratio of 0.44 plane stress elements should not be used. In contrast, for larger units plane stress assumption holds and thus, plane stress shell elements can be employed. It is pointed out that this last statement holds for shell elements imposed to in-plane tensile loads, in contrast to bending dominated loading types. It is believed that for bending, L/t ratio must be larger for successful use of plane stress shell elements. 5

CONCLUSIONS AND DISCUSSION

From the detailed analysis presented above, two main conclusions can be drawn. First, when determining

the mesh size dependency of the fracture strain, stress state should always be considered—the importance of this statement is illustrated in Kõrgesaar & Romanoff (2014). It is shown that for smaller elements, L/t < ∼4, uniaxial tension condition is not anymore applicable. In other words, for these small elements, fracture strain should be always given together with the stress state, since the fracture ductility is strongly influenced by the stress state. This is especially important for practicing engineers who often use equivalent plastic failure strain scaled according to element size. On the other hand, for analysis of large structures where recommended practice prescribes elements L/t > = 5 uniaxial tension remains a valid assumption. Second, analysis of out-of-plane stress component in different averaging units indicates that plane stress condition does not hold for smallest elements, L/t = 0.44. Whether this ratio remains the same under different stress states should be addressed in future studies. ACKNOWLEDGMENTS Thanks are due to Professor Jani Romanoff and Professor Heikki Remes for providing valuable comments and suggestions. The research presented in this paper was partially funded by the FidiProproject called: “Non-linear response of large, complex thin-walled structures” supported by Tekes, Napa, Ruukki, Deltamarin, Koneteknologiakeskus Turku and STX Finland. The author was also supported by the School of Engineering. All the financial support is highly appreciated. REFERENCES ABAQUS 2011. ABAQUS Analysis User’s Manual, Version 6.11, Dassault Systèmes Simulia Corporation. Alsos, H.S., Amdahl, J. & Hopperstad, O.S. 2009. On the resistance to penetration of stiffened plates, Part II: Numerical analysis. Int. J. Impact Eng., 36(7): 875–887. Amdahl, J., Ehlers, S., Leira, B.; (eds.) 2013. Collision and Grounding of Ships and Offshore Structures: Proceedings of the 6th International Conference on Collision and Grounding of Ships and Offshore Structures, ICCGS, Trondheim, Norway, 17–19 June 2013. Choung, J. 2009. Comparative studies of fracture models for marine structural steels. Ocean Eng., 36(15–16): 1164–1174. Choung, J., Shim, C.-S. & Song, H.-C. 2012. Estimation of failure strain of EH36 high strength marine structural steel using average stress triaxiality. Mar. Struct., 29(1): 1–21. Dunand, M. & Mohr, D. 2010. Hybrid experimental– numerical analysis of basic ductile fracture experiments for sheet metals. Int. J. Solids Struct., 47(9): 1130–1143.

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Ehlers, S. & Varsta, P. 2009. Strain and stress relation for non-linear finite element simulations. Thin. Wall. Struct., 47(11): 1203–1217. Hogström, P., Ringsberg, J.W. & Johnson, E. 2009. An experimental and numerical study of the effects of length scale and strain state on the necking and fracture behaviours in sheet metals. Int. J. Impact Eng., 36(10–11): 1194–1203. Kõrgesaar, M., Remes, H. & Romanoff, J. 2014. Size dependent response of large shell elements under inplane tensile loading. Int. J. Solids Struct., 51(21–22): 3752–3761. Kõrgesaar, M. & Romanoff, J. 2014. Influence of mesh size, stress triaxiality and damage induced softening on ductile fracture of large-scale shell structures. Mar. Struct., (38): 1–17.

Samuelides, M. 2012. Designing for protection against collision. Soares, C.G., Garbatov, Y., Fonseca, N. & Texeira, A.P. (eds.), Marine Technology and Engineering: 955–977. Simonsen, B.C. & Lauridsen, L.P. 2000. Energy absorption and ductile failure in metal sheets under lateral indentation by a sphere. Int. J. Impact Eng., 24(10): 1017–1039. Yamada, Y., Endo, H. & Pedersen, P.T. 2005. Numerical study on the effect of buffer bow structure in shipto-ship collisions. Proceedings of the 15th (2005) International Offshore and Polar Engineering Conference (ISOPE), Seoul, Korea June 19–24.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

Dynamic analysis of ring stiffened conical-cylindrical shell combinations with general coupling and boundary conditions X.L. Ma College of Power and Energy Engineering, Harbin Engineering University, Harbin, China Faculty of Engineering and the Environment, Fluid Structure Interactions Research Group, University of Southampton, Boldrewood Innovation Campus, Southampton, UK

G.Y. Jin College of Power and Energy Engineering, Harbin Engineering University, Harbin, China

Y.P. Xiong Faculty of Engineering and the Environment, Fluid Structure Interactions Research Group, University of Southampton, Boldrewood Innovation Campus, Southampton, UK

Z.G. Liu College of Power and Energy Engineering, Harbin Engineering University, Harbin, China

ABSTRACT: Conical and cylindrical shell combinations reinforced by ring stiffeners are widely utilized in practical marine engineering owing to their outstanding mechanical and physical properties. In the paper, dynamic behaviors of ring stiffened conical-cylindrical shell combinations with general elastic coupling and boundary conditions are studied using a modified Fourier series-Ritz method. Under the framework, the non-uniform ring stiffeners are considered as discrete elements while each displacement of the cylindrical and conical shell components is invariantly expressed as the modified Fourier series composed of a standard Fourier series and closed form auxiliary functions in order to satisfy arbitrary elastic coupling conditions between these two components as well as the boundary conditions at the ends. All the series expansion coefficients are determined by Rayleigh-Ritz procedure as generalized coordinates. The convergence of the present method is validated by a number of numerical examples, and also good agreement of results between present method and FEM method confirms the accuracy. The effects of the number and axial position of the stiffeners and the boundary conditions on vibration behavior of the coupled system are investigated. Less computational cost in the calculation demonstrates the effectiveness of the present method compared to the conventional finite element method. 1

INTRODUCTION

Due to excellent mechanical and physical properties, the conical and cylindrical shell combinations reinforced by ring stiffeners are widely used in practical engineering applications such as missiles, torpedoes, rockets and naval hulls of submarines. Since these structures are usually subjected to the intricate environment as well as dynamic loads resulting in fatigue damage, vibration and radiated noise, thoroughly understanding dynamic behavior of the shell combinations is essential to develop adequate vibration control strategy for their fatigue life at the design stage. Much effort was devoted to the vibration theories of thin plates and shells since the 50 s of last century, which has been well summarized and documented by Leissa (1973). Based on these

foundational theories, considerable contributions concentrates on dynamic analysis of elementary shells such as cylindrical, conical and spherical shells. Compared to elementary shell configures, a few but not many publications focused on vibration analysis of shell combinations has been reported in the literature. The Finite Element Method (FEM) computer programs have been well developed in recent years such as ANSYS and NASTRAN to meet the requirement of dynamic analysis on complicated coupled systems. However, large number of interior points at high frequency calculation and inconvenience to identifying certain mode shapes make these programs poor at practical engineering calculation. Therefore, developing an accurate and efficient method is of great significance to characterize the vibration behaviour of coupled structures. The problem in the theoretical formulation of shell

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combinations is mainly caused by the fact that each component of the shell combination finds its natural description in different coordinate systems. Hu & Raney (1967) presented a multi-segmental numerical integration approach to obtain the analytical results for joined con-cylindrical shells. The coupling conditions at the interface are imposed on the segments of the coupled system and the comparison of results to experiment demonstrates the accuracy of the approach. The natural frequencies of cylindrical shells coupled with various types of shells were obtained by Galletly and Mistry (1974) by using variational finite differences and finite elements. A classical approach was adopted by Caresta and Kessissoglou (2010) to investigate the vibration characteristics of isotropic conical and cylindrical shell combinations. In the study, two types of solutions including a wave solution and a power series solution are utilized respectively to solve the equations of the two shell components and coupled together by the means of the continuity conditions at the junction. Kouchakzadeh and Shakouri (2014) investigated free vibration behaviour of coupled cross-ply laminated conical shells. The coupling conditions at the junction were achieved by the extraction of the appropriate expressions among stress resultants and deformations. Ma et al (2014) developed a modified Fourier series method for vibration analysis of coupled conical-cylindrical shells. Rapid convergence as well as excellent accuracy of the improved series is observed while arbitrary elastic coupling and boundary conditions are considered. As far as stiffened shell combinations are concerned, the stiffeners have great impact on vibration behaviour of shell structures. Two main approaches are commonly used depending upon whether the stiffeners are considered by averaging their properties over the surface of the shell or by treated them as discrete elements. The stiffeners are assumed to be smeared over the whole shell in the first approach which is suitable for the shell structures reinforced with a large number of uniform stiffeners. Based on the approach, the vibro-acoustic responses of stiffened submerged vessel are investigated by Caresta (2009). Recently, the second approach attracts increasing attention due to its accuracy and generality for the case with non-uniform space and different eccentricity of ring stiffeners. Wang et al (1997) presented vibration analysis of cylindrical shells with varying ring-stiffener distribution with an automated Rayleigh-Ritz method. Jafari and Bagheri (2006) investigated free vibration of non-uniformly ring stiffened cylindrical shells and compared the results among analytical, experimental and numerical methods. The modified variational approach was developed by Qu et al (2013a, 2013b) for vibration

analysis of ring-stiffened conical-cylindrical shell combinations. The coupled system is partitioned into proper shell segments and all continuity restrains on segments interfaces are treated by means of a modified variational principle and least squares weighted residual method. The primary objective of this study is to develop a uniform, accurate and efficient method for vibration analysis of ring stiffened conical-cylindrical shell combinations with general elastic coupling and boundary conditions, which are always encountered in practical engineering. Regardless of these conditions, each of displacement components of conical and cylindrical shells is invariably expressed in the form of a modified Fourier series consisting of a standard Fourier series and auxiliary functions in order to satisfy arbitrary coupling and boundary conditions. All expansion coefficients are determined using Rayleigh-Ritz method as generalized coordinates. The convergence of present method will be validated by numerical examples for the cases with different truncated configurations. The effects of the number and locations on vibration behaviour of the coupled system are also investigated. 2 2.1

THEORETICAL FORMULATIONS System description

The coordinate systems for the ring stiffened conical-cylindrical-conical shell combination are illustrated in Figure 1. A cylindrical coordinate system ( s ,θ s , s ) is used for the cylindrical shell compo nent with thickness hs , length Ls and radius R, in which xs , θ s and rs depict the axial, circumferential and radial directions respectively. The displacement components of the cylindrical shell in the xs , θ s and rs directions, respectively, are respectively expressed as us , vs and ws . The non-uniform ring stiffeners are arranged within the cylindrical shell at the location xi (ii Nr ) for each stiffener, where Nr is the total number of ring stiffeners.

Figure 1. coordinate systems for the ring stiffened conical-cylindrical-conical shell.

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On the other hand, the conical shell component is described by the ( c ,θ c , c ) coordinate system, where xc is measured along the generator of the cone from its small edge, θ c is the circumferential direction and rc is the normal direction perpendicular to middle surface of the shell. uc , vc and wc are the displacements of the conical shell with respect to its coordinate system. ϕ is the semi-vertex of the conical shell and R0 is the radius at the small end. These two conical shell components described in Figure 1 are of the same geometry properties while all the shell components are made of same material with Young’s modulus E , density ρ and Poisson radio μ for convenience. For both cylindrical and conical shell components, the thickness is assumed to be uniform and rather small compared to other parameters of the shell combination so that the thin shell theory is applicable. The Rayleigh-Ritz energy method is selected to establish the theoretical formulation of the stiffened conical-cylindrical shell with general elastic coupling and boundary conditions. The total energy of the coupled system mainly consists of four components, including (i) the strain energyV Vs and the kinetic energy Ts of the cylindrical shell, (ii) the strain energy Vc and the kinetic energy Tc of the conical shell, (iii) the strain energy Vp and the kinetic energy Tp of the ring stiffener and (iv) the potential energy caused by boundary conditionsV Vb at the ends of the coupled system and coupling conditions V j at the junction. Thus, the Lagrangian energy function Π for the stiffened conical-cylindrical shell combination can be expressed as Π = Vs − Ts + Vc − Tc + Vp − Tp + Vb + V j 2.2

−2

∂vs ∂ 2ws ⎛ ∂v ⎞ +⎜ s ⎟ ∂θ R 2 ∂θ 2 ⎝ R ∂θ ⎠

− 4(1 − μ )

Ts =

ρ hs 2 ∫0

∫0

2

2 ∂vs ∂ 2ws ⎛ ∂v ⎞ ⎪⎫ + 2(1 − μs ) ⎜ s ⎟ ⎬Rdxdθ ⎝ ∂x ⎠ ⎪ ∂x ∂ ∂θ ⎭

(2) ⎧⎛ ∂w ⎞ 2 ⎛ ∂u ⎞ 2 ⎛ ∂v ⎞ 2 ⎫⎪ s s s ⎨⎜ ⎟⎠ + ⎜⎝ ⎟⎠ + ⎜⎝ ⎟⎠ ⎬ Rd dθ ⎝ ∂ t ∂ t ∂ t ⎩⎪ ⎭⎪ (3)

2.3

Energy functional of conical shell component

With regard to the coordinate system introduced for the conical shell component, the strain energy and the kinetic energy can be written as (Leissa, 1973) VC =

Eh hc

2π

2

2(1 − μ )

+

+

Lc

∫0 ∫0

2

{(ε xc 0 ) + (

1− μ (γ xcθc )2 } c ( 2 Ehhc

+

(1)

3

2π

24(1 μ

2

Lc

∫ ∫ ) 0 0

1− μ (τ xc 2

c0

2

c

) + 2 μc ε xc 0ε

c

0

c )d c dθ c

{( kxc

2

2

kθ c 0 ) + 2 μc kxc 0 k

c

0

)2 }Rc ( xc )dxc dθ c (4)

and

Energy functions of cylindrical shell

According to Reissner’s thin shell theory (Leissa, 1973), the strain energy and the kinetic energy of the cylindrical shell component can be express as 2 2π L ⎧ Eh hs ∂vs ws ⎞ ⎪⎛ ∂us Vs = + + ⎟ ∫ ∫ ⎨⎜ 2(1 − μ 2 ) 0 0 ⎩⎪⎝ ∂x R∂θ R ⎠ ∂u ⎛ ∂v w ⎞ − 2(1 − μ ) s ⎜ s + s ⎟ ⎝ ∂x R ∂θ R ⎠ ⎫ 2⎪ ∂us ⎞ ⎬ Rdxdθ ( − μ ) ⎛ ∂vs ∂u + + ⎜ ⎟ 2 ⎝ ∂x R∂θ ⎠ ⎪⎭ 2 ⎧ 2 2π L ⎪⎛ ∂ ws Eh hs3 ∂ 2ws ⎞ + + ⎨ ⎟ ∫ ∫ ⎜ 24(1 μ 2 ) 0 0 ⎪⎝ ∂x 2 R 2 ∂θ 2 ⎠ ⎩ 2 ⎛ ∂ 2ws ⎞ ⎫⎪ ∂ 2w ∂ 2w − 2(1 μ ) 2s 2 s 2 − ⎜ ⎟ ⎬ Rdxdθ ∂x R ∂θ ⎝ ∂ ∂θ ⎠ ⎪ ⎭

2π L ⎧ Eh hs3 ∂v ∂ 2ws −2 μs s 2 2 ∫0 ∫0 ⎨ ∂θ ∂x 2 24R (1 − μ ) ⎩

+

TC =

ρ hc 2

2π

Lc

∫0 ∫0

⎧⎛ ∂wc ⎞ 2 ⎛ ∂uc ⎞ 2 ⎛ ∂vc ⎞ 2 ⎫ ⎨ ⎬Rc(xxc )dxcdθc + + ⎩⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎝ ∂t ⎠ ⎭ (5)

where ε xc 0, εθc 0 and γ xcθc 0 are the normal and shear strains in the middle surface of the conical shell, kxc 0 and kθc 0 are the mid-surface changes in curvature and τ xcθc 0 is the mid-surface twist. 2.4 Energy functional of ring stiffeners The ring stiffeners are treated discrete elements from the foundation cylindrical shell in the study. The space among ring stiffeners is non-uniform while all stiffeners are of same geometry and material properties. Within the coordinate system for the cylindrical shell component, the relationships between displacement components (uri , vri and wri ) of the ith ring stiffener and

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those (us , vs and ws ) of the cylindrical shell are described by ⎛

uri

⎝

us + e

∂ws ⎞ ∂xs ⎟⎠

(6) xs xi

⎡ e e ∂ws ⎤ ⎢vs ( + ) + ⎥ R R ∂θ s ⎦ ⎣ xs

vri wri

ws

(7) xi

(8)

xs xi

With the consideration of effects of stretching, biaxial bending and wrapping, the strain energy of the ring stiffeners are given as (Jafari & Bagheri, 2006) N

Vp =

E r 2π ∑ ⎡I r k12 + I x k22 + Areg 2 2 i =1 ∫0 ⎣ +

1 ⎤ Jrγ 2 ⎥ dθ 2(1 + μ ) ⎦

(9)

2.5 General elastic coupling and boundary conditions Due to the fact that supporting types of practical engineering structures and assembling techniques are always variable in nature, the cases with elastic coupling and boundary conditions are often encountered in practice other than the classical cases. For the sake of simulating general elastic coupling and boundary conditions, artificial stiffness-like spring technique is adopted in the study. Four groups of springs with respect to sub-coordinate systems are used at the junction of two shell components and extreme of the cylindrical shell, including three sets of linear springs respectively along the xs , θ s and rs directions and one set of rotational springs around the rs directions. Similar four groups of springs are located at the extreme of the conical shell component with respect to ( c ,θ c , c ) coordinate system. Under the framework, the boundary conditions for the coupled conical-cylindrical shell combination can be expressed as kx ,s us − N x ,s

where I x and I r are the second moments of areas with respect to x and r directions respectively, Ar is the cross-sectional area of the ring stiffener, Jr is the torsional rigidity and the other symbols are introduced as 1 ⎛ ∂wri 1 ∂ uri ⎞ − + ⎜ R e ⎝ ∂x R + e ∂θ 2 ⎟⎠

kθ ,svs − N xθ ,s

xs =0 xs =0

kr,s ws − Qx ,s −

(10) K r ,s

k2 =

⎛ ∂ 2wri ⎞ ⎜ −wri − ⎟ (R e ) ⎝ ∂θ 2 ⎠

(11)

eg =

1 ⎛ ∂vri ⎞ + wri ⎟ ⎜ ⎠ R e ⎝ ∂θ

(12)

γ =

1

2

1 ⎛ ∂ 2wri + R + e ⎜⎝ ∂ ∂θ

1

∂uri ⎞ ∂θ ⎟⎠

(13)

The kinetic energy of ring stiffeners with effects of triaxial translational inertia and rotary inertia about x and r directions can be expressed as Tp

2 N ρ r 2π ⎪⎧ ⎛ ∂w ⎞ ⎪⎫ Ar (uri + vri + wri ) I p ⎜ ri ⎟ ⎬ dθ ⎨ ∑ ∫ ⎝ ∂x ⎠ ⎪ 2 i =1 0 ⎩ ⎭x x

(15)

=0

(16)

∂M xθ ,s

2

k1 = −

=0

∂ws + M x ,s ∂ xs

R ∂θ s

=0

(17)

xs = 0

=0

(18)

xs =0

where N x ,l and N xθ ,l is the in-plane forces, Qx ,l is the transverse shear force, M x ,l is the bending moment and M xθ ,l is the twisting moment (l c, s ), kx ,s , kθ ,s , kr,s and K r,s respectively, denote the stiffnesses for linear springs in xl , θ l and rl directions and rotational springs around rl direction. As far as the coupling conditions are concerned, the displacement components of the conical shell at the end coupled to the cylindrical shell should be transformed from the the ( c ,θ c , c ) coordinate system into the ( s ,θ s , s ) coordinate system. Thus, the continuity conditions (xc Lc , xs = 0) connecting the two shells are written as

i

(14) It is noted that both number and location of the ring stiffeners are not limited to certain values in the formulation. Thus, present method is of general significance in practical engineering calculations.

ko,x (us

uc cos ϕ + wc sin in ) − N x ,s = 0

(19)

ko, vs

vc ) N xθ ,s = 0

(20)

ko,r (ws uc sinϕ wc coss ) − Qx ,s − ∂w ⎞ ⎛ ∂w K o,rr ⎜ s − c ⎟ − M x ,s = 0 ⎝ ∂xs ∂xc ⎠

∂M xθ ,s R ∂θ c

= 0 (21) (22)

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where ko,x , ko,θ , ko,r and K o,r , respectively, denote the stiffnesses of the springs between the two shell components. As indicated in Eqs. (15)–(22), all of the force and moment resultants of the cylindrical and conical shell components at the junction as well as boundaries are constrained by used springs. Thus, arbitrary coupling and boundary conditions can be generally achieved by varying the value of springs’ stiffness. Thus, the potential energy stored in the coupling and boundary springs can be expressed as Vb

2.6

ws ( xs ,θ s , t )

+∑∑

l =1 n = 0

s  ln A α ξl

∞

πα ⎞ 2 ⎠

⎛

s ∑ ⎨ ∑ ∑ Cmmn nα cos(( λ ssm xs ) cos ⎝ nθ s +

α = 0 ⎩m = 0 n = 0 ∞

2

s

πα ⎞ ⎫ j ⎛ θ + ⎬e ⎝ s 2 ⎠⎭

t

⎧

1

∞

∞

α = 0 ⎩m = 0 n = 0 ∞

2

πα ⎞ ⎫ j ⎛ θ + ⎬e ⎝ c 2 ⎠⎭

c

l =1 n = 0

(23)

2⎫

⎧

1

∞

∞

⎛

c ∑ ⎨ ∑ ∑ Bmmn nα cos(( λccm xc ) cos ⎝ nθ c +

α = 0 ⎩m = 0 n = 0 ∞

2

⎪ ⎬ Rdθ s ⎭⎪ (24)

πα ⎞ ⎫ j ⎛ θ + ⎬e ⎝ s 2 ⎠⎭

t

(28) vc ( xc ,θ c , t )

 clnα ξ +∑∑B l

c

l =1 n = 0

πα ⎞ ⎫ j ⎛ θ + ⎬e ⎝ c 2 ⎠⎭

πα ⎞ 2 ⎠ t

(29) wc ( xc ,θ c , t )

πα ⎞ 2 ⎠

πα ⎞ 2 ⎠

⎛

c ∑ ⎨ ∑ ∑ Ammn nα cos(( λccm xc ) cos ⎝ nθ c +

 clnα ξ +∑ ∑ A l

⎛

s

∞

s  ln + ∑ ∑C α ξl

uc ( xc ,θ c , t )

α = 0 ⎩m = 0 n = 0 ∞

⎧

1

and

s ∑ ⎨ ∑ ∑ Ammn nα cos(( λssm xs ) cos ⎝ nθ s + 2

t

(27)

The admissible function is the key to achieve the unified solution for the coupled system with arbitrary coupling and boundary conditions. The conventional Fourier series can only be used to very simple boundary conditions and lead to discontinuities of the displacement components and their derivatives at the junction and boundaries. The selected admissible function must be close form and sufficiently smooth over the whole coupled structure to satisfy the kinetic relations mentioned above, which means that at least the third order derivatives of displacement components exist and are continuous at any point of the structure especially at the junction and boundaries. Only in this way each of coupling and boundary conditions will be considered in the calculation. With the consideration, the admissible function of displacements for both cylindrical and conical shell is specially selected as ⎧

πα ⎞ ⎫ j ⎛ θ + ⎬e ⎝ s 2 ⎠⎭

(26)

Unified solution for the shell combination

us ( xs ,θ s , t )

s

l =1 n = 0

{

∞

∞

2

1 2π ko,x (us uc cosϕ + wc sin inϕ )2 2 ∫0 + ko,r (ws − uc sinϕ wc cossϕ )2

∞

πα ⎞ 2 ⎠

⎛

α = 0 ⎩m = 0 n = 0

{

1

∞

l =1 n = 0

1 2π kx ,c uc 2 kθ ,cvc 2 kr,cwc 2 2 ∫0 2 ⎛ ∂w ⎞ ⎪⎫ + K r,c ⎜ c ⎟ ⎬ xc = 0 Rdθ c ⎝ ∂xc ⎠ ⎭⎪ 1 2π + ∫ kx ,s us 2 + kθ ,svs 2 + kr,sws 2 2 0 2 ⎛ ∂w ⎞ ⎪⎫ + K r,s ⎜ s ⎟ ⎬ xs = Ls Rdθ s ⎝ ∂xs ⎠ ⎪⎭

⎛ ∂w ∂w ⎞ + ko, vs vc )2 K o,r ⎜ s − c ⎟ ⎝ ∂xs ∂xc ⎠

∞

s  ln +∑∑B α ξl

{

Vj

⎧

1

s ∑ ⎨ ∑ ∑ Bmmn nα cos(( λ ssm xs ) cos ⎝ nθ s +

vs ( xs ,θ s , t )

⎧

1

∞

∞

⎛

c ∑ ⎨ ∑ ∑ Cmmn nα cos(( λccm xc ) cos ⎝ nθ c +

α = 0 ⎩m = 0 n = 0 2

∞

 clnα ξ + ∑ ∑C l l =1 n = 0

c

πα ⎞ ⎫ j ⎛ θ + ⎬e ⎝ c 2 ⎠⎭

πα ⎞ 2 ⎠

t

(30) where ω is the angular frequency, t represents time, s s s s λsm π / Ls , λcm π / Lc , Amn α , Alnα , Bmnα , Blnα , s s c c c c c c     Cmnα ,Clnα and Amnα , Alnα , Bmnα , Blnα , Cmnα Clnα are the expansion coefficients and they can be solved by the Rayleigh-Ritz procedure.ξl ( xi ) ( , 2 ) are the supplementary functions for the in-plane displacements, while ς l ( xi ) ( , 2, 3, 4 ) are the supplementary functions for the radial displacement. In this paper, these supplementary functions are specially selected as 2

ξ1( i )

⎛x ⎞ xi ⎜ i − 1⎟ L ⎝ i ⎠

ξ2 ( i ) =

xi 2 ⎛ xi ⎞ −1 Li ⎜⎝ Li ⎟⎠

ζ1( i ) =

9Li ⎛ π xi ⎞ sin i 4π ⎝ 2 Li ⎠

t

(25)

(31)

(32) Li ⎛ 3π x ⎞ sin ⎜ 12π ⎝ 2 Li ⎟⎠

(33)

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ζ2( i ) = − ζ3( i ) =

9Li ⎛ π xi ⎞ cos 4π ⎝ 2 Li ⎠

Li 3 ⎛ π xi ⎞ sin i ⎝ 2 Li ⎠ π3

ζ4( i ) = −

Li ⎛ π xi ⎞ cos ⎝ 2 Li ⎠ π3 3

Li ⎛ 3π xi ⎞ cos ⎜ 12π ⎝ 2 Li ⎟⎠ Li 3 ⎛ 3π xi ⎞ sin siin ⎜ ⎝ 2 Li ⎟⎠ 3π 3 Li ⎛ 3π xi ⎞ cos cos ⎜ ⎝ 2 Li ⎟⎠ 3π 3

(34)

the rigid body mode of the coupled system will be neglected.

(35)

3.1 Convergence and accuracy study

3

(36)

It can be proved that the requirement for the selection of the auxiliary polynomial function is gguaranteed, since it is easy y to verify y that ς1 ′ 0 ς3 (00)) = ς 2 ′( ) ς 4 ′″( ) =1,ξ1 ξ2 ( ) 1 ξ2 ξ2 ( ) ξ2 ′(0 ) = 0 , ξ1 ξ1( ) ξ1 ′( ) 0 . Substituting Eqs. (25)–(36) into Eq. (1) together with energy equations for each component of the ring stiffened conical-cylindrical shell combination and then setting the variation of the proceeding functional with regard to each expansion coefficients independently to zero, one obtains the equations of motion for the whole structure as (

2

)D

0

(37)

where K and M are respectively the stiffness and mass matrices of the whole structure. Limited to length requirement of the paper, the details of these matrices are not displayed here. The natural frequencies and eigenvectors of the ring stiffened conical-cylindrical shell can be obtained by solving the eigen-problem of Eq. (37). One can also obtain forced vibration solution by replacing the right of Eq. (37) by the column matrix of generalized force composed of the expression series multiplying the force components. 3

NUMERICAL RESULTS AND DISCUSSIONS

In order to illustrate the convergence as well as accuracy of present method and well understand vibration behavior of ring stiffened conical-cylindrical shell combinations, some numerical examples are presented in this section. Unless special specified, the geometric and material parameters of the conical and cylindrical shell combination are given as follows: R0 = 0.4226 4226 m , R Ls = 1 m , hc hs = 0.01 m , E = 211 × 109 Nm −2 , μ = 0.3 and ρ = 7800 Kgm −3 . The cross-section of stiffeners is set to be rectangular with width b = 0.02 m and depth d = 0.04 04 m . The non-dimensional natural E is frequency parameter Ω mn = ω mnR ρ ( μ )/E introduced here for generalization. Conveniently, F, S and C represent free, simple supported and clamped boundary conditions respectively. In addition, zero natural frequency corresponding to

To validate the convergence of present method, natural results of the coupled structure with subject to different truncated configurations are carried out and compared in Table.1. Three ring stiffeners are utilized in this case and they are respectively arranged at locations xs = 0.1 m, 0.3 m and 0.7 m with non-uniform spaces. Since the conical-cylindrical-conical shell combination reinforce by ring stiffeners is symmetrical in circumferential direction, only the meridional truncated number M varies from 11 to 13 with fixed circumferential truncated number N = 10. It can be found from Table 1 that with the truncated number increasing, non-dimensional natural frequencies of the coupled system converge rapidly. One obtains the convergent and stable natural frequencies at the truncated configurations m = 12 and n = 10. Therefore, these truncated numbers are adopted in following numerical examples. As far as the accuracy method is concerned, results from finite element method program ANSYS are selected to confirm the validation of present method since less figures have been reported in the existing literatures for such coupled system. In the calculation model established by ANSYS, 120 × 40 and 120 × 30 finite element mesh of SHELL 63 are used respectively for the conical and cylindrical shell components with 120 finite element mesh of BEAM 188 for ring stiffeners. The convergent natural frequencies agree well with those from ANSYS which confirms the accuracy of present method. Furthermore, only 2961 dofs are used in the calculation by present method with the truncated configurations whereas more than 70,000 dofs are needed for the model established in ANSYS. Consequently, it is clear that one can save much computation cost for vibration analysis of such complex coupled shell combinations by using present method compared to conventional finite element method. 3.2

Effect of location of ring stiffener

The effect of location of ring stiffener on natural behavior of the shell combination is studied in this section. For simplification, only one ring stiffener is used in the calculation and location factor is introduced here as xi/Ls for generalizing the location of stiffener. Figure 2 shows the lowest non-dimensional natural frequencies of the coupled system with F-F boundary conditions respectively corresponding to different circumferential order n = 0, 1, 2, 3 as the

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Table 1. Non-dimensional natural frequencies of the stiffened shell combinations with different truncated configurations under various boundary conditions. Mode No. M×N Boundary conditions n F-F

0

1

2

S-S

0

1

2

C-C

0

1

2

m

11 × 10 12 × 10 13 × 10 ANSYS

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

0.3901 0.5251 0.6244 0.4162 0.4702 0.5670 0.0204 0.0218 0.2810 0.2009 0.3992 0.5281 0.1982 0.4148 0.4754 0.1862 0.2972 0.4154 0.2009 0.3387 0.6041 0.2031 0.4365 0.4993 0.1931 0.3048 0.4199

0.3900 0.5250 0.6244 0.4162 0.4701 0.5670 0.0204 0.0218 0.2809 0.2009 0.3992 0.5280 0.1982 0.4148 0.4753 0.1862 0.2971 0.4154 0.2009 0.3385 0.6041 0.2031 0.4363 0.4993 0.1930 0.3048 0.4199

0.3900 0.5250 0.6244 0.4162 0.4701 0.5670 0.0204 0.0218 0.2809 0.2009 0.3992 0.5280 0.1982 0.4148 0.4753 0.1862 0.2971 0.4154 0.2009 0.3385 0.6041 0.2031 0.4363 0.4993 0.1930 0.3048 0.4199

0.3902 0.5236 0.6247 0.4165 0.4742 0.5715 0.0212 0.0233 0.2820 0.2013 0.3902 0.5236 0.1984 0.4137 0.4788 0.1800 0.2975 0.4176 0.2013 0.3401 0.6176 0.2034 0.4359 0.5045 0.1932 0.3054 0.4225

Figure 2. Non-dimensional natural frequencies corresponding to n = 0, 1, 2, 3 for the shell combination with F-F boundary conditions.

location factor of the ring stiffener varies from 0 to 1. With location factor increasing, the lowest natural frequency corresponding to n = 0 firstly decreases to stable constant in large range of location from 0.2 to 0.8 and then increases to the initial value due to the symmetry of the coupled structure. The similar behavior is also found on the plot of n = 1 with larger range of location for stable natural frequency. On the other hand, the value of non-dimensional natural frequencies corresponding to n = 2 and 3 keep at constants in the whole location range. To well understand the effect of stiffener location, new cases with various boundary condition including F-F, S-S and C-C boundary conditions are implemented. Figure 3 shows the lowest natural frequencies of the shell combinations with these types of boundary conditions. As indicated in Figure 2, the lowest natural frequency for the case with F-F boundary conditions is fixed at a stable value. The plots for the cases with S-S and C-C

Figure 3. Lowest non-dimensional natural frequencies for the shell combinations with F-F, S-S and C-C boundary conditions.

boundary conditions increase firstly and decrease at second half location range. However, larger variation of lowest natural frequency is found in the plot of C-C boundary conditions rather than that of S-S boundary conditions. Generally, the stiffened coupled system with C-C boundary conditions has the highest value of first order natural frequency following with the cases with S-S boundary conditions and F-F boundary conditions. 3.3

Effect of number of ring stiffeners

In practical engineering structures, a large number of ring stiffeners are utilized to reinforce the shell combinations in general. The number of ring stiffeners has great influence on the vibration behavior

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Figure 4. Lowest non-dimensional natural frequencies of the coupled system with F-F, S-S and C-C boundary conditions with the number of ring stiffeners increasing.

of the practical coupled structures. The effect of the number of stiffeners is studied in this section. For simplification and generalization of the effect, the ring stiffeners are arranged with uniform space in the cylindrical shell component. Figure 4 shows the lowest non-dimensional natural frequencies for the coupled structures with F-F, S-S and C-C boundary conditions with the number of stiffeners increasing. The first order natural frequency of the case with F-F boundary condition increases rapidly in the number range from 1 to 9 and keeps on increasing in the following number range, though the speed of increasement is reduced. For the case with S-S boundary conditions, the value of non-dimensional natural frequency increases firstly up to it peak when the number of ring stiffeners increases to 4 and keep stably at the constant in the range from 4 to 15. When one continues to increase the number over 15, the value decreases with the degree as it increases firstly. As far as the C-C boundary conditions are concerned, although higher value of the first order natural frequency is observed from the figure, the general behavior of the first order natural frequency for the case with C-C boundary conditions is similar to that with S-S boundary conditions. The dynamic behabior of first order natural frequencies for cases with S-S and C-C boundary conditions shows that increasing the number of ring stiffeners does not necessary to reinforced the coupled shell structures, which may be due to additional mass caused by ring stiffeners. 4

elastic coupling and boundary conditions are presented in this paper. Rayleigh-Ritz energy method is adopted to establish the theoretical model of the coupled system. The ring stiffeners are arranged on the cylindrical shell component and they are treated as discrete element in the formulation. Regardless of coupling and boundary conditions, each displacement of both conical and cylindrical shell component is invariantly expressed as the modified Fourier series composed of a standard Fourier series and auxiliary functions. Introducing the supplement functions can not only accelerate the convergence of expansion series but also remove all potential discontinuities at the boundaries as well as the junction between these two shell components. The artificial spring systems are utilized at the junction and boundaries to achieve general elastic coupling and boundary conditions. Under the framework, arbitrary including classical coupling and boundary conditions can be easily approached just by varying the stiffness of relative springs. A number of numerical examples are conducted to validate present method as well as investigate the vibration behavior of the coupled shell structures. Fast convergence and good accuracy of present method are observed from comparison of natural frequencies with those obtained by finite element method program. Meanwhile, less cost in the calculation compared to the program demonstrates the efficiency of present method. The effects of location and number of ring stiffeners on vibration behavior of the coupled system are studied. The variation of natural frequencies depends on the type of boundary conditions in a large greed. With the consideration of additional mass with the number of ring stiffeners increasing, more ring stiffeners does not leads to higher natural frequencies and when the number increases over one certain value, the natural frequency decreases. Present method provides a general and accurate algorithm for coupled stiffened shell combinations with general boundary conditions, which is expected to provide accurate results for complicated structures as reference.

ACKNOWLEDGEMENTS This work is supported by the National Natural Science Foundation of China (Nos. 51175098 and 51279035).

REFERENCES

CONCLUSIONS

Free vibration analysis of ring stiffened conical and cylindrical shell combinations with general

Caresta, M. 2009. Structural and acoustic response of a submerged vessel. Ph.D. Thesis, University of New South Wales.

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Caresta, M. & Kessissoglou, N.J. 2010. Free vibration characteristics of isotropic coupled cylindrical-conical shells. J. Sound Vib. 329:733–51. Galletly, G.D. & Mistry, J. 1974. The free vibrations of cylindrical shells with various endclosures. Nucl. Eng. Des. 30:249–68. Hu, W.C.L. & Rancy, J.P. 1967. Experimental and Analytical study of vibrations of joined shells. Am. Inst. Aeronaut Astronaut J. 5:976–80. Jafari, A.A. & Bagheri, M. 2006. Free vibration of nonuniformly ring stiffened cylindrical shells using analytical, experimental and numerical methods. Thin Wall Struct. 44:82–90. Kouchakzadeh, M.A. & Shakouri, M. 2014. Free vibration analysis of joined cross-ply laminated conical shells. Int. J. Mech. Sci. 78:118–25. Leissa, A.W. 1973. Vibration of shells. Washington, DC: NASA SP-288.

Ma, X.L. Jin, G.Y. Xiong, Y.P. & Liu, Z.G. 2014. Free and forced vibration analysis of coupled conicalcylindrical shells with arbitrary boundary conditions. Int. J. Mech. Sci. 88:122–37. Qu, Y.G. Chen, Y. Long, X.H. Hua, H.X. & Meng, G. 2013a. A modified variational approach for vibration analysis of ring-stiffened conical-cylindrical shell combinations. Eur. J. Mech. A Solids 37:200–15. Qu, Y.G. Wu, S.H. Chen Y. & Hua H.X. 2013b. Vibration analysis of ring-stiffened conical-cylindrical shells based on a modified variational approach. Int. J. Mech. Sci. 69:72–84. Wang, C.M. et al 1997. Ritz method for vibration analysis of cylindrical shells with ring stiffeners. J. Eng. Mech. ASCE 123:134–42. Wang, C.M. Swaddiwudhipong, S. & Tian, J. 1997. Ritz method for vibration analysis of cylindrical shells with ring stiffeners. J. Eng. Mech. ASCE 123:134–42.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

The determination of ice-induced loads on the ship hull from shear strain measurements M. Suominen, J. Romanoff, H. Remes & P. Kujala Department of Applied Mechanics, School of Engineering, Aalto University, Espoo, Finland

ABSTRACT: This paper focuses on the determination of ice-induced loading on the ship hull based on strain measurements by implementing a classical beam theory and an influence coefficient matrix. It is shown in the paper that the total loading affecting the frame structure can be determined accurately from the measured shear strains with both methods in an ideal loading case (i.e. calibration pull). However, it is shown that the load location have a significant impact on the determined loading with both methods. The total loading affecting the structure can be severely underestimated or overestimated depending on the load location with respect to the instrumented side of the structure. 1

INTRODUCTION

In the rapidly changing ice conditions, ships are subjected to the locally varying mechanical and physical properties of ice. As the properties of ice and the general ice conditions affect the iceinduced loads on the hull of the ship through the ice breaking process, the magnitude of ice loads may vary significantly in short-term measurements. The ice-induced load is the force induced by the ice on the ship’s shell structure during the ice-breaking process. In order to gain knowledge on ice-induced loads, several full-scale measurements have been conducted with different extents of instrumentations and focusing on different hull areas. In some occasions, the measurements have concentrated on the bow area and strain gauges have been mounted on several adjacent frames at the bow area (see e.g. Kujala & Vuorio, 1986; Ritch et al., 1997; Ship Structure Committee, 1990). In other occasions, the focus has been directed to different hull areas simultaneously. Thus, frames in different hull areas have been instrumented (see e.g. Kujala, 1989; Kotisalo & Kujala, 1999; Suominen et al., 2013). Commonly, the ice-induced load acting on the hull is determined as the force acting on the transverse frames of the shell structure by measuring the shear strain on the upper and lower parts of the transverse frame as it is assumed that the loadtransfer is mainly due to ice-frames. If adjacent frames are instrumented, an influence coefficient matrix can be defined to consider the effect of the adjacent frames on the measurements on the frame in question. Commonly the influence matrix is defined by utilizing a FE model (see e.g. Riska, et al., 1983; Ritch et al., 1999; Ralph et al., 2003).

As the measurements are based on strain measurements, the structural response on loading has to be defined in order to determine the magnitude of the load acting on the frame. The response can be determined by employing a classical beam theory, numerical methods (Finite Element Methods), or by conducting calibration pull tests. Although several instrumentations have been conducted in the past, only a few of them has discussed about the uncertainty or sensitivity of the measurement instrumentation. In a few studies, calibration pulls have been conducted in full-scale on the real structure to validate the FEM models (see e.g. SSC, 1990; Kulaja, 1989; Ritch, et al., 2008). However, the validations have, more or less, been limited in comparing the difference between the measured and modelled strains and reporting the difference in percentages from a few pulls. A comprehensive study on the effect of loading location on the determined ice-induced load would provide a new knowledge on the magnitude of the measured ice-induced loads as the measurement accuracy could be determined. Due to the lack of an inclusive study on the sensitivity of the ice-induced load measurement instrumentation, this paper examines the sensitivity related to the location of the loading in the horizontal direction. The ice loads are determined from the stress-strain relation by employing the basic beam theory with the closed form approach and the influence coefficient matrix. Both methods are validated with a calibration measurement conducted on board S.A. Agulhas II. After the validation, the effects of the location of the loading on the measurement results are studied with both methods.

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2

METHODS

2.1

Principals of ice-induced load measurements

When ships are operating in ice conditions, ice induces a shear force on the frame structure during the ice breaking process. The ice-induced shear force is mainly carried by the frame under loading and a part of the loading is transferred to the adjacent frames through the hull plating. Figure 1 presents a free-body diagram from the frame under loading and the forces affecting the frame. The equilibrium condition of the frame can be expressed with the notations presented in Figure 1 as FICCE

Qx − (Qx

QX ) + Qy

(Q

y

)

+ ΔQy = 0

(1)

FICE = ice-induced load; Qx = shear force carried by the frame; Qy = shear force transferred to the adjacent frames through hull plating; and Δ = changes in Qx and Qy. Denoting the shear force carried by the frame as QFrame and the force transferred by plating as QPlate, Equation (1) can be written as FICCE

Qx + ΔQy

QFrame + QPllate

(2)

Therefore, the ice-induced shear force affecting the frame can be determined from the difference in the shear force at the ends of the frame. Thus, the ice-induced loading is commonly defined by measuring the shear strain occurring at the ends of the frame.

the properties as in the basic beam theory. Figure 1 presents the orientation of the axes. The x-axis of the structure goes along the frame, y-axis is perpendicular to x-axis and goes along the hull plating, while z-axis is parallel to the cross-section of the frame. The stress-strains relation can be expressed with Hooke’s law. It can be show that the shear stress distribution in a cross-section of a beam can be expressed as (Parnes, 2001, p. 259)

τ xz ( z ) =

γ xz ( z )E EI yy b Qx Sz ( z ) or Qx = I yy b 2(( ν ) Sz ( z )

(3)

where γ and τ = shear strain and stress; ν = Poisson coefficient; E = Young’s modulus; Iyy = the second moment of inertia; Sz = the first moment of area; and b = the thickness of the cross-section. The calculation of Iyy and Sz require information on the effective breadth of the hull plating, which affects the location of the neutral axis of the frame. The effective breadth of the hull plating depends on the applied loading and boundary conditions (Schade, 1951). The measurements have shown ice loading to be a line-like lateral pressure loading (Joensuu and Riska, 1989). The effective breadth, be, for a simply supported stiffened plate structure (idealized as a plate-beam combination) under lateral pressure loading can be taken as (Paik, 2008). be =

π (1 + ν ) [

4ω i h 2 (π s/ω ) − s

+

s

] (4)

2.2

Determination of the shear force with beam theory

The structure (a frame consisting of a hull plating and a frame web) is assumed to behave and have

where s = the frame spacing; and ω = the deflection wave-length which depends on the rigidity of the stiffener and load application. ω can be taken as ω = L for stiff transverse frames. Equation (4) takes into account the continuity of the hull plating. Thus no correction for the Young’s modulus of the hull plating is needed. A general expression for the neutral axis, zNA, can be expressed as n

∑z

0i

zNA =

i =1 n

Ai Ei

∑ Ai Ei

(5)

i =1

Figure 1. Free-body diagram of a frame cross-section under loading and the assumed shear stress distribution, τxz, on the web and the axial stress, σx, on the hull plating.

where z0 = the center of the area in z-direction; A = the area of the part in question; and E = Young’s modulus related to the area. The location of the neutral axis is calculated for other directions with the same manner. Now, the bending stiffness, EIyy, can be calculated for the structure by using the parallel axis theorem.

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n

EII yy = ∑ Ei ( I i + ( zi

zNA )2 Ai )

(6)

i =1

The first moment of area is dependent on the location in the z-direction. Considering the coordinate system presented in Figure 1, where origin is placed at the tip of the web, the first moment of area for the frame having T-shape cross-section, see Figure 1, can be taken as z⎞ ⎛ Sz ( z ) = zNA − ⎟ ( zb ) ⎝ 2⎠

(7)

The shear force affecting at the cross-section of the frame can now be solved with Equation (3) from the shear strain. The shear force affecting the frame, QFrame, can be determined by first calculating the shear forces at the ends of the frame based on shear strain measurements, Equation (3), and then calculating the difference in the shear force (recall that ΔQx = QFrame). Equation (3) is good for the calculation or determination of the shear force as long as the force is not affecting close to the measurement location, i.e. close to the strain gauges.

2.3

Determination of the shear force with the influence coefficient matrix

The ice-induced load on the frame system, FICE, can also be determined by utilizing an influence coefficient matrix, a, hence forward refer as ICM { } = [ ]{ γ }

(8)

which is determined using the inverse ICM, c, as follows [ ij ]

Δγ i (

j

)

(9)

where Δγi (i = 1…n) = the shear strain difference measured on the frame i; n = the number of the instrumented frames; and Fj is the force induced on the frame j. The ICM becomes a square matrix where the number of rows and columns is determined by the number of the instrumented frames. The diagonal terms in the matrix define the forcestrain relation of the frame under loading and the off-diagonal terms determine the response of the adjacent frames and surrounding structure. In order to determine the off-diagonal terms of the matrix, the amount of loading carried by the loaded frame and loading carried by the adjacent frames has to be determined. For the studied structure, consider a frame system that compose of five frames, having a frame spacing s, a web thickness tW, a web height hW and a length L, which are

connected with hull plating, which has a breadth of five frame spaces 5s, width of a frame length L and a thickness tp, see Figure 2. The system is loaded at the midpoint in x and y-direction. In order to solve the displacement equation of the plating, the plating is considered as a simply supported plate undergoing cylindrical bending and the frames are considered as springs that induces a line loading, qi, due to the displacement in z-direction, which are parallel to the negative z-axis direction, see Figure 2 for free-body diagram and forces affecting the plating. Due to the symmetric loading q1 L = q5 L and q2 L = q4 L. The origin of the system is placed to the end of the beam at the left-hand-side. The plate is divided into sections based on the location of the frames, see Figure 2, and the displacement equation in z-direction w(y) is solved for each section. Due to the symmetric loading, only three sections are required for solving the displacement equation for the plate. The displacement equation can be solved for each section from the bending moment using EI

d 2w( y ) = −M ( y) dy 2

(10)

The moment equations for each section are solved from the equilibrium condition, i.e. the resultant forces and moments equal zero in each case. After the moment equations are solved for each section, the displacement equations can be formed by integrating Equations (10) twice. Due to double integration, two unknown parameters are formed for each section, six unknown parameters in total. Noting that derivation of the displacement equation once gives the rotation equation, Θ, the parameters can be solved from the boundary conditions and continuity of the plate. From the continuity of the plate it follows that rotation and displacement at the junction of the sections have to be equal [Θ1(s/2) = Θ2(s/2), Θ2(3s/2) = Θ3(3s/2), w1(s/2) = w2(s/2), w2(3s/2) = w3(3s/2)] and from the boundary condition w1(0) = 0 and Θ3(5s/2) = 0.

Figure 2. The frame system considered in Section 2.3 on the top and the idealized free-body diagram.

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The displacement of the individual frames (considered as simply supported) under line loading is solved following the same principles. As the frames are connected to the plate, the displacement of the plate has to equal the displacement of the frame in question at the frame location. Thus, considering the case presented in Figure 2, the displacement of the plate, wP,i(y), and the frame, w F,i(x), has to be equal as follows w P,1(s/2) = wF,1(L/2), wP,2(3s/2) = wF,2(L/2) and wP,3(5s/2) = wF,3(L/2). With these conditions, the shear forces, Q i = q iL, on each frame is solved. As the plating affects the shear strain distribution of the frame, the bending stiffness of the frame, IF, is solved as described in Section 2.2. The shear force on the loaded frame, Q3, is used for calculating the influence coefficients for the diagonal terms, ci = j, Q2 is for the first off-diagonal terms, ci = j ± 1, and Q1 for the second off-diagonal terms, ci = j ± 2. For the structure used in the study Q3 = 0.711FEXT, Q2 = 0.182FEXT and Q1 = −0.021FEXT. The shear force, Q0, used for calculating the third off-diagonal terms, ci = j ± 3, is evaluated to be 0.002FEXT, based on the observed trend. The response of the frame under known loading can be solved with Equation (3). The coefficients of c are then calculated by following a unit load principal. 1 kN external loading is applied on each frame, a frame at a time, and the response of the frames, Δγ, is determined with Equation (3) each time the structure is loaded, see Equation (9). Thus, the loading on the frames determined with ICM from the sear strain difference is in kN. As an example, the coefficients related to the loading on the frame #40 are calculated as follows ⎡ c11 ⎢c ⎢ 21 ⎢c31 ⎢ ⎣c41

c12 c22 c32 c42

c13 c23 c33 c43

⎧ Δγ 441 ( ) ⎫ c14 ⎤ ⎧0 ⎫ ⎪ ⎪ ⎥ ⎪ ⎪ c24 ⎥ ⎪0 ⎪ ⎪Δγ 440 12 ( )⎪ ⎨ ⎬ kN = ⎨ ⎬ c34 ⎥ ⎪1 ⎪ Δγ 440 ( ) ⎪ ⎪ ⎥ ⎪ Δγ 1 (Q2 ) ⎪ c44 ⎦ ⎪⎩0 ⎪⎭ ⎩ 339 2 ⎭

calibration pull measurements are compared to the results obtained with the beam theory, Section 2.2, and with the ICM, Section 2.3. 3.1

Instrumentation

The hull of S.A. Agulhas II was instrumented with strain sensors at the bow, bow shoulder and aft shoulder (Suominen, et. al., 2013). Two types of optical strain sensors were installed, V-shape and one-directional strain sensors. The V-shape sensors were mounted on the upper and lower parts of the frames to measure shear strains occurring in the frame. The one-directional sensors were mounted on the hull plating to measure axial strains. As the focus in this paper is on straight frames, only the frame instrumentation at the aft shoulder is considered. The frame system in the instrumented area in the actual ship consisted of five transverse frames between the web frames. The frame spacing of the structure was 0.4 meters. Figure 3 presents a model of the frame system, limited by two web frames in horizontal direction and two decks in vertical direction, where the actual instrumentation was mounted in the real ships. The space limited by web frames in horizontal direction and decks in vertical direction will henceforward be referred as Void. At the aft shoulder, four adjacent frames were instrumented with eight gauges, (starting from the one closest to the bow) #41, #40½, #40, and #39½, see Figure 3. The gauges on frames #41, #40, and #39½ were mounted on the side of the frame that was towards bow. The sensors on frame #40½ were mounted on the side of the frame that had normal towards aft.

(11)

where the Δγ = calculated response of the structure using Equation (3) with the shear force indicated in the brackets. The subscript of the Δγ denotes the frame in question. After the matrix c has been defined, the influence coefficient matrix a can be obtained by taking an inverse from the matrix c. 3

VALIDATION

This section presents the instrumentation of the ship and the FE model used in the study. In addition, the

Figure 3. A FE model of an instrumented frame (below) and from the instrumented area (on the upper left-hand corner). The stars denote gauges, dlow and dup are the distance of the gauge from the deck and dweb indicates the distance of the gauge centerline from the tip of the web.

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Table 1. strains.

The measured shear strain and pulling force and the forces determined with the beam theory from shear

Frame

#41

Sensor

Name Distance to the tip of the web [m]

Calibration pull

Measured strain [μstrain] Measured force [kN]

#40½

SS16 0.120 0.22

SS17 0.125 0.27

FEM

0.00

0.03

Beam theory closed form

0.06 0.07 −0.01

Modelled [μstrain] Force at the gauge location [kN] Total force on the frame [kN] Total force [kN] Beam theory Force on the frame [kN] influence matrix Total force [kN]

0.71

Increasing numbering was used in the naming of the gauges. The gauges SS16 and SS17 were mounted on the frame #41, SS18 and SS19 on #40½, SS20 and SS21 on #40 and SS22 and SS23 on #39½. The gauges with even number were installed on the upper part of the frames 0.3 meter from the end of the frame, see dup in Figure 3, and the ones with odd number on the lower part of the frames 0.3 meter from the end of the frame, see dlow in Figure 3. Due to the size of the gauges, it was not possible to mount those on the neutral axis. Therefore, the gauges were mounted as close the axis as possible. The distances of the mounted gauge centerline to the tip of the web, see dweb in Figure 3, are presented in Table 1. The hole close to the mid frame is located 0.4 meter above the lower end of the frame. At this location, the thickness of the hull plating changed from 0.021 m to 0.020 m towards the top end of the frame. The geometry and the material parameters of the instrumented frames were as follows: the length, L, of the frames = 1.4 m; the frame spacing, s, = 0.4 m; the web height, hweb, = 0.2 m; the web thickness, tweb, = 0.019 m; the hull plating thickness at the lower part of the frame, tp21, = 0.021 m and at the upper part of the frame, tp20, = 0.020 m; Young’s modulus, E, = 209 GPa; and Poisson constant, ν, = 0.3. 3.2

Figure 4.

#40

#39½

SS18 0.124

SS19 SS20 SS21 SS22 SS23 0.117 0.125 0.120 0.120 0.130 −6.11 4.56 23.29 −23.67 5.10 −3.94 15.55 −3.96 4.35 23.10 −24.82 3.95 −4.31 −1.54 1.18 5.84 −6.06 1.30 −0.98 2.72 11.90 2.28 16.89 17.11 −0.81 −1.24 15.77

The set-up in the calibration pulls.

Figure 5. The measured pulling force and shear strains on frame #40.

Calibration pull

The calibration measurements were conducted by pulling each frame at a time with a winch. The pulling force was measured with a load sensor attached between the winch and the shackles and the strains occurring in the frame were measured with the instrumentation, see Figure 4. The shackle-shackle connection on the other side of the load sensor and the hook-lifting eye connection on the other side ensured that no torsion occurred during the calibration pull. The sampling rates for the

load sensor and the strain gauge instrumentations were 123 Hz and 200 Hz respectively. The load sensor used for the measurements was calibrated with dead weights before the calibration measurements. The pulling force was increased in steps until the upper limit of the measurement range of the load sensor was reached. Figure 5 presents the measured force and shear strains on the frame #40 during the calibration pull conducted on the frame #40. The measured pulling force in Figure 5

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is expressed in kN while the strains are in μstrains. As can be seen from Figure 5, the response of the frame to the pulling is clearly seen from the strain measurements. 3.3

Finite element model

The finite element model used in the study was built with FEMAP and the numerical calculations were conducted with NX Nastran (version 10.3.1). The model was built by using linear plate elements in FEMAP. In order to reduce the effect of the boundary conditions, the FE model consisted of nine Voids, see Figure 6. The Void in the middle of the model was considered as the instrumented area, from which the calculation results were obtained for the study. As the middle Void was used in the study, the mesh size was finer in this region (the mesh size was 0.005 m*0.005 m) and a coarser mesh was used in the eight surrounding Voids. The boundary conditions were considered to be rigid in all the locations where those were applied: the edges of the hull plating, web frames, decks and the end points of the longitudinals and transverse frames. The locations of boundary conditions are indicated with blue lines in Figure 6. 3.4

Comparison of calibration pull to the beam theory and FEM results

In order to compare the measured pulling force to the force determined with the closed form approach and the ICM, the shear strains occurring at the locations of the sensors were determined first. The measured shear strains used in the comparison are taken from the time period when the pulling force was at maximum (15.55 kN) on the frame #40, see time period 90–110 seconds in Figure 5. The measured shear strains at the location of each strain gauge at this time are presented in Table 1.

Figure 6. The whole FE model used in the study with fixed boundary conditions.

To check the validity of the calibration pull and to validate the FE model to be used in Section 4, the measured strains are first compared to the strains calculated with FE model under the same loading conditions. 15.55 kN loading was induced on the FE model in the same location as during the calibration pull on the frame #40 and the shear strains at sensor locations were determined from the model results, see Table 1. The modelled shear strains are in good agreement with the measured strains on the frame #40 where the calibration pull was conducted. The difference between the measured and calculated is increasing when moved further away from the frame #40. On adjacent frames (#40½ and #39½) the difference is between 4 and 36% and on frame #41 over 90%. However, the occurring shear strains are at the lower limit of the measurements system. Therefore, the measurement accuracy set the limitation and the actual difference on frame #41 is within the measurements accuracy. Thus, it is concluded that the FE model is in good agreement with the actual structure and models the calibration pulling with good accuracy. Now the closed form approach is compared to the calibration pull. The shear forces at each gauge location were determined from the measured shear strains as presented in Section 2.2. At first, the effective breadth, the location of the neutralaxis, the bending stiffness and the first moment of area at the sensor location were determined with Equation (4)–(7). The structural parameters at the location of the gauges were taken as presented in Section 3.1. Then the shear force at each gauge location was calculated with Equation (3) with the measured shear strains. The shear force affecting the frame, ΔQx, see Section 2.1, can now be calculated by reducing the shear force affecting the lower part of the frame from the shear force affecting the upper part of the frame (as an example SS20-SS21). As the sensors on the frame #40½ (SS18 and SS19) were mounted on the other side of the frame than the other sensors, the total force occurring on frame #40½ was multiplied with minus one (−1) in order to correct the direction of the force to the same coordinate system. The total force affecting the frame system is determined by summing the total forces affecting individual frames. The results obtained with the closed form approach are presented in Table 1. The measured pulling force on frame #40 was 15.55 kN at the time when the shear strains ware taken. As can be noted from Table 1, the loading on the frame #40 determined with the closed form approach is 11.93 kN, which is 23% smaller than the applied force. However, the loading occurring on the adjacent frames is 2.72 and 2.29 kN. Thus, the total loading affecting the frame system is 16.89 kN which is 7.9%

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higher than the applied loading. Therefore, the loading determined with the closed form approach gives relatively good correspondence, if the adjacent frames are accounted. However, the comparison shows, that a significant amount of loading is carried by the adjacent frames. Next the results determined with the influence coefficient matrix are compared to the measured calibration pull. The ICM was determined as described in Section 2.3 to be −15.4 ⎤ ⎡ 381.9 −110.2 43.5 ⎢ −109.4 415.7 −120.1 43.1 ⎥ ⎥ kN a=⎢ ⎢ 43.1 −120.9 412.7 −108.6 ⎥ ⎢ ⎥ 43.8 −109.4 379.1 ⎦ ⎣ −15.6

(12)

The small variation in the diagonal and offdiagonal terms is due to the variation in sensor locations. The force occurring during the calibration pull was determined by substituting the measured shear strains to Equation (8) together with the matrix a presented in Equation (12). The results obtained with the coefficient matrix are presented in Table 1. Based on the results, the total loading affecting the frame system determined with the ICM is in good agreement with the calibration pulling as the total calculated loading is 1.4% higher than the measured load. The ICM overestimates the loading on the loaded frame but indicates negative loading on the adjacent, which is due to the negative first off-diagonal terms in the matrix a. The negative loading on the adjacent frames is not in accordance with reality, but with the ICM, the total frame system should be accounted.

4

THE EFFECT OF LOAD LOCATION

As show in Section 3.4, the FE model used in the study models the shear strains in good accuracy. Thus, the model is used to study the effect of the load location to the loading determined with the closed form approach, see Section 2.2, and with the ICM, see Section 2.3. As ice-induced loading is line-like loading, a 5 MPa constant pressure loading was utilized. The width of the pressure is one frame spacing (0.4 meter) and the height one element row (0.005 meters). Thus, the total loading affecting the structure is 10 kN. The study is conducted by moving the pressure load on the FE model in the middle of the frame span in vertical direction. At the time instance 0.5, the pressure loading is applied between the frames #42 and #41½. Then the load is moved 0.2 meters (a half frame space) towards stern in half time steps (0.5). This is continued through the instrumented area so that the last applied loading is between frames #39½ and #39 (time instance 5.5). At the full time steps (integers), the midpoint of the loading is on a frame. At each time step, the loading on the instrumented frames is determined from the shear strains occurring at the strain gauge locations, see Table 1, utilizing the closed form approach and ICM. The results are presented in Figure 7. As can be noted from Figure 7, the results vary significantly depending on the location of the loading with respect to the frame. The ICM determines the loading relatively accurately for individual frames when the loading midpoint is on the frame. However, when the load is affecting between the frames at the same side of the frame where the

Figure 7. Forces determined with the closed form approach and the influence coefficient matrix under varying location of the pressure loading. The loading determined with the closed form approach on each frame indicated with black markers whereas the ICM results indicated with grey. Sum BT and ICM are the total loading on the frame system determined with the closed form approach and ICM, respectively.

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gauge was mounted, the method can overestimate the loading significantly, see the results for the frame #41 at time instance 1.5 and for the frame #39½ at time 4.5 in Figure 7. If the loading is affecting between the frames, but on the opposite side of the frame with respect to gauge, the loading can be significantly underestimated, see time instance 2.5 for the frames #41 and #40½. The loading determined with the closed form approach is smaller for individual frames than the loading indicated by the ICM and the applied loading was. This is due to the fact that the closed form approach does not account the adjacent frames. Thus the loading carried by the neighboring frames is not accounted. However, the pattern is similar to the ICM when the loading is affecting between the frames. Minor loading is indicated when the load is affecting on the opposite side of the frame with respect to the gauges and greater loading is indicated when the loading affects the instrumented side of the frame. The total loading affecting the frame system was determined with both methods by summing the loading from all the frames, see Sum BT and Sum ICM in Figure 7. The total loadings defined with both methods are in good agreement with each other, but vary significantly depending on the location of the loading. When the loading is affecting between the frames and the gauges are mounted on the opposite sides of the frames with respect to loading, the total loading can be severely underestimated, see the results at the time instance 2.5 in Figure 7. On the opposite, the total loading can be severely overestimated when the loading is affecting between the frames, but on the instrumented side of the frames, see the results at time instance 3.5 in Figure 7. 5

CONCLUSIONS

In the beginning of the study, two methods (the closed form approach and influence coefficient matrix) to determine the loading affecting the structure based on the measured shear strains were presented. After the presentation, the methods were compared to the calibration pulls and finally the effect of load location on the results was studied with FEM. The comparison of the two methods and calibration pull showed good agreement. The closed form approach indicated 8.9% higher and the coefficient matrix 1.4% higher total loading in comparison to the measured loading. Furthermore, the shear strains modelled with FEM were in good agreement with the measured shear strains. As can be noted from the results obtained from the calibration pulling and from the study on the

effect of load location, a significant amount of loading can be carried by the adjacent frames of the loaded frames. This can be clearly observed when the loading is determined with the closed form approach and is also indicated by the derivation of the ICM, see Section 2.3. This should be taken into account in the analysis of the full-scale measurements. If the ice-induced loading is measured only on one frame in horizontal direction, the total loading affecting the frame system can be underestimated if the loading is determined with the closed form approach and the loading carried by the adjacent frames are not accounted. If calibration pulling is used to determine the response of the structure under loading, the effect of the adjacent frames are taken into account by the actual structure. It was also shown that the results determined with the presented methods differ significantly depending on which side of the frame the loading is affecting with respect to the instrumentation. If the loading is affecting on the side of the frame that is instrumented, the loading can be overestimated and if it is affecting on the other side with respect to the instrumentation, the loading is severely underestimated. This is considered to be due to the rotation of the frames. The best correspondence between the applied loading and the determined loading with the methods is obtained at time instance 3 when the loading is on the frame #40½. This is likely due to the reason that the gauges on the frame #41 are on the same side of the frame as the loading whereas the gauges are on the opposite side with respect to loading on the frame #40. Thus, the rotation of the adjacent frames evens each other out. A detailed study on the effect of rotation to the determined loading should be conducted as a future work. The assumptions made regarding the shape and type of loading and boundary conditions affect significantly to the effecting breadth of the hull plating, which affects the stiffness of the frame, and the structure respond. The effect of the effective breadth and validity of the made assumptions should be studied in the future work. In addition, the approach used in this study assumes the shear stress distribution to be parabolic in the web of the frame, which is not necessarily valid in all the cases. Thus, study on the shear stress distribution is to be conducted as a future work. ACKNOWLEDGEMENTS The writers would like to thank the Finnish Funding Agency for Technology and Innovation (Tekes) and Academy of Finland for funding. Furthermore, all the partners of the Tekes project NB1369

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PSRV full scale ice trial, the University of Oulu, University of Stellenbosch, Aker Arctic, RollsRoyce, STX Finland, Wärtsilä, DNV, the Department of Environmental Affairs of South Africa, and the Finnish Meteorological Institute as a partner of the Academy of Finland ANTLOAD project, are gratefully acknowledged. The financial support of the Lloyd’s Register Foundation while this study was being carried out is greatly acknowledged.

REFERENCES Joensuu, A., Riska, K., 1989. Ice-Structure contact. Helsinki University of Technology, Laboratory of Naval Architecture and Marine Engineering, Report M-88, Otaniemi. 57 p.+ app.154 p. (In Finnish). Kotisalo, K., Kujala, P., 1999. Ice load measurements onboard MT Uikku, Measurements results from the ARCDEV-voyage to Ob-estuary, April-May 1998, Report from WP8 of ARCDEV project supported by the EC Transport programme, Espoo, Finland. Kujala, P., Vuorio, J., 1986. Results and statistical analysis of ice load measurements on board ice breaker Sisu in winters 1979 to 1985, Winter Navigation Research Board, Research report No 43, Helsinki, Finland, ISBN 951-46-9701-4. Kujala, P., 1989. Results of long-term ice load measurements on board chemical tanker Kemira in the Baltic Sea during the winters 1985 to 1988, Winter Navigation Research Board, Research report No 47, Helsinki, Finland, ISBN 951-47-3109-3. Paik, J., 2008. Some recent advances in the concepts of plate-effectiveness evaluation, Thin-Walled Structures 46 (2008), 1035–1046.

Parnes, R., 2001. Solid mechanics in engineering, John Wiley & Sons LTD, Chichester, West Sussex, England, p. 728. Ralph, F., Ritch, R., Daley, C., Broenw, R., 2003. Use of finite element methods to determine iceberg impact pressure based on internal strain gauge measurements, Proceedings of the 17th Int. Conf. on Port and Ocean Engineering under Arctic Conditions, Trondheim, Norway, June 16–19, 2003. Riska, K., Kujala, P., Vuoro, J., 1983. Ice load and pressure measurements on board I.B. Sisu, Proceedings of the 7th Int. Conf. on Port and Ocean Engineering in Artic Conditions, Espoo, Finland, Vol. 2, pp. 1055–1069. Ritch, R., Liljeström, G., Edgecombe, M., 1997. Measurement of the side shell ice loads at full scale: Oden 1996 Arctic Ocean Expedition Final Report, Transport Canada Report TPI12919E, February 1997. Ritch, R, St. John, J, Browne, R, Sheinberg, R, 1999. “Ice Load Impact Measurements on the CCGS Louis S. St. Laurent during the 1994 Arctic Ocean Crossing”, Proceedings of the 18th Int. Conf. on Offshore Mechanics and Arctic Engineering, July 11–16, 1999, St. John’s Newfoundland, paper OMAE99/P&A-1141. Schade, H., 1951. The effective breadth of stiffened plating under bending loads. Transactions, SNAME, Vol. 59. Ship Structure Committee, 1990. Ice Loads and Ship Response to Ice, Summer 1982/Winter 1983 Test Program. Ship Structure Committee report SSC-329. Suominen, M., Karhunen, J., Bekker, A., Kujala, P., Elo, M., von Bock und Polach, R., Endlund, H., Saarinen, S., 2013. Full-scale measurements on board PSRV S.A. Agulhas II in the Baltic Sea., in Proceedings of the 22nd Int. Conf. on Port and Ocean Engineering under Arctic Conditions (POAC), Espoo, Finland, June 9–13, 2013.

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Analysis and Design of Marine Structures – Guedes Soares & Shenoi (Eds) © 2015 Taylor & Francis Group, London, ISBN 978-1-138-02789-3

A study of cross deck effects on warping stresses in large container ships Richard Villavicencio, Shengming Zhang & Jimmy Tong Global Technology Centre, Lloyd’s Register EMEA, Southampton, UK

ABSTRACT: The cross deck structures of container ships suffer diagonal deformations when subjected to torsional moments in waves. This type of deformation provokes concentrated tensile and compressive warping stresses fore and aft of the deck strips, and thus the distribution of these stresses along the ship length is represented as an oscillatory response which may increase the stress range required to evaluate the fatigue strength of the ship structural details. Therefore, the paper presents a simple method to estimate the magnitude of the ‘fluctuation’ of the warping stresses for longitudinals’ end connections in the proximities of the torsion box and within the cargo hold area where the ship cross section remains constant. The method is based on finite element results and uses the rule hydrodynamic torque and hydrodynamic finite element data for its validation. The finite element model represents a constant cross section closed at the fore and aft ends with scantlings and principal dimensions similar to those of an 8,500 TEU Class Container Ship. The approach determines warping coefficients which can be used to evaluate the warping stress fluctuation resulting from wave-induced torsion considering different amplitudes of torsional loadings, phase angles, and wave frequencies so as to account for the variety of loads provided by hydrodynamic analyses. 1

INTRODUCTION

The structural design of container ships requires analysis of the torsional strength since this type of vessels are essentially beams of open cross section, and thus they undergo more rotation and longitudinal deformation of the cross section when subjected to wave-induced torsion moments. As a consequence of this warping deformation, the structural components suffer large longitudinal normal stresses called warping stresses (Hughes 2010). To evaluate the distribution of the warping stresses along the ship length one-dimensional approaches can be used as that summarised by Pedersen (1985). These approaches are based on the assumption that the beam cross sections are rigid in their own plane and require modelling the hull as non-prismatic segments (Lloyd’s Register 1994). Additionally, finite element analyses are required for verification of the global strength, the structural components and details, and the strength of the primary structure, following classification society procedures, for example, Lloyd’s Register (2006). This type of analyses are mandatory for container ships with a beam of a Panamax size or greater and are carried out in view of detailed fatigue strength, such as to evaluate the warping stress concentration around the hatch corners due to the large diagonal deformation of the cross decks.

However, before conducting such finite element analyses, the standard shipyard design and classification process requires ‘simplified spectral direct calculations’ (Lloyd’s Register 2014a) for the fatigue assessment of the longitudinals’ end connections across the outer and inner shell, longitudinal bulkheads and decks, in order to design and approve the key structural drawings. One of the features of this simplified fatigue approach is that determines the loads and stresses for each end connection structural detail arising from the variety of wave loads provided by numerical hydrodynamic analyses, including torsional loadings which are those of interest in the current study. In this type of fatigue analysis, the warping stresses induced by oblique wave encounter can be estimated through the one-dimensional approaches, which provide accuracy similar to that given by finite element results, for those structural elements unaffected by the torsional deformation of the hatch openings, e.g. the outer and inner bottom longitudinals and the bilge stiffeners. However, for the longitudinals’ end connections affected by the large diagonal deformation of the cross decks and the local deformation of the upper and second decks, the beam theory might require additional stress factors since the warping stress distribution along the ship length is manifested as an oscillatory response due to the concentrated tensile and compressive warping stresses fore and aft of the deck strips, as finite element analyses have demonstrated. In this

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paper the amplitude of this oscillatory response is referred to as ‘fluctuation’. Although a number studies have been published for evaluating the torsional strength and fatigue assessment of container ships (e.g., Fricke et al. 2002, Iijima et al. 2004, Li et al. 2013, Lloyd’s Register 2006, 2014c), they are mainly concentrated on the definition of the critical loadings without evaluating the effect of this oscillatory response on the fatigue of the structural details. Therefore, the paper presents a simple procedure to evaluate the amplitude of the fluctuation of the warping stresses along the ship length for longitudinals’ end connections in the proximities of the torsion box and within the cargo hold area where the cross section remains constant. The procedure requires global finite element analyses of a particular container ship design subjected to hydrodynamic torques to determine warping coefficients. Later, these coefficients can be used to estimate the fluctuation induced by the variety of torsional loading obtained from hydrodynamic analyses. The simplified method is based on finite element results of a coarse Box Ship model, closed at the fore and aft ends. Afterwards, it is further validated with the numerical analysis of a Single Island 8,500 TEU Class Container Ship. Discussions on the limitations of the approach as well as its usefulness to evaluate container ships with different structural arrangements are also provided. 2

Figure 1. ship.

General arrangement: 8,500 TEU container

Figure 2.

Midship section: 8,500 TEU container ship.

Figure 3.

Top plan view: Box ship model.

BOX SHIP MODEL

A Single Island 8,500 TEU Class Container Ship is selected to design the Box Ship finite element model. The main particulars of the vessel are summarised in Table 1 and sketches of the general arrangement and the midship section are provided in Figs. 1 and 2. The Box Ship represents the principal dimensions and scantlings of the midship section of the container ship. The Box Ship is modelled as an open cross section constant along the length while the fore and aft ends are closed by upper deck plates (see Fig. 3). The cross decks are modelled as double skin watertight bulkheads equally spaced at 14.44 m. In addition, longitudinal line elements of negligible area and inertia properties are modelled Table 1. ship.

Main particulars: 8,500 TEU class container

Length between perpendiculars Beam Depth Displacement

319 m 42.8 m 24.6 m 135,000 t

at various locations in the cross section in order to obtain the longitudinal distribution of the warping stresses. In general, the design of this global finite element model follows the Lloyd’s Register Structural Design Assessment procedures (Lloyd’s Register 2006). A Box Ship without bulkheads is also modelled so as to evaluate the cross deck effects on the distribution of the warping stresses. In this model, the bulkheads located in the open section are removed while the fore and aft ends remain closed by the upper deck plates and bulkheads. The loading applied in the finite element model represents hydrodynamic torsional moments which are obtained from the rule hydrodynamic torque or from hydrodynamic analyses. The concept used to apply a torsional moment is illustrated in Fig. 4. The difference in the torsional moments between

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that the shape of the beam cross sections does not change during the deformation, see for example Pedersen (1985). The program LR.PASS 20105 (Lloyd’s Register 1994) uses a similar approach to examine torsional loads on open type ships by modelling the entire ship hull as non-prismatic segments. This program predicts the warping stresses distribution with an accuracy that is comparable with finite element analyses, as demonstrated in Fig. 7. While finite element results show that the distribution of the warping stresses along the ship length for structural elements located relatively far from the cross decks is as smooth as the distribuFigure 4. Method of application of torsional moments.

Figure 5. Keel boundary conditions. T represents translational displacements and R represents rotational displacements.

consecutive cross sections is taken, and then a set of equivalent vertical shear forces is applied over the shell at each frame and bulkhead along the depth. In order to maintain the rigid body equilibrium, the aft and forward ends of the keel on the centreline are restrained as illustrated in Fig. 5 (Lloyd’s Register 2014b). These keel boundary conditions provide a symmetric and opposite warping stress response along the ship length (with respect to the midship section) when the Box Ship model is subjected to a constant torsional loading. 3

Figure 6. Distribution of warping stresses around the midship section of a container ship.

FLUCTUATION OF THE WARPING STRESS FORE AND AFT OF THE CROSS DECKS

The torsion induced warping stresses around the cross section of container ships reach the maximum values at the bilge and the outer and inner edges of the upper deck while these stresses are negligible at about the mid-depth, see Fig. 6 for illustration. The elastic warping stress response of hull cross sections to torsion can be approached by onedimensional procedures based on the assumption

Figure 7. Box ship model: comparison of longitudinal distribution of the warping stress at the bilge between LR.PASS program and finite element analysis.

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tion shown in Fig. 7, for elements located in the proximities of the torsion box the numerical predictions indicate that the diagonal deformation of the transverse deck strips, due to warping, provokes an oscillatory response (referred to as ‘fluctuation’) of the warping stress fore and aft of these internal warping restraining structures, see Fig. 8 for illustration. Here, the fluctuation phenomenon is graphically observed when the response of the Box Ship with and without bulkheads is compared (see Fig. 8). The fluctuation of the warping stresses is mainly attributed to the concentrated tensile and compressive warping stresses around the deck strips resulting from the torsional deformations. Therefore, the longitudinals’ end connections located fore and aft of the cross decks are subjected to a stress range larger than those at the web frames. In this paper, the ‘amplitude of the fluctuation of the warping stresses’—or simply fluctuation—is defined as the increased warping stress fore and aft of the cross decks (see Fig. 9). This stresses are determined at the modelled line elements as illustrated in Fig. 9.

Figure 8. Warping stress distribution along the ship length of a longitudinal element located at the outer edge of the upper deck.

Figure 10. Representative hydrodynamic torques applied in the Box Ship model and the resulting longitudinal distribution of the warping stress. Above: hydrodynamic torque (kNm) vs ship length (x/L). Below: resulting warping stress at the upper deck (MPa) vs ship length (x/L). a) Phase angle = 0.0 degrees. b) Phase angle = 72 degrees.

The fluctuation of the warping stresses mainly depends on the magnitude of the hydrodynamic torque and its distribution along the ship length. Fig. 10 shows representative hydrodynamic torque applied in the Box Ship model and the resulting longitudinal distribution of the warping stresses. Generally, the maximum fluctuation occurs at the location with maximum hydrodynamic torque. For example, for the moment distribution shown in Fig. 10(a) the maximum fluctuation occurs at about 0.5 L, while it occurs near 0.3 L for the loading illustrated in Fig. 10(b). This fluctuation also depends on the span of the ship subjected to the torsion moment. It should be mentioned at this point that the fluctuation of the warping stresses is not predicted by the one-dimensional approaches. Therefore, a simplified method to evaluate this phenomenon is required so as to account for this effect when conducting, for example, simplified spectral fatigue analyses. 4

Figure 9. Fluctuation of the warping stress fore and aft of transverse cross decks (Upper deck plan view).

METHOD TO ESTIMATE THE WARPING STRESS FLUCTUATION

In this section the procedure to estimate the amplitude of the fluctuation of the warping stresses (AmpWAR) is summarised. It is proposed that the fluctuation simply depends on the amplitude of the torsional loadings and ‘warping coefficients’ (CWAR). These coefficients, estimated from finite element calculations, are used to determine the fluctuation for a variety of torsional loads. In the procedure, the fluctuation is first evaluated for the outer edge of the upper deck when the rule hydrodynamic torque, using different phase angles, is applied.

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4.1

Fluctuation at the outer edge of the upper deck under the rule hydrodynamic torque

To estimate the fluctuation at the upper deck the following steps are suggested. 1. Based on the rule formulation (Lloyd’s Register 2014c) the design hydrodynamic torques at any position along the ship are estimated (MWTC1 and MWTC2), see Fig. 11(a). Here, the two design torques are summed using Eq. (1), where ωt is denoted here as the phase angle, to determine the ‘total hydrodynamic torque’ (MHYD) along the ship length. Fig. 11(b) shows this summation using a phase of 45 degrees as an example. M HY YD

MWTC1

( t ) MWT i ( t) W C 2 sin( WTC

the fluctuation should be measured where MHYD takes the maximum magnitude. 3. The warping coefficient CWAR (Eq. 2) is defined as the ratio between the mean fluctuation AmpWAR,mean (MPa) and the maximum absolute value of the total hydrodynamic torque MHYD,max (kNm) for the phase angle of 45 degrees. MHYD,max represents the maximum amplitude of the torsional loading along the ship length. For example, in Fig. 11 (b), MHYD,max is about 1.5E6 kNm at 0.4 L. The phase of 45 degrees is simply selected in order to consider the same contribution for each design hydrodynamic torque.

(1)

2. The torsional loading (MHYD) using the full range of phase angles (zero to 360 degrees) is applied to the finite element model of a ‘particular container ship design’ to obtain the fluctuation at the upper deck for each phase angle. This information is used to estimate the ‘mean’ amplitude of the fluctuation. For the purpose, the warping stress distribution for each phase angle can be plotted, as shown in Fig. 10, and

CWAR

AmpWAR,mean M HYD HY YD ,max

4. This warping coefficient can be used to estimate the fluctuation at the upper deck for torsional loadings at frequencies similar to that used in the rule hydrodynamic torque. For this frequency, different amplitudes of the loading and phase angles can be evaluated, see Eq. (3) where AmpWAR(i) represents the corresponding fluctuation when the maximum absolute value of a particular hydrodynamic torque MHYD,max(i) is used. Amp m WAR ( i )

4.2

Figure 11. Rule hydrodynamic torque. (a) Design hydrodynamic torques MWTC1 and MWTC2. (b) Total hydrodynamic torque MHYD.

(2)

CWAR M HYD,max( i )

(3)

Variation of the warping stress fluctuation for different wave frequencies and along the ship depth

The magnitude of the hydrodynamic torques decreases at relatively low and high wave frequencies, and consequently the fluctuation of the warping stresses becomes smaller. In fact, the fluctuation could be neglected for frequencies smaller than ∼0.4 rad/s and those larger than 1.0 rad/s since the torsional loading applied beyond these limits are very small. This aspect is corroborated using finite element tools in which torsional loadings at different wave frequencies are obtained from hydrodynamic analysis. Thus, following a procedure similar to that described in Section 4.1 warping coefficients for torsional loadings at different wave frequencies can be estimated. The fluctuation of the warping stresses reaches its maximum at the upper outer and inner edges of the upper deck. Since the cross decks are connected to the torsion box, their effect on the fluctuation is still evident at the second deck (lower level of the torsion box). However, below the level of the second deck the fluctuation decreases gradually to

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Figure 13. 8,500 TEU Container Ship subjected to a hydrodynamic torque.

Table 2. 8,500 TEU container ship: Phase angle and maximum hydrodynamic torque along the length. Figure 12. Distribution of the warping stress fluctuation along the ship depth.

almost zero at approximately the middepth. These statements are based on the finite elements results. For example, the numerical results of the Box Ship model show that the distribution of the fluctuation of the warping stresses along the ship depth can be idealised as sketched in Fig. 12. It can be seen that the fluctuation at the inner edge of the upper deck is similar to the one at the outer edge while at the level of the second deck it is larger for elements located in the inner hull. The fluctuation at the second deck decreases about 60%, in average, respect to that at the upper deck. Considering the outer and inner hull, it is approximately twice at the inner shell. At the mid-depth the fluctuation of the warping stress could be simply neglected. It should be mentioned that the distribution along the depth is only given as reference. 5

COMPARISON WITH FINITE ELEMENT RESULTS

The amplitude of the fluctuation for the Single Island 8,500 TEU Class Container Ship is estimated following the procedure given in Section 4 and the results are compared with finite elements results. Fig. 13 shows an example of the distribution of the warping stresses along the length of the container ship when subjected to a hydrodynamic torsional loading. First, the rule hydrodynamic torque for different phase angles (Eq. 1) is used to define the loadings in the finite element model. The selected phase angles and the respective maximum magnitude of the loading are given in Table 2. The finite element results indicate that the amplitude of the fluctua-

Phase angle (degrees)

MHYD,max(i) (kNm)

0, 180, 360 36, 216 72, 252 108, 288 144, 324

9.85E5 1.30E6 1.68E6 1.77E6 1.53E5

Figure 14. 8,500 TEU Class Container Ship. Fluctuation of the warping stresses when the rule hydrodynamic torque for different phase angles is applied. Dashed lines: Finite elements results. Continuous lines: Simplified approach using CWAR.

tion varies from about 25 to 45 N/mm2 when the rule hydrodynamic torque is applied for the full range of phase angles. Thus, the mean fluctuation of the warping stresses is considered as 35 N/mm2. Since MHYD,max takes a magnitude of 1.42E6 kNm for the phase angle of 45 degrees, CWAR is estimated at 2.47E-5 m−3. This coefficient times the hydrodynamic torques given in Table 2 (Eq. 3) manages to predict the numerical fluctuation for the different phase angles, as illustrated in Fig. 14. Similarly, the warping coefficients for different wave frequencies can be determined following the procedure described in Section 4.1. These coefficients also manage to predict the amplitude of the

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Figure 16. 13,100 TEU Class Container Ship. Fluctuation of the warping stresses when the rule hydrodynamic torque for different phase angles is applied. Dashed lines: Finite elements results. Continuous lines: Simplified approach using CWAR.

Figure 15. 8,500 TEU Class Container Ship. Fluctuation of the warping stresses (MPa) vs phase angle (degrees) for different wave frequencies.

fluctuation when random torsional loadings at different frequencies are selected from hydrodynamic data (see Fig. 15). Considering the fluctuation along the ship depth, it is found that the distribution shown in Fig. 12 reproduce similar numerical results when, for example, the fluctuation at the level of the second deck is evaluated. 6

DISCUSSION

The procedure described in Section 4 can be used to estimate the amplitude of the fluctuation at the upper deck for container ships with characteristics similar to the one presented here, i.e. single island container ships with capacity until 9,000 TEU. The determination of warping coefficients for container ships with different dimensions and/or structural arrangements may require further analyses. For example, finite element analyses demonstrated that the amplitude of the fluctuation decreases for twin island container ships in spite of the larger size of ships designed with this concept. Fig. 16 compares the simplified and numerical results of the fluctuation for a 13,100 TEU twin island container ship where the warping coefficients are determined as summarised in Section 4. As illustrated in Fig. 16, the numerical results predicts that the amplitude of the fluctuation varies from 10 to 25 N/mm2, thus the mean fluctuation is considered as ∼20 N/mm2.

Most information required to estimate the fluctuation by the simplified method (warping coefficients) is on hand at the early stage of the design. Although hydrodynamic data may not be available, the fluctuation can still be estimated for torsional loading with similar frequencies to that given by the rule, including different amplitudes of the loading and phase angles. The exception is the global finite element model which is required to estimate the mean fluctuation. However, this information could be obtained from previous designs with similar characteristics. It is important to mention that the proposed method considers that the amplitude of the fluctuation is constant along the entire ship length. In practice, however, the fluctuation at the fore and aft ends differs considerably from that observed at, for example, the midship section. In addition, the finite element results demonstrate that the fluctuation could be neglected for structural elements located in the forward holds (∼ 0.8 L to Fore Perpendicular). The characteristic fairing of the container ships’ forward sections and the increased rigidity of these sections, due to the inclusion of stepped longitudinal bulkheads connected with extended intermediate decks, help to reduce significantly the warping stress fluctuation. On the other hand, the sections located aft of the engine room (∼ Aft Perpendicular to 0.2 L) still suffer large fluctuations since these sections remain open. A typical longitudinal tendency of the warping stress fluctuation for a single island container ship is shown in Fig. 17 to illustrate these aspects. Therefore, and considering the concept design of most container ships, it is considered that the approach described herein is fair enough to evaluate the fluctuation of the warping stresses for longitudinals’ end connections in the proximities of the torsion box and within the cargo hold area where the ship cross section remains constant, i.e. where the fluctuation is relatively stable. The amplitude of the fluctuation, estimated in the proximities of the torsion box, can be used, for example, to ‘increase’ the warping stresses provided

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be noted that the increment in the warping stresses, given by the fluctuation, could be largely overestimated for structural elements where the amplitude of the fluctuation decreases. 7

Figure 17. 8,500 TEU Class Container Ship. Longitudinal distribution of the amplitude of the fluctuation.

CONCLUSIONS

The aim of the present study was to investigate the cross deck effects on the warping stresses distribution along the length of large container ships. In the course of this work, a procedure to estimate the amplitude of the fluctuation of the warping stress, based on finite elements results of a Box Ship model, was found to be appropriated when compared with finite element results of the single island container ship. Thus, the warping coefficients determined for this particular ship can be used to evaluate the fluctuation of similar container vessels considering a variety of torsional loadings as those evaluated by hydrodynamic analyses. This information is useful to determine the stress range for conducting spectral fatigue analysis during the initial phase of the design process. In addition, the methodology described here can be used to predict the amplitude of other of container ships structural arrangements, such as the one of a twin island design concept. ACKNOWLEDGEMENTS

Figure 18. Fluctuation of the warping stress fore and aft of transverse deck strips for positive (+) and negative (−) hydrodynamic torques.

by one-dimensional procedures, and thus evaluate the stress range for conducting simplified fatigue analyses. In this respect, it should be considered that the structural elements located forward to the cross deck structures are subjected to compressive warping stresses while those located aft to the cross decks are under tensile stresses when the hydrodynamic torque takes positive values (Fig. 18a). On the other hand, the fluctuation increases, due to tensile stresses, for elements located forward to the cross deck structures and decreases, due to compressive stresses, for elements located aft to the cross deck structures when the hydrodynamic torque is negative (Fig. 18b). Therefore, it should

The authors wish to thank Dr. Sai Wong and colleagues at Lloyd’s Register for their comments and support. The views expressed in this paper are those of the authors and are not necessarily those of Lloyd’s Register. Lloyd’s Register and variants of it are trading names of Lloyd’s Register Group Limited, its subsidiaries and affiliates. Lloyd’s Register EMEA (Reg. no. 29592R) is an Industrial and Provident Society registered in England and Wales. Registered office: 71 Fenchurch Street, London, EC3M 4BS, UK. A member of the Lloyd’s Register group. Lloyd’s Register Group Limited, its affiliates and subsidiaries and their respective officers, employees or agents are, individually and collectively, referred to in this clause as the ‘Lloyd’s Register’. Lloyd’s Register assumes no responsibility and shall not be liable to any person for any loss, damage or expense caused by reliance on the information or advice in this document or howsoever provided, unless that person has signed a contract with the relevant Lloyd’s Register entity for the provision of this information or advice and in that case any responsibility or liability is exclusively on the terms and conditions set out in that contract.

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REFERENCES Fricke W, Cui W, Kierkegaard H, Kihl D, Koval M, Mikkola T, Parmentier G, Toyosada M, Yooni JH. 2002. Comparative fatigue strength assessment of a structural detail in a containership using various approaches of classification societies. Marine Structures. 15: 1–13. Hughes O. 2010. Hull girder response analysis—prismatic beam. In: Ship Structural Analysis and Design, Chapter 3. Published by The Society of Naval Architects and Marine Engineers, New Jersey. Iijima K, Shigemi T, Miyale R, Kumano A. 2004. A practical method for torsional stren