Finite Element Method With Applications in Engineering [PDF]

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Finite Element Method with Applications in Engineering

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Finite Element Method with Applications in Engineering

PREFACE The revolution in computer technology has led to the development of numerous computational techniques. The closed form solutions are either extremely difficult to obtain or not available for many engineering problems. Hence one has to resort to numerical methods. The Finite Difference Method (FDM), Finite Element Method (FEM), Boundary Element Method (BEM) and Mesh Free Method are few examples of important numerical techniques, which are available in the field of computational mechanics. Each of these methods has its own merits and demerits. The choice of an appropriate method for the solution of any particular problem depends on investigator's familiarity of the method and nature of the problem. The aim of this book is to introduce FEM as a general numerical technique for the solution of various engineering problems. The main emphasis in this book is given to important applications in engineering and science. Since majority of the applications of FEM are in the realm of mechanics including solids, fluids and structural systems, descriptions are primarily in terms of these fields of study. This book is written in a simple structured way starting with the basics of various FEM approaches, a description of finite element interpolation functions, FEM formulations and applications in one-, two- and three-dimensional domains and further advanced applications for a few complex problems. Further, some of the commonly used FEM software and web resources available for the solution of various engineering and sciencerelated problems are described briefly. An illustrative and simple computer program, sfeap, with a sophisticated Graphical User Interface for pre-processing and post-processing has been provided for structural and solid mechanics related problems. A few simple MATLAB and FORTRAN-based programs for fluid mechanics related problems in one- and two-dimensional domains are also given. In addition, source codes of the programs are provided so that a reader can easily understand the computer implementation of the method and modify the codes to integrate their own elements and routines. One may take a physical or intuitive approach to learning FEM. On the other hand, one may develop a rigorous mathematical interpretation of the method. A middle path has been adopted in this book with more importance to the physical approach. Efforts have been made to keep the mathematics simple. We feel that the book will be useful for a basic understanding of FEM by beginners, further understanding by FEM practitioners, and to a certain extent for the advanced level understanding by researchers. We encourage readers to convey feedback and suggestions for the improvement of the contents of the book, and point out any necessary corrections in it. Y. M. Desai T. I. Eldho A. H. Shah

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Finite Element Method with Applications in Engineering

Contenido i PREFACE ........................................................................................................................... ii 1 Introduction ..................................................................................................................... 9 1.1 INTRODUCTORY REMARKS .................................................................................. 9 1.2 MATHEMATICAL MODELLING OF ENGINEERING PROBLEMS ............................ 9 1.3 TYPE OF GOVERNING EQUATIONS .................................................................... 12 1.4 SOLUTION METHODOLOGIES ............................................................................. 14 1.4.1 Analytical Method ............................................................................................. 15 1.4.2 Physical Method ............................................................................................... 15 1.4.3 Computational Method ..................................................................................... 15 1.5 NUMERICAL MODELLING..................................................................................... 15 1.6 PRE-PROCESSING AND POST-PROCESSING ................................................... 16 1.7 SCOPE OF THE BOOK.......................................................................................... 18 1.8 HIGHLIGHTS OF THE BOOK ................................................................................ 19 1.9 HOW TO USE THE BOOK? ................................................................................... 19 1.10 CLOSING REMARKS ........................................................................................... 20 REFERENCES AND FURTHER READING.................................................................. 20 2 Approximate Methods of Analysis ................................................................................. 22 2.1 INTRODUCTION .................................................................................................... 22 2.2 APROXIMATING METHODS ................................................................................. 22 2.3 METHOD OF WEIGHTED RESIDUALS ................................................................. 23 2.4 RAYLEIGH–RITZ METHOD ................................................................................... 27 2.5 FURTHER NUMERICAL EXAMPLES .................................................................... 29 2.6 CLOSING REMARKS ............................................................................................. 35 EXERCISE PROBLEMS .............................................................................................. 35 REFERENCES AND FURTHER READING.................................................................. 36 3 Finite Element Method—An Introduction ....................................................................... 37 3.1 GENERAL .............................................................................................................. 37 3.2 WHAT IS FEM? ...................................................................................................... 37 3.3 HOW DOES FEM WORK? ..................................................................................... 38 3.4 A BRIEF HISTORY OF FEM .................................................................................. 40 3.5 FEM APPLICATIONS ............................................................................................. 41 3.6 MERITS AND DEMERITS OF FEM ........................................................................ 42 Merits of FEM ........................................................................................................... 42 Demerits of FEM ....................................................................................................... 43 iii

Finite Element Method with Applications in Engineering 3.7 CLOSING REMARKS ............................................................................................. 43 EXERCISE PROBLEMS .............................................................................................. 43 REFERENCES AND FURTHER READING.................................................................. 43 4 Different Approaches in FEM ........................................................................................ 45 4.1 INTRODUCTION .................................................................................................... 45 4.2 GENERAL STEPS OF FEM ................................................................................... 45 4.3 DIFFERENT APPROACHES USED IN FEM .......................................................... 50 4.3.1 Direct Approach ............................................................................................... 50 4.3.2 Variational Approach ........................................................................................ 52 4.3.3 Energy Approach ............................................................................................. 56 4.3.4 Weighted Residual Approach ........................................................................... 57 4.4 CLOSING REMARKS ............................................................................................. 60 EXERCISE PROBLEMS .............................................................................................. 61 REFERENCES AND FURTHER READING.................................................................. 61 5 Finite Elements and Interpolation Functions .................................................................. 63 5.1 INTRODUCTION .................................................................................................... 63 5.2 INTERPOLATION FUNCTIONS ............................................................................. 64 5.2.1 One-Independent Spatial Variable ................................................................... 64 5.2.2 Two-Independent Spatial Variables .................................................................. 65 5.2.3 Three-Independent Spatial Variables ............................................................... 65 5.3 ONE-DIMENSIONAL ELEMENTS .......................................................................... 65 5.3.1 Line Element: Linear Interpolation Function ..................................................... 65 5.3.2 Quadratic Interpolation Function ...................................................................... 69 5.3.3 Cubic Interpolation Function ............................................................................. 71 5.3.4 Lagrangian Form of Interpolation Function ....................................................... 72 5.3.5 Further Higher Order Elements in One-Dimension ........................................... 74 5.4 TWO-DIMENSIONAL ELEMENTS ......................................................................... 79 5.4.1 Triangular Element: Linear Interpolation Function in Cartesian Co-ordinates ... 79 5.4.2 Triangular Element—Area Co-ordinates........................................................... 83 5.4.3 Integration Formula for Triangular Elements .................................................... 83 5.4.4 Triangular Element—Quadratic Function ......................................................... 84 5.4.5 Triangular Element—Cubic Interpolation Function ........................................... 86 5.4.6 Two-Dimensional Rectangular Elements .......................................................... 87 5.4.7 Rectangular elements—Lagrangian Form in natural and Cartesian co-ordinates ................................................................................................................................. 89 5.4.8 Isoparametric elements .................................................................................... 91 5.4.9 Lagrangian Interpolation Functions for two-Dimensional elements ................... 93 iv

Finite Element Method with Applications in Engineering 5.4.10 Two-Dimensional serendipity elements .......................................................... 96 5.5 THREE-DIMENSIONAL ELEMENTS...................................................................... 99 5.5.1 Tetrahedral Elements ..................................................................................... 100 5.5.2 Tetrahedral elements: Quadratic Interpolation Function ................................. 104 5.5.3 Tetrahedral elements: cubic Interpolation Function ........................................ 104 5.5.4 Three-Dimensional elements—prismatic elements......................................... 104 5.5.5 Three-Dimensional elements in local co-ordinates ......................................... 107 5.5.6 Three-Dimensional serendipity elements ....................................................... 108 5.6 CLOSING REMARKS ........................................................................................... 109 EXERCISE PROBLEMS ............................................................................................ 109 REFERENCES AND FURTHER READING................................................................ 110 6 One Dimensional Finite Element Analysis ................................................................... 112 6.1 INTRODUCTION .................................................................................................. 112 6.2 LINEAR SPRING .................................................................................................. 112 6.2.1 Expressions for Equivalent Spring Constant and Nodal Forces ...................... 123 6.3 TRUSS ELEMENT ............................................................................................... 125 6.3.1 Plane Truss .................................................................................................... 125 6.3.2 Element Equations by Minimizing Potential Energy ........................................ 128 6.3.3 Local and Global Element Equations for a Bar in the X–Y Plane .................... 134 6.4 SPACE TRUSS .................................................................................................... 140 6.5 ONE-DIMENSIONAL TORSION OF A CIRCULAR SHAFT .................................. 147 6.6 ONE-DIMENSIONAL STEADY STATE HEAT CONDUCTION ............................. 150 6.7 ONE-DIMENSIONAL FLOW THROUGH POROUS MEDIA.................................. 153 6.8 ONE-DIMENSIONAL IDEAL FLUID FLOW THROUGH PIPES (INVISCID FLUID FLOW) ....................................................................................................................... 157 6.9 BEAM ELEMENT ................................................................................................. 157 6.9.1 Review of Beam Theory ................................................................................. 158 6.9.2 Finite Element Formulation of a Beam Element ............................................. 160 6.9.3 Illustrative Examples ...................................................................................... 165 REFERENCES AND FURTHER READING................................................................ 211 7 Two Dimensional Finite Element Analysis ................................................................... 213 7.1 INTRODUCTION .................................................................................................. 213 7.3 TWO-DIMENSIONAL STRESS ANALYSIS .......................................................... 222 7.3.1 Review of Theory of Elasticity ........................................................................ 222 7.3.2 Application of Three-Dimensional Equations for Two-Dimensional analysis ... 223 7.3.3 CST Element for Plane Stress and Plane Strain Analyses ............................. 225 7.3.4 Triangular Element for Axisymmetric Analysis ................................................ 232 v

Finite Element Method with Applications in Engineering 7.3.5 Some remarks on Triangular Elements .......................................................... 234 7.3.6 Four-Node Rectangular Element for Plane Problems ..................................... 235 7.4 ISOPARAMETRIC FORMULATION ..................................................................... 238 7.4.1 Two-Node Isoparametric Line Element (Bar Element) .................................... 238 7.4.2 Four-Node Isoparametric Element for Plane Problems (Quadrilateral Element) ............................................................................................................................... 241 7.5 FINITE ELEMENT SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY METHOD OF WEIGHTED RESIDUAL ....................................................................... 249 7.6 FEM FORMULATION BASED ON VARIATIONAL PRINCIPLE ............................ 252 7.7 FINITE ELEMENT SOLUTION OF STOKES FLOW EQUATIONS ....................... 254 7.7.1 Problem Statement ........................................................................................ 254 7.7.2 FEM Solution ................................................................................................. 255 7.8 ILLUSTRATIVE EXAMPLES ................................................................................ 260 8 Three Dimensional Finite Element Analysis ................................................................ 274 EXERCISE PROBLEMS ............................................................................................ 298 9 Computer Implementation of FEM............................................................................... 302 9.3 STATIC CONDENSATION ................................................................................... 304 9.4 COMPUTER IMPLEMENTATION OF FEM–SFEAP ............................................. 308 9.5 STORAGE SCHEMES FOR GLOBAL STRUCTURAL STIFFNESS MATRIX ...... 309 EXERCISE PROBLEMS ............................................................................................ 315 REFERENCES AND FURTHER READING................................................................ 318 10 Further Applications of Finite Element Method .......................................................... 319 10.3 DYNAMICS WITH FINITE ELEMENT METHOD ................................................ 339 10.3.1 Introduction .................................................................................................. 339 10.3.2 Governing Equations .................................................................................... 339 10.4.3 Illustrative Examples .................................................................................... 349 10.4 NON-LINEAR ANALYSIS ................................................................................... 355 10.4.1 Finite Element Formulation for Non-Linear Analysis ..................................... 356 10.4.2 Solution of Non-Linear Equations ................................................................. 357 10.4.3 Illustrative Examples .................................................................................... 358 10.6 HYDRODYNAMICS SIMULATION OF SHALLOW WATER FLOW .................... 364 10.6.1 Introduction .................................................................................................. 364 10.6.2 Governing Equations and Boundary Conditions ............................................... 365 10.6.3 Finite Element Formulation .......................................................................... 366 10.7 FEM–SOFTWARE AND WEB RESOURCES ..................................................... 372 10.7.1 Introduction .................................................................................................. 372 10.7.2 FEM in Structural Engineering...................................................................... 372 vi

Finite Element Method with Applications in Engineering 10.7.3 FEM in Geotechnical Engineering ................................................................ 374 10.7.4 FEM in Fluid Mechanics ............................................................................... 375 10.7.5 FEM in Thermal and Automobile Engineering .............................................. 375 10.7.6 FEM in Physics ............................................................................................ 376 10.7.7 Multi-Field FEM Software ............................................................................. 376 10.7.8 FEM in Other Fields ..................................................................................... 377 10.8 CONCLUDING REMARKS ................................................................................. 377 REFERENCES AND FURTHER READING................................................................ 378 Appendix A .................................................................................................................... 381 A.1 REVIEW OF MATRIX ALGEBRA ......................................................................... 381 A.2 MATRIX CALCULUS............................................................................................ 386 A.3 SOLUTION OF LINEAR SIMULTANIOUS EQUATIONS ...................................... 387 Appendix B .................................................................................................................... 394 B.1 GENERAL ............................................................................................................ 394 B.2 THE EULER–LAGRANGE EQUATION ................................................................ 394 EXAMPLE B.1: ....................................................................................................... 397 B.3 THE VARIATIONAL OPERATOR......................................................................... 398 EXAMPLE B.2 ........................................................................................................ 399 REFERENCES AND FURTHER READING................................................................ 400 Appendix C .................................................................................................................... 401 C.1 GENERAL............................................................................................................ 401 C.2 EXAMPLE ............................................................................................................ 401 Appendix D .................................................................................................................... 406 D.1 GENERAL............................................................................................................ 406 D.2 FUNDAMENTAL CONCEPT ................................................................................ 406 D.3 GAUSS QUADRATURE FOR TWO DIMENSION ................................................ 411 Appendix E .................................................................................................................... 414 E.1 PREAMBLE ......................................................................................................... 414 E.2 PROGRAM LAYOUT ........................................................................................... 415 E.2.1 Global Variables and Arrays .......................................................................... 416 E.2.2 Notes on Element Subroutines ELMTn .......................................................... 419 E.2.3 Input Instructions ........................................................................................... 421 E.3.3 EXAMPLES ................................................................................................... 431 E.3.4 Listing of SFEAP............................................................................................ 442 Appendix F..................................................................................................................... 503 F.1 PROGRAMS ........................................................................................................ 503 F.1.1 Preamble ....................................................................................................... 503 vii

Finite Element Method with Applications in Engineering F.2 EXECUTING PROGRAMS ................................................................................... 504 1. CONTROL INFORMATION................................................................................. 504 2. CO-ORDINATES ................................................................................................ 505 3. ELEMENT DATA ................................................................................................ 507 4. BOUNDARY CONDITIONS AND NODAL DISPLACEMENT DATA .................... 510 5. MATERIAL CHARACTERISTIC DATA ............................................................... 511 6. NODAL LOAD DATA AND DISTRIBUTED LOAD ............................................... 511 Appendix G .................................................................................................................... 515 G.1 COMPUTER PROGRAM FOR THE FLOW NETWORK ANALYSIS .................... 515 G.2 COMPUTER PROGRAM FOR THE ELECTRICAL NETWORK ANALYSIS ........ 518 G.3 COMPUTER PROGRAM FOR THE LAPLACE EQUATION – HOMOGENEOUS AND NON-HOMOGENEOUS MEDIA PROBLEMS .................................................... 520 G.3.1 Example 1: .................................................................................................... 524 G.3.2 Example 2: .................................................................................................... 526 ACKNOWLEDGEMENTS .............................................................................................. 532

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Finite Element Method with Applications in Engineering

1 Introduction 1.1 INTRODUCTORY REMARKS Most practical engineering problems are complex in nature. A model is often needed to solve a system consisting of any domain of interest and various phenomena that take place in it. A model may be defined from the engineering point of view as the selected simplified version of an engineering system that approximately simulates the system's excitation response relations, which are of interest to the user. Information obtained from the model can be used to understand the system, to provide information for design purposes, to provide information to comply with regulations and to, finally, take appropriate management decisions to solve particular problem concerned. Modelling can be done through physical means with laboratory and field experiments or through mathematical means. Physical modelling in laboratory or field can be cumbersome, expensive and time consuming. Due to these difficulties in physical modelling in laboratories or in the real world, use of mathematics in solving the engineering problems has become widespread in the recent times. Use of systems approach, increasing power of digital computers and the development of numerous computational methodologies have made the mathematical modelling inevitable for all branches of learning.

1.2 MATHEMATICAL MODELLING OF ENGINEERING PROBLEMS Mathematical modelling of an engineering problem is a process by which the problem is represented as it appears in the real world and interpreted in an abstract form. Depending on the modelling approach, different mathematical models are possible for tackling the same problem. Also, identical mathematical model may be used to tackle quite different physical situations. Generally, basic elements of modelling can be represented by five components: state variables, external variables, mathematical equations, coefficients or parameters and universal constants. State variables describe the state of the system that is to be modelled. External variables, or the forcing functions, on the other hand, are variables external in nature that can influence the system. Mathematical equations are relations between the external variables and the state variables. Mathematical representation of process in any system contains coefficients or parameters and universal constants. Calibration is often required in modelling to find the best parameters in accordance with the computed and the observed state variables by variation of number of parameters. Modelling procedure and methodology depends on the nature of the problem, aims and objectives of the modelling. Different workable modelling procedures are possible. A typical modelling procedure is shown in Figure 1.1. Following steps are involved in a typical mathematical modelling procedure. Step 1 - Problem Definition: As the first step, it is essential to define the problem clearly with aims and objectives. Sources of facts and data should be known. Step 2 - Problem Simplification: As the real world problems are very complex to any model, various assumptions may be used to make the problem solvable with sufficient accuracy. Step 3 - Problem Solution in Space, Time and Subsystems: Once the simplified form of the problem is available, decision can be made for a three-dimensional, two9

Finite Element Method with Applications in Engineering dimensional or one-dimensional modelling to obtain the required solution. Need for timedependent (dynamic) or time-independent (static) modelling should also be ascertained. Step 4 - Data Requirement: Data collection and handling may be simple or complex, depending on the problem to be solved. Availability and quality of the available data should be checked before proceeding further with modelling. Accuracy of the model prediction depends on the quality of the available data. Step 5 - Conceptual Model: Finally, a conceptual model for the problem should be evolved from the above steps. The conceptual model would tend to reduce the real problem and the real domain to simplified versions, which are satisfactory in view of the objectives of the modelling, available resources and management of the problem. Step 6 - Mathematical Model: Conceptual model can be expressed in the form of a mathematical model. Mathematical model, generally, consists of the definition of domain, the governing equations (algebraic, ordinary differential equations, partial differential equations or integral equations), initial conditions (known state of the system at some initial time) and boundary conditions. For most engineering problems, mathematical representations are possible in terms of equations, depending on varying conditions. Step 7 - Solution Methodology: Solution methodology should be decided once the mathematical model is ready. Depending on the problem and its complexity, solution methodology can be analytical, numerical or physical in nature. Analytical solution is generally possible only for simple problems with regular domain and simple boundary conditions. Physical solution can be obtained through laboratory modelling or field simulations. Generally, physical solutions are time consuming and expensive. If an analytical or easier physical solution is not plausible, numerical process is required. Step 8 - Model Development: A reasonable model to solve the given problem may generally be available as a public domain or commercial model. However, if the model is not available, it should be developed, depending on the objectives of the modelling. If the solution methodology is physical, laboratory or field level investigation would be required. On the other hand, analytical or numerical solution can be developed with the help of a computer. Step 9 - Model Validation: The model cannot be used effectively for the considered problem unless the numerical values of all the coefficients appearing in the model are known. Once a model has been developed or proposed for a particular problem domain, it must be validated for the given conditions. Through model validation, it is ensured that the model includes and describes all the relevant processes, which affect the excitation and model responses of interest to the acceptable degree of accuracy. Further, the solution obtained may be cross checked for few parameters/conditions for which analytical or other standard results are available. Step 10 - Model Sensitivity Analysis: The model sensitivity analysis involves investigations on model behavior for the important model variables. Generally, sensitivity analysis is performed by varying each variable to its upper and lower extremes, while keeping other variables as constant. Sensitivity analysis indicates over all model behavior for extreme conditions of the problem. Step 11 - Model Predictions: Once the model is calibrated and validated for the considered problem, the model is ready for use. Model runs are made to provide required results for the given problem conditions and data.

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Finite Element Method with Applications in Engineering

Figure 1.1 A typical mathematical modelling procedure 11

Finite Element Method with Applications in Engineering Step 12 - Results and Interpretations: Results obtained from the model should be presented in a tabular or graphical form. This includes various post-processing techniques. Results should be interpreted appropriately by considering sensible behavior of the model variables. Step 13 - Summary and Conclusions: At the end, information that the model was expected to provide, including accuracy, uncertainty, suggested follow-up works or other activities should be concluded. Step 14 - Model Reporting: Model reporting should be done in such a way that the important features of the model are clear and the results and interpretations are presented appropriately. Suggested contents of a model report include: abstract, introduction, problem definition, assumptions, simplifications, data, details of the conceptual model, details of the mathematical model, solution methodology, code (program) selection or development, details of model sensitivity analysis, verification, calibration and validation, details of the model runs and predictions, results, discussions, conclusions and references.

1.3 TYPE OF GOVERNING EQUATIONS It is possible to derive mathematical equations governing the problem for most physical problems in nature. Generally, the governing equations can be classified into algebraic equations, ordinary differential equations (ODE), partial differential equations (PDE) and integral equations. Many problems can be represented in the algebraic equation form, which can be generally expressed as where g is a function, x and y are state variables and θ is a set of parameters. The equation is said to be linear if x and y appear linear in g, otherwise the equation is nonlinear. Many engineering problems can be represented by systems of linear or non-linear algebraic equations and generally a set of values are sought as solution that satisfy the system. Large systems of algebraic equations are generated, for example, in the analysis of structures, electric circuits and fluid networks. Most fundamental laws of mechanics, physics, thermodynamics and electricity are based on empirical observations, which explain variations in physical properties and states of the system in terms of spatial and temporal changes. As many physical laws are expressed in terms of the rate of change of a quantity rather than the magnitude itself, the ODE are of great importance in engineering practice. By using ODE, generally two types of problems, i.e., the initial-value and the boundary value problems are addressed. Differential equations are classified according to their order. An equation is said to be of order n if the highest derivative of the dependent variable is of order n. For example, the deflection y of a mass spring system with damping can be expressed with a second order differential equation as

where m is the mass, c is the damping coefficient, k is the spring coefficient and t is the time. Dependent physical quantities vary with two or more independent variables and hence PDE formulations are used in many practical situations. If the dependent variable and its 12

Finite Element Method with Applications in Engineering partial derivatives appear linearly with coefficients, depending only on the independent variable, the equation is said to be linear, otherwise it is termed as a non-linear equation. The order of a PDE is that of the highest-order partial derivative that appears in the equation. For example, consider a second order PDE with two independent variables as given below.

Here a, b and c are functions of x and y and α is a function of x, y, h, and . Such a linear second order PDE can be further classified as elliptic, parabolic and hyperbolic, depending on the values of the coefficients of the second derivative terms a, b and c. An equation is called elliptic if the quantity (b2 – 4ac) < 0. For example, the Laplace equation used in many engineering problems, given below is elliptic.

Elliptic equations are generally used to represent steady-state problems. Steady-state heat conduction problem or groundwater flow in homogeneous isotropic porous media without any pumping and recharge are typical examples where elliptic equations are used. An equation is said to be parabolic if the quantity (b2 – 4ac) = 0. For example, the diffusion equation in one dimension, used in many engineering problems, given below is parabolic.

Parabolic equations typically determine how a dependent quantity varies in space and time. Time-dependent heat conduction problem is an example where parabolic type equations can be used. In the third category, an equation is said to be hyperbolic if the quantity (b2 – 4ac) > 0. For example, the wave equation in one dimension, given below is hyperbolic.

Hyperbolic equations are used to deal with propagation problems and characterized by a second order time derivative as in the case of wave propagation. Sometimes, physical laws are expressed in terms of the summation or integrals of one or more variables over the entire domain or a range, and hence, integral equations are of great importance in engineering practice. Generally, the integral equations can be expressed as follows.

Analytical solutions can be derived only for a small class of formulations involving ODE, PDE or integral equations. Generally, approximate methods such as numerical methods are used to obtain solution. 13

Finite Element Method with Applications in Engineering 1.3.1 Initial and Boundary Conditions Solutions of ODE, PDE or integral equations are attempted in conjunction with initial and boundary conditions. Generally, initial conditions describe values of dependent variable throughout the domain at the given point in time. For example, temperature distribution at the beginning of the calculations should be stated to get appropriate solution in the further time steps, in the transient analysis of heat-conduction problem. Boundary conditions, on the other hand, are described on the boundary of the domain throughout the simulation time, based on which the solution is obtained for the concerned problem. Generally, three types of boundary conditions are used in engineering problems, which are called as essential, natural and the mixed boundary conditions. The essential boundary condition, which is also known as fixed, forced, geometric or Dirichlet boundary condition, describes variation of the state variable (unknown function) concerned on the boundary of the problem domain. For example, h in Eq. (1.3) is the state variable, which can be temperature or potential or any other quantity, depending on the problem. The natural boundary condition, also known as the Neumann boundary condition, describes variation of derivative of the state variable concerned on the boundary of the problem domain. For example, specification of the heat flux rather than the temperature on the boundary is the natural boundary condition for the heated rod problem. Both the state variable and its derivative together are described as the boundary condition in the case of mixed boundary conditions. For example, consider the partial differential Eq. (1.3) in a two-dimensional domain Ω and satisfy the homogeneous conditions on the boundary Γ, the above boundary conditions can be described as the following: Essential or Dirichlet boundary condition Natural or Neumann boundary condition

Mixed boundary condition

where f1, f2 and f3 are known values on the boundary, s is the arc-length along Γ, measured from some fixed point on Γ and outward normal n to Γ.

represents the differentiation along the

In the mathematical modelling, accurate and appropriate representation of the initial and boundary conditions are essential to obtain accurate results.

1.4 SOLUTION METHODOLOGIES It is necessary to decide analysis and solution methodology for the problem concerned, once the mathematical formulation is ready. Analysis of mathematical formulation can be either qualitative or quantitative. Qualitative aspects of the mathematical formulation are 14

Finite Element Method with Applications in Engineering explored without explicitly solving it in qualitative analysis. Quantitative analysis, on the other hand, can be divided into analytical, physical and computational methods. These methodologies are described briefly in the following sections. 1.4.1 Analytical Method An analytical expression involving parameters and the independent variables are obtained in an analytical method using various mathematical procedures. Solution can be exact or approximate depending on how the solution satisfies the formulation. Generally, approximate solutions are sought when it is either impossible or extremely difficult to get an exact solution. Unfortunately, analytical solutions can be obtained only for a small class of mathematical formulations with simplified governing equations, boundary conditions and geometry. 1.4.2 Physical Method A mathematical model represents a real physical system, albeit, on the basis of idealized assumptions, variables and parameters. Thus, such a model can be considered as having physical dimensions and can be analyzed sometimes in the laboratory or in the field itself. Scaling is often required for such models and accuracy depends on scale and simulation of the actual field conditions. Presently, the physical models are used less frequently since they are expensive, cumbersome and difficult to employ. 1.4.3 Computational Method Solution is obtained with help of some approximate methods using a computer in computational method. Solution, thus, obtained would be approximate. Generally, numerical methods are used to obtain solution in computational method. Computational methods of analyses are applicable to a much wider class of mathematical formulations and hence most engineering problems can be solved. With the advent of fast computers, computational models have become very widely used valuable tool for solving engineering problems.

1.5 NUMERICAL MODELLING Variety of numerical methods such as method of characteristics, Finite Difference Method (FDM), Finite Volume Method (FVM), Finite Element Method (FEM), Boundary Element Method (BEM) and so on have been developed by engineers and scientists for solution of engineering problems. Each of these methods has its own advantages and disadvantages, and the choice of method largely depends upon the complexity of the problem and investigator's familiarity with the method. Out of many available numerical methods, FDM, FEM and BEM are the most popular techniques among engineers and scientists. Continuous variation of the function concerned is represented by a set of values at points on a grid of intersecting lines in the FDM. Gradients of the function are then represented by differences in the values at neighboring points and a finite difference version of the equation is formed. This equation is used at points in the interior of the grid to form a set of simultaneous equations, which yield value of the function at a point in terms of values at nearby points. At the edges of the grid, value of the function is fixed, or a special form of finite difference equation is used to provide the required gradient of the function. In FEM, the region of interest is divided in a much more flexible way than what is possible with the FDM. Nodes at which value of the function is found do not have to lie on a grid system but can be on a flexible mesh. Boundary conditions are handled in a more convenient manner so that a standard program can deal very easily with most conditions. The result is a program, which can be used for problems involving many shapes of domain 15

Finite Element Method with Applications in Engineering and types of boundaries. Direct approach, variational principle or weighted residual methods are used to approximate the governing equations. Partial differential equation describing the domain is transformed into an integral equation relating only to boundary values in the BEM. The method is based on Green's integral theorem. Here, the boundary is discretized instead of the domain. Hence, the computational dimension of the problem is reduced by one in BEM. For examples, a threedimensional problem reduces to a two-dimensional problem and two-dimensional problem reduces to a one-dimensional problem. Thus, BEM is ideally suited for solution of many two- and three-dimensional problems in elasticity and potential theory for which other methods are inefficient. However, the method requires explicit expression of Green's functions as kernels appearing in the integral equation. From the above discussion, it is observed that FDM is simple and easy to implement, but difficult to implement for irregular geometry problems. FEM is more versatile and can be applied to most problems compared to the FDM and BEM. BEM has some specific advantages over FEM, such as reduction in computational dimension and ease in handling of data. FDM is confined to the grid system and is not versatile compared to FEM or BEM. Hence, the best choice in many cases is the combination of these three methods in one or other form like FEM-BEM combination or BEM-FDM combination or FEM-FDM combination to suit any particular problem to the benefit, accuracy, efficiency and economy. Discretization of an earthen dam problem using FDM, FEM and BEM is shown in Figure 1.2. Differences in discretization between the FDM, FEM and BEM can be clearly seen from the figure. As discussed earlier, FDM, FEM and BEM have got their own merits and demerits. Choice of the method is largely governed by the complexity of the problem, geometry, investigator's familiarity of the method and other factors such as availability of particular packages, computational facilities and so on. This book deals with the theory and practice of the well-established FEM.

1.6 PRE-PROCESSING AND POST-PROCESSING When numerical methods like FDM, FEM or BEM are used in mathematical modelling, the ensuing computer model generally consists of pre-processing, processing and postprocessing. In the pre-processing part of the model, input required to solve the problem are generated or read in. Input data required include domain geometry, initial and boundary conditions, coefficients and constants for the particular problem, value of universal constants, the grid information (finite difference grid, finite element mesh or boundary element mesh) and various options for the concerned problem such as one-, two- or three-dimensional analysis; steady state or transient analysis. In the processing part, real simulation of the problem with the concerned numerical model is done. For example, processing in the context of FEM includes generation of element matrices, assembly of element equations, and imposition of boundary conditions and solution of system of equations. In the post-processing part, results obtained are processed in terms of tables, charts, graphs, contours, bar chart and so on. Mesh generation is an important task of pre-processing. Mesh or grid generation includes discretization of the concerned domain into elements or cells and determining grid points or nodal positions. Subsequently, nodes and elements are numbered and nodal positions and connectivity between elements are specified such that the processor can identify the problem domain, discretization and nodal position. Mesh or grid generation depends on the type of numerical model used and dimension of the model. Mesh generation is a complex process when numerical methods such as FEM or FVM are used. On the other hand, such process is easier when BEM is used. Mesh or grid can be generated manually 16

Finite Element Method with Applications in Engineering or through computer models. For automatic mesh generation, computer programs can be written and incorporated in the model itself or separate mesh generating packages can be utilized.

Figure 1.2 Flow through an earth dam—discretization using FDM, FEM and BEM Results from a problem analysis consist of numbers pertaining to various unknown variables such as potential, displacements, stresses and so on. It is thus better to present 17

Finite Element Method with Applications in Engineering such details either graphically or in tabular forms. Graphical display may include deformed shape of mesh, vectors, charts, contours, and so on. The post-processing can be done manually or through computer models. For post-processing, computer program can be written such that results are displayed graphically or separate post-processing packages can be used.

1.7 SCOPE OF THE BOOK Purpose of the present book is to introduce FEM as a general numerical technique for the solution of various engineering problems. The main emphasis is given to important applications in engineering and science. An effort has been made to keep the mathematics simple. Since majority of applications of FEM are in the realm of mechanics including solid, fluids, structural and soil, descriptions in this book are primarily in terms of these fields of study. To effectively use the book, necessary pre-requisites are relatively moderate such as understanding of basic mechanics and physics, basic courses in advanced calculus and linear algebra and basic concepts of various engineering theories. For effective study and application of FEM, the reader must have working capability in several fundamental areas such as matrices, algebra, calculus and basic computer skills. The reader is assumed to have these backgrounds. However, to make the text as self-contained as possible, a brief description of these topics are given in appendices. An overview of various approximate methods of analyses has been presented in Chapter 2wherein the basics of FEM have been evolved. An introduction to FEM with history, applications, merits and demerits are detailed in Chapter 3. Important approaches in FEM have been described in Chapter 4. Interpolation functions used in FEM are explained in Chapter 5. Detailed formulation and applications of FEM in one-dimensional for various engineering problems are presented in Chapter 6. Formulation, implementation and applications of FEM in two-dimensional are elaborated in Chapter 7. Three-dimensional formulations of some engineering problems are presented in Chapter 8. Details of the computer implementation of FEM with details of intuitive, teaching purpose program are given in Chapter 9. Further advanced level of FEM formulation and applications are described in Chapter 10. Various FEM software and web resources available are also discussed in this chapter. Brief review of matrix algebra and calculus, elements of calculus of variation, illustration on use of Galerkin's method, review of Gauss quadrature procedure for numerical integration, user's manual for the simplified finite element analysis program (SFEAP), further illustrative one- and two-dimensional computer programs are detailed in various appendices. One may take a physical or intuitive approach to learning of FEM. On the other hand, one may develop a rigorous mathematical interpretation of the method. A middle path has been adopted in this book giving more importance to the physical approach, with essential mathematical basis. The book can be used for basic understanding of FEM by beginners, further understanding by FEM practitioners and to certain extent for the advanced level understanding of researchers. For the benefit of beginners, explanation of FEM with basic theories and fundamental applications in structural, geo- and fluid-mechanics are presented in one-, two- and three-dimensional domains. Once the basic concepts of FEM are understood with illustrative examples, one can study advanced topics and apply the method to complex problems and to other areas.

18

Finite Element Method with Applications in Engineering

1.8 HIGHLIGHTS OF THE BOOK This book is written to introduce the FEM as a numerical method that employs the philosophy of virtual work/equilibrium approach, variational principle and method of weighted residual for solution of various engineering problems. The book is written in a simple structured way starting with the basics of various FEM approaches, description of finite element interpolation functions, FEM formulations and applications in one-, two- and three-dimensional domains and further advanced applications for more complex problems. The book is intended as an introduction to FEM as well as oriented towards various applications in engineering and science. It is envisaged that several practical applications and examples presented in the book will enable an engineer to assess the potential of FEM and its applicability to specific engineering problems. Various exercise problems are also proposed in the book. Further, the important FEM software and Web resources available for solution of various engineering and science related problems are briefly discussed. Emphasis of the book is on development of FEM formulation and implementation of the method to various problems. A very illustrative and simple computer program SFEAP with sophisticated graphical user interface for pre-processing and post-processing are given for structural and solid mechanics related problems. Few simple MATLAB and FORTRAN based programs for the fluid-mechanics related problems in one- and two-dimensional domains are also given. These programs can be operated on desktop PCs. Source codes of these programs are also given so that the reader can easily understand the computer implementation of the method. These codes not only demonstrate the implementation of the methods described in the book, but also help readers who are contemplating in preparing their own computer codes.

1.9 HOW TO USE THE BOOK? A large number of books are available on FEM. Thus, it may be rather difficult for a user (student, teacher or practitioner) to effectively follow-up a plan-of-study. Of course, the plan-of-study may depend on the time available and the objectives of the study such as 1. To study the FEM systematically for the solution of complex problems (students and teachers). 2. To have an in-depth understanding of FEM so as to further pursue research on FEM topics (researchers). 3. To study FEM for minimum understanding to use software for practical design and analysis problems (practitioners). This book has been written primarily to serve the first objective. However, it addresses the other two objectives also. To meet the first objective for students and teachers, the FEM has been presented systematically from the basics to one-, two- and three-dimensional applications. Users may study all chapters systematically from Chapters 2 through 10. For researchers or advanced learners, in addition to the systematic approach, some advanced level topics have been presented in Chapter 10. For FEM practitioners, in addition to understanding of basics of FEM, the book is presented with software with pre- and post-processing graphical user interface, which can be used in practical application of problems and get training. Depending on the interest, to meet the third objective, users may concentrate on Chapters 3, 6, 7, 8and 9. This book has not been written to provide a broad survey of FEM as it may require much more comprehensive volumes. The book has concentrated on more commonly used and 19

Finite Element Method with Applications in Engineering effective FEM techniques that can be applied to engineering problems. As shown in this book, FEM can be systematically programmed to accommodate complex and difficult problems such as linear/non-linear stress–strain behavior, fluid-flow problems, nonhomogeneous materials problems and complicated boundary conditions. It is important, however, that good judgement be developed and exercised concerning the solution of any complex engineering problem to achieve the desired results using FEM. To achieve this, the analyst must be sufficiently familiar with possible available finite element procedure and models. In this direction, the SFEAP software provided with this book will provide sufficient background of learning and in reinforcing the understanding of FEM.

1.10 CLOSING REMARKS Various aspects of mathematical modelling of engineering problems have been discussed in this chapter. Detailed procedures of mathematical modelling are illustrated, which are directly applicable to most engineering problems. Modelling methodology may vary. However, procedure more or less remains unchanged. Further, various types of mathematical equations and boundary conditions used in modelling are discussed. Accurate results can be obtained only if the mathematical model developed is appropriate. Various analyses and solution methodologies for modelling are further illustrated in the subsequent chapters. Since a few analytical solutions are possible for simplified problems and the physical modelling is quite cumbersome and expensive, most engineering problems are solved using numerical methods such as FDM, FVM, FEM and BEM. Various aspects of pre-processing and post-processing are also discussed briefly in the context of various numerical methods. The scope of the book, highlights of the book and details of ‘how to use the book’ are also explained. FEM is elaborated in the subsequent chapters.

REFERENCES AND FURTHER READING Aris, R. (1978). Mathematical Modeling Techniques, Pitman, London. Beer, G., and Watson, J. O. (1992). Introduction to Finite and Boundary Element Methods for Engineers, John Wiley and Sons, Chichestor. Berry, J. S., et al. (eds.) (1987). Mathematical Modelling Courses, Ellis Howard, New York. Boyce, W. E. (ed.) (1981). Case Studies in Mathematical Modeling, Pitman, Boston. Burghis, A. D. Wood (1980). Mathematical Models in the Social, Management and Life Sciences, Elliz Horwood, Chichster. Chapra S. C., and Canale, R. P. (2002). Numerical Methods for Engineers, 4th ed., McGraw-Hill, New York. Clements, R. R. (1989). Mathematical Modeling–A Case Study Approach, Cambridge University Press, Cambridge. Cook, R. D. (1981). Concepts and Applications of Finite Element Analysis, John Wiley and Sons, New York. Dym, C. L, and Ivey, E. S. (1980). Principles of Mathematical Modeling, Academic Press, Boston. Haberman R. (1977). Mathematical Models, Prentice Hall, New Jersey. Kapur, J. N. (1988). Mathematical Modeling, Wiley Eastern, New Delhi. Krishnamoorthy, C. S. (1994). Finite Element Analysis: Theory and Programming, Tata McGraw Hill, New Delhi. 20

Finite Element Method with Applications in Engineering Levine G., and Burkee, C. J. (1992). Mathematical Model Techniques for Learning Theories, Academic Press, New York. Mooney D. D., and Swift R. J. (1999). Course in Mathematical Modeling, Mathematical Association of America, USA. Murray S. K. (eds.) (1987). Mathematical Modeling: Class Notes in Applied Mathematics, Siam, Philadelphia. Reddy, J. N. (1985). An Introduction to the Finite Element Method, McGraw-Hill International Edition, New York. Sanorskii, A. A., and Mikchailov A. P. (2002). Principles of Mathematical Modeling, Taylor and Francis, London. Singh, B. et al. (eds.) (2001). International Conference on ‘Mathematical Modelling’, Tata McGraw Hill, New Delhi. Smith, T. M. (1977). Mathematical Modeling and Digital Simulation for Engineers and Scientists, Wiley Inter-sciences, New York. Zienkiewicz, O. C., and Taylor, R. L. (1989). The Finite Element Methods, 4th ed., Vol. 1, Basic Formulation and Linear Problems, Vol. 2: Solid and Fluid Mechanics; Dynamics and Non-linearity, McGraw Hill, UK.

21

Finite Element Method with Applications in Engineering

2 Approximate Methods of Analysis 2.1 INTRODUCTION Engineering problems can be classified into two fundamental categories on the basis of adopted viewpoints. In one view point called Lagrangian concept, all matter consists of particles that retain their identity and nature as they move through space. The position of particles in space at any instant is obtained by the coordinate systems, which are functions of time. In the other view point, known as Eulerian concept, the problem is considered to be in continuum rather than consisting of particle. All the processes are characterized by field quantities that are defined at every point in space and time and one point is focused to observe the phenomena occurring there. Most engineering problems can be considered as continuum problems and can be approximately represented by differential or partial differential equations. In mathematical physics, the continuum problems are often referred as boundary value problems. Generally, the solution is sought in some domain defined by the known boundary on which the boundary conditions are specified. For the problem concerned, boundary can be closed or open. Boundary is known as closed if the boundary conditions are specified everywhere on the boundary. Boundary is called open if part of the boundary extends to infinity and no boundary conditions can be specified over it. Concept of the mathematical modeling was illustrated in Chapter 1 through governing equations and boundary conditions applicable to the problem domain. Governing equations are required to be solved with the appropriate boundary conditions over the prescribed domain. Depending on the problem, it may be possible to get an exact or approximate solution. An exact solution is one that exactly satisfies the governing differential equation and all the boundary conditions for the problem concerned. On the other hand, an approximate solution may satisfy the differential equation exactly but may not satisfy all the boundary conditions or vice versa. Various approximating methods used in the solution of engineering problems are discussed here to set the stage for the introduction of finite element method (FEM). After discussing some popular approximating methods for different classes of problems, necessary definitions, and terminologies will be established to show FEM as one of the important approximating method.

2.2 APROXIMATING METHODS It is possible to obtain exact analytical solutions for few simplified engineering problems. The exact solution can be obtained by direct integration of the concerned differential equations. Generally, the direct integration is made possible by: (i) separation of variables; (ii) similarity solutions; or (iii) Fourier and Laplace transformations. Since most engineering problems are complex in nature, number of problems having exact solutions is severely limited. Hence, approximate solutions are generally sought. Generally used approximating methods are as follows. 1. 2. 3. 4. 5. 6. 7.

Perturbation Techniques Power Series Solutions Probability Methods Finite Difference Method (FDM) Method of Weighted Residuals (MWR) Rayleigh–Ritz Method Finite Element Method (FEM) 22

Finite Element Method with Applications in Engineering Usefulness of the perturbation method is limited because they are applicable only when the nonlinear terms in the equation are small in relation to the linear terms. The power series method, on the other hand, has been employed with some success. However, as the method requires generation of a coefficient for each term in the series, it is relatively tedious to apply. Also, it is difficult to prove that the series converges. The probability methods are used if the desired variables in the problem concerned are statistical parameters. The probability method is used to obtain statistical estimate of variables by the process of random sampling. The first three methods briefly mentioned above have limited use in engineering problems. Due to wide applicability and simplicity, the most commonly used approximating methods include FDM, MWR, Ritz Method and FEM. Out of various approximating methods, FDM is the most popular method due to its simplicity and wide applications. As discussed earlier, FDM has some merits and demerits compared to FEM. Incorporation of variation of physical properties of the medium in the solution domain is a difficult task in application of FDM to engineering problems. A large number of text books are available describing various applications of FDM such as Forsythe and Wasow (1960), Mitchell and Griffiths (1980), Davis (1986) and Thomas (1995). The MWR, Ritz method and FEM are related methodologies. Hence, discussion here is limited only to methods of weighted residual and Raleigh–Ritz, which are important in obtaining element equations in finite element analysis.

2.3 METHOD OF WEIGHTED RESIDUALS The MWR is a very useful tool to approximate solutions to differential equations. Here the differential equation is considered directly rather than its variational forms. An approximate solution has some error with respect to the exact solution. Depending on the problem, error may be in the domain or on the boundary or in both. The objective of the method is to reduce the error. The procedure through which the error minimization is implemented is called the weighted residual technique. When method of weighted residual is used, the concerned problem is represented by differential equation valid over the domain Ω with the prescribed boundary conditions on the boundary Г. Initially, a trial function is assumed as solution. The function may not satisfy the differential equation and the boundary conditions exactly. By substituting the trial function in the differential equation and boundary condition, an error known as residual will be produced. The unknown parameters in the trial function are determined in such a way that the residual is made as small as possible. A more detailed treatment of weighted residual techniques can be found in Finlayson and Scriven (1966) and Finlayson (1972). Some of the most commonly used weighted residual techniques in engineering applications are Method of Point Collocation, Method of Least Squares, and Galerkin's Approach. These important weighted residual techniques are elaborated next with examples. 2.3.1 Method of Point Collocation A number of discrete points referred to as collocation points are chosen in the collocation method and the error is forced to become zero at these points. As a result, the differential equation is satisfied at these chosen points. Let Lɸ = 0 be the differential equation under consideration. Approximate solution should be such that the polynomial or any other function chosen should satisfy the boundary conditions of the problem, i.e., ɸ can be expressed by 23

Finite Element Method with Applications in Engineering

Here, ɸi are undetermined parameters and Ni are known functions linearly independent in dimensionless forms. Parameters ɸi are determined by enforcing the error or residual by ε{=L(ɸ)} = 0 at n points in the domain resulting in n equations to determine n values of ɸi. The number of collocation points depends on the number of unknowns in the assumed function. Depending upon the problem, the point collocation technique can be classified into three methods. In interior collocation method, the admissible function satisfies all the boundary conditions but not the differential equation in the domain. Here, errors arising are set to zero at chosen points in the interior of the domain. In boundary collocation method, the admissible function satisfies the differential equation exactly in the domain, but does not satisfy the boundary conditions. Here the collocation points are chosen at the boundary. Inmixed collocation method, the admissible function does not satisfy the differential equation and boundary conditions exactly. Thus, collocation points are chosen in the domain as well as on the boundaries. 2.3.2 Method of Collocation by Sub-Regions This method is very similar to point collocation method. However, error over the integral of the function, rather than collation points is made zero over different regions. Hence if R is the residual function,

Example 2.1 Solve following differential equation using point collocation method.

Use boundary conditions ɸ(x = 0) = 0 and ɸ(x = 1) = 0. Choose x = 0.25 and x = 0.5 as collocation points. Solution: The exact solution for the expression (2.3) is Let the approximate solution that satisfies the boundary conditions be of the form

Considering only the first two coefficients,

24

Finite Element Method with Applications in Engineering are derived. Substitution of Eqs. (2.6) and (2.8) into Eq. (2.3) yields an error, ε, where

Choosing x = 0.25 and x = 0.5 as collocation points, where the error is made equal to zero, following expressions are obtained.

Solving Eqs. (2.10) and (2.11) simultaneously, it can be shown that α1 = −0.1459 and α2 =−0.15263. Hence, the solution is

Here it should be noted that the number of collocation points should be equal to the number of unknown parameters αi. 2.3.3 Method of Least Squares In the method of least squares, the sum of squares of the residuals are minimal (or made zero) after substitution of the approximate solution in the differential equation. That is, the sum of the squares of the error is minimized as

For F to be minimum ɸi.

that yields n equations to determine n values of

Thus, the final equations are

where

are the weights.

Method has been demonstrated in the following example. Example 2.2 Solve the following differential equation using least square method.

Use boundary conditions ɸ(x = 0) = 0 and ɸ(x = 1) = 0. Solution: Here the approximation function is assumed to be the same as given in Eq. (2.6). Then the error is given by

25

Finite Element Method with Applications in Engineering

Simplifying Eqs. (2.19) and (2.20), the resulting equations are

2.3.4 Galerkin's Method In the Galerkin's method, approximating or trial functions are considered to be the weighting functions. Consider a system with homogeneous boundary conditions

An approximate function that satisfies these conditions can be used such that

Here Ni are the assumed functions and Ψi are either the unknown parameters or unknown functions of one of the independent variables. Substituting this approximating function in Eq. (2.24) may produce a residual as In the Galerkin's method, the error is made orthogonal to the same trial function Ψi i.e.,

This method also yields n linear equations to determine n values of Ψi. Galerkin's method may produce symmetrical coefficients in many cases since the weighting function and approximating functions are same. Many finite element formulations are based on the Galerkin's type approximating method. In the later chapters, use of Galerkin's method in development of FEM will be described in details. Example 2.3 Solve the following differential equation using Galerkin's method. 26

Finite Element Method with Applications in Engineering

Use boundary conditions ɸ(x = 0) = 0 and ɸ(x = 1) = 0. Solution: The functional form of error is given earlier by Eq. (2.16). Here the error ε is made orthogonal to the trial functions,

Simplifying equations (2.30) and (2.31), resulting equations are

Solving equations (2.32) and (2.33) yields α1 = −0.14588 and α2 = −0.16278 Hence,

Finally, the results of examples 2.1, 2.2 and 2.3 are compared in the following table. Table 2.1 Comparison of Results Obtained by Different Methods

The above comparison shows that Galerkin's method yields better results in comparison with the Point collocation or least square method.

2.4 RAYLEIGH–RITZ METHOD Form of the unknown solution is assumed in terms of known functions (trial functions) with unknown adjustable parameters in the Rayleigh–Ritz Method. From the family of trial functions, the function that renders the functional stationary are selected and substituted into the functional, which is a function of the functions. Thus, the functional is expressed in terms of the adjustable parameters. The resulting functional is differentiated with respect to each parameter and the resulting equation is set equal to zero. If there are n unknown parameters in the functional, there will be n simultaneous equations to be solved for the parameters. Thus the best solution is obtained from the family of assumed solutions. Accuracy depends on the trial functions chosen. Generally, the trial function is constructed

27

Finite Element Method with Applications in Engineering from the polynomials of successively increasing degree. It may be noted that the method requires the knowledge of the functional. Thus the main aim of Rayleigh–Ritz method is to replace the problem of finding the minima and maxima of integrals by finding the minima and maxima of functions of several variables. For example, consider search of a function L(x) that will extremize certain given functional (L). As mentioned, L(x) can be approximated by a linear combination of suitably chosen coordinate functions c1(x), c2(x),………cn(x). Then L(x) can be written as where gi are unknown constants to be found. Since each of ci(x) is an admissible function, the functional I(L) becomes a function of g. By taking the differential of the function, unknown g can be determined as follows.

Using Eq. (2.36), n algebraic equations are obtained from which the unknown constants gj are determined. Method has been explained in the following example. Example 2.4 Solve the following differential equation using Rayleigh–Ritz Method.

Use boundary conditions ɸ(x = 0) = 0 and ɸ(x = 1) = 0. Solution: Solution is obtained from the corresponding variational functions. Variational function for the above differential equation can be written as

F satisfies the Euler–Lagrangian equation of variational calculus given as

yielding the differential equation chosen. Further,

substituting all these into Eq. (2.39), , can be obtained, which is the given differential equation. Substituting Eqs. (2.6) and (2.7) into

28

Finite Element Method with Applications in Engineering and integrating separately term by term within the limits x = 0 to 1,

can be derived. Finally, the variational function can be written as

Minimizing Ψ with respect to undetermined coefficients α1, and α2

Solving equations (2.42) and (2.43) simultaneously, α1 = –0.1460 and α2 = –0.1628. Hence, the expression is, This result indicates that whenever a functional exists for any differential equation, both Galerkin and Raleigh–Ritz methods give almost identical results. 2.4.1 Relation Between FEM and Rayleigh–Ritz Method From the mathematical point of view, the FEM can be considered as a special case of Rayleigh–Ritz Method. Both FEM and Rayleigh–Ritz Method use a set of trial functions as the starting point for obtaining the approximate solution, take linear combinations of trial functions and try to get combinations of trial functions, which make a functional stationary. However, the trial function is considered for the entire domain in the Rayleigh–Ritz Method and hence simple domain only can be modeled. On the other hand, functions are not defined over the whole solution domain in FEM. The trial function is defined element wise and they do not have to satisfy boundary conditions.

2.5 FURTHER NUMERICAL EXAMPLES Example 2.5 Solve the differential equation ɸ″ - ɸ = x2 with boundary conditions ɸ(x = 0) = 0 and ɸ(x = 1) =0, by Point Collation, Least Square, Galerkin and Rayleigh–Ritz Method and compare results with the exact solution.

Solution: 1. 2.

Point Collocation

Solution that satisfies boundary conditions is

29

Finite Element Method with Applications in Engineering Limiting it to two terms,

Then

and

Substituting values of

in original equation, error ε is found to be

Simplifying, Choosing x = 0.25 and x = 0.5 as collocation points,

Simplifying and making error = 0 at x = 0.5,

Solution of Eqs. (2.50) and (2.51) yields

Therefore,

Method of Least Squares

Assuming same trial function as above, Minimize the square of error with respect to α1 and α2 over the domain to get α1 and α2. with respect to α1 and α2. Therefore,

This can be achieved by differentiating

30

Finite Element Method with Applications in Engineering

Solving Eqs. (2.53) and (2.54) for α1 and α2, α1 = –0.06284 and α2 = –0.17903. Hence the solution is

Galerkin's Method

In this method, the error is made orthogonal to the trial functions over the domain. That is,

Integrating and substituting the limits, it can be shown that

Raleigh–Ritz Method

Solution is obtained from the corresponding variational functional, which can be written as

When this equation is substituted into the Euler–Lagrangian equation, the original differential equation should be obtained.

31

Finite Element Method with Applications in Engineering

Substituting

all

these

into

Eq.

(2.60),

the

governing

differential

equation

or ɸ″ – ɸ = x2 is obtained. Substituting Eqs. (2.46) and (2.47) into

and integrating separately term by term, following equations are obtained.

Solving, α1 = –0.05496 and α2 = –0.16279 Therefore,

Exact Solution The exact solution of differential equation ɸ″ – ɸ = x2 can be derived as: Table 2.2 shows a comparison of results by different methods with the exact solution. It can be seen that Galerkin’s and Rayleigh–Ritz Methods give better results. Table 2.2 Comparison of Results Obtained by Different Methods

32

Finite Element Method with Applications in Engineering Example 2.6 Solve ɸ″ + ɸ' – 2ɸ = 0, 0 ≤ x ≤ 1 with boundary conditions ɸ(x = 0) = 0, ɸ(x = 1) = 1 by using (i) Point Collocation and (ii) Least Square Method. Solution: Let ɸ = α1x + α2x2 + α3 x3 The boundary conditions are: at Now, ɸ' = α1 + 2α2x + 3α3x2 and ɸ″ = 2α2 + 6α3x Therefore, ε = 2α2 + 6α3x + α1 + 2α2x + 3α3x2 – 2α1x – 2α2x2 – 2α3x3 Or ε = α1 (1 – 2x) + α2 (2 + 2x – 2x2) + α3(6x + 3x2 – 2x3) (i) Collocation Method: Choosing x = 0.25 and 0.5 as collocation points,

On solving, α1 =1.136, α2 = –0.4765, α3 = 0.3404. Hence the solution is ɸ = 1.136x – 0.4765x2 + 0.3404x3. (ii) Least Squares Method:

Solving (2.66), (2.67), and (2.68),

Example 2.7 Solve ∇2h= f(x,y) in the region (x, y) є (0, a); x (0, b) and f(x, y) = c, with boundary condition h(x = 0) = 0, h(x = a) = 0, h(y = 0) = 0, h(y = b) = 0, by Galerkin's method. Solution: Let h = α x (x – a) y (y – b)

33

Finite Element Method with Applications in Engineering

According to Galerkin criterion, or,

which yields

Example 2.8 Solve the problem of laminar flow through a rectangular duct, wherein the governing differential equation for flow is of the form

with boundary conditions h = 0 at x = ± a, y = ± b, using the Galerkin's method. Solution: Assume the trial function of the form

Substitution of Eq. (2.71) into Eq. (2.70) yields

By using Galerkin's method,

The resulting equations are

34

Finite Element Method with Applications in Engineering From which α1, α2, α3 can be calculated. For a = 1, b = 1, s = slope of the pressure gradient = 0.001, g = 9.8, υ = 0.01 m2/sec (kinematic viscosity), the equations are as follows:

The solution is: α1 = –0.6521; α2 = –1.0485; α3 = –0.3434. Finally, h = (a2 – x2)(b2 – y2)(–0.6521 – 1.0485x – 0.3434y).

2.6 CLOSING REMARKS In this chapter, various approximating methods are introduced. Procedures for applications of various approximating methods are discussed in detail. As FEM is also one of the approximating methods, it is essential to understand how various methods can be applied to solve differential equations. The MWR and the Rayleigh–Ritz Method based on the variational principles are the most important approximating methods. The basic principles used in FEM are very similar to that used in weighted residual and Rayleigh–Ritz Methods. Generally, most FEM formulations are either based on the variational principles (similar to Rayleigh–Ritz Method) or Galerkin's method based on weighted residuals technique. Approximate methods described in this chapter are applied to the entire domain of the problem. On the other hand, it will be shown later that approximation is applied initially element wise in the FEM. Total system is then assembled for all elements to obtain solution. Various FEM principles, methodologies and applications are elaborated further in the subsequent chapters.

EXERCISE PROBLEMS 1. Solve y” – x = 0, 0 ≤ x ≤ 1 with the boundary conditions y(x = 0) = 0 y(x = 1) = 0 by using (i) Collocation method; (ii) Least square method, and (iii) Galerkin's method and compare the results. 2. Solve y” + y + x = 0, 1 ≤ x ≤ 2 with boundary condition y(x = 0) = 0, y(x = 1) = 0 by using (i) Point collocation, (ii) Least square and (iii) Galerkin method. 3. Solve x2y” – 2xy’ + 2y = 0, 1 ≤ x ≤ 4 with following boundary conditions y(x = 1) = 0, y(x= 4) = 12 by using (i) Point Collocation; (ii) Least square, and (iii) Galerkin's method and compare the results. 4. Solve x2y” + 6xy’ + 6y = 1/x2, 1 ≤ x ≤ 2 with boundary conditions y(x = 1) = 1, and y(x = 2) = 2. The exact solution is Use (i) Collocation method, (ii) Least square, and (iii) Galerkin's method and compare the results. 5. Solve y”’ – 6y” + 12y’ – 8y = 0, 0 ≤ x ≤ 1 with boundary conditions y(x = 0) = 0, y(x = 1) =e2. The general solution is y = (1 – 2x + 2x2) e2x. Use (i) Collocation method, (ii) Least square, and (iii) Galerkin's method and compare the results. 6. Solve y” + 4y’+ 4y = cos(x), 0 ≤ x ≤ 1 with boundary conditions y(x = 0) = 0 and y(x = 1) = 1. The general solution is y = (A + Bx) e–2x + 1/25 (3cos x + 4sin x). Use (i) Collocation method, and (ii) Least square and compare the results. 7. Solve y” + 4y’+ 3y =10e–2x, 0 ≤ x ≤ 1 with boundary conditions y(x = 0) = 0, y(x = 1) = 1. Use (i) Collocation method, (ii) Least square, and (iii) Galerkin's method and compare the results.

35

Finite Element Method with Applications in Engineering 8. Solve (1–x2)y”–xy' = 0, 0 ≤ x ≤ 1 with boundary conditions y(x = 0) = 0, y(x = 1) = π/2 by (i) Least square, and (ii) Galerkin's method and compare the results. 9. Solve y”+ y = x, 0 ≤ x ≤ 2 with boundary conditions y(x = 0) = 0, y'(x = 0) = 2, by Rayleigh–Ritz method. 10. Solve y” + 2y + x2 = 0, 0 ≤ x ≤ 1 with boundary condition y(x = 0) = 0, y(x = 1) = 0 by Rayleigh–Ritz method. 11. Solve x2y”– 2y – 1 = 0, 1 ≤ x ≤ 2 with boundary conditions y(x = 1) = 0 and y(x = 2) = 0 by using (i) Point collocation, (ii) Least Square method, and (iii) Galerkin's method.

REFERENCES AND FURTHER READING Bathe, K. J. (2001). Finite Element Procedure, Prentice Hall of India, New Delhi. Chandrupatla, T. R., and Belegundu, A. D. (2003). Introduction to Finite Elements in Engineering, Prentice Hall of India, New Delhi. Davis, J. L. (1986). Finite Difference Methods in Dynamics of Continuous Media, Macmillan, New York. Finlayson, B. A. (1972). The Method of Weighted Residuals and Variational Principles, Academic Press, New York. Finlayson, B. A., and Scriven, L. E. (1966). The Method of Weighted Residuals: A Review, Applied Mechanics Review, 19(9). Forsythe, G. E., and Wasow, W. R. (1960). Finite Difference Methods for Partial Differential Equations, John Wiley, New York. Huebner, K. H. (1975). The Finite Element Method for Engineers, John Wiley and Sons, New York. Hilderband, F. B. (1965). Methods of Applied Mathematics, Prentice-Hall, Englewood Cliffs. Mitchell, A. R., and Griffiths, D. F. (1980). Finite Difference Method in Partial Differential Equations, John Wiley, Chichester. Reddy, J. N. (1985). An Introduction to the Finite Element Method, McGraw-Hill International Edition, New York. Thomas, J. W. (1995). Numerical Partial Differential Equations: Finite Difference Methods, Springer-Verlag, New York. Zienkiewicz, O. C. (1980). Finite Element Method, 3rd ed., McGraw Hill, New York.

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Finite Element Method with Applications in Engineering

3 Finite Element Method—An Introduction 3.1 GENERAL In the last few decades, revolution in the computer technology has led to development of numerous computational techniques for solving many engineering problems. As mathematical modelling became an integral part of analysis of engineering problems, a variety of numerical methods such as finite difference method (FDM), finite element method (FEM), finite volume method (FVM) and boundary element method (BEM) have been developed. Each of these methods has its own merits and demerits depending on the problem to be solved. Out of the available numerical techniques, FEM is one of the most flexible and versatile method for solving engineering problems. FEM is used widely for solving engineering problems in solid mechanics, heat transfer, structural mechanics, aerospace, automobiles, biomechanics, fluid mechanics, etc. Development of FEM for solution of practical engineering problems began with the advent of the digital computers. FEM envisions the solution process as built up of many small interconnected sub-regions or elements and gives a piecewise approximation to the problem concerned. Hence, most engineering problems, which can be considered to be in a continuum, can be solved using FEM by the piecewise approximation.

3.2 WHAT IS FEM? In the most numerical methods, the unknown state variables are solved at a discrete number of points in the problem domain considered to obtain approximate solutions. Process of dividing the problem domain into an equivalent system of smaller domains or units and selecting a discrete number of points is called discretization. Once a problem domain is discretized, solution can be obtained for each of the smaller domains or units considered. Finally, such domain wise solutions can be combined together to obtain solution for the entire domain. Hence, the solution is obtained from the approach known as ‘going from part to whole’. Through this approach, analysis is simplified even though large amount of data may have to be handled. Basic idea of FEM is developed from the above principle. Discrete points considered in the domain are called nodes and the smaller domains or units considered are called elements. Elements and nodes together constitute the mesh. Fineness of the mesh increases accuracy of the solution but at the cost of computation time. On the other hand, the number of unknown parameters at each node, determines the degrees of freedom. FEM offers a way to solve wide variety of complex continuum problems by sub-dividing them into a series of simpler interrelated problems. Essentially, it provides a consistent technique for modelling whole system as assemblages of discrete parts or finite elements. The whole system may be a body of matter or a region of space in which some phenomenon of interest is occurring. The degrees of freedom to which the assemblage of elements represents the whole system usually depend on the number, size and type of elements chosen for the representation. Sometimes it is possible to choose elements in such a way that they lead to exact representation. However, this occurs in some special cases only. Most often, the choice of an element is a matter of engineering judgment based on accumulated experience. Fundamental idea of discretization in the finite element method stems from the physical procedures used in the network analysis or framed structures. An engineering problem may be solved in one-dimensional, two-dimensional or three-dimensional. Accordingly, the FEM discretization can also be in one-dimensional, two-dimensional or three-dimensional 37

Finite Element Method with Applications in Engineering as shown in Figs. 3.1, 3.2 and 3.3. In mathematical modelling, the problem to be solved can be generally represented by mathematical equations. As FEM includes discretization of the domain into elements, the finite element solution gives a piecewise approximation to the governing mathematical equations.

3.3 HOW DOES FEM WORK? In engineering problems of continuum in nature, the field variable (such as displacement, potential, pressure, velocity and temperature) possesses infinitely many values because it is a continuous function of generic point in the body or solution domain. Hence, the problem becomes one with an infinite number of unknowns. The discretization procedure reduces the problem to one of a finite number of unknowns by dividing the solution domain into elements. Then the unknown field variable is expressed in terms of assumed approximating functions within each element. Approximating functions (or interpolating functions) are defined in terms of values of the field variables at specified nodes or nodal points. Nodes usually lie on the element boundaries (boundary nodes) where adjacent elements are considered to be connected or inside the element as interior nodes. Behavior of the concerned field variable is defined by the nodal values of the field variable and the interpolation functions for the element within the elements. The nodal values of the field variable become the new unknowns for the finite element representation of the problem. Once the nodal unknowns are obtained, the interpolation functions define the field variable throughout the assemblage of elements of the problem. Thus, solutions are formulated for individual elements before putting them together to represent the entire problem. Nature of the solution and degree of accuracy depend on the size and number of elements and the kind of interpolation function used. The interpolation functions are chosen such that the field variable or its derivatives are continuous across adjoining element boundaries. For example, Figure 3.1 shows the FEM discretization for an axially loaded bar member, which is considered as a one-dimensional problem. The physical system includes the bar subjected to an axial load g(x) and the problem to be solved is to determine the displacement u = u(x). The problem can be mathematically represented by the following equation:

where AE represents the axial rigidity of the bar assumed to be constant. To obtain a solution using FEM, the bar considered is discretized into elements using connecting nodes as shown in Figure 3.1.

Figure 3.1 Finite element discretization of a bar in one dimensional domain Here, the unknown variable is the displacement u. The unknown variable is approximated by a set of piecewise continuous functions (referred to as interpolation or shape function) assembled over the elements at the nodes. Then the equations governing the behavior of the system are derived by assembling the equations formulated for all the elements. Finally, the response of the system (here the bar deflection) is obtained by solving the approximated system of equations.

38

Finite Element Method with Applications in Engineering Consider another example of a two-dimensional steady flow through a homogeneous isotropic earth dam resting on an impermeable formation. The problem domain and FEM discretization is shown in Figure 3.2. The physical problem includes the ‘potential’ distribution in the domain of the dam due to the differential heads at upstream and downstream and determination of the free surface position in the dam. The upstream head value is known as h1(=10 m) and the downstream head value is known as h2 (=2 m). The problem can be mathematically represented by the Laplace equation as

where h(x,y) is the head or ‘potential’. To get a solution, initially a free surface is assumed and the dam domain including the free surface considered is discretized into triangular elements using connecting nodes as shown in Figure 3.2. Here, the unknown variable is the head or ‘potential’ (h) at various nodes. The unknown variable h is approximated by the two-dimensional interpolation or shape functions assembled over the triangular elements at the nodes. Then, the equations governing the behavior of the problem are derived by assembling the equations formulated for all the triangular elements. Finally, the ‘potential’ is obtained by solving the approximated system of equations after introducing the upstream and downstream head values. As the location of free surface is not known a priori for this problem, the final solution is obtained through an iterative procedure which is discussed later in the book. For demonstration purpose, consider a problem dealing with steady-state threedimensional heat conduction in an isotropic homogeneous plate. The problem domain and three-dimensional FEM discretization are shown in Figure 3.3. The domain is discretized using prism elements. The physical problem includes the ‘temperature’ distribution in the plate due to the differential temperature from left to right, with values T1 and T2. The problem can be represented mathematically by the Poisson equation as

Figure 3.2 Earth dam discretization using two-dimensional finite elements

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Finite Element Method with Applications in Engineering

Figure 3.3 Three-dimensional finite element discretization of a plate with prism elements where T is the temperature, k is the thermal conductivity and Q is the heat source or sink. The domain is discretized into prism elements using connecting nodes to obtain solution as shown in Figure 3.3. Here, the unknown variable is the temperature at various nodes. The unknown variable ‘temperature’ is approximated by the three-dimensional interpolation or shape functions assembled over the prism elements at the nodes. Then, the equations governing the behavior of the problem are derived by assembling equations formulated for all the prism elements of the problem. Finally, the ‘temperature’ distribution is obtained by solving the approximated system of equations after introducing the boundary conditions. Above mentioned problems indicate a broad perspective of working of FEM in the onedimensional, two-dimensional and three-dimensional problems. Further chapters will explain in detail how FEM is used in the solution of engineering problems.

3.4 A BRIEF HISTORY OF FEM Basic idea of FEM originated from analysis procedure used in framed structures like trusses, aircraft structural analysis and flow network analysis. For the origin and development of FEM, there are different perspectives from the viewpoint of a mathematician, a physicist and an engineer. From the mathematician's perspective, solutions to boundary value problems of continuum mechanics were sought by finding approximate upper and lower bounds for eigenvalues. The physicists were trying to solve continuum mechanics problems by means of piecewise approximating functions. The engineers were investigating methodologies for the solution of complex aero-elasticity problems such as stiffness of shell type structures reinforced with ribs. In the 1930s, when civil engineers dealt with truss analysis, they identified the solution procedure by solving the component stresses and deflections as well as the overall strength of the system. They recognized the truss as an assembly of members of rods whose force deflection characteristics can be easily obtained. By combining these individual characteristics using laws of equilibrium and solving the resulting system of equations, the unknown forces and deflections for the overall truss were obtained. Efforts of mathematicians, physicists and engineers finally resulted in the development of basic ideas of finite element method in 1940s. Hrenikoff (1941) proposed the ‘frame work method’ for the solution of elasticity problems. On the other hand, Courant (1943) presented an assemblage of piecewise polynomial interpolation over triangular elements and the principle of minimum potential energy to solve torsion problems. Some mathematical aspects related to eigenvalues were 40

Finite Element Method with Applications in Engineering developed for boundary value problems by Poyla (1954), Hersch (1955) and Weinberger (1958). Foundation of finite element was laid by Argyris in 1955 through his book on ‘Energy Theorems and Structural Analysis’. Turner et al. (1956) developed the stiffness matrices for truss, beam and other elements for engineering analysis of structures. For the first time in 1960, the terminology ‘Finite Element Method’ was used by Clough (1960) in his paper on plane elasticity. In 1960s, a large number of papers appeared related to the applications and developments of the finite element method. Engineers applied FEM for stress analysis, fluid flow problems and heat transfer. A number of international conferences related to FEM were organized and the method got established. The first book on FEM was published by Zienkiewicz and Cheung in 1967. With the advent of digital computers and finding the suitability of FEM in fast computing for many engineering problems, the method became very popular among scientists, engineers and mathematicians. By now, a large number of research papers, proceedings of international conferences and short-term courses and books have been published on the subject of FEM. Many software packages are also available to deal with various types of engineering problems. As a result, FEM is the most acceptable and well-established numerical method in engineering sciences.

3.5 FEM APPLICATIONS FEM is now applicable to a wide range of engineering problems. Majority of applications of FEM are in the realm of solid mechanics and fluid mechanics. In the last two decades, FEM has also been used in electrical and electromagnetic problems as well as in bioengineering problems. Categories of problems that can be solved using FEM can be divided into equilibrium, eigenvalue and transient problems. The equilibrium problems are generally steady-state problems such as determination of stress and displacement in solid mechanics-related problems, determination of temperature distribution in thermal problems, estimation of potential, velocity and pressure in fluid mechanics problems. The eigenvalue problems are also steady state in nature but include estimation of vibration and natural frequencies in solids and fluids. In the transient problems, FEM is used in propagation problems of continuum mechanics with respect to time. A brief description of the applications of FEM in various engineering fields is given below. Structural mechanics and aerospace engineering: FEM applications in equilibrium conditions include analysis of beams, plates, shell structures, stress and torsion analysis of various structures. On the other hand, eigenvalue analyses include stability of structures, viscoelastic damping, vibrations and natural frequency analysis of structures. The transient or propagation analysis using FEM includes dynamic response of structures to periodic loads, viscoelastic and thermo-elastic problems and stress wave propagation. Geotechnical engineering: FEM applications include stress analysis, slope stability analysis, soil structure interactions, seepage of fluids in soils and rocks, analysis of dams, tunnels, bore holes, propagation of stress waves and dynamic soil structure interaction. Fluid mechanics, hydraulic and water resources engineering: FEM applications include solutions of potential and viscous flow of fluids, steady and transient seepage in aquifers and porous media, movement of fluids in containers, external and internal flow analysis, seiche of lakes, ocean and harbors, salinity and pollution studies in surface and sub-surface water problems, sediment transport analysis and water distribution networks. Mechanical engineering: In mechanical engineering, FEM applications include steady and transient thermal analysis in solids and fluids, stress analysis in solids, automotive design and analysis and manufacturing process simulation. 41

Finite Element Method with Applications in Engineering Nuclear engineering: FEM applications include steady and dynamic analysis of reactor containment structures, thermo-viscoelastic analysis of reactor components, steady and transient temperature-distribution analysis of reactors and related structures. Electrical and electronics engineering: FEM applications include electrical network analysis, electromagnetics, insulation design analysis in high-voltage equipment’s, thermosonic wire bond analysis, dynamic analysis of motors, molding process analysis in encapsulation of integrated circuits and heat analysis in electrical and electronic equipment. Metallurgical, chemical and environmental engineering: In metallurgical engineering, FEM is used for the metallurgical process simulation, molding and casting. In chemical engineering, FEM can be used in the simulation of chemical processes, transport processes (including advection and diffusion) and chemical reaction simulations. FEM is used in environmental engineering widely in the areas of surface and sub-surface pollutant transport modelling, air pollution modelling, land-fill analysis and environmental process simulation. Meteorology and bioengineering: In the recent times, FEM is used in climate predictions, monsoon prediction and wind predictions. FEM is also used in bioengineering for the simulation of various human organs, blood circulation prediction and even total synthesis of human body.

3.6 MERITS AND DEMERITS OF FEM As discussed above, FEM can be applied to almost all branches of engineering. The fact that FEM can be used to solve a particular problem does not mean that it is the most ideal solution technique. To solve a given problem, often several attractive numerical techniques are available. Each method has its own merits and demerits. Depending on the problem, ‘the best’ method should be chosen by comparing the merits and demerits of the method. Here, the merits and demerits of FEM are discussed. Merits of FEM Compared to other numerical methods some of the merits of FEM are as follows. Modelling of complex geometries and irregular shapes are easier as varieties of finite elements are available for discretization of domain. Boundary conditions can be easily incorporated in FEM. Different types of material properties can be easily accommodated in modelling from element to element or even within an element. Problems with heterogeneity, anisotropy, nonlinearity and time-dependency can be easily dealt with. The systematic generality of FEM procedure makes it a powerful and versatile tool for a wide range of problems. FEM is simple, compact and result-oriented and hence widely popular among engineering community. FEM can be easily coupled with computer-aided design (CAD) programs in various streams of engineering. An FEM model can be developed at different levels and it is possible to interpret the method in physical terms.

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Finite Element Method with Applications in Engineering In FEM, it is relatively easy to control the accuracy by refining the mesh or using higher order elements., Availability of large number of computer software packages and literature makes FEM a versatile and powerful numerical method. Demerits of FEM Some demerits of FEM are as follows. Closed-form expressions in terms of problem parameters are not available in FEM. Numerical solution is obtained at one time for a specific problem case only. Hence, unlike analytical solutions, there is no advantage of flexibility and generalization. Large amount of data is required as input for the mesh used in terms of nodal connectivity and other parameters depending on the problem. Generally, voluminous output data must be analyzed and interpreted. Experience, good engineering judgment and understanding of the physical problems are required in FEM modelling. Poor selection of element type or discretization may lead to faulty results.

3.7 CLOSING REMARKS In this chapter, the finite element method is introduced. The basic idea in FEM ‘going from part to whole’ of the problem concerned is explained. Various aspects of FEM discretization are elaborated. Domain can be discretized in one-, two- or three-dimensions. Various aspects of working of FEM in one-dimension, two-dimensions and threedimensions are explained with example problems. Further, history of FEM is briefly reviewed by mentioning the important developments. FEM is now used in almost all branches of engineering. The important applications of FEM in different branches of engineering are further described briefly. As a numerical method, FEM has its own merits and demerits, which are briefly outlined. Various FEM principles, methodologies and applications are elaborated in subsequent chapters.

EXERCISE PROBLEMS With the help of appropriate domain discretization, governing equations and boundary conditions, explain the one-dimensional application of FEM in heat conduction through cylindrical bar of 5 m length and 5 cm diameter. Use linear line elements for discretization. With the help of appropriate domain discretization, governing equations and boundary conditions, explain the two-dimensional application of FEM for deformation and stress distribution in a cantilever bar of 1 m and 10 cm and 2 cm thickness and a load of 1 KN at the free end. Use triangular elements for discretization. With the help of appropriate domain discretization, governing equations and boundary conditions, explain the three-dimensional application of FEM for potential flow in a cubic cavity of size 1 m. Use linear prism elements.

REFERENCES AND FURTHER READING Argyris, J. H., (1955), Energy theorems and structural analysis, Aircraft Engineering, Vol. 27. Bathe, K. J. (2001). Finite Element Procedure, Prentice Hall of India, New Delhi. Chandrupatla T. R., and Belegundu, A. D. (2003). Introduction to Finite Elements in Engineering, Prentice Hall of India, New Delhi. 43

Finite Element Method with Applications in Engineering Clough, R. W. (1960). The finite element method in plane stress analysis, Proceedings of ASCE, 2nd Conference on Electronic Computation, Pittsburgh, 23: 345–378. Courant, R. (1943). Variational methods for the solution of problems of equilibrium and vibrations, Bulletin of the American Mathematical Society, 49: 1–23. Hersch, J. (1955). Equations Differentielles et Fonctions de Cellules, C. R. Acad. Sci., Vol. 240. Hrenikoff, A. (1941). Solution of problems in elasticity by the frame work method, Journal of Applied Mechanics, Transactions of the ASME, 8: 169–175. Huebner, K. H. (1975). The Finite Element Method for Engineers, John Wiley and Sons, New York. Poyla, G. (1954). Estimates for Eigenvalues: Studies Presented to Richard von Mises, Academic Press, New York. Reddy J. N. (1985). An Introduction to the Finite Element Method, McGraw-Hill, New York (Int. ed.). Turner, M. J., Clough, R. W., Martin, H. C., and Topp, L. J. (1956). Stiffness and deflection analysis of complex structures, Journal of Aeronautical Science, 23(9): 805–824. Weinberger, H. F. (1958). Upper and lower bounds for eigenvalues by finite difference methods, Commun, Pure Appl. Math., Vol. 9. Zienkiewicz, O. C., and Cheung, Y. K. (1967). The Finite Element Method in Structural and Continuum Mechanics, Mac-Graw Hill, London. Zienkiewicz, O. C. (1980). Finite Element Method, 3rd ed., Mac-Graw Hill, New York.

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Finite Element Method with Applications in Engineering

4 Different Approaches in FEM 4.1 INTRODUCTION Most conventional methods are cumbersome, uneconomical and time consuming to apply to many engineering applications, due to introduction of new materials such as composites, fiber reinforced materials etc. and complicated geometries to be modelled. FEM has become one of the most popular numerical methods to deal with complex geometries and material properties with the availability of high speed digital computers. Generally, FEM can be considered as a variant of Rayleigh–Ritz method in combination with the variational principle applied to continuum mechanics. In the Rayleigh–Ritz method, trial functions are used to represent the entire domain. On the other hand, the trial functions used in FEM are approximated over a sub-domain, referred to as an element. Hence, FEM can better handle any material inhomogeneity within the domain and can model any irregular geometry by using varieties of shapes of elements. General steps used in FEM are explained next.

4.2 GENERAL STEPS OF FEM Even though FEM procedure may vary depending upon problem and approach used for solution, general steps in the FEM analysis remain more or less the same. The first step to the solution of the considered problem is to define the problem in detail with the available data. Based on the problem statement and available data, a mathematical model is developed defining the geometry of the problem, material properties, assumptions and simplifications used, governing equations, boundary and initial conditions. In this section, the general steps used in the solution using the finite element method are described with explanations. Step 1: Discretize and select element types This step involves subdividing body into an equivalent system of small bodies, called ‘finite elements’. Points at which the primary unknowns are required to be evaluated, are called ‘nodes’ or ‘nodal points’, and the interfaces between the elements are called nodal lines (or nodal planes/surfaces). Number of unknowns at a node is termed as ‘nodal degrees-offreedom (DOF)’. Most appropriate element type is chosen for the analysis required. Total number of elements used and their variation in size and type within a given body are primarily matters of engineering judgment. One may choose one-dimensional (1D), twodimensional (2D), three-dimensional (3D) or an axisymmetric element, depending upon the physical system under consideration. These elements are shown in Figure 4.1 through Figure 4.4.

Figure 4.1 One-dimensional (line) elements Axisymmetric elements are used when the geometry and material properties of the physical system are axisymmetric. An axisymmetric element is actually a degenerated,

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Finite Element Method with Applications in Engineering three dimensional element and is constructed by rotating a two-dimensional finite element about the axis of symmetry by 360°. Discretization is the most important step in the finite element analysis. In a typical problem, a substantial amount of the time is spent in this phase of the analysis. As a result, many commercially available finite element software packages have sophisticated preprocessing part with graphical user interface to expedite data preparation. It may be noted that accuracy of results depends upon the details of finite element discretization. Hence, sound engineering judgment and understanding of physics of the problem are essential to get meaningful results from the finite element analysis.

Figure 4.2 Two-dimensional elements

Figure 4.3 Three-dimensional elements Step 2: Select approximation functions This step involves choosing a pattern or shape for the distribution of the unknown quantity u, within each element. Unknown quantity can be displacement for Figure 4.4 Axisymmetric element stress-analysis problems, temperature in heat flow problems, fluid pressure and/or velocity for fluid flow problem, and both temperature (fluid pressure) and displacement for coupled problems involving effects of both flow and deformation. Approximation function is defined within the element using the nodal values of the element. Linear, quadratic and cubic polynomials are frequently used functions because they are simple to work with in the finite element formulation. However, trigonometric series could also be used. For an n-node element, the approximation function can be expressed as

46

Finite Element Method with Applications in Engineering where u1, u2, …, un are the values of the unknowns at the nodal points and N1, N2, …, Nn are the interpolation functions or shape functions. Accuracy of a solution depends greatly on the selection of approximation function. Linear polynomials N1 and N2 are shown in Figure 4.5for a one-dimensional two-node element.

Figure 4.5 One-dimensional approximation of u

Figure 4.6 Constitutive laws and gradients of primary unknowns Step 3: Define the gradients of the unknown quantity and constitutive relationships These relationships are necessary for deriving the equations for each finite element. In stress-analysis problems, gradients and constitutive relationships are simply the straindisplacement and the stress–strain expressions, respectively, as illustrated in Figure 4.6(a). The examples are given below. (a) One-dimensional–Stress analysis

Strain/displacement relation: Stress-strain relation: σx = E εx (Hooke's Law) 47

Finite Element Method with Applications in Engineering where, εx = strain in the x-direction, σx = stress in the x-direction and E = modulus of elasticity (material property) (b) One-dimensional flow through porous media (Figure 4.6(b)) Fluid gradient = Constitutive relation: vx = − kxx gx (Darcy's Law) where kxx = coefficient of permeability and vx = velocity. (c): One-dimensional heat flow temperature gradient = constitutive relation:

(Fourier's law of heat conduction)

where qx is the heat flow and kxx is the thermal conductivity. Step 4: Derive element equations In this step, equations governing the behavior of a generic (typical) finite element are obtained by invoking available laws and principles. These equations describe a relationship between the nodal DOF and the nodal forcing parameters for the generic element. This relationship can be written in compact matrix form as where [ke] e

= element property matrix (or element stiffness matrix)

{q }

= element vector of unknown DOF

{fe}

= vector of element nodal forcing parameters

Basic approaches to derive the element equations will be explained in later sections of this chapter. Step 5: Assemble element equations to obtain the global or total equation and introduce boundary conditions Repeated application of generic element equations results in the element equations for other elements. Then, element equations are added together using a method of superposition to obtain global or total equations for the entire body. This process of superposition is called ‘assembling’. Implicit in the assembly process is the concept of continuity, or compatibility, which requires that the body remains together and that no tears occur anywhere in the structure. The assembled equation can be written in matrix form as where [K]

= assembled (global) property (stiffness) matrix (assembly of [ke])

{q}

= assembled (global) vector of nodal unknowns (assembly of {qe})

{F}

= assembled (global) vector of nodal forcing parameters (assembly of {f e}).

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Finite Element Method with Applications in Engineering

Figure 4.7 General process of finite element analysis 49

Finite Element Method with Applications in Engineering Above equations indicate the capabilities of a body to withstand applied forces. To evaluate performance of a body, certain boundary conditions need to be introduced. Boundary conditions are the physical constraints or supports that must exist so that the structure or body is not mobile. These conditions are commonly specified in terms of known values of unknowns on a part of the surface or boundary of the body. At this stage, it is sufficient to note that the global equations are modified when the boundary conditions are incorporated. Step 6: Solve for the unknown DOF (primary unknowns) The assembled equations (after the modification for the boundary conditions) are solved for the q's by using the Gauss elimination or iterative methods. The q's are called the ‘primary unknowns’ or primary quantities because they are the first quantities determined using the FEM. Step 7: Solve for secondary quantities In most problems, it is necessary to compute secondary quantities from the primary quantities. In the case of stress-analysis problems such quantities can be strains, stresses, moments and shear forces; for fluid flow problems they can be velocities and discharges. Once the primary quantities are known, in most cases, the relationship defined in Step 3 can be employed to find the secondary quantities. Step 8: Interpret the results Steps 7 and 8 are essentially post-processing parts of a finite element analysis. Usually, a tabulated or graphical presentation of results helps in making the design/analysis decisions. General process of finite element analysis is shown in Figure 4.7.

4.3 DIFFERENT APPROACHES USED IN FEM There are number of ways in which one can formulate the properties of individual elements of the domain. Most commonly used approaches to formulate element matrices are: (1) direct approach, (2) variational approach, (3) energy approach and (4) weighted residual approach. 4.3.1 Direct Approach As discussed earlier, the basic idea of finite element method was conceived from the physical procedure used in framed structural analysis and network analysis (pipe network and electric network). It may be possible to choose elements in a way that leads to an exact representation of the problem in certain applications. Hence, element properties are derived from the fundamental physics and nature of the problem in the direct approach. Analysis based on stiffness method is an example of this approach in structural mechanics. Main advantage of this approach is that an easy understanding of techniques and essential concepts is gained without much mathematical illustrations. In solving a problem using the direct approach, first, the elements are defined and then their properties are determined. Once the elements have been selected, direct physical relationships are used to establish element equations in terms of concerned variables. Finally, element equations for various elements or members are combined to generate a system of equations which are solved for the unknowns.

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Finite Element Method with Applications in Engineering Some engineering problems which can be solved using the direct approach include spring systems, trusses, beams, fluid flow in pipe networks, electric resistance networks etc. Here, the linear spring system problem is analyzed to demonstrate the development of finite element technique using the direct approach formulation. 4.3.1.1 Linear Spring System A system of linear springs is used in many engineering problems. To demonstrate the finite element procedure using the direct approach, a system of three linear springs as shown in Figure 4.8 is analyzed. The system is connected in series with springs of different stiffness coefficients. One end of the spring on left-hand side (LHS) is rigidly fixed, while the other end is free to move, as shown in Figure 4.8a. All springs can be subjected to either tension or compression. Physical parameters involved in this system are forces, displacements and spring stiffnesses. In the finite element procedure, each spring is considered as an element and the connection as nodes. The system considered here has three springs and hence consists of three elements and four nodes as shown in Figure 4.8b.

Figure 4.8a A linear spring system with three springs

Figure 4.8b Finite element discretization of linear spring system

Figure 4.8c Single spring element Considering only one spring as shown in Figure 4.8c, which is being treated as isolated from system, force displacement equations can be developed as follows. Forces and displacements are defined at each node and field variable is the displacement. There is no 51

Finite Element Method with Applications in Engineering need for an interpolation function to represent variation of field variable over the element because an exact representation is available from Hooke's law, which relates the nodal displacements and the applied nodal forces F given as

where k is the spring stiffness constant, δ is the displacement, F is the force applied and C is the deflection coefficient. In other words, k can be treated as the force required to produce unit deflection or C as the deflection caused by a unit force. Force displacement equations for node 1 and node 2 are

In matrix form, Eqs. (4.2) and (4.3) can be written as

The square matrix [ke] is known as the element stiffness matrix, the column vector {qe} is the nodal displacement vector, and {f e} is the resultant nodal force vector for the element. Depending on the problem, the stiffness coefficients of matrix [ke] are determined exactly or approximately. In the present case, a typical stiffness coefficient kij of [ke] is defined as the force required at node i to produce a unit deflection at node j. It should be noted that the left hand side element is constrained to have zero displacement at one node which does not influence derivation of element matrix. Constraint conditions or boundary conditions are taken into account only after element equations are assembled to form system of equations. 4.3.2 Variational Approach Problems in continuum mechanics can be represented by two different but equivalent formulations—the variational formulation and the differential formulation. Unknown function or functions are formulated such that extremize (maximize or minimize) or make stationary a functional or system of functionals subject to the given boundary conditions in the variational formulation. On the other hand, a differential equation or a system of differential equations, are integrated in the differential formulation, subject to the known boundary conditions. A close observation of both methods indicates that both are equivalent because the functions that satisfy differential equations and boundary conditions also extremize or make the functional stationary. Variational approach relies on the calculus of variations and involves extremizing a functional. A functional is a quantity whose value depends on entire shape of some functions rather than a number of discrete variables. Functional involves unknown functions and the aim of calculus of variation is to find conditions which are imposed when integrals attain stationary values. It can be observed in variational approach that function that is extreme of the functional is also solution of the corresponding operator equations. Also, it is easy to prove the 52

Finite Element Method with Applications in Engineering existence of a solution and the conditions involved at the interfaces can be easily tackled. Variational formulation has also got the advantages that more complex boundary conditions can be modelled. Moreover, the functional contains lower order derivatives than the differential operator and the problem may possess reciprocal variational formulation (Zienkiewicz, 1980). The main task in using variational principle is to find the variational functional for the particular problem concerned. It may be possible to find functional for most problems in engineering. However, mathematical manipulation may be required to get the functional, if a classical variational functional is not available for the problem concerned. These mathematical manipulations, whenever required, make the approach more complex. Depending on the problem, functional could be an integrated quantity which may be characteristic of the problem. Generally, it can be a physically recognizable quantity such as energy or a mathematically defined entity without physical interpretation. If an example from solid mechanics is considered, the functional turns out to be the potential energy or some derivative of it. In fluid mechanics problems, the functional can be kinetic energy. If only the differential equation and the boundary conditions are known and no physical principle giving a variational functional is apparent in a problem, it should be first decided whether a classical variational principle exists before the problem can be attempted using the variational approach. Different methodologies are available in literature to decide existence of the variational principle and determination of the functional (Mikhlin, 1964; Vainberg, 1964; Finlayson, 1972). Most commonly used methods include (Heubner, 1975): formulation of variational principle based on classical procedure available in mathematical literature (Mikhilin, 1964), derivation of variational principle by mathematical manipulations and use of Frechet derivatives (Tonti, 1969, Finlayson, 1972). Consider a differential equation of the form L(u) = 0. For this differential equation if functional I exists, a function u(x) is required in the variational approach, such that the integral within the range a and b

is extreme (maximum or minimum) for given functional ƒ. Function u(x), which maximizes or minimizes ƒ, is called the stationary function. In the calculus of variational principle, it may be shown that the stationary function ∫ƒ [x,u,u′(x)] dx must satisfy the Euler– Lagrangian equation given by (refer to Appendix B)

Similar equations can be obtained for higher order functions (Mikhlin, 1964; Vainberg, 1964; Finlayson, 1972). This will yield the given differential equation. For example, consider the potential flow in terms of h in a domain Ω with boundary Г = Г1 + Г2. The governing equation is Laplace equation with boundary conditions

53

Finite Element Method with Applications in Engineering In variational approach, the above problem can be represented by an equivalent problem of finding function h that minimizes

subject to the same boundary conditions. Eq. (4.10) can be used instead of the original Laplace equation when the variational approach is employed. FEM procedure using the variational approach includes following steps. First step is to discretize the domain into elements and substitute the trial functions into the functional for each element. The functional is then differentiated with respect to each parameter and the resulting equation is set equal to zero. Then the systems of equations are assembled for all elements and a system of equations are formed which are solved for unknowns after substituting boundary conditions. In the variational approach, nodal values of u are required to be found so as to make the functional I(u) stationary such that

where n is the number of discrete solution u assigned to the domain. From Eq. (4.11), it follows that

Then as per the variational procedure for FEM, for whole domain discretization

where m is the total number of elements, and the superscript (e) denotes an element in the domain. Adding contribution of every element in the domain

can be obtained. From Eq. (4.14), it can be shown that

where p is number of nodes per element. These p equations give characteristics of the element. Finally, similar equations are written for all elements and assembled. Solution of this system of equations after application of boundary conditions gives solution to the problem. For structural mechanics problems, variational functional or some kind of relationships useful in FEM analysis can be derived using the principle of virtual work, principle of minimum potential energy and Castigliano theorem (Shames and Dym, 1991). These relations are mainly used in stress analysis problems. In structural mechanics problem, the minimum potential energy method is most commonly used (Shames and Dym, 1991; Zienkiewicz, 1980). Principle of minimum energy is based on the idea of finding the consistent equilibrium states of body or structures associated with stationary values of a 54

Finite Element Method with Applications in Engineering scalar quantity assumed by the loaded bodies. This scalar quantity is referred as ‘functional’. This approach is similar to the variational approach discussed in this section. Following numerical example demonstrates the variational approach in FEM. Example 4.1 Using FEM based on the variational approach develop the stiffness matrix for the following differential equation. Use one-dimensional linear elements for discretization.

in the range 0 ≤ x ≤ 1, with the boundary conditions h(x = 0) = 0 and h(x = 1) = 0. Solution: Variational functional for the above differential equation that should be minimized can be written as

Here, the highest order derivative appearing in I(h) is a first order derivative and hence the simplest interpolation functions satisfying the compatibility and completeness requirements are linear functions (explained in Chapter 5). As minimum 2 nodes are required for linear variation to be represented, two node linear elements are considered here. Let the domain be divided into n elements and hence the solution can be written as

where Ni (discussed in Chapter 5) is the interpolating function Ni = Ni(x) and hi is the unknown nodal value which is to be determined. For linear interpolation function, at any node i, Ni = 1 and at other nodes Ni = 0. Within any element, assuming linear variation of h, the expression for h can be written as

where and L is the length of the element and x is the distance measured from node i within the element. For nodes 1 and 2 in the element concerned, variation of h can be written as

and the variation of x can be written as Then the derivative of h can be obtained as

Since the interpolation functions for approximating h satisfy the compatibility and completeness requirements, the functional in Eq. (4.17) can be written as 55

Finite Element Method with Applications in Engineering

where m is the number of elements and (e) is the element concerned. To find I(e)(h)(e), the equation for h given in Eq. (4.19) is substituted in the expression for I(h) as follows.

Here the prime indicates differentiation with respect to x. Equations for an element are obtained by performing minimization of I(e)(h)(e) or differentiation of I(e)(h)(e) with respect to nodal values of h and equating to zero as follows.

Both the equations can be written together as

Using linear interpolation functions, Now the element property matrix equations can be written as

In a similar way, element matrices can be written for all other elements. Finally, element property matrices are assembled for all the elements. After introducing the boundary conditions, system of equations is solved for the unknown nodal values of h. 4.3.3 Energy Approach Principle of virtual work, the principle of minimum potential energy theorem are used frequently to derive equations of elements used problems. Here, the principle of minimum potential energy, which is conservative system, will be considered since it is probably the most three approaches mentioned above.

and Castigliano's in stress analysis applicable only to well-known of the

Principle of minimum energy is based on the idea of finding the consistent states (equilibrium states) of the body or structure associated with stationary values of a scalar quantity assumed by the loaded bodies. This scalar quantity is generally referred to as a ‘functional’ (defined to be a function of another function). For example, let f (x) be a function of the variable x and π be the functional defined such that π = π[f (x)]. Then, π is a function of the function f. Stationary values are given by (see Figure 4.9)

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Finite Element Method with Applications in Engineering

Figure 4.9 Stationary values of a functional In structural analysis problems, π is the potential energy of the body or structure. In this case, element equations can be obtained by invoking the principle of minimum potential energy (minimizing the potential function). FEM can be applied generally to any problem provided this scalar function is available. As an illustration, consider one spring of Section 4.3.1.1. The potential energy for this elastic system is

4.3.4 Weighted Residual Approach In many engineering problems, it is difficult to obtain the functional based on variational principles. As discussed earlier, method of weighted residual is a technique for obtaining approximate solutions to linear and non-linear partial differential equations. It offers another means to formulate finite element equations. In Chapter 2, various methods of weighted residuals techniques have been discussed. In FEM using the weighted residuals approach, the domain considered is first discretized with suitable elements. Then, general functional behavior of dependent field variable is assumed so as to approximately satisfy given differential equations over elements. Initially, approximation is applied over each element and subsequently this approximation is substituted into the original differential equation and assembled for the whole domain. Approximation results in some error called a ‘residual’, which is required to vanish in some average sense over the entire solution domain. Resulting system of equations is solved after substitution of boundary conditions to get approximate solution. This approach is advantageous because it thereby becomes possible to extend FEM to problems where no functional is available. 57

Finite Element Method with Applications in Engineering Use of weighted residual technique is explained here with a sample problem. The aim is to find an approximate functional representation for a field variable h governed by the differential equation in the domain Ω bounded by surface Г with appropriate boundary conditions. Function g is a known function of the independent variables. Finite element procedure using the method of weighted residual is illustrated in the following steps. In the first step, the unknown exact solution h is approximated by approximating functions . Here, it should be noted that the functional behavior of is completely specified in terms of unknown parameters or the functional. The functional dependence on all but one of the independent variable is specified while the functional dependence on the remaining independent variable is left unspecified. Accordingly, the dependent variable can be approximated as

Here, Ni an assumed functions and Di are the unknown parameters or the unknown functions of one of the independent variables. As is the approximated functions, it may not satisfy the equation when it is substituted into Eq. (4.23). Hence, it can be written as

Here, R1 is the residual or error that results from approximating h by . In the method of weighted residuals, the m unknowns Di are determined in such a way that the error R1 over the entire solution domain vanishes or becomes very small. In the weighted residual approach, the error is minimized by forming a weighted average of the error and specifying that this weighted average vanish over the solution domain. It is achieved in the weighted residual technique by using m linearly independent weighting functions Ψi. Finally, it can be written as

After this approximation, the residual approaches to zero in weighted sense. Here, the error depends on the choice of the weighting functions. After the weighting functions are specified, Eq. (4.26) is solved for the Di to get an approximation of the unknown field variable h given in Eq. (4.24). As discussed in Chapter 2, there are a variety of weighted residual techniques. FEM procedure using the weighted residual method can be built on these techniques. One of the most commonly used weighted residual techniques is Galerkin's method. Details of the Galerkin method as an approximating method have been given in Chapter 2. As stated, the weighting functions are chosen to be the same as the approximating functions used to represent h according to Galerkin's method. In other words, Ψi = Ni for i = 1, 2,….. m. Thus, Eq. (4.26) can be written as

As the domain is discretized into elements and nodes, initially the weighted residual technique based on Galerkin's approach can be formulated for an element and then it can be assembled for the entire domain. Ni are the interpolation functions, Ni (e) defined over the element and Di are the undetermined parameters, which may be the nodal values of 58

Finite Element Method with Applications in Engineering the field variable or its derivatives. For the element considered, equations governing the behavior of the element can be written as

Here superscript (e) refers to an element, h(e) = [N(e)]{h}(e), g(x,y)(e) is the forcing function defined over element (e) and r is number of unknown parameters assigned to the element. In FEM using the Galerkin's technique, a set of equations, like Eq. (4.28), is obtained for each element of the whole assemblage. The choice of approximating function Ni should guarantee the inter-element continuity necessary for the assembly process. While many interpolation functions provide continuity of value, fewer provide continuity of slope. By applying integration by parts to the integral expression of Eq. (4.28), expressions containing lower-order derivatives can be obtained (called weak formulation). Hence, approximating functions with lower order inter-element continuity can be used. Such a process offers a convenient way to introduce the natural boundary conditions that must be satisfied on some portion of the boundary. Finally, the system is assembled for all the elements, boundary conditions are applied and the system equations are solved for the unknowns. Example 4.2 By using the Galerkin's weighted residual FEM approach, derive the element equations for the second order differential equation

where a, b and c are constants. Solution: For an element of length L, according to Galerkin's criterion in FEM,

The first term can be written as

The second term can be written by using h = [N]{h} as

The third term can be written as

Within any element, assuming linear variation of h, the expression for h can be written as 59

Finite Element Method with Applications in Engineering

where

and

L is the length of the element and x is the distance

measured from node i within the element. Therefore, noting

and

expressions

4.4 CLOSING REMARKS In this chapter, different approaches and detailed analysis procedure in finite element methods are introduced. Procedures for applications of various FEM approaches are discussed in detail. After the problem domain is discretized in FEM, variables are approximated in each element using some relationships or trial functions to form element property matrix. Most commonly used FEM approaches for derivation of element property matrix are direct approach, variational approach, energy approach and method of weighted residual. As discussed in this chapter, each FEM approach has its own advantages and limitations. Depending upon the problem to be solved, suitable FEM approach can be chosen for the solution. Use of direct approach is possible only if a direct relationship based on fundamental physics and nature of the problem is available for the problem concerned. Applicability of variational approach depends on the possibility of deriving a variational functional for the problem concerned. In most structural mechanics problems, either direct approach or variational approach based on the potential energy can be used to derive element property matrix. The energy approach has limited applications mainly in thermo-mechanic problems. It is observed that the FEM based on weighted residual approach has wide applications in most engineering problems. Further, a detailed description of finite element analysis procedure is presented in this chapter. Finite element analysis is deliberated 60

Finite Element Method with Applications in Engineering essentially in eight steps. In further development of FEM, these steps will be followed for the systematic approach.

EXERCISE PROBLEMS Using the direct approach, develop element stiffness matrix for a cantilever beam of length L subjected to a concentrated load W at the free end and find the expression for the deflection and slope at the free end. Using FEM based on the variational approach develop element property matrix for the following differential equation. Use one-dimensional linear elements for discretization. ɸ″− ɸ = x2 with boundary conditions ɸ(x = 0) = 0 and ɸ(x = 1) = 0. Using FEM based on the weighted residual approach (Galerkin's method), develop element property matrix for the following differential equation. Use one-dimensional linear elements for discretization. y (x = 0) = 0 and y (x = 1) = 0.

in the range 0 ≤ x ≤ 1, with the boundary conditions

Using FEM based on the weighted residual approach (Galerkin's method), develop element property matrix for the following differential equation. Use one-dimensional linear elements for discretization. ɸ″ + ɸ′ − 2ɸ = 0, 0 ≤ x ≤ 1, with boundary conditions ɸ(x = 0) = 0, ɸ(x =1) =1

REFERENCES AND FURTHER READING Bathe, K. J. (2001). Finite Element Procedure, Prentice Hall of India, New Delhi. Finlayson, B. A. (1972). The Method of Weighted Residuals and Variational Principles, Academic Press, New York. Huebner, K. H. (1975). The Finite Element Method for Engineers, John Wiley and Sons, New York. Mikhlin, S. G. (1964). Variation Methods in Mathematical Physics, Macmillan, New York. Oden, J. T. (1969a). A general theory of finite elements I: Topological considerations, International Journal for Numerical Methods in Engineering, 1(3). Oden, J. T. (1969b). A general theory of finite elements II: Applications, International Journal for Numerical Methods in Engineering, 1(3). Oden, J. T. (1970). Finite element analogue of the Navier stokes equations, Proceedings of ASCE, Journal of the Engineering Mechanics Division, 96(EM4). Rao, S. S. (1989). The Finite Element Method in Engineering, Pergamon Press, Oxford. Reddy J. N. (1985). An Introduction to the Finite Element Method, McGraw-Hill International, New York. Segerlind, L. J. (1987). Applied Finite Element Analysis, John Wiley and Sons, New York. Shames, I. H., and Dym, C. L. (1991). Energy and Finite Element Methods in Structural Mechanics, Wiley Eastern, New Delhi. Stasa, F. L. (1985). Applied Finite Element Analysis for Engineers, CBS Publishers Japan, New York. Tonti, E., (1969). Variational formulation of nonlinear differential equations, I, II, Bull. Acad. Roy. Belg. (Classe Sci.) 55(5): 137–165, 262–278. 61

Finite Element Method with Applications in Engineering Vainberg, M. M. (1964). Variational Methods for the Study of Nonlinear Differential Operators, Holden-Day, San Francisco. White, R. E. (1985). An Introduction to the Finite Element Method with Applications to Nonlinear Problems, John Wiley and Sons, New York. Zienkiewicz, O. C. (1980). Finite Element Method, 3rd ed., McGraw Hill, New York.

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5 Finite Elements and Interpolation Functions 5.1 INTRODUCTION As discussed earlier, domain of the problem is discretized into sub-domains or elements as the first step in finite element modelling. Each sub-domain is referred to as finite elements. A field variable is approximated in the next step by using trial functions, which are also known as interpolation or shape functions. These functions are used to form the element property matrix. A subject of utmost importance in a finite element analysis is the selection of a specific finite element for a domain discretization and a definition of the appropriate trial functions within each element. Selection of a particular shape or configuration of element depends on the problem and its dimension. For example, a line element is used for one-dimensional problem. On the other hand, triangular, quadrilateral or rectangular elements are employed for two-dimensional problems. Similarly, tetrahedron or any shape of a prism can be used to discretize three-dimensional problems. Interpolation functions can be linear, quadratic, cubic, quartic and quintic depending upon the desired accuracy and anticipated variation of the primary field variable. For example, field variable ɸ can be expressed in terms of interpolation function N of any element and its nodal value ɸ i as: ɸ = ΣNiɸi = [N] {ɸ}, where Ni = Ni(x) or Ni(x, y) or Ni(x, y, z) for one-, two- and three-dimensional analysis, respectively. Variable ɸi is the nodal value of the field variable at any specified point i. [N] is the array of interpolation functions and {ɸ} is the array of nodal variables. A problem is said to have C0 continuity if the field variable ɸ is continuous at the element interface. Additionally, if the first derivatives of the field variable are continuous across element interfaces, the problem is said to have C1 continuity. Similarly, Cr continuity can be defined if the r-th derivatives are continuous at the element interfaces. Conditions of compatibility and completeness are required in a finite element analysis. If functions appearing under integrals in the element equations described earlier contain derivatives up to the (r + 1)-th order, the compatibility requirement states that Cr continuity must be satisfied at the element interfaces and Cr + 1 continuity must be satisfied within the element. These requirements hold regardless of whether element equations (integral expressions) are derived using variational, Galerkin or any other method. Following information is required in the finite element discretization: shape of the element, number of nodes per element, type of nodal variables and type of the interpolation function used in analysis. As discussed earlier, various shapes of elements can be used depending on dimension of the problems concerned. In an analysis, number of nodes assigned to a particular element depends on the type of the nodal variable, type of interpolation function and the degree of continuity required. For domains with curved boundaries and nonlinearity, the linearly varying elements may not accurately represent variation of different field variables. It would be better to use higher order finite elements for accurate representation of field variables in many non-linear problems and irregular boundary problems with a finite element grid. Any element with a second or higher order interpolation function is called higher order element. With higher order elements such as quadratic or cubic, primary variable will vary non-linearly within the element. Total number of unknowns necessary to completely specify variation of primary unknown field in an element is referred to as the degrees of freedom of the element. Degrees of freedom are specifically identified with a single nodal point and represent field variables having a clear physical interpretation. Degrees of freedom occurring at the external and 63

Finite Element Method with Applications in Engineering internal nodes are distinguished by referring to them as nodal degrees of freedom and internal degrees of freedom. The minimum number of degrees of freedom (or generalized co-ordinates) necessary for a given element is determined, according to the completeness requirements for convergence, geometric isotropy requirement and the necessity of an adequate representation of the field variable in the formulation (See Chapter 10 for details). Beyond the minimum number of degrees of freedom, additional degrees of freedom may be included for a finite element by adding secondary external nodes or by specifying degrees of freedom of higher order derivatives of the field variable at the primary nodes. Such higher order elements with additional degrees of freedom may increase the complexity of the individual element properties. Use of higher order interpolation functions, generally, results in more accurate result in those application areas where the gradient cannot be properly approximated by a set of constant values. Fewer higher order elements may be required to obtain the same degree of accuracy in the final answers. A reduction in number of elements reduces the number of element data, which in turn decreases possibility of errors in preparing the element data. Use of higher order elements does not, however, always lead to a reduction in the total computation time. Numerical integration techniques are needed to obtain element matrices. Hence, these techniques can involve a large number of calculations by increasing the total computation time per element. It has been observed in many cases that the numerical integration itself may constitute about 70% to 80% of the total computation time. Characteristics of some commonly used simple linear type of finite elements as well as higher order finite elements in one-, two- and three-dimensions will be discussed in this chapter.

5.2 INTERPOLATION FUNCTIONS Functions used to represent behavior of a field variable within an element are called interpolation functions. These functions are also called shape functions or approximating functions. Different types of functions could be used as interpolation functions such as polynomials and trigonometric functions. As polynomials are easy to manipulate, they are used most widely as the interpolation functions. In this book, polynomials will be mainly used as the interpolation functions. Mixed or hybrid formulations will not be discussed (refer to Zienkiewicz and Taylor, 1989 for details). Depending on the problem dimension, polynomials in one, two or three independent variables may be used in the interpolation functions. 5.2.1 One-Independent Spatial Variable A general complete nth order polynomial may be written for one-dimensional analysis as

where Pn(x) is the nth order polynomial in x, and ai (i = 0, 1, … n) are the coefficients of xi. Some examples of polynomials in one-independent variable are as follows.

64

Finite Element Method with Applications in Engineering 5.2.2 Two-Independent Spatial Variables A general polynomial for two-dimensional analysis may be written as

Some examples of polynomials in two-independent variables are,

Any higher order polynomial can be written in a similar manner. 5.2.3 Three-Independent Spatial Variables A polynomial for three-dimensional analysis may be written as

Some examples of polynomials in three-independent variables are

Any higher order polynomials for three-independent variables can be written in a similar manner. A polynomial series in one or more variables is complete if it is of a high enough degree and if no terms are omitted in the series. Thus, polynomials of Eqs. (5.1), (5.5) and (5.8) are complete. Coefficients aij or aijk in a polynomial series representing the field variable are referred to as the generalized co-ordinates. These generalized co-ordinates are independent parameters that specify or fix the magnitude of the prescribed distribution of the polynomial. On the other hand, the nature of the prescribed distribution is determined by the order of the polynomial selected. These co-ordinates have no direct interpretation but are rather the linear combinations of the physical nodal degrees of freedom. In addition, they are not identified with any particular node.

5.3 ONE-DIMENSIONAL ELEMENTS Generally, line elements are used with different order of interpolation function for onedimensional finite element modelling. Various properties of linear line elements and related interpolation functions are discussed in this section. 5.3.1 Line Element: Linear Interpolation Function As discussed earlier, line element is used to discretize one-dimensional problems. Let the field variable ɸ be a function of x, i.e., ɸ = ɸ(x). Let B and D be the points on the x-axis with co-ordinates xi and xj, respectively and the field variable values be ɸi and ɸj. It is required to evaluate the value of ɸ at any point between B and D. Let the co-ordinate of point G be x, where the ɸ value is to be determined. From geometry in Figure 5.1,

65

Finite Element Method with Applications in Engineering

Figure 5.1 Line element with linear interpolation function

Expression (5.11) can also be derived from the generalized polynomial expressed as follows. Let ɸ = ɸ(x) be expressed in linear form as ɸ(x) = a1 + a2 x, or

At x = x1, ɸ(x) = ɸ1 and at x = x2, ɸ(x) = ɸ2. Hence, Eq. (5.14) can be expressed in the following form.

66

Finite Element Method with Applications in Engineering Combining Eqs. (5.14) and (5.17),

is obtained; where

Figure 5.2 Line element representation In a similar manner, location of any point xp within the element as shown in Figure 5.2 can be expressed as Derivatives of the field variable ɸ can be obtained as

Linear variation of Ni and Nj over the element is shown in Figure 5.3. Example 5.1 Find the value of the field variable ψ and its first derivative at a place x = 0.6. Consider xi= 0, xj = 2, ψi = 1.2 and ψj = 1.5. Solution: Length of the element, L = xj- xi = 2. Value of Ni at x = 0.6 is Ni = 1 – x/L = 1 – (0.6/2) = 0.7. Value of Nj at x = 0.6 is Nj = x/L = 0.6/2 = 0.3. Thus, the value of ψ(x = 0.6) = Ni ψi + Nj ψj = 0.7(1.2) + 0.3(1.5) = 1.29. The first derivative of the field variable over the element is dψ/dx = (ψj – ψi)/L = (1.5 1.2)/2 = 0.15. It may be noted that the first derivative is constant throughout the element.

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Finite Element Method with Applications in Engineering

Figure 5.3 Variations of Ni and Nj over the element

Figure 5.4 Length co-ordinates 5.3.1.1 Integration formula for length co-ordinates When the interpolation functions Ni and Nj are linear functions of length co-ordinates only, they can be treated as length co-ordinates. For example, consider Figure 5.4 where distance ij = L, p is an intermediate point, ip = Lj, pj = Li. Now Ni and Nj can be written as Ni = Li/L and Nj= Lj/L. If L = L, then Ni = 1. If Lj = L then Nj = 1 which is true. Hence, (Li/L) and (Lj/L) can be treated as length co-ordinates. Integration of length co-ordinates over length of an element can be done with the aid of following formula (Heubner, 1975)

Eq. (5.21) is applicable only if Ni and Nj are linear functions of x and the exponents a and b are integers. Example 5.2 Evaluate Solution:

The integral I can be written as + 1)! =L/6.

. Here, a = 1, b = 1, therefore, I = (1! 1! L)/(1 + 1

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Finite Element Method with Applications in Engineering 5.3.2 Quadratic Interpolation Function For a line element with linear interpolation function, 2 nodes are sufficient. However, 3 nodes are needed for a line element with quadratic interpolation function. Similarly, if n is the order of interpolating polynomials for a line element, n + 1 nodes are needed. For an element shown in Figure 5.5, variation of a field variable ɸ over the element can be written as

The boundary conditions over the element are At node 1: ɸ = ɸ1 and s = 0 At node 2: ɸ = ɸ2 and s = L/2 At node 3: ɸ = ɸ3 and s = L

Figure 5.5 Line element–quadratic type Putting the boundary conditions in the polynomial Eq. (5.22),

is obtained. Thus,

From Eqs. (5.22) and (5.25), the polynomial ɸ can be expressed as Polynomial ɸ can also be written as Comparing Eqs. (5.27) and (5.28), it can be seen that

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Finite Element Method with Applications in Engineering

Example 5.3 Find the value of ɸ and its derivative at a point P(s = 0.25 L) shown in Figure 5.3 given that ɸ1= 4, ɸ2 = 4.5 ɸ3 = 6.

Figure 5.6 Example 5.3

Figure 5.7 Line element with 4 nodes Solution: Variation of field variable over the element is ɸ = [N]{ɸ}, where {ɸ}T = [4 4.5 6]. Shape functions N1,N2,N3 at the point P are required to be computed. Using Eq. (5.31), N1(s = 0. 25 L) = 1 – 3(0. 25L)/L + 2(0. 25 L)2/L2 = 3/8 N2(s = 0. 25 L) = 4(0. 25 L)/L – 4(0. 25 L)2/L2 = 3/4 N3(s = 0. 25 L) = – (0. 25 L)/L + 2(0. 25 L)2/L2 = –1/8 Hence, ɸ(s = 0.25 L) = Σ Ni ɸi = 4.125 Derivative of the field variable at the given point P(0.25 L) is dɸ/ds = Σ ɸi (dNi /ds). From Eq. (5.32), [dNi /ds] at the given point P is [–2/ L, 2/L, 0]. Hence, dɸ/ds = – (4)(2)/L + (4. 5) 2/L = 1/L.

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Finite Element Method with Applications in Engineering 5.3.3 Cubic Interpolation Function The cubic interpolation function for a line element shown in Figure 5.7 can be derived in a similar manner as that of quadratic function. In this case, 4 nodes are needed and the polynomial for ɸ will be of the form

The boundary conditions are as follows,

Putting these conditions in polynomial in Eq. (5.33),

is obtained. Thus, the polynomial can be written as

From Eq. (5.37),

From Eqs. (5.36) and (5.38), the polynomial ɸ becomes The polynomial ɸ can also be written as Comparing Eqs. (5.40) and (5.41), it can be shown that

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Finite Element Method with Applications in Engineering

The first derivative of these functions Ni with respect to s are

Example 5.4 Evaluate the field variable ɸ at a point P(s = L/6) for a line element with cubic interpolation function and also its first derivative at the same point, given that {ɸ}T = [4 4.5 6 9]. Solution: From Eqs. (5.43) and (5.44), at s = L/6; [N] and [dN/ds] values are obtained as [0.3125 0.9375 –0.3125 0.0625] and [–2.875 2.625 0.375 –0.125]/L, respectively. Hence, ɸ(L/6) = [N]{ɸ} = 4.15625 and (dɸ/ds) = [dN/ds]{ɸ} = 1.4375/L. 5.3.4 Lagrangian Form of Interpolation Function It has been shown in the preceding section that one has to resort to matrix inversion to derive shape functions Ni. Such a process can be quite cumbersome for higher order polynomials. Same functions can be derived by using Lagrangian polynomial easily for any order.

Figure 5.8 Line element The Lagrangian polynomial of order n is given by

It can be easily seen that Ni(x) = 1 for x = xi and Ni(x) = 0 for x = xm, m ≠ i. a) Linear Function: For linear function, n = 2. Thus,

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Considering Figure 5.8, s = x – x1 and L = x2 – x1. Thus, from Eqs. (5.46) and (5.47), N1 = (s + x1– x2)/{–(x2 – x1)} = 1 – (s/L) and N2 = s/L. It may be noted that identical expression has been derived earlier in Eq. (5.18). b) Quadratic Function: For quadratic function, n = 3. By using Eq. (5.45),

can be obtained. Further, it can be shown from Figure 5.9 that

Thus, the interpolation function can be written from Eqs. (5.48), (5.49), (5.50), (5.51) and (5.52) as

Figure 5.9 Line element with 3 nodes

Figure 5.10 Line interpolation function

element

with

linear

It can be noticed that Eqs. (5.53), (5.54) and (5.55) are identical to Eq. set (5.31). In a similar manner higher order interpolation function can be derived easily using Lagrangian interpolation polynomial.

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Finite Element Method with Applications in Engineering 5.3.5 Further Higher Order Elements in One-Dimension Further higher order interpolation functions are discussed here in one-dimension with more number of nodes in each element. Procedure of deriving interpolation functions and its properties is very similar to that derived for quadratic type interpolation functions earlier. Here a special indexing scheme is used. The scheme would be initially demonstrated for linear, quadratic and cubic type line elements before explaining further to higher order elements (Heubner, 1975). Consider the linear type line element shown in Figure 5.10. Let L1, and L2 be length co-ordinates defined as

Nαβ(L1,L2) are interpolation functions, where α + β = n for nth order polynomial.

With the use of Eq. (5.58), interpolation functions are derived for various line elements. Linear Interpolation Function: From Figure 5.10.

Figure 5.11 Quadratic type line element Quadratic Interpolation Function: Refer to Figure 5.11.

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Cubic Interpolation Function: Referring to Figure 5.12, α = 3, β = 0 at node 1; α = 2, β = 1 at node 2; α = 1, β = 2 at node 3; and α = 0, β = 3 at node 4.

Figure 5.12 Line element with cubic interpolation function

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Quartic Interpolation Function: Quartic type line element has 5 nodes in one element as shown in Figure 5.13. Using the formula

Figure 5.13 Line element with quartic interpolation function 76

Finite Element Method with Applications in Engineering Thus, At Node 1:

At Node 2:

At Node 3:

At Node 4:

At Node 5:

Quintic Interpolation Functions: Quintic type line element has got 6 nodes in one element as shown in Figure 5.14.

From Figure 5.14, using the formula

Figure 5.14 Line element with quintic interpolation function 77

Finite Element Method with Applications in Engineering Similarly,

Then, the interpolation functions can be obtained as follows. At Node 1:

At Node 2:

At Node 3:

At Node 4:

At Node 5:

At Node 6:

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In a similar way, further higher order interpolation functions for other one-dimensional elements and its properties can be easily derived.

5.4 TWO-DIMENSIONAL ELEMENTS Triangular, rectangular and quadrilateral elements are most widely used for twodimensional finite element modelling. Various properties of these elements and related interpolation functions are discussed in this section. 5.4.1 Triangular Element: Linear Interpolation Function in Cartesian Co-ordinates The triangular element was formulated by Turner et al. (1956). It is commonly used element shape for two-dimensional problems. Triangular elements can be used when the field variable is a function of two independent variables, namely x and y. Variation of field variable ɸ over the domain of an element can be written in polynomial form as where α1, α2 and α3 are the coefficients to be evaluated. Let the nodes of the element be designated as 1, 2, 3, as shown in Figure 5.15, in counterclockwise direction with known co-ordinates (x1, y1), (x2, y2) and (x3, y3), respectively. Let where [P] = [1 x y] and {α}T = [α1 α2 α3]. If ɸ has values ɸ1, ɸ2 and ɸ3 at nodes 1, 2 and 3,

Combining Eqs. (5.84) and (5.86), is obtained. Field variable ɸ can also be written as Here, [N] = [P] [G]–1 and

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Figure 5.15 Triangular element As shown in Figure 5.16, variation of Ni, I = 1, 2 and 3 is such that Ni at node i is 1 and zero at the other nodes. Independent variables x and y can be expressed in terms of nodal co-ordinates as where {x}T = [x1 x2 x3] and {y}T = [y1 y2 y3].

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Figure 5.16 Variations of N1, N2 and N3 over the triangular domain Example 5.5 The triangular element used for ground water flow simulation is as shown in Figure 5.17. The nodal co-ordinates are (x1 = 1, y1 = 2), (x2= 4, y2 = 0.5) and (x3 = 3, y3 = 4). The nodal values of hydraulic heads {ɸ} at different nodes are [3.5, 2.2, 4.4], respectively. Find the value of hydraulic head ɸ at point (2.5, 2.5).

Figure 5.17 Element for example 5.5 Solution: The value of ɸ at point p, ɸp can be written as

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Example 5.6 A point sink (well) is located at a point p(x = 2.5, y = 2.5) whose strength Q is 0.2 m3/min in a region as given in Example 5.5. Distribute the strength of the sink proportionately to the nodes 1, 2 and 3. Solution: From Example 5.5, values of N1, N2 and N3 are 3.25/9, 2/9 and 3.75/9, respectively at point P. Contribution of sink at node 1 is N1Q = 3.25(0.2)/9 = 0.07222 m3/min. Similarly, at node 2 it isN2Q = 0.0444 m3/min. and at node 3 the contribution is N3Q = 0.08333 m3/min. Note that (N1 + N2 + N3)Q = 0.2. First Derivatives of the Field Variable It was shown in Eq. (5.88) that ɸ = N1 ɸ1 + N2 ɸ2 + N3 ɸ3. Then, the first derivatives of the field variable ɸ with respect to x and y can be written as

Now

Therefore,

Further, it can be deduced from Eq. (5.95) that Thus,

82

Finite Element Method with Applications in Engineering In a similar manner,

Example 5.7 Find the values of (∂ɸ/∂x) and (∂ɸ/∂y) for the data given in Example 5.5. Solution:

5.4.2 Triangular Element—Area Co-ordinates Generation of interpolation functions for a triangular element can be greatly simplified if the area co-ordinates are chosen. Let A be the area of the triangle 123 shown in Figure 5.18. If p is the point inside the triangle, A1 = area (p23), A2 = area (p13), A3 = area (p12). Further, by letting L1, L2 Figure 5.18 Area co-ordinates and L3 to be the area coordinates, L1 = A1/A, L2 = A2/A and L3 = A3/A. If point p lies on edge 23, A1 = 0, hence L1 = 0. If point p coincides with point 1, L1 = 1. These two conditions satisfy that L1qualifies for N1. In a similar way, it can be shown that L2 and L3 satisfy conditions for N2 andN3. Field variable ɸ can, thus, be written as

This is true only for linear interpolation functions. Derivative of field variable with respect to x and y can be written as

5.4.3 Integration Formula for Triangular Elements Following formula can be used to evaluate integrals usually required in finite element formulations using triangular elements (Connors and Brebbia, 1976)

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where N1, N2 and N3 are the linear functions of (x, y) and i, j, k are integers. Example 5.8 Evaluate I = ∫∫N2dA over the triangular element. Solution: This integral can be written as ∫∫ N11 N 22 N30 dA. Here, i = 0, j = 1 and k = 0. Using Eq. (5.106), I = (0! 1! 0!)(2A)/(0 + 1 + 0 + 2)! = A/3. Example 5.9 Evaluate I = ∫∫N1 N2 N2 dA over the triangular element. Solution: This integral can be written as ∫∫N11 N22 N30 dA. Here, i = 1, j = 2 and k = 0. Using Eq. (5.106), I = (1! 2! 0!)(2A)/(1 + 2 + 0 + 2)! = A/30. 5.4.4 Triangular Element—Quadratic Function If the procedure given in section 5.3.5 is followed to derive shape functions for a triangular element with quadratic interpolation functions, evaluation of resulting [G]–1 would be very tedious. Hence, a different strategy is to be used. A convenient method of obtaining interpolating polynomial of higher order was suggested by Silvester (1969) by considering triple index numbering scheme: Nαβγ(L1, L2, L3) such that α + β + γ = n, n is the order of the interpolating polynomial; and L1, L2, L3 are the area co-ordinates, as shown in Figure 5.19(Huebner, 1975).

Figure 5.19 Triangular element with triple index numbering scheme

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Integration of area co-ordinates over the area of an element can be done with the aid of the formula (Huebner, 1975)

Example 5.10 Evaluate following integrals over a triangle with 6 nodes where Ni is quadratic in nature.

Solution:

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Finite Element Method with Applications in Engineering

5.4.5 Triangular Element—Cubic Interpolation Function It can be shown by using the triple index numbering scheme as shown in Figure 5.20 that the cubic interpolation functions for 10 nodded triangular elements are

Figure 5.20 Triangular element with 10 nodes 86

Finite Element Method with Applications in Engineering Example 5.11 Evaluate and at point P (1, 4) for a triangular element shown in Figure 5.21 using cubic interpolation function. Solution: For the given data, ai, bi, ci, i = 1,2,3 and A can be Figure 5.21 Example problem 5.11 obtained from Eq. (5.90), (5.91), (5.92) and (5.93) as a1 = 16, a2 = 0, a3 = 0; b1 = –4, b2 = 6 and b3 = –2; c1 = –2, c2 = –1, c3 = 3 and A = 8. Hence, L1, L2, L3 are

5.4.6 Two-Dimensional Rectangular Elements Consider a rectangular element as shown in Figure 5.22. Variation of field variable ɸ over the domain of the rectangular element can be written in polynomial form as where αi, i = 1,2,3,4 are the coefficients to be evaluated. By letting values of ɸ to be ɸ1, ɸ2, ɸ3and ɸ4 at nodes 1, 2, 3 and 4, respectively, matrix equation can be written as

where {ɸ}T = [ɸ1 ɸ2 ɸ3 ɸ4], {α}T = [α1 α2 α3 α4].

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Finite Element Method with Applications in Engineering

Figure 5.22 Rectangular element Further, {α} = [G]-1 {ɸ}, where

Thus,

Value of N1 at node 1 is 1 and 0 at other nodes. Similarly, value of N2 at node 2 is 1 and 0 at other nodes. At any point within the element, N1(p) + N2(p) + N3(p) + N4(p) = 1 should be satisfied. Derivatives of interpolation functions can be written as

Field variable ɸ can be expressed in terms of nodal ɸ values as

Since ɸ is at a particular node, it is not varying with respect to x and y. Hence, the first derivative with respect to x and y are

88

Finite Element Method with Applications in Engineering Similarly,

Sometimes, it is more convenient to use non-dimensional interpolation functions of the form

where Ni is a function of (ξ, η) or Ni = N(ξ, η) at node i, ξ and η are the local co-ordinates such that -1 ≤ ξ ≤ 1, -1 ≤ η ≤ 1, ξi and ηi are the local co-ordinates of the nodes such that ξ = ±1 and η = ±1. For example, ξ 1 = -1 and η1 = -1 at node 1 and ξ2 = 1 and η2 = -1 at node 2; ξ3 = 1 and η3 = 1 at node 3 and ξ4 = -1 and η4 = 1 at node 4, as shown in Figure 5.23. Thus,

Figure 5.23 Rectangular element, local co-ordinates

Local co-ordinates are often referred to as natural or intrinsic co-ordinates. Derivatives of interpolation functions with respect to ξ and η are

5.4.7 Rectangular elements—Lagrangian Form in natural and Cartesian co-ordinates Interpolation functions for rectangular elements with sides parallel to the global axes can be developed using Lagrangian interpolation function. Consider a line segment of length 2 units with 3 nodes as shown in Figure 5.24(a). By applying Lagrangian interpolation function,

These functions in global co-ordinates can be converted into natural co-ordinates by noting that

Figure 5.24a Line segment with 3 nodes in x–direction Figure 5.24b Line segment with 3 nodes in y-direction 89

Finite Element Method with Applications in Engineering

Similarly, choosing η as natural co-ordinate in the y-direction as shown in Figure 5.24(b),

By using Eqs. (5.135), (5.136), (5.137) and (5.138), interpolation functions for a rectangular element with 9 nodes can be constructed.

Figure 5.25 Two-dimensional Lagrangian type element

Interpolation functions at different nodes of the element can now be written as node 1 of the quadrilateral that is formed by using nodes 1 and 3 of the line elements shown in Figure 5.25. Hence,

In a similar way, other interpolation functions can be derived. Rectangular elements—Lagrangian Interpolation Function in Cartesian Co-ordinates Consider the 9 nodded rectangular elements (Figure 5.26) with Lagrangian interpolation function in Cartesian co-ordinates. The Lagrangian interpolation function, Ni in Cartesian co-ordinates can be derived in a very similar manner as done for rectangular elements with Lagrangian form in natural coordinates as discussed earlier. Interpolation functions can be written

Figure 5.26 Rectangular elements with 9 nodes

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Finite Element Method with Applications in Engineering as

5.4.8 Isoparametric elements Domains with curved irregular boundaries are often encountered in FEM analysis where straight edged elements are difficult to use in discretization. Elements with curved edges as shown in Figure 5.27 can be used advantageously for accurate representation of the domain. Curve-edged elements of quadrilateral type were first introduced in FEM by Taig (1961). Later many other researchers extended the idea of curved elements. Curvededged elements are often formulated by transforming simple geometric shapes in some local co-ordinate system into distorted shapes in the global Cartesian co-ordinate system and then by evaluating element properties for the resulting curve-sided element (Huebner, 1975). Mapping functions are used to transform simple geometric shapes to curved one. If the functional representation of field variable and mapping functions are expressed by interpolation functions of same order, the curve-edged element used is referred to as an isoparametric element. If element geometry is described by lower order polynomial than used for the field variable, the element is termed as sub-parametric element. On the other hand, element is called super-parametric element if the element is described by higher order polynomial compared to the field variable. Out of three categories of curve-edged elements, isoparametric element is most commonly used. Hence, properties of quadrilateral isoparametric elements are discussed here. To demonstrate properties of isoparametric element in two-dimensions, consider a parent element of four nodded rectangular elements and its corresponding curve edged isoparametric element as shown in Figure 5.27. As discussed earlier, field variable ɸ can be expressed in terms of nodal ɸ values as

Nodes in the ξ—η plane may be mapped into corresponding nodes in the x—y plane by defining 91

Finite Element Method with Applications in Engineering

Since ɸ is specified as a function of the local co-ordinates ξ, and η, it is necessary to express derivatives of ɸ as a function of ξ and η. This can be done as follows.

Figure 5.27 Element with curved edge and its mapping to natural co-ordinate system Further, x and y are functions of local co-ordinates ξ and η such that x = x(ξ, η) and y = y(ξ, η). Thus, following transformations can be used to formulate element matrices.

Eqs. (5.147) and (5.148) can be written in matrix form as

92

Finite Element Method with Applications in Engineering These derivatives as given by Eqs. (5.145) and (5.146) are possible only when inverse of the Jacobian, [J]-1exists. This inverse exists provided interior angles of the quadrilateral element are less than 180o. Numbering of nodes must follow the counter–clockwise direction to produce positive |J| for the coordinate system shown in Figure 5.27. Since,

can be written as

Figure 5.28 Cubic interpolation function in one-dimension In the above discussion, two-dimensional quadrilateral element has been used to illustrate the concept of isoparametric element. The same concept can be extended to other type of elements or for one-dimensional and three-dimensional domains. 5.4.9 Lagrangian Interpolation Functions for two-Dimensional elements Here, the Lagrangian interpolation function elaborated in Section 5.4.7 is further discussed. The Lagrangian interpolation function can be expressed, with reference to Figure 5.28, as

Using the expression (5.160), interpolation functions for nodes 1, 2, 3, 4 can be written as

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Finite Element Method with Applications in Engineering

Figure 5.29a Natural co-ordinate system in ξ direction

Interpolation functions in natural co-ordinates can be derived using the above relations, with reference to Figure 5.29(a) as follows.

Figure 5.29b Natural co-ordinate system in η-direction

Similar interpolation functions in natural co-ordinates in the η direction (Figure 5.29b) can be described as

94

Finite Element Method with Applications in Engineering Using the above expressions, interpolation functions for a Lagrangian element with 16 nodes (Figure 5.30) can be derived as

Figure 5.30 Lagrangian element with 16 nodes

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Finite Element Method with Applications in Engineering

5.4.10 Two-Dimensional serendipity elements It can be seen from Sections 5.4.4 and 5.4.5 that internal nodes are necessary to maintain completeness of polynomial for higher order triangular element. Further, it can be inferred from Section 5.4.9 that a number of identical nodes on two opposite sides are required for higher order Lagrange elements. However, nodes on sides cannot be added without introducing additional interior nodes. A sample of some Lagrange elements is shown in Figure 5.31. Polynomials are cubic in both directions as shown in Figure 5.31(a). On the other hand, polynomial is quadratic in η direction (Figure 5.31b) and is linear in η direction (Figure 5.31c). It may be remarked that Lagrange polynomials are never complete and they possess geometric isotropy only when equal number of nodes are present in both directions. It is possible to generate interpolation functions dependent on nodal values placed on element boundary (Zienkiewicz and Taylor, 1989). These elements, known as the serendipity elements, were derived by chance. They are very useful since their construction can be formalized to a concise numerical algorithm. Here step by step construction of interpolation function is demonstrated through an 8-node quadratic element in natural co-ordinates (ξ, η). To facilitate the construction, following onedimensional interpolation functions are used for s = ξ or η. Figure 5.31 Sample Lagrange elements (a) Cubic–cubic (b) Cubic– quadratic (c) Cubic – linear 96

Finite Element Method with Applications in Engineering

The basic 4-node rectangle and the targeted 8-node element with node numbering are illustrated in Figure 5.32 in ξ – η coordinates. For the basic 4-node linear element, interpolation functions are

Figure 5.32 Serendipity elements (a) Basic 4-node element (b) Targeted 8node element

It is observed from Figure 5.33 that N5(ξ, η)/2 needs to be subtracted from N1(ξ, η) to construct 5-node element. Here the revised N1(ξ, η) can be then evaluated as

Thus, the field variable To extend this construction to 6-node element, it is observed from Figure 5.34 that N5/2 andN6/2 need to be subtracted from N2 and N6/2 from N3, where N6 = n2(1)(ξ)n2(2)(η), which yields

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Figure 5.33 Generation of 5-node serendipity element

Similarly,

Figure 5.34 Generation of 6-node serendipity element 98

Finite Element Method with Applications in Engineering

By redefinition, Continuing the process, 8-node quadratic element can be generated with interpolation functions

The above equations for linear and quadratic elements in compact form can be listed as follows. 1. Linear element

2. Quadratic element Corner nodes:

Mid-side nodes:

5.5 THREE-DIMENSIONAL ELEMENTS Generally, three-dimensional elements maintain C0 continuity. Thus, continuity of field variable along faces of an element is the basic requirement. Also, it needs to be ensured that nodes on element interfaces uniquely define variation of function on interfaces. As compared to the basic two-dimensional triangle and quadrilateral elements, the analogous elements in three-dimensions are tetrahedral and hexahedral. In this section, characteristics of basic three-dimensional elements such as tetrahedral and prism elements are discussed. 99

Finite Element Method with Applications in Engineering 5.5.1 Tetrahedral Elements Four-nodded tetrahedral is the simplest element in three-dimensions (Figure 5.35). Tetrahedral elements can be used when field variable is a function of three independent variables, namely x, y and z. Variation of field variable ɸ over the domain of element can be written as

where [P] = [1 x y z] and {α}T = [α1 α2 α3 α4]. Figure 5.35 Tetrahedral element

By letting nodes of the element designated as 1, 2, 3 and 4 with known co-ordinates (x1, y1, z1), (x2, y2, z2), (x3, y3, z3) and (x4, y4, z4), respectively, it can be shown that

By combining, Eqs. (5.198) and (5.199), is obtained. The field variable ɸ can also be written as where [N] = [P] [G]–1 with

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Finite Element Method with Applications in Engineering Other constants can be obtained through a cyclic permutation of subscripts 1, 2, 3 and 4. By using Eqs. (5.198), (5.202) and (5.203), it can be shown that

Therefore,

Let p be any point within the tetrahedral chosen as shown in Figure 5.36. Let V1, V2, V3 and V4be volumes of tetrahedrals p234, p134, p124 and p123, respectively. Then the volume co-ordinates Li can be expressed as In this case, Ni = Li = Li(x, y, z) are linear interpolation functions.

Figure 5.36 Tetrahedral element—volume co-ordinates Derivatives of field variable ɸ can be written as

Integration formula over an element for linear tetrahedral element can be shown to be (Huebner, 1975)

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where ! is for factorial, i, j, k and l are integers, N1, N2, N3 and N4 are linear functions of x, y, z. Example 5.12 Nodal co-ordinates of tetrahedral element are (x1 = 0, y1 = 0, z1 = 0), (x2 = 1, y2 = 1, z2 = 3), (x3= 2, y3 = 2, z3 = 0), (x4 = 0, y4 = 3, z4 = 0). Find linear interpolation functions and value of field variable ɸ at a point (1.5, 1.5, 2), given that {ɸ}T = [3 2 2.5 3.5]. Solution: From Eq. (5.207),

Then, [N] = [P] [G]–1. Thus, N1 = (18 – 3x – 6y – 3z)/18; N2 = 6z/18; N3 = (9x – 3z)/18; N4 = (–6x + 6y)/18; At point (1.5, 1.5, 2), [N] = [–0.0833 0.667 0.4167 0], therefore, ɸ = [N] {ɸ} = 2.124. Example 5.13 Nodal co-ordinates of tetrahedral element are: (x1= 0, y1= 0, z1= 0), (x2= 1, y2= 1, z2= 3), (x3= 2, y3= 2, z3= 0), (x4= 0, y4= 3, z4= 0). Determine linear interpolation functions in terms of volume co-ordinates and find value of ɸ at a point (1, 1, 2), given that {ɸ}T = [3 2 2.5 3.5]. Solution: From Example 5.12, volume of tetrahedral is V = 3. Various required volumes are:

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Thus, L1 = V1/V = 1/6; L2 = V2/V = 2/3; L3= V3/V = 1/6; and L4 = V4/V = 0; Finally, [N] = [L] = [1/6, 2/3, 1/6, 0]; Thus, ɸ = [L] {ɸ} = 2.25. Example 5.14 Evaluate ∂ɸ/∂x, ∂ɸ/∂y, ∂ɸ/∂z at point p (1.5, 1.5, 2) for the data given in Example 5.12. Solution: From Example 5.12, N1 = (18 – 3x – 6y – 3z)/18; N2 = (0 + 0 + 0 + 6z)/18; N3 = (9x + 0 + 0 – 3z)/18; N4 = (–6x + 6y + 0 + 0)/18; and 6V = 18. Further, {b}T = [–3 0 9 –6]; {c}T = [–6 0 0 6]; {d}T = [–3 6 –3 0] and {ɸ}T = [3 2 2.5 3.5] From Eq. (5.211),

From Eq. (5.212),

From Eq. (5.213),

Example 5.15 Evaluate following integrals over tetrahedral element.

Solution: From the integration formula (5.214),

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5.5.2 Tetrahedral elements: Quadratic Interpolation Function Quadratic interpolation functions for tetrahedral elements can be easily derived by following the procedure, which is analogous to the one used for triangular elements (Silvester, 1969). Thus, referring to Section 5.4.4 the interpolation functions for the tetrahedral element shown in Figure 5.37 are

Here Li = Li (x, y, z) = (ai+ bix + ci y + diz)/6V, where ai, bi, ci, di and V are as defined by Eq. (5.205) and (5.206). 5.5.3 Tetrahedral elements: cubic Interpolation Function Cubic interpolation functions for tetrahedral elements can be easily derived by following the procedure, which is analogous to the one used for triangular elements (Silvester, 1969). Thus, referring to Section 5.4.5 the interpolation functions for the tetrahedral element shown in Figure 5.38 are

5.5.4 Three-Dimensional elements—prismatic elements A variety of prismatic elements such as triangular prisms, rectangular prism and hexahedral can be used in the three-dimensional discretization. Shape of a typical undistorted prismatic element is shown

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Figure 5.37 Tetrahedral elements with quadratic interpolation function

Figure 5.38 Tetrahedral element with cubic interpolation function in Figure 5.39. By choosing the co-ordinate system and node numbering as shown, interpolation functions for different nodes can be written as

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Figure 5.39 Rectangular prism element

First derivatives of interpolation functions with respect to x, y and z are as follows.

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5.5.5 Three-Dimensional elements in local co-ordinates Interpolation functions for a prismatic element in the local (or natural or intrinsic) dimensionless co-ordinates ξ, η and ζ in x, y and z directions can be defined in a manner similar to the one used for two-dimensional system. Let ξ = (x – xc)/a; η = (y – yc)/b and ζ = (z – z)/c in Figure 5.39, where –1 ≤ ξ ≤ 1; –1 ≤ η ≤ 1; –1 ≤ ζ≤ 1. Then, interpolation function for any node i can be written as where ξ i, η i, ζ i are the local co-ordinates. Or,

Further,

For a distorted element, x = x(ξ, η, ζ), y = y(ξ, η, ζ) z = z(ξ, η, ζ). Then,

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In matrix form, Eq. (5.231) can be written as

5.5.6 Three-Dimensional serendipity elements By following procedure and analogous notations described in Section 5.4.10, interpolation functions for three-dimension rectangular prismatic elements shown in Figure 5.40 can be derived. The origin of the ξ,η,ζ, co-ordinate system is at the centroid of the element. Here, these functions are listed in compact forms.

Figure 5.40 Rectangular prismatic serendipity elements (a) 8-node element (b) 20-node element 1. Linear element

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Finite Element Method with Applications in Engineering 2. Quadratic element Corner nodes:

Typical mid-side node:

with the remaining mid-side node expression obtained by changing variables.

5.6 CLOSING REMARKS Different types of finite elements in linear and higher order forms suitable for a variety of problems are presented in this chapter. Characteristics of various polynomials are also discussed and their properties are presented. With the basic idea of polynomial interpolation functions and Lagrangian functions, general procedures for constructing finite elements and characteristics have been discussed. Using the linear interpolation concepts, element properties for line, triangular, rectangular, tetrahedral and prism elements have been elaborated. Using the higher order interpolation concepts, element properties for one-dimensional line element, two-dimensional triangular and rectangular elements are discussed. Each case has been explained in local and natural co-ordinate systems. Also, the basics of isoparametric element and its properties are introduced. Further, two-dimensional and three-dimensional serendipity elements are derived. In finite element analysis, use of higher order elements can reduce the number of elements to be used for a particular problem but more computational time is required for the higher order elements. Depending upon the problem to be solved, suitable elements can be chosen for discretization, and element properties can be derived for solution of the problem. Details of interpolation functions and element properties given in this chapter can be used as general guidelines for all types of elements used for discretization.

EXERCISE PROBLEMS 1. Using the linear interpolation function for line element, find the value of the field variable ɸ and its first derivative at a place x = 0.8 given that xi = 0, xj = 5, ɸi = 2.5 and ɸj = 7.0. 2. Using the integration formula for linear type line element, evaluate 3. Temperature distribution in a steel plate is simulated using the linear type triangular element with the nodal co-ordinates of (x1 = 1, y1 = 1), (x2 = 8, y2 = 0.5) and (x3 = 4, y3 =5). The nodal values of temperature {ɸ} at different nodes are {25, 27, 23}, respectively. Find the value of temperature ɸ at point (3.5, 3.5). 4. The triangular element is used for ground water flow simulation. The nodal coordinates are (x1 = 1, y1 = 1), (x2 = 8, y2 = 0.5) and (x3 = 4, y3 = 5). The nodal values of hydraulic heads {ɸ} at different nodes are {12.5, 12.2, 12.8}, respectively. A point source (recharge well) is located at a point p(x = 3.5, y = 3.5) whose strength Q is 0.3 m3/min Distribute the strength of the source proportionately to the nodes 1, 2 and 3. 5. Find the values of (∂ɸ/∂x) and (∂ɸ/∂x) for the data given in problem 3. 6. Using the integration formula for linear type triangular element evaluate I = ∫∫ N12 N2N3 dA over the triangular element.

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Finite Element Method with Applications in Engineering 7. Evaluate (∂Ns/∂x) and (∂Ns/∂y) at a point P(1, 4) for the triangular element with quadratic interpolation function. 8. The co-ordinates of the linear tetrahedral element are: (x1 = 2, y1 = 2, z1 = 0), (x2 = 3, y2= 3, z2 = 4), (x3 = 3, y3 = 3, z3 = 0), (x4 = 5, y4 = 4, z4 = 0). Determine the linear interpolation functions and the value of field variable ɸ at a point (2, 2,1.5), given that {ɸ}T = [5.5 7 6.7 3.5]. 9. Evaluate ∂ɸ/∂x, ∂ɸ/∂y, ∂ɸ/∂z at point p(2, 2, 3) for the data given in problem 7. 10. Using the integration formula for linear type tetrahedral element, evaluate the following integrals over the tetrahedral element: (1)

11. Using the quadratic interpolation function for line element, find the value of the field variable ɸ and its first derivative at a place x = 0.8 given that xi = 0, xj = 5, ɸi = 2.5 and ɸj = 7.0. 12. Calculate the field variable ɸ at a point P(s = L/3) for a line element with cubic interpolation function and also its first derivative at the same point, given that {ɸ}T = [2 3 5 5 7.5]. 13. Using the integration formula for quadratic type triangular element with 6 nodes, evaluate I = ∫∫N12 N2 N3 dA over the triangular element. 14. The temperature distribution in a steel plate is simulated using the quadratic type triangular element with the nodal co-ordinates of (x1 = 1, y1 = 1), (x2 = 8, y2 = 0.5) and (x3= 4, y3 = 5). Find the value of (∂Ns/∂x) and (∂Ns/∂y) at a point P (3.5, 3.5). 15. Assuming cubic interpolation functions, for the problem in question 4, determine (∂Ns/∂x) and (∂Ns/∂y) at a point P(3, 4). 16. Using the special indexing scheme for line element, derive the interpolation functions for an element with 8 nodes.

REFERENCES AND FURTHER READING Bathe, K. J. (2001). Finite Element Procedure, Prentice Hall of India, New Delhi. Connors, J. J., and Brebbia C. A., (1976). Finite Element Techniques for Fluid Flow, Newnes-Butter worths, London. Chapra S. C., and Canale R. P. (2002). Numerical Methods for Engineers, 4th ed. Tata McGraw-Hill, New Delhi. Desai, C. S., and Abel J. F., (1987). Introduction to the Finite Element Method – A Numerical Method for Engineering Analysis, Wads Worth, California. Huebner, K. H. (1975). The Finite Element Method for Engineers, John Wiley and Sons, New York. Rao, S. S. (1989). The Finite Element Method in Engineering, Pergamon Press, Oxford. Reddy J. N. (1985). An Introduction to the Finite Element Method, McGraw-Hill International, New York. Segerlind, L. J. (1987). Applied Finite Element Analysis, John Wiley & Sons, New York. Silvester, P. (1969). Higher order polynomial triangular finite elements for potential problems, International Journal of Engineering Science, 7:849–861. Taig, I. C. (1961). Structural analysis by the matrix displacement method, English Electric Aviation Report, SO-17. 110

Finite Element Method with Applications in Engineering Turner, W. J., Clough, R. W., Martin, H. C., and Topp L. S. (1956). Stiffness and deflection analysis of complex structures, Journal of Aeronautical Science,23:805–823. White, R. E. (1985). An Introduction to the Finite Element Method with Applications to Nonlinear Problems, John Wiley & Sons, New York. Zienkiewicz, O. C. (1980). Finite Element Method, 3rd ed., Mac-Graw Hill, New York. Zienkiewicz, O. C., and Taylor R. L. (1989). The Finite Element Method, Vol 1, Mac-Graw Hill, London.

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6 One Dimensional Finite Element Analysis 6.1 INTRODUCTION Many practical problems can be modelled by employing one-dimensional finite elements (also referred to as line elements). The one-dimensional discretization can give good results when the primary unknown quantity (i.e., unknown continuous field to be modelled) vary mainly in one direction and remains constant in the other two dimensions. Typical examples of situations, where the one-dimensional analysis may be adequate, are framed (skeletal) structures subjected to external forces; torsion of circular shafts; flow through pipes, seepage through narrow, confined soil mass; heat conduction across a fire wall; flow analysis in channels etc. When FEM is employed for one-dimensional analysis, 2node elements are used extensively. The FEM analyses of various one-dimensional problems are illustrated in this chapter with various formulations and examples. When FEM is employed for one-dimensional analysis, different types of 2-node elements, as given in Table 6.1, are extensively used. All the elements summarized in Table 6.1 [except space frame elements (Gere 1965)] are elaborated in the following sections.

6.2 LINEAR SPRING The linear spring–the simplest kind of line element–is considered first because it provides a simple, yet generally instructive, tool to illustrate the basic concepts of the FEM. For the illustration, consider the two spring assemblage as shown in Figure 6.1. It is required to calculate the displacements at points (nodes) B and C, force each spring and for the reaction at A by employing FEM. This problem can be solved by following the steps outlined in Chapter 4. Step 1: Discretize and select element types The two spring assemblage shown in Figure 6.1 is discretized by employing two-node spring elements, as shown in Figure 6.2(a). The discretized model has three nodes–two externally applied nodal forces F2 = FB and F3= Fc, and two spring elements. A typical spring element is shown in Figure 6.2(b). A typical spring element has two nodes. The variation of displacement along x direction is expressed in terms of axial displacements u1 and u2 at nodes 1 and 2 of the element, respectively. In Figure 6.2(b), f1 and f2 are the elemental nodal forces. Note the sign convention for nodal forces and displacements. The lower case x represents the local or element co-ordinate system. It will be assumed that tensile forces are positive and compressive forces are negative. Table 6.1 Summary of 2-node finite elements used in one-dimensional analysis

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Figure 6.1 Two spring assemblage Step 2: Select approximation functions A line can be defined uniquely by specifying two points on it. Hence, displacement function for u(x) is assumed to vary linearly along the x-axis of the element because the element has two nodes (and hence, two degrees of freedoms (DOF)). Therefore, assume

Figure 6.2a Finite element discretization of two spring assemblage

Figure 6.2b Typical spring element 113

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Figure 6.3 Assumed variation of displacement field where a1 and a2 are unknown coefficients. The u(x) can be expressed as a function of nodal displacements u1 and u2 by employing conditions (Figure 6.3):

Shape Functions N1 and N2 are called ‘shape functions’ (also known as interpolation or basic functions), shown in Figure 6.4, because Ni's express shape of the assumed displacement function over the domain of the element when the i-th element degree of freedom has unit value and all other degrees of freedom are zero. A property of shape functions is that their sum equals unity. Such a property is essential to ensure proper simulation of rigid body displacements. Hence,

By expressing Eq. (6.3) in matrix notation,

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Figure 6.4 Shape functions is obtained where

Step 3: Define the strain-displacement and constitutive relationships Tensile force T produces a total elongation (deformation) δ of the spring. T and δ are related through Hooke's law for the linear spring by where δ = u2 – u1 and K is the spring constant (stiffness of the spring). Step 4: Derive element equations Since the physics of a spring is simple to understand, the direct equilibrium method can be employed to derive element equations expeditiously. A spring element subjected to tensile (positive) force T is shown in Figure 6.5. By comparing Figures 6.2(b) and 6.5, it can be easily seen that

For the equilibrium of generic element (in the x direction), therefore, It can be shown from Eqs. (6.7), (6.8) and (6.9) that

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Figure 6.5 Spring element in equilibrium Alternatively, in the matrix notation,

i.e., the element equation for a generic element is Step 5: Assemble the element equations to obtain the global equations and introduce boundary conditions By employing Eq. (6.11) to elements 1 and 2, equations

and

can be obtained. Note that throughout the text, the superscript in parentheses refers to the element number. To derive global (or the system) equations, consider the free-body diagram of individual spring elements as shown in Figure 6.6. Here, the nodal forces are shown to be consistent with the element force-sign convention.

Figure 6.6 Free-body diagram of spring assemblage The concept of continuity (compatibility) is employed implicitly in the assembly process as shown in Figure 6.7. This concept requires that the body remains together and that no tearing occurs anywhere in the system.

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Figure 6.7 Inter-element compatibility Therefore, elements 1 and 2 must remain connected at the common node 2 throughout the deformation process. This continuity requirement yields

Furthermore,

In the above Eqs. (6.14) and (6.15), u1, u2 and u3 are displacements of nodes 1, 2 and 3, respectively. Assembly can be performed in two ways: Assembling by nodal equilibrium conditions; and Assembling by direct superposition (direct stiffness method). Assembling by Nodal Equilibrium Conditions From Figure 6.6, the nodal equilibrium conditions at nodes 1, 2 and 3 can be written, respectively, as

In view of Eqs. (6.12), (6.13), (6.14) and (6.15), Eq. (6.16) can be written as

Alternatively, in the matrix notation,

or

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where

= total or assembled stiffness matrix.

Assembling by Direct Superposition (Direct Stiffness Method) Element equations for elements 1 and 2 are given by Eqs. (6.12) and (6.13), respectively. These element matrices cannot be added directly together because they are not associated with the same DOF's. An easy way of adding these matrices is to rewrite them in an expanded form as

By adding Eqs. (6.21) and (6.22) together, the global (assembled equations) matrix can be obtained as

From Eq. (6.16), Eq. (6.23) can be written as

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For problems involving a large number of DOF's, it is tedious (and a waste of computer resources) to expand each element matrix to the order of total stiffness matrix. To avoid this expansion, a shortcut is employed in the assembly process where the columns of each element matrix are labelled according to the DOF's associated with them as follows:

The system is then constructed simply by directly adding terms associated with DOF's in (1) and K(2) into their corresponding identical DOF locations in as

The {q} and {F} are then simply arranged in the order of DOF locations as in

The determinant of

. Thus,

can be obtained as

It can be noticed that is symmetric and singular. Since is singular and F1 is an unknown nodal force (reaction), Eq. (6.23) cannot be solved for {q}. The mathematical singularity of is related to the physical reality that the spring assemblage cannot stand in the space uniquely without having proper boundary conditions. Boundary Conditions There are two types of boundary conditions: (i) geometric, forced, essential or kinematics boundary condition (also known as Dirichlet conditions in the context of seepage), and (ii) the natural (sometimes referred to as Neumann conditions in the context of seepage) boundary conditions. In the context of structural analysis, the latter type of boundary condition may be, say, zero bending moment at the hinge support of a beam. The assembled equations need to be modified for forced boundary conditions only. (There is no need to satisfy natural boundary conditions. They will be satisfied automatically.)

119

Finite Element Method with Applications in Engineering Forced-boundary conditions are further classified into two categories of homogeneous and non-homogeneous boundary conditions. The homogeneous or the zero-boundary conditions occur at locations that are completely restrained (i.e., prevented from moving). On the other hand, non-homogeneous boundary conditions occur where non-zero values are specified for some selected DOF (e.g. support settlement). In general, forcedboundary conditions can be treated mathematically by partitioning the global equations as follows:

Here, 1 contains the unconstrained DOF's and By expanding Eq. (6.27),

2

contains specified (constrained) DOF's.

are obtained. For homogeneous-boundary conditions,

= {0}. Therefore,

For the two spring assemblage problem of Figure 6.2(a), suppose forced-boundary condition is specified at node 1 (i.e., u1 is specified). Then, the assembled equations can be partitioned as

For homogeneous-boundary conditions (i.e., u1 = 0),

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Finite Element Method with Applications in Engineering is obtained. Note that Eq. (6.35) could have also been obtained directly by deleting the row and column of Eq. (6.23), corresponding to the zero-displacement DOF. To illustrate the case of non-homogeneous–boundary conditions, let u1 = δ, where δ is a known (specified) displacement. In view of Eq. (6.28), Eq. (6.32) can be written as

Hence, the row and column corresponding to the non-zero DOF of assembled equations cannot be simply deleted when dealing with non-homogeneous–boundary conditions. Step 6: Solve for the unknown DOF Eqs. (6.35) and (6.36) are the global equations modified for boundary conditions. By solving Eq. (6.35) u2 and u3 can be obtained as follows.

Step 7: Solve for secondary quantities The secondary quantities of interest in this problem are the reaction at node 1 and the element forces. The reaction at node 1, F1, can be found from either Eq. (6.29) or Eq. (6.31) depending on the type of boundary condition. Finally, element forces are determined by back-substitution, applied to each element equation.

Figure 6.7a Spring assembly in series Step 8: Interpret the results For spring assemblage, the computed global nodal forces and element nodal forces can be shown on a free-body diagram for easy interpretation. Illustrative Example 1 For the spring assemblage shown in Figure 6.7(a), obtain the assembled stiffness matrix; the displacements of nodes 2 and 3; the reaction forces at nodes 1 and 4; and the forces in each spring.

121

Finite Element Method with Applications in Engineering Nodes 1 and 4 are fixed. Element equation for spring element is given by

The global

is given by

The global equation has the form

Boundary conditions for this problem are u1 = u4 = 0. By partitioning the global equation (or deleting the first and the fourth rows and columns from ) and by setting F2 = 0 and F3 = 1,000, the modified equations are obtained as

Figure 6.7b Free body diagram of series spring assemblage

By solving for u2 and u3,

The global nodal forces at nodes 1 and 4 can then be computed as

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Use element equations to obtain element forces.

The free body diagram of the spring assemblage is shown in Figure 6.7(b). 6.2.1 Expressions for Equivalent Spring Constant and Nodal Forces In situations where springs are arranged parallel to each other or in series, an equivalent element can be formed. This equivalent element can be employed to reduce the size of the problem at hand. Expressions for equivalent spring constants and nodal forces for parallel springs and springs in series are derived below. 6.2.1.1 Parallel Springs Figure 6.8(a) shows a spring assembly in parallel conditions. The element equations can be derived as given below.

Figure 6.8a Spring assembly in parallel The global equations can be obtained by following the assembly procedure as:

By comparing the global equations with equations for a generic spring element, it can be seen that

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Finite Element Method with Applications in Engineering 6.2.1.2 Springs in Series Figure 6.8(b) shows spring system in series. The element equations can be obtained as follows

The global equations can be obtained by following the assembly procedure as:

By condensing u3 (refer to Chapter 10 on computer implementation of FEM for static condensation process) from the global equations,

Figure 6.8b Spring system in series

By comparing Eqs. (6.40) and (6.41) with equations for a generic element, it can be seen that

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6.3 TRUSS ELEMENT Lattice transmission towers, steel truss bridges or roof support systems can be modelled accurately by employing truss elements. In this section, the element stiffness matrix is derived for a linear-elastic truss (or bar) element. Local and global co-ordinate systems are introduced to facilitate the formulation. Each element has its own local co-ordinate system which is designated by lower case letters, x, y, z whereas the global co-ordinate system (i.e., reference co-ordinate) is indicated by upper case letters X, Y, Z (Figure 6.9(a)). Similarly, the displacement components with respect to local co-ordinate system are indicated by the lower case letters u, v, w and those in the global co-ordinate system are shown by the upper case letters U, V, W. Moreover, the element level matrices and vectors are shown by lower case and upper case letters, respectively, when referred to local and global co-ordinate system. Thus, ke and Ke represent element stiffness matrix in the local and global co-ordinate systems, respectively. However, the element DOF vector is referred to as {q1e} in the local co-ordinate system, and as {q1e}, in the global co-ordinate system. (Note that in the previous section on analysis of spring assemblage, the local coordinate system was assumed to coincide with the global co-ordinate system. Hence, both {qle} and {qe} had the same meaning). 6.3.1 Plane Truss A typical example of a plane truss is shown in Figure 6.9(a).

Figure 6.9a A typical example of plane truss

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Finite Element Method with Applications in Engineering Figure 6.9b Two-node truss element subjected to tensile force T Note the global co-ordinate system (X, Y) and local co-ordinate systems (x, y) for elements I and j. There is only one global co-ordinate system to represent the entire structure, whereas each element has its own local co-ordinate system. The generic element of length L from this plane truss is shown in Figure 6.9(b). The following assumptions are made to formulate element equations for a truss member. Displacements and forces pointing in the positive directions of axis are positive. Tensile forces are positive. Truss (bar) is linearly elastic and has a constant cross-sectional area. (Later, a truss member having varying cross-section will be considered.) By definition, a truss member cannot sustain shear force. In other words, f1y = f2y = 0. Consequently, effect of transverse displacement (along local y-axis) is ignored. Hooke's law applies: that is σx = Eεx. Thus, it can be shown easily that

where A is the cross-sectional area of bar, E is the modulus of elasticity of bar material and δ = deformation = u2 – u1. Eq. (6.45) leads to

where K = (AE/L). Hence, a truss member and a spring member are analogous as shown in Figure 6.9(c). By following the procedure adopted for spring element, element equations (in local coordinate system) for a truss member can be obtained easily. Thus, approximation function can be written as

Figure 6.9c A truss member and spring member

Similarly, the strain–displacement relation and the constitutive law can be expressed as 126

Finite Element Method with Applications in Engineering

respectively, where K = (AE/L). Equilibrium condition for the element (in x direction) gives

which leads to element equations.

Assembly process and other steps are similar to those for the spring element. Illustrative Example 2 For the three-bar assemblage shown in Figure 6.10, determine (a) the assembled stiffness matrix, (b) the displacements of nodes 2 and 3, and (c) the reactions at nodes 1 and 4. By using Eq. (6.51), the element stiffness matrices are obtained as

Figure 6.10 Three-bar assemblage Therefore, the assembled equations are

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Applying homogeneous boundary conditions u1 = u4 = 0, the above assembled equation set (6.52) reduces to

By solving the above 2 × 2 equation set, the unknown displacements are obtained as

Back substitution of Eq. (6.54) in Eq. (6.52) yields

6.3.2 Element Equations by Minimizing Potential Energy One of the alternative methods often used to derive element equations for an elastic body is based on the principle of minimizing potential energy which is one of the energy (or variational) methods. In this text, the method of minimizing potential energy will be referred to as the energy method. This method is widely used in finite element formulations, especially those involving two- and three-dimensional domains. As mentioned in the Chapter 4, the energy method is based on the idea of finding equilibrium states of the body associated with stationary values of a scalar quantity. This scalar quantity is the potential energy, πp, in structural analysis. The potential energy πp is given by

where the strain energy, ø, is the capacity of internal forces (or stresses) to do work through strains in the body. Work done by the external forces, Ω, on the other hand, is the capacity of forces such as body forces, surface traction forces, and applied nodal forces to do work through deformations of the body. Expressions of ø are given below for the one-, two- and three-dimensional deformations (Refer to Zienkiewicz, 1980 for details.) One-Dimensional Deformations in the x Direction It can be shown that

where σx and εx are, respectively, the stress and the strain along x. In the matrix notation, Eq. (6.57) can be shown as 128

Finite Element Method with Applications in Engineering

Here, {σ} = {σx} and {ε} = {εx}. Two-Dimensional Deformations in the X–Y Plane It can be shown that

where σx, σy are normal stresses in the X and Y directions, respectively εx, εy are normal strains in the X and Y directions; and τxy and γxy are the shear stress and strain, respectively, in the X-Y plane. In the matrix notation,

where

Three-Dimensional Deformations The strain energy expression for a body experiencing three-dimensional deformation is given by

Here, σx, σy and σz are normal stresses in the X, Y and Z directions, respectively; εx, εy and εz are normal strains in the X, Y and Z directions; and τxy, τyz, τzx and γxy, γyz, γzx are, respectively, shear stresses and strains in the X, Y and Z direct strains. In the matrix notation,

where

6.3.2.1 General Forces Acting on one-Dimensional Truss Element For the forces shown in Figure 6.11(a), the Ω is given by 129

Finite Element Method with Applications in Engineering

Where xb

=

body force (weight) per unit volume

Tx

=

surface loading or traction per unit surface area

V

=

volume

S1

=

part of the surface on which surface loading acts

M

=

number of points at which joint forces are applied.

Figure 6.11a Bar element subjected to loads 6.3.2.2 Derivation of Stiffness Matrix by Energy Method By ignoring body forces and surface tractions, πp for a truss element is obtained as

By using Eq. (6.47), the axial displacement function can be expressed in terms of the shape functions and nodal displacements as

The axial stress–strain relationship is given by linear Hooke’s law 130

Finite Element Method with Applications in Engineering

where

Eq. (6.64) can be written in matrix notation as

By substituting Eqs. (6.68) and (6.71) into Eq. (6.73),

is obtained. Here,

By expanding Eq. (6.75), following equation can be derived

By applying the principle of minimum potential energy, the equilibrium state corresponds to

Therefore,

Hence, the element equation for a truss member in the local co-ordinate system is

131

Finite Element Method with Applications in Engineering Therefore,

Note that A or E have not been assumed to be constant. Therefore, any variation in A or E with respect to x-direction can be included in [ke]. In view of Eq. (6.69),

By substituting Eq. (6.82) into Eq. (6.81), the element matrix for a truss member having constant A and E can be derived to be

Therefore, element equations take the form

Figure 6.11b Bar element with varying cross section Illustrative Example 3 For the bar element with varying cross-section given in Figure 6.11(b), derive the element matrix. Consider the problem shown in Figure 6.11(b) for which E is constant and A varies linearly, The element stiffness matrix can be derived as

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Hence, the element equations are given by

6.3.3.3 Equivalent Nodal Loading Due to Surface Forces and Body Forces It was shown previously that

Therefore,

From the principle of minimum potential energy,

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After assembling,

6.3.3 Local and Global Element Equations for a Bar in the X–Y Plane Eq. (6.80) relates the displacement components in the local direction to the forces in the local x direction. Therefore, Eq. (6.80) is called ‘the local element equation’. In plane truss problems, it is often necessary to relate global displacement components of an element to its respective global force components. This relationship is called the ‘global element equation’. The local element equations are

where

It is desired to write the global element equations as

where

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Finite Element Method with Applications in Engineering

It is now required to establish a relationship between [ke] and [Ke]. For this purpose, the relationships between {qle} and {qe}, and the relationship between {fe} and {Fe} have to be established first. As shown in Figure 6.12(a), the components of displacement vector in the local x – y and the global X–Y directions are, respectively, u, v and U, V. It can be seen that

Similarly, by looking at Figure 6.12(b), it can be shown that

Application of Eq. (6.94) to the displacement components associated with local nodes 1 and 2 results in

Eqs. (6.95) and (6.96) can be written in matrix form as

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Finite Element Method with Applications in Engineering

Figure 6.12a Co-ordinate transformation

Figure 6.12b Components of force in the x–y and X–Y directions By letting

It can be shown that

and

Eq. (6.100) can be written in view of Eq. (6.80) as

where

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Finite Element Method with Applications in Engineering

Here, C=cosθ and S=sinθ. 6.3.4 Computation of Stress for a Bar in the X–Y Plane The axial tensile stress (σe) in element ‘e’ is given be

It can be extracted from Eq. (6.80) that

By combining Eqs. (6.99), (6.104) and (6.105),

is obtained where

Illustrative Example 4 For the plane truss shown in Figure 6.13, determine the displacement components at nodes, 137

Finite Element Method with Applications in Engineering determine the stresses in each bar, and verify the nodal equilibrium at node 1. Solution: Let the direction of x for each element is taken in the direction from node 1 to other node. Once the direction is chosen, the angle θ is established as positive when measured counter clockwise from the positive X to x. It is convenient to construct a table as an aid in determining each element stiffness matrix.

Figure 6.13 Plane truss–Example 4

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Finite Element Method with Applications in Engineering

Therefore,

Eq. (6.106) gives

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Finite Element Method with Applications in Engineering

Hence, axial stress in each element is given by

and

By summing the forces in the global X and Y directions,

6.4 SPACE TRUSS Derivation of the global stiffness matrix for a space truss is analogous to the plane truss case. An element of constant cross-section as shown in Figure 6.14 is considered here. Element equilibrium equation and stiffness matrix for an element are given in Eqs. (6.84) and (6.83), respectively.

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Figure 6.14 (a) Two-node space truss element subject to tensile force T (b) Global displacements Displacement u and the force f in the local co-ordinate have the components u(x) = (U, V, W) and f = (Fx, Fy, Fz) in the global co-ordinates (X, Y, Z). It remains now to find the transformation matrix [T*] to transform from the local to the global co-ordinate system. It can be seen from Figure 6.15 that an arbitrary vector

has the components

Figure 6.15 Co-ordinate transformation for space truss element and the direction cosine

cosθx, cosθy, and cosθz are components of a unit vector in the x-direction. From Figure 6.14(b)and Eq. (6.108), it can be shown that 141

Finite Element Method with Applications in Engineering

which in matrix notation can be written as

Further, the displacement vectors u1 and u2 can be shown form Figure 6.14(b) and Eq. (6.108) to be Multiplying Uj by cosθx, Vj by cosθy, and Wj by cosθz, adding them and using Eq. (6.109), the following equation is obtained.

Eq. (6.113) can be written in the matrix notation as

Substituting Eq. (6.114) in Eq. (6.84), and this in turn in Eq. (6.111), the following equation can be derived.

Then matrix [Ke] can be obtained as

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where

Force and stress in space truss element can be obtained in an analogous way to computation of stress for a bar in the X-Y plane Illustrative Example 5 For the space truss shown in Figure 6.16, Determine the displacement components at node A in terms of AE. Forces in each bar For all bars AE = constant. Note: This problem is statically determinate. By taking the equilibrium of joint A, the answers for the forces in bars are: TAB = 15,094.77 N; TAC = 13,856.95 N; TAD = 15,577.84 N. All forces are tensile. Solution: Let the direction of x for each element is taken in the direction from node 1 to the other node as shown in Figure 6.16(b) above. The direction cosine of each member is shown in the table below. Element 1 is from node i = 1 to node j = 2. Element 2 is from node i = 1 to node j = 3. Element 3 is from node i = 1 to node j = 4.

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Figure 6.16 Space truss–example

From Eq. (6.116), the matrix equation for each element in global co-ordinates is

144

Finite Element Method with Applications in Engineering

145

Finite Element Method with Applications in Engineering

By assembling the matrices of the three members, the nodal equations are obtained as

Application of boundary conditions, U2 = 0, V2 = 0, W2 = 0, U3 = 0, V3 = 0, W3 = 0, U4 = 0, V4= 0, W4 = 0, yields the first three rows (1 – 3) and the first three columns as

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Finite Element Method with Applications in Engineering

Thus,

By solving above equation AEU1 = 0.0940 × 104; AEV1 = –2.8942 × 104; AEW1 = –1.4315 × 104. Again from the assembled equation

substituting the values of U1, V1 and W1 F2x = 9,038.0 N; F2y = 12,048.0 N; F2z = 0. The resultant force for element 1,

= 15,060. N

Similar operations for elements 2 and 3 yields the values of the element forces as F3x = 0.0 N; F3y = 12,198.0 N; F3z =6,506.0 N and F4x = –9,036.0 N; F4y = 10,843.0 N; F4z = 6,506.0 N and

= 13,825.0 N = 15,542.0 N.

An alternate way of obtaining the forces is to use the transformation of displacement components from the global axis to local axis and use the fundamental definition of a force in a truss element. Consider the element 1, as an example. For this element the nodes are from 1 to 2 and the node 2 is pinned. Therefore, u2 = 0.0 and u1 = U1, cosθx + V1 cosθy + W1 cosθz Thus u1 can be determined as AEu1 = [(0.0940 × 0.6) + (–2.8942 × 0.8) + (–1.4315 × 0.0)] × 104

6.5 ONE-DIMENSIONAL TORSION OF A CIRCULAR SHAFT The one-dimensional torsional elements can be used to analyze an assemblage of circular shafts. Procedure is explained in the following steps. Step 1: Select element type A two-node torsion element is shown in Figure 6.17(a). The T1 and T2 are the torques at nodes 1 and 2, respectively, whereas ø1 and ø2 are angular rotations at nodes 1 and 2, respectively, of the element. The sign convention is shown in the Figure 6.17(a). 147

Finite Element Method with Applications in Engineering

Figure 6.17a One-dimensional torsion element Step 2: Select approximation function Let the rotation about x-axis at a section located at distance x units from node 1 can be written as Because ø(x = 0) = ø1 and ø(x = L) = ø2,

where

Step 3: Define constitutive law and gradient of the unknown Constitutive Law The constitutive law is given by

where G = shear modulus and J = polar moment of inertia of the cross-sectional area about the axis of the element. Alternatively, the stress–strain relationship can be expressed as

Here, A represents the cross-sectional area. Gradient or unknown rotation It can be shown easily that 148

Finite Element Method with Applications in Engineering

where

Step 4: Derive element equations The total potential energy of a torsion element is given by

By substituting Eqs. (6.121) and (6.123) into (6.124), it can be shown that

Hence, element equations are given by

Illustrative Example 6 A circular shaft as shown in Figure 6.17(b) is subjected to torques T2 and T3 as shown in the diagram. By employing one-dimensional torsion elements, compute angular rotations at nodes 2 and 3 and reactive torques at nodes 1 and 4. Solution: The element matrices can be obtained as follows.

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Figure 6.17b Circular shaft

The system can be assembled as

By applying homogeneous boundary conditions ø1 = ø4 = 0, following equation is obtained.

After solving, the solution is obtained as,

6.6 ONE-DIMENSIONAL STEADY STATE HEAT CONDUCTION A 2-node heat flow element as shown in Figure 6.18 can be utilized to analyze onedimensional steady state heat conduction problems. Note that convection is neglected in the formulation. The solution is obtained in the following steps. Step 1: Select element type Sign convention: Heat flow entering a control volume is positive. Outgoing heat flow is negative. 150

Finite Element Method with Applications in Engineering Let q be the heat flow (per unit area) and t be the temperature. Then, qi, i = 1, 2 are the nodal heat flow rates and ti, i = 1, 2 are the nodal temperatures.

Figure 6.18 One-dimensional heat flow Element Step 2: Select approximation function Let the temperature at a section located at distance x units from node 1 be Note that

Step 3: Define gradient of the unknown and the constitutive law Gradient of Unknown Temperature The gradient is obtained as

Constitutive Law: Fourier's Law of Heat Conduction According to Fourier's law, the heat flow in the x direction is given by

where and Kxx = Thermal conductivity in x-direction [KW/m.°C in SI units]. The Kxx measures the amount of heat energy (W.hr) that flows through a unit length of a given substance in a unit time (hr.) to raise the temperature 1 degree (°C). Step 4: Derive element equations Since physics of the element can be understood easily for the one-dimensional heat flow element, direct equilibrium approach can be employed to derive element equations. By employing Eq. (6.131), the heat flowing in the x-direction at node 1 is

151

Finite Element Method with Applications in Engineering From the principle of conservation of energy, the net heat flow into the system must be zero for a steady state condition. Hence, the element equations are given by

Illustrative Example 7 The furnace wall shown in the Figure 6.19 consists of 25 cm of fire brick [K1 = 0.012 W/(cm °C)], 10 cm of insulation brick [K2 = 0.0014 W/(cm °C)], and 20 cm of red brick [K3 = 0.0086 W/(cm °C)]. The specified inner and outer temperatures are 500°C and 150°C, respectively. Determine the internal temperature distribution.

Figure 6.19 Cross section of furnace wall and its finite element model Solution: This problem can be treated as a one-dimensional model. A constant arbitrary heat conduction area of unit cm2 is assumed for all elements. Element conduction matrices are given by

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Finite Element Method with Applications in Engineering

Assembly

Figure 6.20 Internal temperature distribution By applying the non-homogeneous boundary conditions, the following equation can be obtained.

Hence,

Figure 6.20 shows the temperature distribution.

6.7 ONE-DIMENSIONAL FLOW THROUGH POROUS MEDIA Analysis for one-dimensional flow through porous media is similar to the heat flow analysis (e.g., ground water flow, seepage). A 2-node element as shown in Figure 6.21 is the

153

Finite Element Method with Applications in Engineering simplest possible element can be used to analyze one-dimensional seepage problems. Formulation is given in the following steps. Step 1: Select element type Sign convention: Fluid flow entering a control volume is positive. Outgoing fluid flow is negative. Let vx, q and h be the flow velocity, the volumetric flow rate and the hydraulic head, respectively. Then, qi, i = 1, 2 and hi, i = 1, 2 shown in Figure 6.21 are the nodal fluid flow rates and the nodal hydraulic heads, respectively. Step 2: Select approximation function Let the hydraulic head at a section located at distance x units from node 1 be

Figure 6.21 One-dimensional fluid flow element Note that

Step 3: Define gradient of the unknown and the constitutive law Gradient of hydraulic head The gradient is obtained as:

Constitutive Law: (Darcy's law) According to the Darcy's law; the fluid velocity vx in the x direction is given by

where Kxx is the coefficient of permeability in the x-direction. Consequently, the volumetric flow rate q can be obtained from

Here, A represents cross-sectional area of the element. Step 4: Derive element equations 154

Finite Element Method with Applications in Engineering The element equations can be derived easily by adopting the direct equilibrium approach. From Eq. (6.146), fluid flow entering the element in the x-direction at node 1 is where

From the principle of conservation of mass, the net fluid flow into the system must be zero for a steady state condition. Hence, the element equations are given by

or

Figure 6.22 One-dimensional flow through porous media and its finite element model Illustrative Example 8 In a laboratory, experiment of one-dimensional flow through porous media over the section shown, the following data are known. Determine the distribution of hydraulic head over the length of the section, the flow rate and fluid velocities in each element. Solution: Element matrices are given by

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Finite Element Method with Applications in Engineering

Assembly

Hence, or The hydraulic head distribution is shown in Figure 6.23.

Flow rate = 1.333 cm3/s. By making use of Eq. (6.145), fluid velocity in each element can be obtained as

Figure 6.23 Hydraulic head distribution

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6.8 ONE-DIMENSIONAL IDEAL FLUID FLOW THROUGH PIPES (INVISCID FLUID FLOW) The one-dimensional ideal fluid flow analysis is analogous to analysis for one-dimensional flow through porous media. Let A = cross-sectional area, vx = fluid velocity in the x direction øi (I = 1, 2) = velocity potentials at nodes 1 and 2, and qi (i = 1, 2) = fluid flow rates at nodes 1 and 2 in Figure 6.24. It can be shown that

Hence, set Kxx = 1 in the one-dimensional porous media problem to get the element equations. Then,

where

.

Figure 6.24 A finite element to analyze one-dimensional ideal fluid flow through pipes

6.9 BEAM ELEMENT A beam is a long, slender structural member generally subjected to transverse loading that produces significant bending effects as opposed to twisting or axial effects. A continuous beam can be modelled by employing 2-node beam elements. To derive equations of a beam element, it is important to briefly review classical beam theory.

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Figure 6.25 (a) Beam subjected to transverse loading, (b) Beam theory sign conventions 6.9.1 Review of Beam Theory Consider the simply supported beam shown in Figure 6.25. The beam is of length L with axial co-ordinate x and transverse local co-ordinate y. In Figure 6.25, fy(x) = transverse loading, v = transverse displacement, V = shear force, and M= bending moment. The equilibrium of the differential element in the y direction is given by Therefore, in the limit Δx → 0.

The moment equilibrium about x-axis results in Σ M at A = 0. Thus,

Referring to (Crandall et al. (1978)) and Figure 6.25, it can be shown for a shallow beam, that

where ρ = radius of curvature, E = Young's modulus of elasticity and I = moment of inertia. 158

Finite Element Method with Applications in Engineering

Figure 6.26 Neutral axis before and after deformation The axial strain developed at any point on a cross section, located at distance x from the left support is given by

where u is axial displacement. The u at any point on a cross section, on the other hand, can be shown to be of the form (refer to Figure 6.27)

Thus,

Figure 6.27 u variation at a typical cross section 159

Finite Element Method with Applications in Engineering Stress–strain relation (Hooke's law) From the Hooke's law, the axial stress σx is related to εx by Note that tensile stresses are considered positive. Therefore, from Eqs. (6.165) and (6.166), moment M about z-axis can be expressed as

Here, I = moment of inertia of the cross-sectional area about the z-axis. It is to be noted here that EI has not been assumed to be constant. By combining Eqs. (6.159), (6.161) and (6.167), Eq.

can be obtained. Thus, the governing differential equation for bending of a beam is

If EI is constant, the above Eq. (6.169) takes the form

If fy(x) = 0, the governing equation becomes

General solution of Eq. (6.171) is given by

where C1, C2, C3, C4 are unknown, constant terms. They can be determined by applying boundary conditions. 6.9.2 Finite Element Formulation of a Beam Element A finite element formulation of a beam element is presented below in a step-by-step manner. Step 1: Discretize and select element type A continuous beam can be discretized into number of 2-node beam elements. A typical beam element is shown in Figure 6.28 with positive nodal displacements, rotations, forces and moments.

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Figure 6.28 Typical beam element In the formulation, deformation due to bending is measured in terms of a transverse displacement v and a rotation θ about z-axis. Hence, there are two DOF vi and θi at i-th node of the element. As a result, the formulation pertains to C1 continuum (in all previous cases, formulation for C0 continuum was considered). Let element DOF vector be given by

Step 2: Select approximation function Solution of the homogeneous Eq. (6.171) can be chosen as basis for selecting approximation function for a beam element. In conjunction with the solution given by Eq. (6.172), following four geometric boundary conditions need to be satisfied.

Therefore, the four unknown constants Ci, i = 1, 2, 3, 4 in Eq. (6.172) can be obtained in terms of element DOF by making use of boundary conditions of Eq. (6.168). By substituting Ci's back into Eq. (6.172), equation describing variation of v (the primary unknown quantity) over x in terms of element DOF, as

Here, [N] = [N1 N2 N3 N4] where

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Finite Element Method with Applications in Engineering

N1, N2, N3 and N4 are the shape functions for a beam element. In the literature, Ni's are called beam functions, Hermite polynomials, or cubic splines. Variations of Ni, i = 1, 2, 3, 4 over the length of element are sketched in Figure 6.29. Here, it may be noted that

Figure 6.29 Shape functions for a beam element At node 1, v = v1. Hence, from Eq. (6.177), it can be seen that at x = 0. N1 = 1 and Ni = 0, i= 2, 3, 4. Similarly, at node 2, v = v2, and N3 = 1 and Ni = 0 for i = 1, 2, 4. At node 1,

. Thus, at x = 0,

, i = 1, 3, 4.

Similarly, at node 2, θ = θ2 and

, i = 1, 2, 3.

Step 3: Define strain-displacement relation and constitutive law Strain-Displacement Relation From Eqs. (6.165) and (6.175), the strain-displacement relation can be defined, in terms of {qe}, as

where 162

Finite Element Method with Applications in Engineering

Here, Alternatively, [B] can be shown as

where

Constitutive Law Following expressions can be easily derived from Eqs. (6.166) and (6.178):

Figure 6.30 A beam element subjected to distributed and concentrated point loads Step 4: Derive element equations In the formulation presented below, the element equations are derived by employing the principle of minimum potential energy. (Note that equations can also be derived by employing direct equilibrium approach. Refer to texts (e.g., Logan (1993); Gere (1965)) for details.) Consider the beam element shown in Figure 6.30 for deriving element equations. Following notations are used in Figure 6.30. fy(x) is distributed load Pn are concentrated point load applied at distance xn from node 1 (n = 1,2,3,…) 163

Finite Element Method with Applications in Engineering Mm are concentrated moment applied at distance xm from node 1 (m = 1,2,3,…) Equating the work done by these loads undergoing assumed displacement field v(x) with the work done by the nodal loads fiy0 and nodal moments mi0 (i = 1, 2) undergoing displacements vi and rotations θi (i = 1,2), respectively, expression for the equivalent load vector {f0e} can be derived. These equivalent load vectors are subsequently used in the analysis. The total potential energy of a beam element is given by

In Eq. (6.183), fiy and mi are (nodal components), respectively, the shear force and the bending moment at the i-th node. Moreover, v′ indicates slope of v with respect to x. In other words, v′ represents rotation. By letting

and by substituting Eqs. (6.176), (6.178) and (6.182) into Eq. (6.183),

can be obtained. Here,

and

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Finite Element Method with Applications in Engineering

The {f0 e} in Eq. (6.188) is equivalent concentrated nodal load vector minimizing the potential energy πp, the element equations are obtained as

. By

Note that variation in E and I along x can be considered easily in Eq. (6.186). For constant EI, Eq. (6.186) simplifies to

By substituting Eq. (6.181) in Eq. (6.190) and performing the matrix multiplication, [ke] takes the form

All the terms of the element stiffness matrix [ke] can be evaluated explicitly by substituting Ni's from Eq. (6.177). The resulting matrix is given below.

Step 5: Assemble element equations and apply boundary conditions By applying Eq. (6.189) to all the elements of a discretized continuous beam, the system (or global) equations can be obtained as

Here, [K], [F] and {F0} are assembled stiffness matrix, the load vector consisting of loads/moments applied directly at the joints (it also includes support reactions), and the assembled, equivalent nodal load vector. Note that the size of the assembled matrix is reduced after applying the geometric (or forced) boundary conditions. Step 6, 7 and 8: Solve for primary and secondary unknowns and interpret results The assembled equations, which have been modified by applying boundary conditions, can be solved for unknown displacement components contained in {q}. Once the primary quantities ({q}) are known, the secondary quantities like reactions and element forces can be computed by applying Eqs. (6.193) and (6.189), respectively. Shear force and bending moment diagrams can then be constructed for design purposes. 6.9.3 Illustrative Examples Example 1 Consider a beam subjected to a uniformly distributed load of intensity w per unit length (in the –y direction). Derive equivalent nodal force vector. 165

Finite Element Method with Applications in Engineering Solution: By applying Eq. (6.188), the equivalent concentrated nodal force vector is given by

Example 2 Consider a beam subjected to concentrated load P (in –y direction) at a distance ‘a’ from node 1. Let L = a + b. Derive equivalent nodal force vector. Solution: By applying Eq. (6.188), the equivalent nodal force vector can be computed easily as

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Example 3 Consider a fixed beam subjected to distributed and/or concentrated loads. Derive equivalent nodal force vector. Solution: Note that for a fixed beam, {qe} = 0. Then, from Eq. (6.189), the actual (or corrected) element nodal forces are given by

In other words,

Therefore, equivalent concentrated nodal forces can be treated as fixed-end reactions (forces and moments) when both ends of the beam element are fixed. Fixed-end reactions for various transverse loading cases can be found in Gere (1965). Example 4 Analyze beam shown in the figure by using FEM.

Solution: Consider the beam to be made up of the two elements 1 and 2. The bou ndary conditions are

167

Finite Element Method with Applications in Engineering V2 and θ2 are required to be computed. Element matrices are given by

After performing assembly, following equation is obtained.

Substituting l = L/2 and applying boundary conditions, following equation is obtained.

Thus, from the assembled equation set,

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Finite Element Method with Applications in Engineering

From the element equations,

Letting l = L/2 and simplifying the element force vector is obtained as

Similarly,

Above results can be represented graphically as shown below.

Example 5 Analyze beam system shown by using FEM. Consider P = wL, M = PL = wL2 and EI = constant.

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Solution: Consider the beam to be made up of two elements

and

The boundary conditions are V1 = θ1 = V2 = V3 = θ3 = 0 Element matrices are given by

After performing assembly, following equation is obtained.

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Finite Element Method with Applications in Engineering

Now applying boundary conditions and solve for θ2

or

From the assembled equation set,

or

By making use of the element equation, the element force vectors are obtained as

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Finite Element Method with Applications in Engineering

Above results can be represented graphically as shown below.

Example 6 Analyze beam shown in the figure by using FEM. Consider constant EI and

Solution: Consider beam elements and as well as spring element The boundary conditions are V1 = θ1 = V3 = θ3 = V4 = 0 The element equations are obtained as shown below. Element

172

to discretize the domain.

Finite Element Method with Applications in Engineering

Element

Element

The assembled equations are given below.

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Force vector for each beam member as well as force in the spring can be computed by making use of respective element matrices. 6.10 ANALYSES OF PLANE FRAMES AND GRIDS Many civil engineering structures like buildings and bridges consist of frames and/or grids. In this section, finite element analysis procedures will be formulated for such structures. Generally, a plane frame is a series of slender members rigidly connected to each other such that the original angles made between elements at their joints remain unchanged from one element to another at the joints. In addition, the element centroids, as well as the applied loads, lie in a common plane. Since applied loads may have components along an element’s local x-axis, unlike beam element, a frame element may be subjected to flexural as well as axial effects. A grid, on the other hand, is a structure on which loads are applied perpendicular to the plane of the structure, as opposed to plane frame, where loads are applied in the plane of the structure. Since applied loads do not have any component along a grid element's local x-axis, no axial force can be developed in a grid member. A typical plane frame and grid are shown in Figure 6.31. 6.10.1 Plane Frame Analysis 174

Finite Element Method with Applications in Engineering Elements in plane frames are subjected to both lateral loading and axial loading. Therefore, axial deformations need to be included in the analysis. As described earlier, all the distributed and point loads (which are not applied at the nodes) are replaced by equivalent nodal loads. Local effective nodal forces (actual element nodal forces plus equivalent concentrated loads), which act on an arbitrarily oriented member of a plane frame, are shown in Figure 6.32. The plane frame structure consisting of prismatic members is assumed to lie in the X–Y plane. The local axes x and y are located along the plane frame element’s length (node 1 → 2) and transverse to the element, respectively, and the global axes X and Y are chosen with respect to whole structure.

Figure 6.31 Typical plane frame and grid structures

Figure 6.32 A typical member in a plane frame By considering strain energy in axial deformation (i.e., strain energy considered in the formulation of truss element), it can be shown that the axial effects are governed by

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Finite Element Method with Applications in Engineering

where C1 = (AE/L). Here, f1x and f2x are axial forces at nodes 1 and 2 in the local x direction. The u1 and u2, on the other hand, are axial displacement components along local x at nodes 1 and 2, respectively. Similarly, by considering strain energy in bending, (strain energy expression considered in the formulation of beam element), equations governing the transverse effects (bending and shear) can be derived to be

where C2 = (EI/L3). Here, fiy and mi are the transverse shear force and bending moments, respectively at i-th node. By combining the axial effects from Eq. (6.194) and flexural effects from Eq. (6.195), following equation can be derived for a plane frame element.

In other words, a plane frame member's governing equations in the local direction are given by

By looking at Figure 6.12(a), it can be easily proved that

where C = cos α, S = sin α. The C and S can be obtained from the global co-ordinates of the element's nodes as

where

176

Finite Element Method with Applications in Engineering Note that since the local z-axis coincides with the global Z axis, no transformation is required for the rotation θ along z-axis. It can be shown that

and since z coincides with Z, Application of Eq. (6.198) to nodes 1 and 2 results in

or,

where

Similarly, by applying Eqs. (6.200) and (6.201) to local nodes 1 and 2, the following equation Eq. (6.205) is obtained.

is obtained where

In view of Eqs. (6.197), (6.198) and (6.203), Eq. (6.205) can be written as

Therefore, the global equation for a plane frame element is given by

where

177

Finite Element Method with Applications in Engineering After performing the triple matrix product, Eq. (6.209) can be written as

where

Thus, for each element,

which can be written in the global co-ordinate system. Moreover, the effective element force vector can be written as In Eq. (6.213), the {Fee} and {F0e} are, respectively, the element force vector (which contains details like shear force and bending moments) and the equivalent nodal force vector (corresponding to transverse loading) in the global co-ordinates. By combining Eqs. (6.212) and (6.213), it can be shown that

By applying Eq. (6.214) to all the plane frame elements in the structure and assembling the resulting equations can be formed, where {F} = external nodal force vector; {F0} = equivalent nodal force vector due to transverse loads; [K] = assembled stiffness matrix; and {q} = assembled (global) displacement vector. Summary of plane frame analysis: The plane frame analysis can be summarized in the following steps. For each element, determine [ke], {f0e} [Ke] = [T]T [ke] [T] {F0e} = [T]T{f0e} 178

Finite Element Method with Applications in Engineering By assembling element equations, obtain system equations Apply boundary conditions Solve for {q} Solve for unknown reactions For each element, determine local effective forces by employing

Local actual forces by employing

These steps are demonstrated through an illustrative example given below. Illustrative Examples Example 1 Analyze the plane frame shown in the figure by using FEM. Consider E = 200 GPa, I = 1 × 10−4m4 and A = 1 × 10−2 m2.

Solution: Let element axis be directed from node 1 to node 2 for element and from node 2 to node 3 for element . Various coefficients required in analysis are given below in the tabular form.

For analysis of plane frame, it is useful to write global element equations in symbolic, partitioned form. Thus,

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Finite Element Method with Applications in Engineering

After Assembly,

Here, By substituting boundary conditions, following equation is obtained.

Calculation of element force vectors Element 1

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Finite Element Method with Applications in Engineering

Element 2

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Finite Element Method with Applications in Engineering

Example 2 Analyze the plane frame shown in the figure by using FEM. Consider E = 200 GPa, I = 2 ×10−4 m4 and A = 1 × 10−2 m2.

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Finite Element Method with Applications in Engineering

Solution: Let element axis be directed from node 1 to node 2 for element 1 and from node 2 to node 3 for element 2. Various coefficients required in analysis are given below in the tabular form.

Write global element equations in symbolic, partitioned form. Thus,

After assembly,

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Finite Element Method with Applications in Engineering

Here, By substituting boundary conditions, following equation is obtained.

Calculation of element force vectors Element 1

184

Finite Element Method with Applications in Engineering

Element 2

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Finite Element Method with Applications in Engineering

6.10.2 Grid Analysis In this section, element equations are derived for a grid element. A typical grid element used to discretize a grid structure is shown in Figure 6.33. Since a grid structure is assumed to lie in the X–Z plane, the local y-axis coincides with the global Y-axis. The degrees of freedoms at the i-th node of a grid element, in the local 186

Finite Element Method with Applications in Engineering directions, are a deflection viy (normal to the grid), a torsional rotation θix about the x axis, and a bending rotation θiz about the z-axis. The nodal effective (actual plus equivalent concentrated) forces at the i-th node consists of a transverse force fiy normal to the grid, a torsional moment mix about the x-axis, and a bending moment miz about the z-axis. As mentioned before, a grid member is not subjected to axial force components. Loading in the direction perpendicular to the plane of grid may induce a torsion and bending in a grid element. Thus, one can consider strain energy in torsion and flexure to derive equations for a prismatic grid element. By considering strain energy in torsion, it can be shown (from the one-dimensional torsion element formulation) that

Figure 6.33 A typical grid element

where C3 = . On the other hand, by minimizing potential energy expression for flexure (refer to formulation of beam element), the following equation

is obtained where C2 = (EI/L3). 187

Finite Element Method with Applications in Engineering By combining Eqs. (6.218) and (6.219), the local element equations for a grid element can be obtained as

Or,

By adopting the procedure used to obtain Eq. (6.207), the local degrees of freedoms can be expressed in terms of global degrees of freedoms by putting

where

In Eq. (6.226), the global displacement components Vi, θiX and θiZ, respectively, are the deflection along Y, rotation about X and rotation about Z axes at the i-th node. Note that α is measured in the X–Z plane from the local x to global X (and not from the global X to local x, as done for plane truss and frame) in the counter clockwise direction. By following the formulation of a plane frame element, the global equations for a grid element can be derived as where

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Finite Element Method with Applications in Engineering

By performing the triple matrix multiplication in Eq. (6.228), the global stiffness matrix for a grid element is obtained as

For the grid member also, the steps summarized for plane frame analysis in Section 6.10.1 can be followed. Illustrative Example Analyze the grid shown in the figure by using FEM. Consider E = 200 GPa, G = 80 GPa, I = 4 × 10−4 m4, J = 2 × 10−4 m4

Solution: Consider element axes for both the elements as shown in the figures. Note that local xaxis for each element is defined as directed from node 1 to node 2 of the element

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Finite Element Method with Applications in Engineering

Element properties and direction cosines are shown in the tables presented below.

Element matrices in the global directions are shown below.

After assembly, following equation is obtained.

By applying boundary conditions

the assembled equations reduce to

Element forces are calculated as below. Element 1

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Element 2

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Finite Element Method with Applications in Engineering

6.11 FURTHER ONE-DIMENSIONAL APPLICATIONS 6.11.1 Flow Network Analysis Flow of fluid through pipes is often required to be examined for municipal water supply, sewage networks, chemical flows in factories etc., where usually a network consisting of a large number of pipes, pumps, valves etc. are involved. A large number of pipe elements of uniform cross sections and pipe junctions and other elements are usually utilized in a flow network. Flow in a pipe network can be considered as one-dimensional as the flow is predominant in the direction of flow. Simple pipe networks can be analyzed using analytical/approximation based methods like Hardy-Cross method and Newton Raphson 192

Finite Element Method with Applications in Engineering method (Streeter et al., 1998). However, for complex pipe networks FEM can be very effectively used by considering networks as a system of one-dimensional finite elements consisting of the pipe elements. Here, application of FEM in fluid flow networks is discussed.

Figure 6.34 Pipe element Consider a typical pipe element shown in Figure 6.34 which can be used to analyze simple network without any booster pumps like the one shown in Figure 6.35. The flow takes place only due to the existing pressure gradient. For simplicity of problem formulation, the flow in these pipes is assumed to be laminar. Problem here, is to find rate of flow in each of the links and pressure at each node. This network can be treated as a collection of finite element by considering each flow path between two junctions as an element. The network shown in Figure 6.35 is having 6 nodes and 9 elements. Element characteristic would then be pressure head loss versus flow rate relationships. If the flow is assumed to be fully developed laminar flow and the flow paths are circular pipes of constant cross-sectional area, the pressure head loss H1 – H2 between nodes 1 and 2 due to flow rate is given by (Baker, 1983)

where Q is flow through pipe, L is the length of the pipe, μ is the dynamic viscosity of the fluid, and D is the internal diameter of the pipe. For a typical element shown in Figure 6.34, the flow entering the element at the nodes 1 and 2 in terms of the pressure heads can be written as

for node 1 and

Figure 6.35 Flow network

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Finite Element Method with Applications in Engineering

for node 2. In other words,

where

In matrix notation, these equations can be written as

For the configuration shown, in Figure 6.35, the element to node connection relationship is as follows: Element, e

Node, i

Node, j

1

1

2

2

2

4

3

1

3

4

4

6

5

3

5

6

5

6

7

1

4

8

3

4

9

3

6

where e denotes element, 1, 2–denote connecting nodes of the element e, Q1(e) is the flow entering the element at node 1, Q2(e) is the flow entering the element at node 2. For elements 1 to 9, the element equations are written for the network in Figure 6.35. 194

Finite Element Method with Applications in Engineering

The nodal flow continuity equations can be set up as follows.

195

Finite Element Method with Applications in Engineering

Here, ci, i = 1,2,…,6 are consumptions at the nodes or the rate of outflow from the network with proper sign. On the other hand, Qi, i = 1,6 are flow through the pipe. The ci can be termed as loads on the network. The assembled matrix is as shown below, which is of the form [K]{H} ={q}, where [K] is the fluid stiffness matrix, {H} = column vector of nodal pressure heads and {q} is the column vector of nodal consumptions.

In the above equations,

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Finite Element Method with Applications in Engineering

Note that the matrix shown above is applicable for only laminar flow and this does not include the pressure losses associated with the fittings, valves and pumps that are the part of the most fluid network. These pressure losses are normally taken into account by introducing an equivalent length of pipe (Streeter et al., 1998), that will cause the same loss. In many practical applications, the fluid flow in network is turbulent. It is still possible to define an element as a length of fluid carrying conduit, but the element equations are no longer linear. The pressure head-loss versus flow rate relationship is given by DarcyWeisbach, in the following form (Streeter et al., 1998).

Here, H1 – H2 is the pressure head drop in meters between section 1 and 2 which are L (meter) and f is the friction factor. As before, Eq. (6.246) can be used to write two equations for flow at the nodes in terms of pressure at the nodes. Flow ‘entering’ the element at its ends are obtained from Eq. (6.246) as

where h=|H1−H2|=pressure head loss. Now the flow Q is,

where Te is the transmittance of the element in turbulent flow case, which is defined as

Then from Figure 6.34, the turbulent flow equations are

for node 1,

for node 2. In the matrix form, above equations can be written as

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Finite Element Method with Applications in Engineering

As in the laminar flow case, for flow network in Figure 6.35 the individual element equations from Eq. (6.251) can be written as follows.

The global matrix can be written as follows:

where

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Finite Element Method with Applications in Engineering

The global assembly of these element matrices is given above. The individual elements in the global matrix on the left hand side are not independent of the pressure head H. These matrix elements are to be updated often in each iteration. The number of iterations depends on the convergence criteria. Hence, values of pressure heads are required to be determined by an iterative process. At the first instance, pressure heads at different nodes are assumed and the global transmittance matrix is formulated. The ensuing set of linear equations can be solved for pressure head and the transmittance matrix can be updated again. This procedure can be repeated till convergence is achieved. Illustrative Example Find the flow rates in each of the elements (pipes) in the network as shown in Figure 6.35. The diameters of the pipe are 10, 10, 10, 10, 10, 10, 12, 12, 12 mm, respectively. The lengths are 1,000, 1,000, 1,000, 1,000, 1,000, 1,000, 1,500, 1,500, 1,500 mm, respectively. The flow rates at each node out of the network are [+125, –25, –25, –25, –25, –25] cubic cm/min. The liquid flowing is water. If the pressure head at node 1 is 2 m expressed in terms of the height of liquid that is flowing, find the pressure heads, (H), at each of the remaining nodes and the flow rates in the pipe. The friction factor may be taken as 0.02. Solution: The given data has been tabulated as follows.

The problem has been solved by using MATLAB program as given in Appendix G (Section G.1). Results are as shown below. Pressure head values at different nodes in (m) are: Node

H (m)

1

2.00000

2

1.99999

3

1.99995

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Finite Element Method with Applications in Engineering

4

1.99997

5

1.99999

6

1.99995

Discharge values in different pipes, on the other hand, are Pipe No.

Discharge (lit/s)

1

0.00061

2

0.00019

3

0.00065

4

0.00031

5

0.00027

6

−0.00016

7

0.00082

8

−0.00029

9

0.00028

6.11.2 Electrical Network Analysis In the previous section, the fluid flow network was discussed, wherein flow path was considered as a finite element of fluid network. Similarly, the direct-current electrical networks can be considered as a finite element network. The current carrying member of the electrical network can be taken as a finite element. The Ohm's law can be used for establishing the element characteristics. Procedure of finite element formulation is very similar to those used for fluid flow network. Voltages V1 and V2 play the same role as the nodal pressures and a current I replaces the flow rate Q. Basic relationship between the current flowing and voltage difference as given by Ohms' law is (see Figure 6.36)

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Finite Element Method with Applications in Engineering

Figure 6.36 A typical element in an electrical network

Figure 6.37 Basic electrical circuit where Ge is the conductance of the element e and Ie is the current flowing in the resistance (element) e, Re is the resistance of the element, Ii(e) is the current entering the element e through node i and Ij(e) is the current entering the element e through node j, such that Ii(e) = Ie and Ij(e) = −Ie. It can be shown that

Consider the basic electrical circuit shown in Figure 6.37. Element equations for each resistor can be written as follows. Element No. 1 (connecting nodes are 1 and 2): The element equation is

Element No. 2 (connecting nodes are 2 and 3): The element equation is

Element No. 3 (connecting nodes are 2 and 3): The element equation is

Element No. 4 (connecting nodes are 3 and 4): The element equation is

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Finite Element Method with Applications in Engineering

Element No. 5 (connecting nodes are 3 and 4): The element equation is

Element No. 6 (connecting nodes are 4 and 5): The element equation is

Here, Gi = 1/Ri, i = 1,2,…6. All these element equations are assembled into the global form. There are 6 elements and 5 nodes. Thus, the global assembly will be 5 by 5 size matrix given as

From Figure 6.37, it may be noted that

From these equations, the right hand side of the global matrix can be written as

Thus,

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Finite Element Method with Applications in Engineering

Proper boundary conditions must be substituted before this problem can be solved. Ensuing set of linear equations can be solved for voltage at various other nodes. Illustrative Example For the electrical network shown in Figure 6.37, the voltage difference between nodes 1 and 5 is 10 V. The resistance at various resistors are: 250, 100, 200, 300, 400, 500 Ohm, respectively. Calculate the voltage at various nodes and the current in each element? Solution: It can be seen in Figure 6.37, that are 6 elements and 5 nodes. By substituting G1 = 1/250, G2 = 1/100, G3 = 1/200, G4 = 1/300, G5 = 1/400 and G6 = 1/500, V1 = 10 V and V5 = 0, in Equation (6.273), following ensuing equation is obtained.

This equation set has been solved by using the Matlab program in the Appendix G, Section G.2. The current values are as follows. Element No.

Currents

1

0.01012

2

0.00675

3

0.00337

4

0.00578

5

0.00434

6

0.01012

On the other hand, voltages at nodes are as follows.

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Finite Element Method with Applications in Engineering

Node No.

Voltage

1

10.00000

2

7.46988

3

6.79518

4

5.06024

5

0.00000

6.12 SUMMARY ELEMENTS

OF

ELEMENT

MATRICES

FOR

ONE-DIMENSIONAL

FINITE

Element matrices of one-dimensional finite elements for various cases considered can be summarized as follows. Element stiffness matrix for a truss/spring element: (in the local co-ordinate system)

Element stiffness matrix for a plane truss element: (in the global co-ordinate system)

Element stiffness matrix for a space truss element in the global co-ordinate system:

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Finite Element Method with Applications in Engineering

Element stiffness matrix for a torsion element:

Element conductivity matrix for a heat flow element:

Element permeability matrix for a fluid flow element:

Element stiffness matrix for a beam element:

Element stiffness matrix for a plane frame element: (in the global co-ordinate system)

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Finite Element Method with Applications in Engineering

The C, S and L in above equations are:

Element stiffness matrix for a grid element: (in the global co-ordinate system)

206

Finite Element Method with Applications in Engineering

where

Element matrix for an element in flow network analysis

where

for laminar flow and

for turbulent flow.

Element matrix for a resistor element

6.13 CLOSING REMARKS In this chapter, one-dimensional finite element analysis of engineering problems is described. The direct as well as minimum potential energy methods in Chapter 4 are used in the analysis. Linear type line elements are used in the problem analysis. Engineering problems of axial bar, plane and space truss, beam, frame, grid, pipe flow, electrical networks, porous media flow, torsion problem and heat conduction are discussed as sample applications. Few sample problem analysis are also demonstrated. Sample Matlab programs are presented in the Appendix, which can be used to solve similar types of 207

Finite Element Method with Applications in Engineering problems. Even though the problems discussed in this chapter are related to onedimensional FEM analysis using direct and variational approach, method of weighted residual approach can also be applied for various one-dimensional problems in a very similar way as discussed in Chapter 4. The two-dimensional FEM analysis of engineering problems will be discussed in the next chapter, i.e. Chapter 7. EXCERCISE PROBLEMS 1. Compute the assembled 8 × 8 matrix for the spring assemblage shown in the figure.

2. Analyze the composite axial bar shown in the figure when F2 = 10 kN; F2 = 10 kN and U4 = 2 mm; and F2 = 10 kN and U1 = U4 = 2 mm. Consider Est = 200 GPa, Ast = 2 × 10−2 m2; Eal = 70 GPa, Aal =

1 × 10−2 m2

3. Analyze the plane truss structure shown in the figure using finite element method. Let E = 200 GPa, A = 1 × 10−2 m2 and L = 1 m.

4. Analyze the plane truss shown in the figure by using finite element method. 208

Finite Element Method with Applications in Engineering Let E = 200 GPa, A = 2 × 10−2 m2

5. Analyze the space truss shown in the figure by using finite element method. Let E = 200 GPa, A = 4 × 10−2 m2

6. Analyze the shaft shown in the figure by using finite element method. Let G = 70 GPa, J = 4 × 10−2 m4, L = 2 m.

7. Analyze the temperature distribution through the composite wall shown in the figure using finite element method. Also determine the head flux entering and exiting the system.

8. Analyze the one-dimensional flow through porous media over the section shown in the figure by using finite element method. Also compute the flow rate and fluid velocity in each element. Let A = 4 cm2.

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Finite Element Method with Applications in Engineering

9. Analyze the beam system shown in the figure by using finite element method.

10. Analyze the beam supported by a vertical spring by using finite element method.

11. Analyze the beam system shown in the figure by using finite element method.

12. Analyze the plane in the figure by using finite element method.

13. Analyze the plane frame shown in the figure by using finite element method.

210

Finite Element Method with Applications in Engineering 14. Analyze a horizontal elastic member supported by a wire by using finite element method.

15. Analyze the grid system shown by using finite element method.

16. Find the flow rates in each of the elements (pipes) in the network as shown in Figure 6.35. The diameters of the pipe are 10, 10, 10, 10, 10, 10, 12, 12, 12 cm, respectively. The lengths are 100, 100, 100, 100, 100, 100, 150, 150, 150 m, respectively. The flow rates at each node out of the network are [+50, −10, −10, −10, −10, −10] lit/sec. The liquid flowing is water. The friction factor may be taken as 0.02. If the pressure head at node 1 is 15 m, expressed in terms of the height of liquid that is flowing, find the pressure heads, (H), at each of the remaining nodes and the flow rates in the pipe. [Hint: The problem can be solved by suitably modifying the MATLAB program given in Appendix G, Section G.1.]

REFERENCES AND FURTHER READING Backstrom, G. (1999). Fluid Dynamics by Finite Element Analysis, Studentlitteratur, Lund. Baker, A. J. (1983). Finite Element Computational Fluid Mechanics, McGraw Hill, Hemisphere, New York. Bathe, K. J. (2001). Finite Element Procedure, Prentice Hall of India, New Delhi. Bear, J. (1972). Dynamics of Fluids in Porous Media. Elsevier, New York. Chandrupatla, T. R. (2004). Finite Element Analysis for Engineering and Technology, University Press, Hyderabad. Connor, J. J., and Brebbia, C. A. (1976). Finite Element Techniques for Fluid Flow, Butterworth, London. Chung, T. J. (1978). Finite Element Analysis in Fluid Dynamics, McGraw Hill, New York.

211

Finite Element Method with Applications in Engineering Crandall, S. H., Dahl, N. C., and Lardner T. J. (1978). An Introduction to the Mechanics of Solids, Mechanics of Solids, McGraw Hill, New York. Desai, C. S., and Abel J. F. (1987). Introduction to the Finite Element Method—A Numerical Method for Engineering Analysis, Wadsworth, California. Gere, J. M. (1965). Analysis of Framed Structures, Princeton, New Jersey. Huebner, K. H. (1975). The Finite Element Method for Engineers, John Wiley and Sons, New York. Logan, D. L. (1993). A First Course in Finite Element Analysis, PWS Publishing, Boston. Mohr, G. A. (1992). Finite Elements for Solids, Fluids and Optimization, Oxford University Press, New York. Pironneau, O. (1989). Finite Element Methods for Fluids, John Wiley and Sons, Chichester. Rao, S. S. (1999). The Finite Element Method in Engineering, Pergamon Press, Oxford. Reddy, J. N., and Gartling, D. K. (1994). The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, London. Segerlind, L. J. (1985). Applied Finite Element Analysis, John Wiley and Sons, New York. Shames, I. H., and Dym, C. L. (1995). Energy and Finite Element Methods in Structural Mechanics, Wiley Eastern, New Delhi. Stasa, F. L. (1985). Applied Finite Element Analysis for Engineers, CBS Publishing Japan, New York. Streeter, V. L., Wylie, E. B., and Bedford, K. W. (1998). Fluid Mechanics, WCB/McGraw Hill, Boston. Visser, W. (1985). A finite element method for the determination of non-stationary temperature distribution and thermal deformations, Proceeding of Conference on Matrix Methods in Structural Mechanics, Air Force Institute of Technology, Ohio. White, R. E. (1985). An Introduction to the Finite Element Method with Applications to Nonlinear Problems, John Wiley and Sons, New York. Zienkiewicz, O. C. (1980). Finite Element Method, 3rd ed., McGraw Hill, New York.

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Finite Element Method with Applications in Engineering

7 Two Dimensional Finite Element Analysis 7.1 INTRODUCTION One-dimensional finite element analysis of various engineering problems was discussed in the previous chapter. It was discussed in the introductory chapter that even though most engineering problems are three-dimensional in nature, a two-dimensional analysis can give sufficiently accurate and satisfactory results in many occasions. Since threedimensional analysis is generally cumbersome and time consuming, majority of engineers and scientists prefer two-dimensional analysis to the maximum extent possible. Twodimensional finite element analysis of various engineering problems is discussed in this chapter. Several practical engineering problems can be defined conceptually in twodimensions and can be represented in terms of two-dimensional governing equations. Generally, two-dimensional FEM analysis and discretization can render good results when the primary unknown quantities vary mainly on two directions while keeping constant in the third direction. Application of finite element analysis to two-dimensional problems will be discussed in this chapter. In particular, the following formulations will be considered. 1. 2. 3. 4. 5. 6.

Triangular element for two-dimensional fluid flow analysis Triangular element for two-dimensional stress analysis Triangular element for axisymmetric analysis Quadrilateral element for two-dimensional stress analysis Isoparametric element for two-dimensional stress analysis Application of method of weighted residual for various partial differential equations

Because the scalar functionals (strain energy) are easy to define, especially for two- and three-dimensional analyses, they will be applied to the fluid flow and stress analysis initially. Further, method of weighted residual is used for the solution of various partial differential equations. 7.2 TWO-DIMENSIONAL FLOW THROUGH POROUS MEDIA (SEEPAGE FLOW) Formulation presented in this section is focused on the two-dimensional fluid flow analysis. However, the same formulation is applicable straightforwardly to two-dimensional torsion and two-dimensional steady state heat flow (in the absence of convection). At steady state, the governing differential equation for these problems can be written as

Table 7.1 Analogy Between Two-Dimensional Seepage, Heat Flow and Torsional Analyses

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Finite Element Method with Applications in Engineering

The primary unknown field ɸ is function of (X,Y). The ɸ and other terms appearing in Eq. (7.1) are explained in Table 7.1 in the context of two-dimensional seepage, heat flow and torsional analyses. Eq. (7.1) can be solved by modelling the problem with two-dimensional finite elements. The two-dimensional elements are thin plate-like (plane) elements such that two coordinates define a position on the element surface. The ɸ appearing in Eq. (7.1) is given approximately by

for two-dimensional seepage analysis, where h indicates elevation (or fluid head), p represents pore pressure and w is the specific weight of the fluid (water). The velocity head term v2/(2g) has been customarily omitted from the above equation. The two-dimensional steady state seepage problems can be categorized into (i) unconfined seepage problem and (ii) confined seepage problem. Unconfined seepage problem Seepage through an earthen dam can be classified as unconfined seepage. Unconfined seepage is distinguished from the confined seepage by the presence of a free or phreatic surface. At the phreatic surface, the potential ɸ equals the fluid head h measured from a datum. Moreover, there cannot be flow in the direction normal to the free surface. (Also, note that flow cannot occur through an impervious bed. In other words, for the example shown in Figure 7.1, ∂ɸ/∂Y = 0 at the impervious bed.) Mathematically, these two conditions can be written at free surface as

where n indicates normal to the free surface. Both the conditions mentioned above need to be satisfied simultaneously. The mixed boundary condition problem introduces nonlinearity in the system and an iterative approach is required to solve the problem. Such a case is discussed later in the chapter. Confined seepage problem In the case of a confined seepage, the flow occurs in the absence of a free surface, through a saturated media subjected to prescribed boundary conditions (usually ɸ). The confined steady-state seepage requires a linear analysis for solution. An example of confined seepage flow is shown in Figure. 7.2. 214

Finite Element Method with Applications in Engineering

Figure 7.1 Unconfined seepage through a dam The porous, saturated media in Figure 7.2 can be modelled by employing a variety of twodimensional elements. A typical constant strain triangular (CST) element used in discretization is shown in the figure. The finite element formulation for CST element is elaborated below. 7.2.1 Step-by-step Formulation for the CST Element for Two-Dimensional Confined Seepage Analysis In the formulation presented below, steps outlined in Chapter 3 are followed. Step 1: Discretize and select element type A typical i-th CST element from the discretized media is shown in Figure 7.3.

Figure 7.2 Confined seepage flow

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Finite Element Method with Applications in Engineering

Figure 7.3 CST element The CST element has 3-nodes. Choose any apex as node 1, number the remaining 2nodes in the counter-clockwise direction, starting from node 1. At each node, there is one primary unknown fluid potential ɸ. Thus, the nodal DOF (degree of freedom) for the CST element for seepage analysis are ɸ1, ɸ2 and ɸ3 at nodes 1, 2 and 3, respectively. Let the nodal co-ordinates in the X –Y plane be defined as (X1, Y1), (X2, Y2), and (X3, Y3). Step 2: Select approximation function Unknown continuous field ɸ can be expressed in terms of nodal quantities ɸi, i = 1, 2, 3. Hence, three terms can be chosen uniquely to form an approximation function. In many two-dimensional situations, Pascal's triangle is used as the basis in choosing an approximation function of the form

Here, dij represents a constant term. Pascal's triangle is shown in Figure 7.4. By selecting the first three terms from the Pascal's triangle, the approximation function can be constructed as where d00, d10 and d01 are unknown constants. Letting for convenience, Eq. (7.4) can be written as Application of the above equation at nodes 1, 2 and 3 results in

216

Finite Element Method with Applications in Engineering Figure 7.4 Pascal's triangle Alternatively, in the matrix form,

Then,

where |P| is the determinant of [P] and

The [P] can be obtained to be To further simplify the expression for |P|, consider Figure 7.5.

Figure 7.5 Area calculation for a CST element 217

Finite Element Method with Applications in Engineering From Figure 7.5,

Thus, Similarly, it can be proved that

It can be seen from Eqs. (7.10) and (7.14) that

By substituting the above equations into Eq. (7.6) and by collecting the coefficients of ɸ1, ɸ2and ɸ3, the following equation can be derived. Here,

The N1, N2 and N3 are called the shape functions for the CST and [N] denotes the shapefunction matrix (Ni are also called local area co-ordinates). Properties of shape functions It can be easily verified that 1. Ni = 1 at the i-th node and Ni = 0 at the other 2-nodes (i = 1, 2, 3). 2. Variations of Ni, i = 1, 2, 3, over an element are shown in Figure 7.6.

218

Finite Element Method with Applications in Engineering By referring to the second property of shape functions, it can be noted that at any point (X,Y),

Figure 7.6 Shape functions for a CST element Alternatively, from the divided areas A1, A2 and A3 in Figure 7.7(a), it can be proved that

In many finite element books, Ni are designated as local, area or triangular co-ordinates. The triangular co-ordinate system (N1, N2, N3) is defined such that the nodal co-ordinates of nodes 1, 2 and 3 are respectively (1, 0, 0), (0, 1, 0) and (0, 0, 1), as shown in Figure 7.7(b). Step 3. Define constitutive law and gradients. Gradient of Unknown Field Let εx and εy be the gradients (slopes) of ɸ in the X and Y directions, respectively. Then,

219

Finite Element Method with Applications in Engineering

Figure 7.7 (a) Another interpretation of shape functions and (b) Triangular co-ordinate system From Eqs. (7.6) and (7.22), it can be easily seen that

Thus, the gradients (strains) are constant over the element and, hence, a 3-node triangular element is referred to as constant strain triangle (CST) element. By substituting Eq. (7.17) into Eq. (7.22), it can be shown that

Constitutive Relations (Darcy's Law) Noting that flow takes places from a higher fluid potential to a lower one, the velocities vx, vy of the seepage flow in the X and Y directions are given, respectively, by

Therefore,

or where

Step 4: Derive element equations

220

Finite Element Method with Applications in Engineering As mentioned earlier, energy method is applied to derive element equations. Application of the energy method requires a scalar functional πp to be defined. The πp for seepage analysis is given by

Assuming unit thickness in the Z-direction, Eq. (7.31) can be written, in view of Eqs. (7.25) and (7.29), as

where

Minimization of πp results in the element equations Note that [B] and [D] are constant matrices. Therefore, Eq. (7.34) can be written as

Steps 5, 6 and 7: Assemble element equations and apply boundary conditions, solve for primary unknowns, calculate secondary unknowns Assembling element equations given by Eq. (7.35) for the whole system, equation can be derived, where [K], {q} and {F} represent the assembled permeability matrix, the potential head vector and the externally applied fluid flux vector, respectively. After applying geometric boundary conditions (specified potential heads corresponding to Dirichlet conditions) Eq. (7.37) can be solved for the unknown {q}. The secondary quantities can be obtained by equations developed previously. For example, flow velocity components can be computed from Eqs. (7.26) and (7.27). 221

Finite Element Method with Applications in Engineering Illustrative examples dealing with confined seepage are discussed later in this chapter.

7.3 TWO-DIMENSIONAL STRESS ANALYSIS Two-dimensional finite elements can be used for two-dimensional stress analysis problems. A summary of pertinent equations from the theory of elasticity is presented below in order to better understand finite element formulation for two-dimensional stress analysis. 7.3.1 Review of Theory of Elasticity Three-Dimensional Elastic Body (Crandall et al., 1978) Consider an elastic body in space. Any particle within the elastic body may experience displacements U, V, W in the X, Y, Z directions, respectively. For such displacement components, under the assumption of isotropic, elastic material properties, following expressions are well established. 1. Equations relating strains to displacements Normal strains:

Shear strains:

2. Stress-strain equations Normal strains:

Shear strains:

3. Generalized Hooke's law

222

Finite Element Method with Applications in Engineering 7.3.2 Application of Three-Dimensional Equations for Two-Dimensional analysis In reality, a physical system has two-dimensional domain. However, wherever possible, engineering approximations are introduced on the basis of physics of the problem to reduce the dimension of a problem at hand. For example, in the previous chapter, onedimensional analysis was presented for a truss, beam and a frame member under the assumption of slender body (i.e., small cross sectional dimensions compared to member length). Similarly, in many three-dimensional situations (even when all three dimensions are comparable), the domain of the problem can be considered to be a two-dimensional one. Three common situations are: (i) plane stress, (ii) plane strain and (iii) axisymmetric solids. Typical examples of plane stress, plane strain and axisymmetric situations are shown in Figure 7.8. These three analyses procedures are elaborated next. 7.3.2.1 Plane Stress Situation It is that state of stress in which the normal stress and the shear stresses directed perpendicular to the plane X-Y (in which loads are applied) are assumed to be zero. This approximation can be applied when two dimensions of a physical system are larger than the third one (say, thickness). A typical example of plane stress situation is a thin plate with a hole subjected to in-plane tensile loading. Note that if the loads act normal to the plate, plate bending theory (e.g. Bathe, 2001) is required in the analysis.

Figure 7.8 Typical examples of plane stress, plane strain and axisymmetric situation For the plane stress condition, it is assumed that

Hence, it can be seen from Eq. (7.42) that

223

Finite Element Method with Applications in Engineering

and where {σ} and {ε} are

and

7.3.2.2 Plane Strain Situation It is a state of strain in which the strain normal to the X-Y plane (the plane in which loads are applied), i.e., εz, and the shear strains γxz and γyz are assumed to be zero. The plane strain approximation can be applied when the third dimension (say length) is much larger than the dimensions of the cross section, and the loading is uniform over the length. A typical example where the plane strain approximation may be adequate is a gravity dam. For plane strain case, following strains are assumed to be zero.

In view of Eq. (7.42), it can be seen from the above equation that

or,

where

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Finite Element Method with Applications in Engineering

7.3.2.3 Axisymmetric Situation As the name ‘axisymmetric’ suggests when the geometry, boundary conditions, material properties and loading are identical with respect to axis of symmetry (i.e., independent of angle θ), the three-dimensional problem can be reduced to an analogous two-dimensional problem. Typical examples, where the axisymmetric analysis may be sufficient, are pressure vessels, pistons, rotors, water tanks and circular column foundations. General loading will be considered later in Chapter 8. By using cylindrical co-ordinate system (r, θ, Z) and denoting the displacement components in radial, tangential and vertical directions as U, V and W (see Figure 7.9), it can be shown for axisymmetric case that

Here, εr, εθ and εz are the components of strain in the radial, tangential and vertical directions, respectively.

Figure 7.9 Axisymmetric analysis From Eqs. (7.53) and (7.42), it can be easily shown that

7.3.3 CST Element for Plane Stress and Plane Strain Analyses Steps outlined in Chapter 3 are followed in the finite element formulation. Step 1: Discretize and select element type 225

Finite Element Method with Applications in Engineering A typical i-th CST element used to discretize a two-dimensional domain is shown in Figure 7.10. A typical CST element has 3-nodes. Unknown continuous field consists of displacement components U and V in the X and Y direction, respectively. Step 2: Select approximation function By following the formulation of CST element for seepage analysis, it can be shown that

Figure 7.10 Typical CST element for plane stress/strain analysis where N1, N2, N3 are shape functions for the CST element and U1, U2, U3, V1, V2, V3 are nodal displacement components. Therefore,

or Here,

Step 3: Define Strain-displacement relations and constitutive law Strain–Displacement (Gradients) Relations

226

Finite Element Method with Applications in Engineering

Therefore,

Moreover,

where

Note that {ε} is a constant vector. Therefore, strains are constant. Constitutive Relations

where [D] is given by Eq. (7.47) for plane stress conditions and by Eq. (7.52) for plane strain conditions. By substituting Eq. (7.61) into Eq. (7.63),

can be obtained. Step 4: Derive element equations By employing energy method, the element equations can be derived easily. The potential energy πp is defined for two-dimensional analysis as

Here, {X}, {fee} and {T} represent body force (per unit volume) vector, the element nodal force vector and the surface traction (per unit surface area) vector, respectively. In view of Eqs. (7.61) and (7.64), Eq. (7.65) can be written as

227

Finite Element Method with Applications in Engineering where

and {P} represents the element nodal force vector. Minimization of πp results in the element equations

Element Stiffness Matrix Since [B] and [D] are constant matrices, for an element with constant thickness t, Eq. (7.67) becomes

Assume t = 1 for plane strain analysis. Body Forces The equivalent nodal loads due to body forces are given by

where

and Xb and Yb are the body forces (weight densities) in the X and Y directions, respectively. Hence,

Consider

Note that

228

Finite Element Method with Applications in Engineering By definition, co-ordinates of the centroid are Thus,

it

can

be

observed

that

Similarly, by integrating other terms in Eq. (7.73),

The above equation indicates that each node shares 1/3 of total body force. Element Surface Forces The equivalent nodal loads due to surface forces are given by

Steps 5, 6 and 7: Assemble element equations and apply boundary conditions, solve for primary unknowns, calculate secondary unknowns Assembling the contributions from all elements of discretized domain, equation

can be derived, where [K], {q} and {F} represent the assembled stiffness matrix, the nodal displacement vector and the externally applied, equivalent nodal load vector, respectively. After applying geometric boundary conditions (specified displacement components), Eq. (7.80) can be solved for the unknown {q}. The secondary quantities can be obtained by

229

Finite Element Method with Applications in Engineering equations developed previously. For example, the principal stresses can be computed from

where σx, σy, τxy and be calculated from Eq. (7.64) (θp in the above equation indicates principal angle). Illustrative Example Two-dimensional model of an anchor plate of a communication tower's guy cable is shown in the illustrative figure. The anchor consists of a triangular steel plate, which is subjected to force of 20 kN, as shown in the figure. Analyze the anchor plate. Thickness of the plate is 8 mm. Consider E = 200 GPa and v = 0.3.

Solution: Consider reference axes X and Y as shown in the figure. Further for illustration, consider one CST element to model entire plate, with nodes as shown in the figure.

From Eq. (7.11),

Considering the state of plane stress, the elasticity matrix [D] is obtained from Eq. (7.47) as 230

Finite Element Method with Applications in Engineering

Further, Eq. (7.69) can be written in partitioned form as

where

Due to homogeneous boundary conditions at nodes 2 and 3,

Thus, only ḵ11 is required for computation of Q1. By substituting values of t, A,D11,D12, D33,b1 and a1, the following equation is obtained.

By making use of Eq. (7.64),

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Finite Element Method with Applications in Engineering

Since only one CST element is used to model the entire plate, the stress field is constant throughout the plate.

Figure 7.11 Typical triangular element to analyze axisymmetric solids 7.3.4 Triangular Element for Axisymmetric Analysis Step-by-step formulation of a triangular element is presented below. Step 1: Discretize and select element type A typical 3-node, triangular element from the discretized domain of an axisymmetric solid is shown in Figure 7.11. An axisymmetric solid is modelled by rotating such elements through 360°. At each node, there are two DOF corresponding to the displacements in the radial and vertical directions. Thus,

Step 2: Select approximation function It can be shown that

where shape functions N1, N2 and N3 are given by

Alternatively,

232

Finite Element Method with Applications in Engineering

or

Here,

Step 3: Define strain–displacement relations and constitutive law Strain-Displacement (Gradients) Relations It can be shown from Eqs. (7.53) and (7.86) that

i.e.,

where

Note that the strain vector is no longer constant. Also, [B] involves singular terms when r = 0. Constitutive Relations where [D] is given by Eq. (7.54). By substituting Eq. (7.89) into Eq. (7.91), can be obtained. Step 4: Derive element equations 233

Finite Element Method with Applications in Engineering By minimizing the potential energy, element equations can be derived as Here,

and

Integrations appearing in Eqs. (7.94) and (7.95) can be evaluated by several ways. 1. Integrate all the terms explicitly. However, the explicit evaluation may be tedious and complicated due to (1/r) terms in the integrands. 2. Use numerical integration. Usually, a Gauss-quadrature procedure is employed for numerical integration. Refer to Appendix D for details. 3. Integrations can be obtained approximately by evaluating integrands at the centroid of an element. This approach is reasonable when ‘small’ elements are employed in discretization. At the centroid,

and thus, say, [Ke] can be approximated as

where [ ] is evaluated at the centroid. Steps 5, 6 and 7: Follow steps outlined for plane stress/plane strain analyses by using CST element Axisymmetric formulations with both axisymmetric as well as non-axisymmetric forces have been further elaborated later. 7.3.5 Some remarks on Triangular Elements In order to improve accuracy of results, mesh can be refined by reducing the size of 3node, triangular elements. This approach is often known as h-version of finite element analysis. Alternatively, in the p-version, instead of refining the mesh, higher order elements (where strains can vary linearly, parabolically, cubically, etc.) can be used to improve accuracy. Higher order elements are presumed to closely model the strain and stress distribution in a continuum. Triangular elements having 6- and 10-nodes are shown in Figure 7.12. Since the strains vary linearly and quadratically for 6- and 10-node elements, they are, respectively, referred to as linear strain triangle (LST) and quadratic strain triangle (QST) elements.

234

Finite Element Method with Applications in Engineering 7.3.6 Four-Node Rectangular Element for Plane Problems In this section, a step-by-step, finite element formulation of a 4-node rectangular element is presented. Step 1: Discretize and select element type A rectangular element shown in Figure 7.13 can be used to discretize a domain having a regular geometric shape. However, in practice, quadrilateral elements are used more extensively. Note that a rectangular shape is a degenerated quadrilateral.

Figure 7.12 Higher order triangular elements

Figure 7.13 Typical, 4-node, rectangular element A typical 4-node, rectangular element from the discretized domain is shown in Figure 7.13. The element DOF vector for this element is given by

Step 2: Select approximation function Since there are 4-nodes (and, hence, four unknowns per global direction), one can select only four terms from the Pascal's triangle (Figure 7.14). Assume

235

Finite Element Method with Applications in Engineering By eliminating ai's and bi's, it can be shown that

where

Figure 7.14 Criterion for selecting approximation function The displacement vector {U (X, Y)} can be written as

In other words,

where

Step 3: Define strain–displacement relations and constitutive law Strain-Displacement Relation It can be easily shown that

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Finite Element Method with Applications in Engineering

Let

Then,

where

Note that, unlike in the CST formulation, the strains are not constant. Constitutive Relation As shown earlier,

where 3 × 3 elasticity matrix [D] is given by Eq. (7.47) for plane stress condition and Eq. (7.52) for plane strain conditions. Step 4. Derive element equations. By following the procedure adopted for the CST formulation, in conjunction with energy method, the element equations are given by

where

237

Finite Element Method with Applications in Engineering

Here, {X} represents the body force vector, {P} indicates the element nodal force vector and {T} represents the surface traction vector. Analytical evaluation of integrations in Eqs. (7.111) and (7.112) is tedious. Therefore, numerical integration is usually employed. The most widely used integration scheme in the FEM is the Gauss-quadrature. The Gauss-quadrature is defined in a natural (or intrinsic) co-ordinate system. Therefore, the integrals expressed in the X-Y co-ordinate system need to be transformed to the natural co-ordinates ξ-η. The numerical integration is discussed later in the context of isoparametric formulation. Step 5, 6 and 7: Follow procedures outlined previously

7.4 ISOPARAMETRIC FORMULATION Term ‘isoparametric’ is derived from the use of the same shape functions (or interpolation functions) to define the element's geometric shape as those used to define the variation of unknown field within an element. Isoparametric element equations are formulated using a ‘natural’ (or ‘intrinsic’) co-ordinate system defined by the element geometry. The intrinsic co-ordinates normally vary from −1 to +1, regardless of the element's dimensions. Thus, use of such a co-ordinate system becomes very attractive, especially when all the matrices need to be integrated numerically. The concept of isoparametric formulation is explained below in the context of a line (bar) element and is extended later to a two-dimensional, quadrilateral element. 7.4.1 Two-Node Isoparametric Line Element (Bar Element) Consider a 2-node bar element of length L as shown in Figure 7.15. Refer to the finite element formulation presented in the Chapter 6 for a bar element. In this section, this formulation is revisited in the context of natural co-ordinate system. For the element shown in Figure 7.15, the nodal co-ordinates can be written in global, local or intrinsic co-ordinate system. At node 1: X = X1; x = 0; ξ = −1. At node 2: X = X2; x = L; ξ = +1. X global co-ordinate (X1 ≤ X ≤ X2) x local co-ordinate (0 ≤ x ≤ L) ξ natural or intrinsic co-ordinate (−1 ≤ ξ ≤ 1) Co-ordinate Transformation Global co-ordinate (X) is related to natural co-ordinate (ξ) by

Therefore,

238

Finite Element Method with Applications in Engineering

or

where

Variation of shape functions over an element are shown in Figure 7.16. Approximation Function

Figure 7.15 Two-node, isoparametric line element

Figure 7.16 Shape-function variation with respect to natural co-ordinate Strain-Displacement Relation

From chain rule of differentiation,

239

Finite Element Method with Applications in Engineering

Hence,

Constitutive relation: Stiffness matrix:

Note that X = X1 + x and dX = dx. From calculus

where is called the ‘Jacobian matrix’ of the co-ordinate transformation. | | is determinant of . For one-dimensional case, is a scalar, equal to L/2. From Eq. (7.114),

Therefore,

240

Finite Element Method with Applications in Engineering It can be seen from the above equations that each element of the stiffness matrix is of the form . In general, these integrations can be performed by employing Gaussquadrature procedure. (Refer to Appendix D for details.) However, for a linearly varying function, these expressions can be derived explicitly. Thus, it can be seen for 2-node element that

7.4.2 Four-Node Isoparametric Element for Plane Problems (Quadrilateral Element) Formulation presented for rectangular element is extended for a more general 4-node element in this section. A typical quadrilateral element from the discretized domain is shown in Figure 7.17. Intrinsic co-ordinates ξ, η are also shown in this figure. Note that regardless of the size or shape of the quadrilateral element, the intrinsic co-ordinates always vary from −1 to +1. Thus, a given quadrilateral element is effectively mapped into a rectangular element in the natural co-ordinate system.

Figure 7.17 Quadrilateral element in the X-Y and the ξ-η co-ordinate systems Co-ordinate Transformation By considering four terms from the Pascal's triangle shown in Figure 7.14, X and Y coordinates can be expressed in terms of ξ and η

Co-ordinates of all nodes in the global and natural system are:

241

Finite Element Method with Applications in Engineering Hence, the application of Eq. (7.126) to all the nodes yields

By solving the above equation set for ai's and by substituting the results in Eq. (7.126), is obtained, where

or

where ξi and ηi are natural co-ordinates of i-th node. Similar expression for Y can be derived as Thus, the co-ordinate transformation is given by

or

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Finite Element Method with Applications in Engineering

Figure 7.18 Shape functions for a quadrilateral element Notes: 1. Ni(i = 1, 2, 3, 4) is equal to one at node i and equal to zero at all other nodes. 2. For all values of ξ and η,

Variations of Ni, i = 1, 2, 3, 4 over an element are shown in Figure 7.18. Approximation Function In an isoparametric formulation, the same shape functions are employed to describe the variations in the global co-ordinates and the unknown field. Thus,

Alternatively, 243

Finite Element Method with Applications in Engineering

where

Strain-Displacement Relation It can be easily shown that

where

Here,

The chain rule of differentiation yields

244

Finite Element Method with Applications in Engineering

In the above equations, and

and

are unknowns. They can be obtained, in terms of

, as

Where

Hence

Constitutive Relation The constitutive relation is given by where [D] is given by Eqs. (7.47) and (7.52) for plane stress and plane strain conditions, respectively. Derivation of Element Equations By minimizing the potential energy, the element equations can be obtained, where

Assume t = 1 for plane strain situation. Computations of [Ke], {fb} and {fs} are discussed next. 1. Stiffness matrix: From calculus, for an arbitrary function f(x, y) 245

Finite Element Method with Applications in Engineering

Hence,

Where

Here,

Therefore, It can be seen from the derivation that each element of [Ke] is of the form

These expressions can be evaluated by employing Gauss-quadrature procedure for area integration, as shown in Appendix D. 2. Body forces:

One can use Gauss-quadrature rule to compute {fb}. 3. Surface forces: Consider edge η = 1 with overall length L.

Gauss-quadrature rule can be used easily to compute {fs}. Illustrative Example 1 For the element shown, evaluate k11 of the element stiffness matrix [Ke]. Assume plane strain condition. Let E = 200 GPa, v = 0.25. The global co-ordinates (in m) are shown in the figure.

246

Finite Element Method with Applications in Engineering Solution: From Eq. (7.155), the expression for k11 is obtained as

Where

Therefore,

Hence,

Where

1.

1 × 1 Gauss-quadrature

2.

2 × 2 Gauss-quadrature

3.

3 × 3 Gauss-quadrature

4.

Exact value

Illustrative Example 2

247

Finite Element Method with Applications in Engineering A cantilever beam of span 9 m is subjected to a tip load of 40 kN. Analyze the beam by employing 4-node isoparametric elements under plane stress condition. Assume E = 200 GPa, v = 0.25 and width = 0.3 m.

Solution: The finite element representation of the problem is shown in the figure above. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. The problem has been solved by using program sfeap given in the Appendix F. The nodal displacements are tabulated below. NODE

U -DISP (m)

V -DISP (m)

1

.0000000E+00

.0000000E+00

2

.0000000E+00

.0000000E+00

3

.0000000E+00

.0000000E+00

4

–.1940316E–04

–.6343784E–04

5

.0000000E+00

–.6298111E–04

6

.1940316E–04

–.6343784E–04

7

–.3112135E–04

–.2198400E–03

8

.0000000E+00

–.2197260E–03

9

.3112135E–04

–.2198400E–03

10

–.3502455E–04

–.4232532E–03

11

.0000000E+00

–.4231881E–03

12

.3502455E–04

–.4232532E–03

248

Finite Element Method with Applications in Engineering It can be noted that nodes 1, 2 and 3 are completely restrained and, thus, both U and V displacements are zero at these nodes. Further, the U displacement is zero for all nodes on the neutral axis (viz., nodes 5, 8 and 11). Also, as expected, U displacements at the nodes are anti-symmetric with respect to the reference neutral axis. On the other hand, V displacements at the nodes are symmetric with respect to the neutral axis. Stresses at the centroids of the elements are tabulated below.

7.5 FINITE ELEMENT SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY METHOD OF WEIGHTED RESIDUAL In this section, the finite element formulations of some important steady-state partial differential equations governing the field problems in two dimensions are presented. In particular, Laplace equation, Poisson equation and Helmholtz equation are considered. Laplace equation is an elliptic equation used to characterize steady-state systems (Chapra and Canale, 2002). Thus, the two-dimensional form of Laplace equation can be used to determine the steady-state distribution of an unknown function in two spatial dimensions. The Laplace equation can be used to model a variety of engineering problems involving potential of an unknown such as heat conduction problems, seepage problems, aquifer problems, distribution of an electric field, potential flow problems, etc. Poisson equation is another class of partial differential equations similar to Laplace equation with a constant term such as a source or sink. The two-dimensional form of Poisson equation can be used to determine steady-state distribution of an unknown function in two spatial dimensions with a source or sink term in the domain. The Poisson equation can be used to model a variety of engineering problems involving heat transfer problems, torsion of constant cross section member, steady-state aquifer problems with source or sink, electrostatics, magnetostatics, irrotational flow, etc. Helmholtz equation is another class of steady-state partial differential equations closely related to the quasi-harmonic equations. The Helmholtz equation can be used to model a variety of engineering problems involving standing waves on a shallow body of water (seiche motion) (Taylor et al., 1969), electromagnetic waves (Arlett et al., 1968), acoustic vibration (Kinsler and Fray, 1950), fluid structure interaction (Brebbia and Walker, 1979), etc. For example, the Helmholtz equation with appropriate boundary conditions can be used to model the fluid vibrating in a closed volume or the propagation of electromagnetic waves in a waveguide filled with a dielectric material (Huebner, 1975). These equations can be solved by using method of weighted residual technique or classical variational principle. Here, the Galerkin’s based method of weighted residual has been used in the formulation of these equations for illustration. A brief discussion of the FEM based application of classical variational principle is also presented. 7.5.1 Governing Equations and Boundary Conditions 1. Laplace Equation For homogeneous isotropic media field problems, the Laplace equation in two-dimensions can be written as

249

Finite Element Method with Applications in Engineering where ɸ is the potential function, x and y are the Cartesian co-ordinates. 2. Poisson’s Equation The Poisson's equation can be used to represent the steady state of an unknown variable such as heat transfer with a heat source, groundwater flow with recharge or pumping, electrostatics or magnetostatics with a charge density, a torsion problem with an initial twist etc. For homogeneous media problems the Poisson equation in two dimensions can be written as where ɸ is the potential function and C is the source or sinks term in the domain considered. 3. Helmholtz Equation General form of Helmholtz equation in two dimensions can be written as (Brebbia and Walker, 1979)

where Kx and Ky are media properties (for example, in case of heat transfer problems, Kx and Ky represent the thermal conductivities), λ is a constant and Q is a term such as source or sinks in the domain considered. The commonly used boundary conditions are Dirichlet and Neumann type boundary conditions. The boundary conditions can be generally represented as (refer Figure 7.19)

where n is the unit outward normal to the boundary, α is the transfer coefficient (e.g., heat transfer) and (ɸ - ɸ0) denotes the difference between the potential within the medium; and p are known values and Γ = Γ1 + Γ2.

Figure 7.19 Definition of problem domain 7.5.2 FEM Formulation It can be seen from the above equations that the Helmholtz, Poisson and Laplace equations describe similar class of steady-state field problems with some variations from each other. When Kx = Ky = 1; λ = Q = 0 in Eq. (7.164), Laplace equation is obtained. On the other hand, for Kx = Ky = 1; λ = 0; Q = C, Poisson equation is obtained from Eq. (7.164). Here, the detailed FEM formulation of Helmholtz equation is presented. The FEM formulation for other equations can be easily derived in a similar way. 1. Helmholtz’s Equation Let the domain be divided into n CST elements. By using the Galerkin's approach, first approximate the unknown exact solution ɸ as 250

Finite Element Method with Applications in Engineering

where N is the shape function or interpolation function and {ɸ}i is the array of field variable at specified nodes of the element of the discretized domain. Applying the Galerkin's criteria, the Helmholtz Eq. (7.164) can be reduced to a set of linear equations as follows:

The first term can be integrated by parts as

Similarly, the second term can be written as

By using Eq. (7.168), the original equation can be written as

By substituting for ɸ using the relationship (7.166) and rearranging, following equation is obtained.

Element Eq. (7.170) can be written in a matrix form as

where

251

Finite Element Method with Applications in Engineering As shown in Section 7.2 and Figure 7.3, using CST elements or any other two-dimensional elements and assembling the element matrices and applying the boundary conditions, a linear system of equations can be formed, which are solved for the unknowns. 2. Laplace Equation For the Laplace Eq. (7.162), in a similar way as demonstrated for Helmholtz equation, the element equation can be derived. As given in Section 7.2 and Figure 7.3, using the CST element and shape functions given in Eq. (7.18), the element property matrix can be written with kx = ky = 1 as

Here, p1, p2 and p3 are known values, if any. Eq. (7.173) can be written in the compact form

Eq. (7.174) can be assembled into global matrix over all elements and can be solved to get the potential values ɸ at different nodes, after application of the global boundary conditions. 3. Poisson Equation For the Poisson Eq. (7.163), the element equation can be derived in a similar way as demonstrated for Helmholtz equation. As given in Section 7.2 and Figure 7.3, using the CST element and shape functions given in Eq. (7.18), the element property matrix can be written as

Eq. (7.175) can be written in the form

Eq. (7.176) can be assembled into global matrix over all elements and got solved to get the potential values ɸ at different nodes.

7.6 FEM FORMULATION BASED ON VARIATIONAL PRINCIPLE In the previous section, the FEM formulation for various field problems using Galerkin's based method weighted residual technique has been presented. In this section, the application of variational principle for a field problem is demonstrated by considering a non-homogeneous steady-state field problem. The governing differential equation for nonhomogeneous steady-state media problem such as seepage flow problems, can be expressed as

Here, ɸ is the potential function and k represents the media property such as hydraulic conductivity. The boundary conditions can be generally represented as (refer Figure 7.19) 252

Finite Element Method with Applications in Engineering

where kx and ky are the hydraulic conductivities in x and y directions and n is the unit outward normal to the boundary. As discussed in the previous section, the governing equation can be reduced to a system of equations by using the FEM technique. In the previous section, the method of weighted residual approach for the FEM formulation has been used. Here, the variational formulation approach for the derivation of the element matrices is used. From Eq. (7.177) and (7.178), using equivalent variational formulation, equation reduces to minimization of the functional (Huebner, 1975)

over the whole domain Ω. In the variational approach, the functional has to be minimized so that

Here, Galerkin's technique can also be used to formulate the problem, which is very similar to that given in Section 7.5 for homogeneous steady-state problems. As given in Section 7.2 and Figure 7.3, the CST element and shape functions given in Eq. (7.18) has been used so that

with usual notation as given in Section 7.2. From the variational principle, Eq. (7.180) can be written as

Therefore,

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Finite Element Method with Applications in Engineering

Combining the above three Eqs. (7.183–7.185) and substituting boundary prescribed values p1,p2, p3 etc. if any, the following element matrix will be obtained.

For isotropic case kx = ky = k. Eq. (7.186) can be written in the form, Eq. (7.187) can be assembled into global matrix over all elements and solved to get the potential values ɸ at different nodes after application of boundary conditions.

7.7 FINITE ELEMENT SOLUTION OF STOKES FLOW EQUATIONS In the preceding sections, the FEM formulations of various types of partial differential equations such as Laplace equation, Poisson equation and Helmholtz equations were considered. In this section, the simplest form of two-dimensional steady-state flow called Stokes flow or creeping flow, which has a large number of applications in civil, mechanical, metallurgy and chemical engineering is considered. In many engineering fluid flow problems, the Reynolds number, which represents the ratio of inertia force to viscous force in a fluid motion, can be very small (less than one) and, hence, the inertia forces are insignificant compared to viscous force and can be omitted from the concerned momentum equations (Navier Stokes equations) (Granger, 1985). Here, the flows with small Reynolds number such as slow-moving flows and the flows of very viscous fluids such as polymer processing flows are considered. 7.7.1 Problem Statement The governing equations for Stokes or creeping flows in two-dimensions are given by (Granger, 1985) the following momentum equations.

and the continuity equation is

Here, p is the pressure, u and v are velocity components in the x and y directions and μ is coefficient of dynamic viscosity. Boundary conditions may include specified pressure, velocity or velocity gradient. The u, v and p are solved within the problem domain considered.

254

Finite Element Method with Applications in Engineering 7.7.2 FEM Solution Here, the Galerkin's method based on the weighted residual approach is used. The domain (Figure 7.20) is divided into n CST elements (Figure 7.21). Using the Galerkin's approach, the unknown exact solution ɸ is approximated as

where N is the shape function or interpolation function and {ɸ}i is the array of field variable at specified nodes of the element of the discretized domain, say here, (u, v and p). Considering a two-dimensional domain A bounded by curve S and applying Galerkin's approach on the Eq. (7.188), the following equation is obtained.

As demonstrated in the previous sections, after integration by parts, the first term can be converted as

Similarly, the second term can also be written. Hence, Eq. (7.192) can be written as

Here, lx = nx and ly = ny, as defined early.

Figure 7.20 Problem domain for Stoke's Flow

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Finite Element Method with Applications in Engineering

Figure 7.21 FEM Discretization of the Problem Domain (not to scale) Similarly, Eq. (7.189) can be written as

Using the Galerkin's approach, the continuity Eq. (7.190) can be written as

or,

Consider the Stokes flow problem in a rectangular domain. The domain and boundary conditions are shown Figure 7.20. The domain is divided into CST elements as shown in Figure 7.21 and the numbering is done. As discussed earlier, the interpolation function can be written as

256

Finite Element Method with Applications in Engineering Figure 7.22 (a) Triangular element For the shape shown in Figure 7.22 with 1 unit as the side length of the triangle and 1/6 unit as height of the triangle, the various element properties can be written as follows.

For the following element shape:

257

Finite Element Method with Applications in Engineering

Figure 7.22 (b) Triangular element

For the first elemental shape shown in Figure 7.22(a), Eqs. (7.193), (7.194) and (7.196) can be shown to be

258

Finite Element Method with Applications in Engineering

Where For the second elemental shape shown in Figure 7.22(b), Eqs. (7.193), (7.194) and (7.196) can be shown to be

Using the above equations, the system of elemental equations can be assembled in the form,

259

Finite Element Method with Applications in Engineering where {ɸ} represent the field variables (u, v and p). After substitution of boundary conditions, the system of equations can be iteratively solved to find the unknown velocities and pressure.

7.8 ILLUSTRATIVE EXAMPLES Based on the Galerkin FEM formulation given in Section 7.5, a computer program has been developed using the FORTRAN for the Laplace equation. The program can be used for homogeneous and non-homogeneous media problems. The program is tested with the problem mentioned below. The detailed program listing is given in Appendix G. Example 1–Potential Variation in Porous media Determine the potential variation in a homogeneous isotropic porous media of length 100 m and width 30 m. As shown in Figure 7.23, the potential at one side of the porous media is 300 m and the opposite side is 100 m. Other two sides of the section are considered as impervious. Solution: The problem is solved as a two-dimensional problem using the Laplace equation with the FEM. The domain is discretized using 24 CST elements and 20 nodes as shown in Figure 7.23. This problem is solved using the FORTRAN Computer program given in Appendix G, Section G3. The input file used and results are also given Appendix G. Results are plotted in Figure 7.24(a)and (b). Results are compared with analytical solution ɸ = (ɸ2 – ɸ1) x/L + ɸ1; where ɸ is the head at distance x, ɸ1 is the head at the beginning, ɸ2 is the head at the end and L is the total length. Example 2–Flow net analysis below a Dam Consider the flow beneath a concrete dam on a homogeneous isotropic permeable foundation. As shown in Figure 7.25, there is an impermeable boundary at the upstream and the downstream. Boundary conditions with the potential function and stream function are shown in the figure. Determine the potential and stream function distribution in the considered permeable foundation and draw the flow net.

Figure 7.23 Flow domain and FEM discretization for Example 1

260

Finite Element Method with Applications in Engineering

Figure 7.24 (a) Potential variation–Example 1

Figure 7.24 (b) Potential variation in the aquifer media–Example 1

Figure 7.25 Problem domain and boundary conditions - Example 2 Solution: Consider the Laplace equation in potential function ɸ and stream function Ψ. Both the equations are solved separately with the given boundary conditions. The domain in Figure 7.25is discretized using 44 CST elements and 39 nodes, as shown in Figure 7.26. This problem is solved using the FORTRAN Computer program given in the Appendix G. The

261

Finite Element Method with Applications in Engineering input file used and results are also given in the same Appendix. Results are plotted in the form a flow net in Figure 7.27. Example 3–Flow Beneath a Spillway-Case with three Zones Here, the potential variation beneath a spillway is analyzed. There are three zones with permeabilities 1 m/hr for zone 1, 2 m/hr for zone 2 and 3 m/hr for zone 3. Potentials on the upstream and the downstream are respectively 20 m and 2 m. Domain size is 100 m × 20 m. On both the sides and bottom, the media is impermeable and hence interzonal faces, there are two cut-off walls of size 5 m × 1 m as in Figure 7.28.

Figure 7.26 FEM discretization–Example 2

Figure 7.27 Flow net-flow beneath a dam - Example 2

262

. At the

Finite Element Method with Applications in Engineering

Figure 7.28 FEM discretization-flow beneath a spillway - Example 3 The FEM discretization using CST elements is shown in Figure 7.28. The domain is discretized using 96 triangular elements and 69 nodes as in Figure 7.28. The input data can be prepared in the similar way as in the previous case studies. This problem is solved using the FORTRAN Computer program given in Appendix G. The potential variation at various internal nodes is given in Table 7.1. Example 4–Seepage Problems in Earth Dams Location of free surface is of paramount importance in the case of earth dams, embankments, canals etc. For homogeneous media problems, even though some approximate solution like Casagrande's solution exists, exact location of free surface for heterogeneous or zoned media is still a challenge. The finite element formulation given in Section 7.5 is applied here for the determination of free surface of zoned media. The top boundary must satisfy two constraints: (1) The surface is a no flow boundary and (2) The head at each point on the boundary must be equal to its elevation. With the above conditions and using the FEM formulation for the zoned media, following additional steps are used in the free surface determination. 1. 2. 3. 4.

Assume an initial free surface curve y = f (x). Discretize the domain. Identify the free surface nodes. Using the formulation given in Section 7.5, determine the potential at the nodes. The boundary head values on the nodes on the assumed free surface are compared against the assumed y co-ordinates values of the respective nodes.

Table 7.1 Flow Beneath a Spillway–Potential Variation

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Finite Element Method with Applications in Engineering

5. If the difference obtained in Step 4 is within the permissible tolerance, the obtained potentials are assumed as free surface. 6. If the discrepancy is too large, go for another iteration with Y co-ordinates of free surface node as the potential values obtained in Step 3. The procedure is repeated in each iteration, until the discrepancies are within the permissible limit. In each iteration, the mesh used should be adjusted to relocate the new free surface position. Example: The location of free surface in a zoned earth dam is required to be determined. There are three zones with different permeabilities. It is assumed that permeabilities of zone 1, 2 and 3 are 2 m/hr, 1 m/hr and 3 m/hr, respectively. Upstream head is 10 m and the downstream head is 2 m. Bottom face,

264

Finite Element Method with Applications in Engineering

Figure 7.29 FEM Discretization–Free Surface in a Zoned Earth Dam–Example 4 where the dam is placed is impermeable. On both the impermeable face and the free surface, . Dimensions are as shown in Figure 7.29. Discretization of the domain is as shown in Figure 7.29. A total of 26 CST elements and 24 nodes are used for discretization. Number of prescribed boundary nodes are 5. A computer program, in a very similar way that is given in the Appendix G, can be constructed for this problem also. Results along with the location of free surface is plotted in Figure 7.30. Example 5–aquifer Problem Here, a steady ground water recharge problem is analyzed using the FEM formulation for Poisson equation given in Section 7.5. A domain of 2,400 m length and 1,200 m width is considered. There is a recharge basin of size 200 m × 100 m at the central portion. The recharging rate is 0.2 m/day/m. The thickness of the aquifer is assumed to be 100 m. The transmissibility is 1,000 m2/day. Configuration of the aquifer is shown in Figure 7.31. At the center of each side of the aquifer, the potential is known. At the center of BC, ɸ = 97 m. At the center of each DE and HA, ɸ = 90 m. At the center of FG =, ɸ = 92 m. The aim is to compute the head variation at various points in the domain. Sides AB, CD, DE, EF, GH and HA are impermeable so that

prevails on these sides.

For a steady state recharge problem in porous media, the governing differential equation can be written as (Bear, 1972)

Figure 7.30 Free Surface Profile in a Zoned Earth Dam - Example 4

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Finite Element Method with Applications in Engineering

Figure 7.31 Aquifer recharge - Problem Domain (not to scale) where T is the transmissibility of the porous media, R is the source or sink term. For isotropic media, Tx = Ty = T. Therefore, the equation becomes

Here, the FEM formulation for the Poisson equation described in Section 7.5 is used. The domain is discretized using CST elements as shown in Figure 7.32. The domain is divided into 136 CST elements and 95 nodes. Number of recharging nodes are 9. It can be seen in Figure 7.32 that the recharge zone is discretized with a finer mesh and other regions using a coarser mesh. A computer program for the Poisson equation can be developed in a very similar way as demonstrated for the Laplace equation. The difference is only with respect to the recharge term. Using a similar computer program given in Appendix G (for Poisson equation after adding the recharge effects), the above problem is solved. The potential variation at some selected points is given in Table 7.2.

266

Finite Element Method with Applications in Engineering

Figure 7.32 FEM Discretization-Aquifer Problem (not to scale) Table 7.2 Aquifer Problem-FEM Results

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Finite Element Method with Applications in Engineering

7.9 CLOSING REMARKS In this chapter, two-dimensional finite element analysis of engineering problems are described. The energy method, method of weighted residual based Galerkin approach and variational principles in FEM described in Chapter 4 are used in the analysis. Various types of elements such as linear type triangular elements, rectangular elements and quadrilateral elements are used in the analysis of various problems. Basic concepts of isoparametric formulation are also elaborated. FEM formulation for stress-strain analysis, Laplace equation, Poisson equation, Helmholtz equation and Stokes flow equations are discussed in this chapter. Engineering problems of flow through porous media, stress 268

Finite Element Method with Applications in Engineering analysis, seepage below hydraulic structures, earth dam seepage analysis and aquifer recharge are discussed using the FEM. A sample two-dimensional computer program based on FORTRAN is also presented and used in the solution of some engineering problems. This program can be used to solve any similar types of problems, as demonstrated. Even though the problems discussed in this Chapter are related to twodimensional FEM analysis using energy method, method weighted residual and variational approaches, the three-dimensional FEM analysis can be done in a similar way. EXERCISE PROBLEMS 1. Two-dimensional model of an anchor of a communication tower's guy cable is shown in Figure E1. The anchor consists of a triangular steel plate, which is subjected to force P as shown in the figure. Thickness of the plate is 8 mm. The maximum displacement of the node, where the force is applied, has to be restricted to 0.1 mm. Compute the largest force P that can be supported by the anchor. Use only one appropriate CST element to model the plate for expeditious analysis. Consider E = 200 GPa and v = 0.25.

Figure E1 Exercise Problem 1 2. The 8-node square element of size 10 cm × 10 cm shown in Figure E2 is subjected to uniformly distributed pressure load of magnitude 10 MPa, horizontal point load of 10 kN applied at the centroid of the element and vertically downward uniformly distributed body force of magnitude 100,000 kN/m3. Determine the equivalent nodal forces at nodes 1, 3 and 6. Consider thickness of the element to be 2 cm.

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Finite Element Method with Applications in Engineering

Figure E2 Exercise Problem 2 3. Compute co-efficient k34 of 8 × 8 element property matrix for the 4-node rectangular element shown in Figure E3. Consider state of plane stress. Consider thickness of the element to be 2 cm. Consider E = 200 GPa and v = 0.25.

Figure E3 Exercise Problem 3 4. Derive a 2-node element for axisymmetric analysis of a long cylinder by considering state of plane strain in the length direction of the cylinder. Elaborate all steps clearly. 5. Analyze cross section of a long culvert shown in Figure E4 by using 4-node elements. Use program sfeap. Consider state of plane strain the length direction of the culvert. Assume E= 20 GPa, v = 0.30 and uniform thickness of 0.3 m.

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Finite Element Method with Applications in Engineering

Figure E4 Exercise Problem 5 6. For the corner flow (flow between two parallel plates) shown in the Figure E5, equal amounts of inflow and outflow giving a zero net flow rate takes place. Using the FEM formulation for Laplace equation (in stream function) and suitable mesh with linear triangular elements, determine the stream function variation across the corner line AB. (FORTRAN program given in Appendix G can be used for solution).

Figure E5 Problem 6 7. For the earth dam with the assumed free surface shown in Figure E6, using the free surface algorithm and FEM formulation given for the Laplace equation, determine the final position of the free surface. Develop a computer program for the given algorithm. Use a suitable mesh with linear triangular elements.

271

Finite Element Method with Applications in Engineering Figure E6 Problem 7 8. For the flow beneath weir problem shown in the Figure E.7, determine the potential variation. The FEM formulation for Laplace equation and the FORTRAN program given in Appendix G can be used for the solution.

Figure E7 Problem 8 9. For the aquifer problem with a recharging river and 2 pumping wells shown in Figure E8, calculate the head distribution using a suitable mesh. The stream recharges 0.2 m3/(day.m) and pump P1 pumps 800 m3/day and P2 pumps 600 m3/day. 10. For the Stokes flow problem given in Figure 7.20, assume the domain is of dimensions 6 m × 36 m and determine the velocity distribution in the domain.

Figure E8 Problem 9 REFERENCES AND FURTHER READING Arlett, P. L., Bahrani, A. K., and Zienkiewicz, O. C. (1968). Application of Finite Element Method to the Solution of Helmholtz's Equation, Proc. IEE, 115(12):1762–1766. Backstrom, G. (1999). Fluid Dynamics by Finite Element Analysis, Studentlitteratur, Lund. Baker, A. J. (1983). Finite Element Computational Fluid Mechanics, McGraw Hill/Hemisphere, New York. Bathe, K. J. (2001). Finite Element Procedure, Prentice Hall of India, New Delhi. 272

Finite Element Method with Applications in Engineering Bear, J. (1972). Dynamics of Fluids in Porous Media. Elsevier, New York. Brebbia, C. A., and Walker, S. (1979). Dynamic Analysis of Offshore Structures, NewnesButterworths, London. Chapra S. C., and Canale R. P. (2002). Numerical Methods for Engineers, 4th ed., Tata McGraw Hill, New Delhi. Connor, J. J., and Brebbia, C. A. (1976). Finite Element Techniques for Fluid Flow, Butterworth, London. Chung, T. J. (1978). Finite Element Analysis in Fluid Dynamics, McGraw Hill, New York. Crandall, S.H., Dahl, N.C. and Lardner, T.J. (1978). An Introduction to the Mechanics of Solid, McGraw Hill, New York. Desai, C.S., and Abel J.F., (1987), Introduction to the Finite Element Method–A Numerical Method for Engineering Analysis, Wads Worth Publishing Co., California. Gere, J. M. (1965). Analysis of Framed Structures, Princeton, New Jersey. Granger R. A. (1985). Fluid Mechanics, Holt, Rinehart & Winston Publications, New York. Huebner, K. H. (1975). The Finite Element Method for Engineers, John Wiley and Sons, New York. Kinsler, L. E., and Frey, A. R. (1950). Fundamentals of Acoustics, John Wiley and Sons, New York. Logan, D. L. (1993). A First Course in Finite Element Analysis. PWS Publishing, Boston. MacCamy, R. C., and Fuchs, R. A. (1945). Wave Forces on Piles: A Diffraction Theory, US Army Coastal Engineering Center, Tech. Mem. No. 69. Mohr, G. A. (1992). Finite Elements for Solids, Fluids and Optimization, Oxford University Press, New York. Pironneau, O. (1989). Finite Element Methods for Fluids, John Wiley and Sons, Chichester. Reddy, J. N., and Gartling, D. K. (1994). The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, London. Segerlind, L. J. (1985). Applied Finite Element Analysis, John Wiley and Sons, New York. Stasa, F. L. (1985). Applied Finite Element Analysis for Engineers, CBS Publications Japan, Ltd., New York. Srinivas K. (1988). Finite Element Analysis of Wave Forces on Offshore Gravity Structures, M. Tech Thesis, Supervisor: Prof. B.V. Rao, Department of Civil Engineering, IIT Bombay. Streeter, V. L., Wylie, E. B., and Bedford, K.W. (1998). Fluid Mechanics, WCB/ McGraw Hill, Boston. Taylor, C., Patil, B. S, and Zienkiewicz, O. C. (1969). Harbor Oscillation: A Numerical Treatment for Undamped Natural Modes, Proc. Inst. Civil Eng., 43:141–155. White, R. E. (1985). An Introduction to the Finite Element Method with Applications to Nonlinear Problems, John Wiley and Sons, New York. Zienkiewicz, O. C. (1980). Finite Element Method, 3rd ed., McGraw Hill Book, New York. 273

Finite Element Method with Applications in Engineering

8 Three Dimensional Finite Element Analysis 8.1 INTRODUCTION Two-dimensional finite element analysis of various engineering problems was discussed in the previous chapter. Analysis of axisymmetric solids for axisymmetric loading was also discussed in the previous chapter by using 3-node triangular elements. In this chapter, analysis of axisymmetric solid is considered in detail for axisymmetric as well asymmetrical loading by using semi-analytical approach. Loads are expanded in the circumferential direction using Fourier series and thus the finite element discretization is still considered in the two-dimensional domain. Three-dimensional stress analysis is also considered by using 8-node element. Energy methods have been used in the formulations. Illustrative examples are also presented to demonstrate the formulations. 8.2 AXISYMMETRIC SOLIDS This section deals with analysis of axisymmetric solids (solids of revolution) subjected to axisymmetric and non-axisymmetric (asymmetric) loadings. Solid is formed by rotating a plane about an axis of rotation, z (see Figure 8.1) and is assumed to have material properties and boundary conditions independent of the circumferential co-ordinate, θ. The axisymmetric solid structures have wide applications in civil and mechanical applications, like water tanks, cooling towers, circular columns, silos, rotors, pressure vessels, etc. Though the structure is three-dimensional, it can be mathematically decomposed into a series of two-dimensional problems. If the loading is axisymmetric, the problem reduces to a two-dimensional problem analogous to the plane stress/strain case. For the nonaxisymmetric case, loading is represented by the Fourier series in the circumferential direction, and for each harmonic, n, a two-dimensional problem is solved. Final solution is determined by adding solutions of each harmonic. A brief description of the Fourier series for determination of the Fourier coefficients of the loading is presented followed by the isoparametric finite element formulation for structures under axisymmetric and non-axisymmetric loadings. 8.2.1 Determination of the Fourier Coefficients Assume that a pressure distribution, p(θ), along the circumference of the solid can be expressed in a finite Fourier series as

274

Finite Element Method with Applications in Engineering

Figure 8.1 Cross-section of a cylinder showing typical finite elements where (2N + 1) number of coefficients (a0, a1, a2, …., aN; b1, b2, ….,bN) are to be determined; an and bn are the amplitudes of the corresponding harmonic. Coefficient b0 is immaterial since the term bn sin nθ gets eliminated for n = 0. Each term of the series is called a harmonic and n is the harmonic number. Multiplying both sides of Eq. (8.1) first time by cosmθ and next by sinmθ and integrating along the circumference for each case gives

Following orthogonality conditions for sine and cosine can be used.

From Eq. (8.2), the amplitude coefficients an and bn can be determined as 275

Finite Element Method with Applications in Engineering

It may be noted that only a0 exists for the axisymmetric loading. On the other hand, all the Fourier coefficients exist for non-axisymmetric loading. Coefficients an and bn are coefficients of the harmonics, which are symmetric and anti-symmetric, respectively, with respect to θ = 0. Values of an are the amplitudes at θ = 0 and that of bn are the amplitudes at θ =90°. If the closed form answers of Eq. (8.3) are not available, integration is evaluated by numerical means. For example, the trapezoidal or Simpson's rule for the numerical integration may be used. It now remains to consider determination of number of terms N for the convergence. For this purpose, coefficients of the Fourier expansion need to be examined. If the coefficients are small, the chosen number of terms N is considered to be adequate. The smallness of the last three coefficients in each series (i.e., aN, aN-1, aN-2 and bN, bN-1, bN-2) may be assessed for the truncation of the Fourier series. 8.2.2 Isoparametric Finite Element Formulations Axisymmetric structure is formed by rotating curve(s) about z-axis as shown in Figure 8.1. Finite elements are now the nodal rings whose cross-section can be the usual twodimensional elements discussed in the previous chapters. The figure shows typical 3-node and 4-node elements in the θ = 0 plane in a cylinder. In the cylindrical co-ordinates at each node of the element, (u, w, v) are displacements in the r (radial), z (axial) and θ (circumferential) directions for non-symmetric loading, and (u, w) for symmetric loading. Though the formulations are valid for materials having θ as symmetry axis, only isotropic materials will be considered here. Isoparametric formulations for these two cases will be discussed separately. 8.2.2.1 Axisymmetric Loading For definiteness, a 4-node element as shown in Figure 8.2 is considered as an example, but the formulation is general. The displacement field for this element is given by

Here, {q} is the vector of nodal displacements (u1 w1 u2 w2 u3 w3 u4 w4). For isoparametric formulation, displacements and geometry are interpreted using the same shape functions

276

Finite Element Method with Applications in Engineering

Figure 8.2 Four-node element for axisymmetric loadings

Strain-displacement relations for this problem can be written as

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Finite Element Method with Applications in Engineering

Stress-strain relations for isotropic material in cylindrical co-ordinates are

where

The 2×2 element stiffness matrix [kij] can be evaluated in the usual manner as

Here determinant of the Jacobian matrix is obtained as discussed for plane stress/strain case. For numerical integration, the terms r in matrix [Bi] and integrand of [kij] are evaluated at the Gauss points to avoid singularities. 8.2.2.2 Non-axisymmetric Loading Loads applied to geometrically and materially symmetric structures are seldom distributed in an axisymmetric form. The problem is the three-dimensional one and all three displacement components u, w and v exist. An arbitrary non-axisymmetric loading may be represented by its Fourier series expansion as

278

Finite Element Method with Applications in Engineering

Figure 8.3 Symmetric and anti-symmetric loadings fr, fz and fz are the load components in r, θ and z-directions. In Eq. (8.9), the single over bar denotes the amplitudes of the force components symmetric about θ = 0 and the double over bars denote anti-symmetric parts. Examples of symmetric and anti-symmetric loadings are shown in Figure 8.3. Resulting displacements have the corresponding variation

Summations are over the integral values of n from zero to infinity. Introduction of the negative sign in the terms for v has the effect of yielding the same stiffness matrix for both symmetric and anti-symmetric components. Amplitudes of displacement components are primary unknowns, which are functions of r and z only and do not depend on θ. Thus, what was originally a three-dimensional problem, is reduced to a two-dimensional problem due to orthogonality of the trigonometric functions. Furthermore, the symmetric and antisymmetric components of the displacements are also uncoupled because of orthogonality. For isotropic material, the Fourier harmonics are uncoupled. This implies that different numerical values of n present different problems that do not interact. Moreover, the linearity of the problem enables the super-position of separate solutions. Thus, the need for a division into finite elements in the θ direction is replaced by the need to superpose separate solutions.

279

Finite Element Method with Applications in Engineering Since the symmetric and anti-symmetric components of the displacements are uncoupled, only symmetric components are considered here for nth harmonic (n ≠ 0). Again a 4-node element, shown in Figure 8.4, is considered for definiteness. Displacement field for this case is written as

Figure 8.4 Four-node element for non-axisymmetric loadings Amplitudes of displacements

are interpolated from the nodal amplitudes

which is

Substitution of Eq. (8.13) into Eq. (8.11) results in

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Finite Element Method with Applications in Engineering

which can be written as

The strain–displacement relations for the problem is

Strains can now be expressed in terms of the nodal amplitudes by substituting Eq. (8.14) into Eq. (8.15) as

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Finite Element Method with Applications in Engineering

Stress-strain relationship for an isotropic material is given by which can be written as

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Finite Element Method with Applications in Engineering

Noting that the 3×3 element stiffness matrix can be evaluated in the usual manner as

where

is the determinant of the Jacobi matrix.

r in the matrix

and in the integrand of [kij] is evaluated at Gauss points by using

. Integration with respect to ξ and η is done similar to plane strain/stress problems. 8.2.2.3 Traction on the Edge Let the edge 2–3 be subjected to an externally applied distributed load{T} as shown in Figure 8.5. For axisymmetric loading, the surface nodal forces {Q} are evaluated as

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Finite Element Method with Applications in Engineering

Figure 8.5 Element with traction on edge 2–3

In Eq. (8.19), dl is along edge 2–3

is given by Eq. (8.5), the over bar on the shape

functions indicates that they are to be evaluated on the edge ξ = 1 and . Since variables in the integrand of Eq. (8.19) are independent of θ, the integration results in

Similarly, for non-axisymmetric loading, expression for the surface nodal force {Qn} for nth harmonic can be written as

where

is given by Eq. (8.13) and

8.2.2.4 Boundary Conditions As can be observed from strain-displacement relations Eq. (8.6) and Eq. (8.15) that term (1 / r) must be eliminated for r → 0, otherwise the strains become unbounded (Belytschko, 1972). This yields the following boundary conditions, which are applicable to those nodes lying on the z-axis.

Furthermore, the rigid modes must be suppressed in the global stiffness matrix for axisymmetric loading. Because of the axial symmetry, translation in the z-direction is the 284

Finite Element Method with Applications in Engineering only possible rigid body mode. It can be restrained by prescribing w on a single node lying on the r-axis. For the harmonic number n = 1, imposition of the boundary conditions for the points (nodes) on the z-axis needs special considerations. Designating from

Eq.

(8.23)

the

boundary

conditions for n = 1 can be written as

The first condition is not difficult to impose. It is more appropriate to refer the second one as a ‘constraint equation’; the imposition of which is not straight forward. This can be achieved by defining a transformation for the nodal unknowns of the nodes lying on the zaxis as

The element stiffness is, therefore, formulated with nodal unknowns

for the

nodes on the z-axis and for nodes not located on the z-axis. The conditions are then imposed in addition to other boundary conditions. 8.2.2.5 Illustrative Examples Illustrative examples on application of formulation presented above have been presented next. Example 8.1 A hollow column of length 0.8 m having inner radius 0.2 m and outer radius of 0.4 m is loaded by a uniformly distributed load of 1.0 GPa as shown in the figure. The column rests on ground at the base and is free at the top. Analyze this column using 8-node isoparametric elements as shown. Assume E = 20 GPa, v = 0.20 and restrain node 1 in the radial direction.

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Finite Element Method with Applications in Engineering

Solution: This example has been analyzed by using program sfeap discussed in Appendix E on the basis of the formulation presented in this section. The example corresponds to axisymmetric solid with axisymmetric loading. The output for nodal displacements has been presented below. NODE

U (m)

W (m)

1

.0000E100

.0000E+00

2

–.1180E –02

.0000E+00

3

–.2296E–02

.0000E+00

4

–.2794E–02

.0000E+00

5

–.3271E–02

.0000E+00

6

–.1991E–02

.9670E–02

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Finite Element Method with Applications in Engineering

7

–.2581E–02

.9965E–02

8

–.3686E–02

.1013E–01

9

–.1488E–02

.1992E–01

10

–.2361E–02

.1991E–01

11

–.3247E–02

.1986E–01

12

–.3633E–02

.1993E–01

13

–.4022E–02

.2002E–01

14

–.2261E–02

.2992E–01

15

–.2933E–02

.2997E–01

16

–.4051E–02

.2990E–01

17

–.1506E–02

.4015E–01

18

–.2352E–02

.3992E–01

19

–.3179E–02

.3983E–01

20

–.3559E–02

.3995E–01

21

–.3945E–02

.4002E–01

Note that U and W, respectively, represent radial and vertical displacements. As expected, the maximum displacements occur at the nodes located on the top free surface. Example 8.2 The circular tank of inner radius 5 m having thickness 0.25 m and height 4 m is shown in Figure Example 8.2(a). It is partially filled with water up to 2 m as shown. It is assumed that the wall is fixed at the base and free at the top. Assume E = 20,000 kN/m2, v = 0.15, p = γw(L1— z), L1 = 2m and γw = 9.81 kN / m3. Analyze the tank by employing equal 8-node isoparametric elements as shown in the finite element idealization of Figure Example 8.2(b). 287

Finite Element Method with Applications in Engineering

Example 8.2 Solution: This example has also been solved by using program sfeap detailed in Appendix E. Both the geometry and loads are axisymmetric. The nodal displacements obtained from program sfeapare tabulated below. NODE

U (m)

W (m)

1

.0000E+00

.0000E+00

2

.0000E+00

.0000E+00

3

.0000E+00

.0000E+00

4

.4394E–02

.2790E–02

5

.4234E–02

–.2744E–02

6

.1151E–01

.3260E–02

7

.1139E–01

–.1037E–04

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Finite Element Method with Applications in Engineering

8

.1137E–01

–.3279E–02

9

.1773E–01

.2411E–02

10

.1752E–01

–.2631E–02

11

.2159E–01

.9413E–03

12

.2156E–01

–.2541E–03

13

.2139E–01

–.1448E–02

14

.2255E–01

–.5590E–03

15

.2234E–01

–.2796E–03

16

.2102E–01

–.1766E–02

17

.2096E–01

–.5792E–03

18

.2084E–01

.6009E–03

19

.1780E–01

–.2506E–02

20

.1766E–01

.1054E–02

21

.1383E–01

–.2800E–02

22

.1378E–01

–.8379E–03

23

.1373E–01

.1117E–02

24

.9918E–02

–.2756E–02

25

.9847E–02

.9027E–03

289

Finite Element Method with Applications in Engineering

26

.6476E–02

–.2530E–02

27

.6437E–02

–.9809E–03

28

.6429E–02

.5634E–03

29

.3694E–02

–.2250E–02

30

.3667E–02

.2153E–03

31

.1529E–02

–.1986E–02

32

.1512E–02

–.1032E–02

33

.1517E–02

–.8110E–04

34

–.1342E–03

–.1784E–02

35

–.1332E–03

–.2892E–03

36

–.1467E–02

–.1652E–02

37

–.1465E–02

–.1028E–02

38

–.1456E–02

–.4061E–03

39

–.2637E–02

–.1586E–02

40

–.2618E–02

–.4418E–03

41

–.3761E–02

–.1553E–02

42

–.3747E–02

–.9892E–03

43

–.3734E–02

–.4277E–03

290

Finite Element Method with Applications in Engineering 8.3 EIGHT-NODE ISOPARAMETRIC ELEMENT FOR THREE-DIMENSIONAL STRESS ANALYSIS Formulation presented for 4-node isoparametric element presented in the previous chapter for two-dimensional stress analysis has been extended here to 8-node element for threedimensional stress analysis. A typical element from the discretized domain is shown in Figure 8.6. Intrinsic co-ordinates ξ, η and ζ are also shown in this Figure. Note that regardless of the size or shape of the element, the intrinsic co-ordinates always vary from –1 to +1. Thus, a given 8-node element is effectively mapped into a cube in the natural coordinate system. At each i-th node, Ui, Vi and Wi are degrees of freedom, where i = 1,2, …, 8.

Figure 8.6 Eight-node element in the X-Y-Z and the ξ-η-ζ co-ordinate systems Co-ordinate Transformation Following relations can be used for co-ordinate transformation

By following the process used in the previous chapter, the co-ordinates can be expressed as

291

Finite Element Method with Applications in Engineering

where

Approximation Function In an isoparametric formulation, same shape functions are employed to describe variations in the global co-ordinates and the unknown field. Thus,

Alternatively,

where

and

292

Finite Element Method with Applications in Engineering

Strain–Displacement Relation It can be easily shown that

or,

where

The chain rule of differentiation yields 293

Finite Element Method with Applications in Engineering

or

In the above equations,

are unknowns. They can be

obtained in terms of

by inverting the matrix.

Constitutive Relation The constitutive relation is given by

where the constitutive matrix [D] is given by Eq. (7.42). Derivation of Element Equations By minimizing the potential energy, the element equations

can be obtained, where

Computations of [K e], {fb} and {fs}are discussed next. 1. Stiffness matrix 2. From calculus, for an arbitrary function f(x, y, z)

294

Finite Element Method with Applications in Engineering Hence,

where

It can be seen from the derivation that each element of [Ke] is of the form

These expressions can be evaluated by employing Gauss quadrature procedure. 3. Body forces

One can use Gauss quadrature rule to compute {fb}. 4. Surface forces 5. Surface forces can be evaluated by extending formulation discussed in the previous chapters. Illustrative Example A cantilever beam of span 120 cm is subjected to an end-load of 40 kN as shown in the figure below. Analyze the beam by employing 8-node brick elements. The beam is 10 cm deep and 2 cm wide. Assume for all members, Young's modulus, E, as 20,000 kN/m2 and Poisson's ratio v = 0.3.

295

Finite Element Method with Applications in Engineering The finite element representation of the problem is also shown in the figure. Node numbers are indicated by numbers outside the circle, while element numbers are denoted by encircled Arabic numbers. Solution: Nodes 1, 10, 19 and 28 are constrained to represent fix boundary. Generally, threedimensional stress analysis using finite element is performed using computer program. The problem described above has been analyzed by using program sfeap detailed in the Appendix E. The outputs for displacements at different nodes are presented below.

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Finite Element Method with Applications in Engineering

It can be seen from the output that V displacements of all the nodes on the top surface are equal to those at the analogous nodes on the bottom surface of the beam. Also, the U displacements at the nodes on the top surface are equal and opposite to those at the 297

Finite Element Method with Applications in Engineering analogous nodes on the bottom surface of the beam. Such a trend can be attributed to the fact that plane X–Z is plane of anti-symmetry. Planes of symmetry and anti-symmetry and ensuing boundary conditions are further elaborated in the next chapter. 8.4 CLOSING REMARKS Three-dimensional stress analysis has been presented in this chapter. For axisymmetric loading, it has been shown that the three-dimensional problem can be effectively reduced to two-dimensional one by expanding the loading using Fourier series. Two-dimensional analysis can be performed for each term in the Fourier series by taking one term at a time. The final solution can then be obtained by superposing results obtained for individual terms. Isoparametric formulations have been presented for axisymmetric solids. Formulation for an 8-node isoparametric element has also been presented for analysis of three-dimensional solids. Formulations have been implemented in computer program sfeap, which has been discussed later. Few illustrative examples have been considered to demonstrate applicability of the formulations.

EXERCISE PROBLEMS 1. Analyze the column shown in the Illustrative Example 1 when it is loaded by a triangular load of zero at inner radius (r = 0.2 m) to 0.4 GPa at the outer radius (r = 0.4 m). 2. Consider an assembly of two heavy circular shafts. The top shaft has inner radius of 0.5 m and outer radius of 0.75 m, while the bottom one has inner radius of 0.5 m and outer radius of 1.0 m. Shafts are restrained at the top and bottom surfaces to movement in the vertical direction. A uniform load of 1 GPa is acting as shown in Figure E1. Analyze the assembly employing 8-node isoparametric elements. Assume E = 20 GPa, v = 0.20 and restrain node 1 in the radial direction.

Figure E1 Assembly of two shafts 3. Analyze the tank shown in the Illustrative Example 2 when the water tank is completely filled to the top. 4. The circular tank of inner radius 5 m having thickness 0.25 m and height 4 m is shown in Figure E2(a). It is assumed that the wall is fixed at the base and free at the top. The tank is subjected to the wind pressure p as p = p0 (–0.387 + 0.338 cosθ), p0=0.6×v2 where v is the wind velocity in m/sec. Assume E = 20,000 kN/m2, v = 0.15 and v = 30 m/sec.

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Finite Element Method with Applications in Engineering Employing equal 8-node isoparametric elements as shown in the finite element idealization of Figure E2(b) analyze the tank for p = p1 = –0.387 p0 p = p2 = 0.338 p0 cosθ

Present the results for p = p1 + p2.

Figure E2 Water tank subjected to wind pressure p 5. A circular tank of inner radius 4 m has a stepwise variation of thickness whose cross section is shown in Figure E3. The tank is assumed to be fixed at the base and free at top. Analyze the tank under hydrostatic loading when it is filled to the top with water. Note that the triangular load is not shown in the figure. Assume E = 20,000 kN/m2 and v = 0.15 and γw = 9.81 kN/m3.

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Figure E3 Stepwise variation of tank wall 6. Compute the nodal force vector for a typical element considered in the illustrative example of Section 8.3.1 for a constant body force in the x-direction of magnitude 10 kN/m3. 7. Compute the nodal force vector for the element shown in Section 8.3.1 (illustrative example) for uniformly distributed pressure loading of 10 kN/cm2 on surface formed by nodes 26, 27, 36 and 35 of element number 8.

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Finite Element Method with Applications in Engineering 8. For an 8-node isoparametric element used to model three-dimensional solid cube of side 5 cm, the nodal displacements are U1 = 0.0 cm, V1 = 0.0 cm, W1 = 0.0 cm; U2 = 0.0 cm, V2= 0.0 cm, W2 = 0.0 cm U3 = 0.0 cm, V3 = 0.0 cm, W3= 0.0 cm; U4 = 0.0 cm, V4= 0.0 cm, W4= 0.0 cm U5 = 0.001 cm, V5 = 0.001 cm, W5 = 0.001 cm; U6 = 0.002 cm, V6 = 0.001 cm, W6 = 0.001 cm U7 =0.001 cm, V7 = 0.002 cm, W7 = 0.001 cm; U8 = 0.001 cm, V8 = 0.001 cm, W8 = 0.002 cm Evaluate state of strain and the state of stress at the centroid of the element. Consider E =200 GPa and v = 0.3. 9. Analyze the illustrative example considered in Section 8.3.1 for uniform axial load at the free end of magnitude 40 kN. Use program sfeap with the discretization, boundary conditions and material data considered in the illustrative example. Compare results with one-dimensional axial bar. REFERENCES AND FURTHER READING Belytschko, T. (1972). Finite elements for axisymmetric solids under arbitrary loading with nodes on origin, AIAA Journal, 11(9):1357–1358. Chandrupatla, T. R., and Belegundu, A. D. (2003). Introduction to Finite Elements in Engineering, Prentice Hall of India, New Delhi. Cook, R. D. (1981). Concepts and Applications of Finite Element Analysis, 2nd ed., John Wiley and Sons, New York. Krishnamoorthy, C. S. (1996). Finite Element Analysis Theory and Programming, 2nd ed., Tata McGraw Hill, New Delhi. Wilson, E. L. (1965). Structural analysis of axisymmetric solids, AIAA Journal, 3(12):2269– 2274. Zienkiewicz, O. C. (1980). Finite Element Method, 3rd ed., McGraw Hill Book, New York. Zienkiewicz, O. C., Taylor, R. L., and Zhu, J. Z. (2007). The Finite Element Method Its Basis and Fundamentals, 6th ed., Elsevier, Oxford.

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9 Computer Implementation of FEM 9.1 GENERAL Computer implementation of the FEM is discussed in this chapter. FEM can be applied more elegantly by reducing the size of the problem. To this effect, one-dimensional and two-dimensional analyses were discussed in the earlier chapters, wherein dimension of three-dimensional problems were reduced by making engineering approximations on the basis of geometry, loads and boundary conditions. The size of the problem can be further reduced when certain conditions are met. Reduction in size of the problem has been discussed in this chapter by (i) considering application of conditions of symmetry and antisymmetry and (ii) employing the static condensation procedure. Computer implementation of the FEM has been discussed with the help of simplified finite element analysis program (sfeap). Various storage schemes for the global stiffness matrix and application of boundary conditions in computer code are also discussed. 9.2 USE OF SYMMETRY AND ANTI-SYMMETRY CONDITIONS IN REDUCING A PROBLEM In many problems, conditions of symmetry (or anti-symmetry) can be employed to reduce the size of the problem. Symmetry symbolizes correspondences in size, shape, position of loads; material properties and boundary conditions that are on opposite sides of a dividing line or plane, which are often referred to as a line or a plane of symmetry (or antisymmetry). Condition of symmetry: All displacement components normal to the plane of symmetry are zero. Condition of anti-symmetry: All displacement components in the plane of anti-symmetry are zero. Note that the discussion on conditions of symmetry and anti-symmetry is not restricted to the structural analysis. These conditions can be applied to any finite element analysis. Conditions of symmetry and anti-symmetry are applied symbolically in Figure 9.1 to reduce the size of problem. Illustrative example The bar elements 1, 2, 7 and 8 of the plane truss shown below have axial rigidity of 400,000 kN and bar elements 3, 4, 5 and 6 have axial rigidity of × 200,000 kN. The vertical plane perpendicular to the plane truss passing through nodes 2, 3 and 4 is the plane of symmetry because identical geometry, material, loading and boundary conditions occur at the corresponding locations on the opposite side of this plane. In other words, the geometry, material, loading and boundary conditions on the opposite side of the plane are ‘mirror images’. The loads and the area in the plane of symmetry must be halved in the reduced structure. Furthermore, all the displacements normal to the plane of symmetry must be set to zero in the reduced structure. Thus, U2 = U3 = U4 = 0. All the boundary conditions applied to the reduced plane truss and the resulting displacements and forces are shown in Figure 9.2.

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Figure 9.1 Conditions of symmetry and anti-symmetry It can be seen from the figure that analysis performed on the reduced structure yields the same displacements and element forces as those prevalent in the actual structure, with one exception compared to the actual plane truss–forces in elements 3 and 4 in the reduced truss are half in magnitude. This difference is expected because the crosssectional areas of these elements were halved in the analysis. It is possible to compute displacements and member forces in the real structure by analyzing only a quarter portion of the actual truss, as shown in Figure 9.3. However, two separate analyses are required [Cases (A) and (B)].

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Figure 9.2 Analysis of a plane truss by employing conditions of symmetry Results from these two analyses are tabulated in Table 9.1. Final results, which are presented in Figure 9.2, can then be obtained by superimposing results from Cases (A) and (B), by appropriately employing the conditions of symmetry and anti-symmetry.

9.3 STATIC CONDENSATION An application of FEM to a problem results in simultaneous equation set If size of K is too large to fit into the CPU memory of the computer at hand, it is necessary to reduce the size of the matrix to solve the problem. Static condensation is a process of eliminating or condensing some unknown DOF (known as secondary or slave DOF) and re-cast the original Eq. set (9.1) in terms of fewer unknowns (termed as primary unknowns or master DOF).

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Figure 9.3 Further reduction in the size of the problem Table 9.1 (a) Nodal displacements from the analysis of the quarter structure

Table 9.1 (b) Element forces stemming from the analysis of the quarter structure

9.3.1 Applications of Static Condensation The concept of static condensation is utilized in the following. 305

Finite Element Method with Applications in Engineering Forming a ‘super element’. Super elements are used quite frequently in the analysis of complicated structures or geometries, where several smaller elements are condensed to reduce the number of unknowns without sacrificing the accuracy. Sub-structure analysis. For example, soil structure interaction problems. 9.3.2 Static Condensation Procedure To eliminate the secondary DOF, matrix Eq. (9.1) is partitioned as

where K11(s×s), K12(s×p), K 12(p×s) and K22(p×p) are sub-matrices of matrix K (n×n). The T

, on the other hand, are the sub-vectors of the partitioned vectors and contains secondary DOF and contains primary DOF. Note that n = s + p.

and such that

The aim of performing static condensation is to re-cast Eqs. (9.1) into reduced equation set as

By expanding the partitioned Eqs. (9.2), the following equations are obtained.

Hence, from Eq. (9.4),

By substituting Eq. (9.6) in Eq. (9.5), the following equations can be derived.

=>

=>

where

Illustrative Example Given

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Derive

In order to use Eq. (9.5), Eq. (9.10), the 3rd and 4th rows must be inter-changed with the 1st and 2nd rows as well as the 3rd and 4th columns with the 1st and 2nd columns so that the symmetric matrix in Eq. (9.11) can be partitioned into the form given by Eq. (9.2). Basically, the original matrix has to be re-arranged such that the ‘secondary’ variables (i.e., the variables to be eliminated) x3 and x4 appear in the first sub-vector, , and the ‘primary’ variables (i.e., the variables to be retained) x1 and x2 appear in the second subvector, . After re-arranging the equations, the following equation is obtained.

so that

From Eq. (9.10), it can be noticed that the following matrix operations are required.

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Finally,

and

9.4 COMPUTER IMPLEMENTATION OF FEM–SFEAP In order to illustrate computer implementation of FEM, simplified Finite Element Analysis Program (sfeap) has been developed in FORTRAN language. Following different elements have been incorporated in the program. 1. 2. 3. 4. 5. 6.

2-node axial bar/plane truss element 2-node space truss element 2-node beam/plane frame element CST element for two-dimensional stress analysis 3- to 9-node element for two-dimensional stress analysis Axisymmetric isoparametric element for axisymmetric load (Circumferential harmonic = 0) 7. Axisymmetric isoparametric element for asymmetrical load (Circumferential harmonic > 0) 8. 8-node isoparametric element for three-dimensional stress analysis In addition, few dummy routines are kept, which can be used to add user defined elements. The program has been developed such that steps followed in the method are easily understood while going through the organization of the program. Details of the program including overall flowchart, description of various sub-routines, major arrays and variables are provided more conveniently in the Appendices E and F. The program can be used to learn the implementation of method on computer and also for solving a variety of problems. In order to facilitate data preparation, a graphical user interface has been developed. The entire software has been given in the accompanying CD. Source code of the program has also been given in the CD for teaching and research purpose.

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Finite Element Method with Applications in Engineering Storage scheme for the assembled matrix and application of the boundary conditions in program sfeap are discussed next.

9.5 STORAGE SCHEMES FOR GLOBAL STRUCTURAL STIFFNESS MATRIX There are several ways in which the global stiffness matrix can be stored in the computer memory. 1. Full form: The matrix is stored in its full (natural) form. This requires large storage and running time. 2. Upper triangular form: Since the global stiffness matrix [K] is symmetric, it is necessary to store only the elements in the upper triangle of the [K] matrix.

If N is the number of equations, this scheme requires ½ N × (N + 1) storage locations. Note that the element Kij will be found at the [1/2{j(j −1)} + i]-th location. In other words, K(I, J) ⇒ S (L), where L = 0.5 × J × (J - 1) + I 3. Banded form: Bandedness of the stiffness matrix has little to do with the formulation of finite elements. It results from an efficient nodal numbering scheme. Refer to Figure 9.4. As a thumb rule, number the joints (nodes) in a grid in the narrow direction to minimize the bandwidth. MBAND = (maximum node difference + 1) × NDF Elements of the stiffness matrix that correspond to the second numbering scheme can be stored in the upper triangular, banded form as shown in Eq. (9.27).

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Program sfeap stores the global stiffness matrix in the banded form. 4. Skyline form: This also results from an efficient nodal numbering. Consider the stiffness matrix shown in Eq. (9.28).

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Figure 9.4 Banded assembled equations

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Finite Element Method with Applications in Engineering Stiffness matrix shown in Eq. (9.28) can be written in a one-dimensional array as shown in Eq. (9.29).

9.6 APPLICATION OF BOUNDARY CONDITIONS There are two ways to handle boundary conditions associated with a constrained node. 1. Specify a very large number on the diagonal associated with the corresponding displacement component. 2. Do not assemble the equation corresponding to the specified displacement component. The second approach has been discussed next as the same has been employed in program sfeap. Consider example of two span continuous beam shown in Figure 9.5. The total DOF for this example are

Specifying the unknowns in this manner is not convenient for computer implementation. Thus, let the DOF be referred to as [r1 r2 r3 r4 r5 r6]T. Then, the boundary conditions are For the computer implementation, these DOF's are given a zero ID number as follows. DOF

ID#

r1

0

r2

0

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Finite Element Method with Applications in Engineering

r3

0

r4

1

r5

0

r6

2

Figure 9.5 Application of boundary conditions Note that r4 and r6, respectively, correspond to unconstrained DOF θ2 and θ3. Thus, it is seen that only four coefficients of the system level matrix are required to be assembled by assigning location numbers corresponding to unconstrained DOF to reduce number of storage locations in the computer's random access memory (RAM). For illustration, consider the element stiffness matrix for element 2. Coefficients of this matrix are required to be assembled at locations 3, 4, 5 and 6 as shown below. Further, since r3 = r5 = 0, all elements of rows marked 3 and 5, as well as columns marked 3 and 5 are not required to be assembled and, thus, the same have been renumbered as 0. The original rows marked 4 and 6 as well as columns marked 4 and 6 are re-numbered as 1 and 2, respectively.

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Figure 9.6 Application of boundary conditions (a) two-dimensional domain; (b) DOF prior to application of boundary conditions and (c) unknown DOFs Consider the example of two-dimensional analysis using four CST elements shown in Figure 9.6 (a). For CST element, there are two DOF per node. These DOF have been numbered sequentially as shown in Figure 9.6 (b). Note that U1 = V1 = U2 = V4 = 0. Thus, equations pertaining to these DOF are not required to be assembled. The resulting re-numbered DOF, which are required to be computed, are shown in Figure 9.6 (c). In program sfeap, the above mentioned steps are implemented as follows. 1. Initialize all elements of array ID with dimensions NDF × Number of nodes (NUMNP) to zero.

2. Read boundary conditions. Note that constrained DOF are assigned value unity in the input whereas free (unconstrained) DOF are assigned value zero.

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3. Generate equation numbers. Start the counter from 1 and increment by 1 for each unconstrained DOF sequentially.

Finally, assembly takes place at locations 1 through 6, and six simultaneous equations in banded form are solved by using Gauss elimination method discussed in Appendix A. Further details of computer implementation of FEM through program sfeap are given in Appendices E and F. 9.7 CLOSING REMARKS Details on application of conditions of symmetry and anti-symmetry have been given in this chapter. It has been emphasized that with the help of such boundary conditions, the size of the problem can be reduced significantly. Static condensation process has also been elaborated in the chapter, which can be used to reduce size of the problem and also to form a super-element. The salient features of program sfeap are presented through which computer implementation of FEM can be studied. Various storage schemes used for assembled matrix are discussed and the process of application of boundary conditions and assembly are explained. Further details of program sfeap are given in Appendices E and F.

EXERCISE PROBLEMS 1. It has been decided to analyze the plane truss shown in Figure E1(a) by using symmetric and anti-symmetric components of loads. By applying appropriate boundary conditions at nodes 4 and 5, the reduced half portion of the truss has been analyzed for four cases. Cases 1 and 2 correspond to unit horizontal load applied at node 2, as shown in Figure E1(b). The displacements at all joints and member forces are summarized in Tables E1.1 and E1.2, respectively, for Cases 1 and 2. On the other hand, Cases 3 and 4 correspond to unit vertical load applied at node 3, as shown in Figure E1(c). The displacements at all joints and member forces are summarized in Tables E1.3 and E1.4, respectively, for Cases 3 and 4. Identify each case and using the data of these four cases, compute, for the truss shown in Figure E1(a), 1. displacements at all nodes; and 2. force in all elements. Consider the axial rigidity AE of all the elements to be 10,000 kN in the actual truss.

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Finite Element Method with Applications in Engineering Figure E1(a)

Figure E1(b)

Figure E1(c)

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2. Matrix Eq. [K]4×4 {q}4×1 = {f}4×1 has been defined where

Condense q2 and q3, and modify the equation to [K*]2×2 {q*}2×1 = {f*}2×1 where {q*} = [q1 q4]T.

REFERENCES AND FURTHER READING Bathe, K. J. (1982). Finite element procedures in engineering analysis, Prentice Hall India, New Delhi. Crandall, S. H., Dahl, N. C., and Lardner, T. J. (1978). An Introduction to the mechanics of solid, McGraw-Hill, New York. Desai, C. S., and Abel, J. F. (1972). Introduction to the finite element method, Van Nostrand Reinhold, New York. Gere, J. M. (1965). Analysis of framed structures, Princeton, New Jersey. Logan, D. L. (1993). A first course in finite element analysis, PWS Publishing, Boston.

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10 Further Applications of Finite Element Method 10.1 INTRODUCTION Typical finite element formulations for various engineering applications were discussed in the earlier chapters. Details presented in the preceding chapters are sufficient to provide essential ideas and procedures for the development of finite element formulations and computer models for most engineering problems in one, two and three dimensions. However, each complex engineering problem differs in one way or another and there are many additional ideas and difficulties that need to be addressed, which are related to formulation of the problem using finite element method (FEM), development of computer models and applications to field problems. A few more advanced important engineering problems and their related FEM formulations and applications are discussed in this chapter. Discussions are focused on demonstration of applications of FEM to complex engineering problems. There are a number of books on the topics covered in this chapter. Detailed description of these topics is beyond the scope of the current book. However, it is envisaged that the readers will be able to get general ‘feel’ of the topics and can use the details presented herein as a step in the direction of further development of FEM formulation and modelling of the engineering problems concerned with the area of their interest. 10.2 FINITE ELEMENT ANALYSIS OF PLATES 10.2.1 Introduction A plate is a three-dimensional structure bounded by two parallel planes separated by a distance, h, denoting thickness of the plate. The x–y plane at z = 0, which is equidistant from two bounding planes, is called the middle surface of the plate. In the linear elastic plate theory for isotropic material, the problem uncouples into in-plane and out-of-plane problems, depending on whether the loading is parallel to the x–y plane or normal to it. Finite element formulation for the parallel loading has already been presented in Chapter 7 for plane stress or plane strain situation. In this section, only the bending of plates is presented when the loading is normal to the x–y plane. An interested reader can refer to the book on plates by Timoshenko and Woinowsky-Krieger (1959) and Reddy (1999) for detailed studies. If the ratio h/L where L is the least dimension on the plan in x–y plane, is less than1/10, it is normally classified as a thin plate. On the other hand, the plate is classified as a thick plate if the ratio exceeds 1/10. Procedure for deriving the pertinent finite element formulation for plate theory will follow the pattern presented in Chapter 6 for the beam theory. First, the plate theories will be presented. This is followed by the finite element formulation for thick and thin plates.

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Figure 10.1 Forces on a plate element: (a) Element detail and (b) Forces on the middle plane of the plate 10.2.2 Review of Plate Theories Consider an infinitesimal plate element in the x–y plane as shown in Figure 10.1 (a) and (b). The x–y plane is located at the middle surface of the plate and the thickness of the plate is assumed to be h (– h/2 ≤ z ≤ h/2). Load q(x, y)(N/m2) acts normal to the plate. 10.2.2.1 Equilibrium Equations The stress resultants, moments (Mx,Mxy,My) and the shear forces (Qx,Qy) shown in Figure 10.1 (a) and (b) are defined as

Moments and shear forces have dimensions of Nm/m and N/m, respectively. Mx, My are the bending moments and Mxy is the twisting moment. Moments and shear forces on the positive faces are expressed in the Taylor series, retaining only the first two terms as

As presented in the beam theory, the static equilibrium equations are written by considering force equilibrium in the z direction and moment equilibrium equations about the x and y directions, respectively, and neglecting higher order terms as 320

Finite Element Method with Applications in Engineering

10.2.2.2 Thin-plate (Classical plate) Theory Geometry: Following assumptions are made for deformation in the classical plate theory (CPT). The middle surface of the plate is unstrained. The normal to the middle surface remains normal after the deformation. This assumption Kirchhoff's hypothesis, is analogous to the one made in the thin-beam theory, where the plane normal to the neutral axis remains plane and normal after deformation. Normal displacement, w, is independent of z, which implies w = w(x, y). Deflection (the normal component of the displacement vector) of the mid-plane is small compared with the thickness of the plate. The slope of the deflected surface is also very small so that the square of the slope can be neglected in comparison with unity. The strain-displacement relations for linear elasticity in three dimensions are given by

where u, v and w are the displacements in x, y, and z directions, respectively. However, the displacements u and v can be expressed from assumption (ii) as

Thus, the strain components can be written as

Stress–strain relations and manipulations: Stress–strain relations for the plate theory are assumed to be those for the plane strain case (see Chapter 7), since εz = 0. They are

321

Finite Element Method with Applications in Engineering From Eq. (10.6), these relations can be written as

Since plate is assumed to be thin, the normal stress σz is assumed to be negligible. Substituting Eq. (10.8) into Eq. (10.1), the stress resultants can be evaluated after simple integration as

where D is the plate rigidity given by Plate rigidity D plays similar roll to the beam rigidity EI. For a beam of unit thickness and depth h, the moment of inertia I = h3/12. It can be seen that D = EI if v = 0. Shear forces can be evaluated from the equilibrium Eqs. (10.3) to be

where the Laplacian

which can be written as

where the bi-harmonic operator,

4

, is given by

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It may be mentioned here that though γxz = 0 and γyz = 0, it is assumed that τxz and τyz exist. Shear stresses cannot be evaluated from the theory of elasticity. In practice, guided by elementary beam theory, they are approximated as

Though there is an inconsistency in the theory, it produces acceptable results for thin plates. Because equation for the deflection w is a fourth order partial differential equation, two boundary conditions on the edge are required. There are three moments and two shear forces. Therefore, the situation is different compared to the beam theory. Kirchhoff derived the consistent boundary conditions (Timoshenko and Woinowsky-Krieger, 1959) using from the calculus of variation given below for a rectangular plate of dimension a × b.

10.2.2.3 Thick-plate (Shear deformation) Theory Geometry: Shear deformation is neglected in the thin-plate theory resulting in a governing differential equation in one dependent variable w(x, y). This is justifiable for thin plates. However, the shear stresses exist for thick plates. Thus, there will be deformations on the x and y faces. When there is variation of shear forces along the x and y directions, there will be a stretch in the x and y directions, resulting in the longitudinal and shear strains and stresses. Thus, it is desirable to include the effect of shear deformation when the plate is thick. This will require increase in number of dependent variables to three to account for the shear deformation. Following assumptions are made for deformation in the first-order shear deformation plate theory (SDT). The middle surface of the plate is unstrained.

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Finite Element Method with Applications in Engineering The normal to the middle surface before deformation remains straight but is not necessarily normal to the middle surface after the deformation. Normal displacement, w, is independent of z, which implies w = w(x, y). Deflection (the normal component of the displacement vector) of the mid-plane is small compared with the thickness of the plate. The slope of the deflected surface is also very small such that the square of the slope can be neglected in comparison with unity. Here, the procedure followed by Mindlin (1951) is described, which is simple and mathematically elegant. The assumed displacement distributions are taken as (see Figure 10.2)

where w is the transverse displacement, βx is the rotation of the normal to the undeformed middle surface in the x–z plane and βy is the rotation of the normal to the undeformed middle surface in the y–z plane. Note that in the Kirchhoff's hypotheses βx = −∂w / ∂x and βy = ∂w; / ∂y. Strain components can be obtained by substituting the assumed displacements Eq. (10. 12) in Eq. (10.4) as

Stress–Strain Relations and Manipulations: Stress–strain relations given in Eq. set (10.7) can be written as

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Figure 10.2 Rotations of the normal about x and y axes

s Shear stresses τxz and τyz in Eq. set (10.14) represent average stresses over the sections. Stress resultants can now be written from Eq. (10.1) as

In order to account for the non-uniform shear stress distribution over the cross-section, the shear correction factor K is introduced. The correction factor K is normally assumed to be 5/6 (statics) or π/12 (dynamics).

325

Finite Element Method with Applications in Engineering Substitute Eq. (10.15) into Eq. (10.3) to obtain the displacement equations of motions. From the first of the Eq. (10.3),

which simplifies to

From the second of Eq. (10.3),

After the manipulation of this equation, the second of Eq. (10.3) can be written as

From the third of Eq. (10.3),

which after some manipulation can be written as

Thus, the three coupled equations for variables w,βx, and βy are

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10.2.3 Finite Element Formulations Finite element formulation is straightforward for a thick plate and is presented first followed by formulation for thin plate. 10.2.3.1 Thick Plate Only energy due to bending is stored for a thin plate, since shear deformation is neglected. However, energy due to shear deformation has also to be included for a thick plate. The normal and shear stresses and strains are uncoupled for an isotropic material. Therefore, energy due to each of them can be added separately. Bending strains εxx,εyy, γxy and the transverse shear strains γyz,γxz are

The constitutive relations are given by

The potential energy π, on the other hand, is given by

which can be expanded to

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where K is the shear correction factor, p(x, y) is the distributed load and Fi are the concentrated forces. After integrating over the thickness h, Eq. (10.23) yields

where

{εb} and {εs} contain only the first-order derivatives of the dependent variables βx,βy and w. Procedure presented earlier for the plane stress/strain problems can be followed to formulate the stiffness matrix of the thick plate element. Assuming the displacement and rotation distribution as

where wi,θyi,θxi are the nodal values, Ni (ξ,η) are the shape functions associated with the ith node, and Ne is the number of nodes in that element. Eq. (10.26) can be written as

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where

Therefore,

where

where and

Minimizing π of Eq. (10.24), symmetric matrices yields

= 0, and noting

and

are

or

where

Once the formulation for [K] and {R} are completed, analysis of the plate structure follows the standard procedure.

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Finite Element Method with Applications in Engineering It may be mentioned here that this formulation is restricted to thick plates, since w, βx, and βyare treated as independent quantities. The differential operators [L]b and [L]s contain only the first-order derivatives of the shape functions, which requires only C0 continuity. From Eqs. (10.29) and (10.24), bending stiffness matrix stiffness matrix

and shear

where

and

From these equations, it is inferred that the relative magnitude of the shear stiffness becomes of order 1/(h/L)2 times that of bending stiffness, as plate thickness is reduced. For example, if h/L = 0.01, 1/(h/L)2 = 104. Here, L is a characteristic length. This means that if thick plate formulations are used to analyze thin plates, the solution will be governed according to the shear stiffness. However, shear (transverse) strains are zero and shear stiffness is neglected in thin-plate theory. The dominance of the shear stiffness in C0 formulation in thin-plate range is known as ‘shear locking’; that is, the element becomes stiffer and deforms only in the shear mode and flexural deformations are negligible. To remedy the problem, some techniques are proposed (Zienkiewicz and Taylor, 1989): (a) impose as constraints and develop a constrained variational formulation (penalty function approach; discreet Kirchhoff's hypothesis); (b) use reduced integration technique making shear stiffness singular through a selective integration scheme or (c) formulate element stiffness using thin-plate theory as discussed in the following section. 10.2.3.2 Thin Plate From Equations (10.6) and (10.8), one can write strain–displacement and stress–strain as

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The potential energy π is given by

which can be expanded to

Substituting Eq. (10.31) in Eq. (10.32) and integrating over the thickness of the plate, the first term results in

Assume the displacement distribution for w, the only degree of freedom (DOF) at any node, as

Here, wi are the nodal displacements, Ni are the shape function associated with node i and Ne is the number of nodes in that element. Substituting Eq. (10.34) in the curvature vector {K} yields

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Finite Element Method with Applications in Engineering

Thus, the potential energy can be written as

Minimizing the potential energy of Eq. (10.36), following equations are obtained.

Evaluation of stiffness matrices [K]b and [K]s in Eq. (10.29) and [K]p in Eq. (10.37) requires integration. It can be seen from Eqs. (10.28) and (10.35) that the integrands of [K]b and [K]s contain first-order derivatives of shape functions while those of [K]p contains secondorder derivatives. As pointed out in Chapter 5, the shape functions need to be continuous only at the interface (C0 continuity) for evaluation of [K]b or [K]s. However, evaluation of [K]p also requires the continuity of the derivative (C1 continuity) at the interface. For onedimensional problem, the beam functions (Hermit polynomials), which satisfy C1 continuity at the nodes have been encountered in Chapter 6. However, it is difficult in two dimension to construct shape functions, which not only satisfy C1 continuity but are also computationally efficient. FEM is a numerical method where the domain is an assemblage of elements. Procedure to form the global matrix for the domain is to evaluate the potential energy π(e) of each element and then summing total energy π of the system as

and then minimizing π. Summation of Eq. (10.38) presumes that the displacements and rotations are continuous at the inter-element boundaries. If the continuity is not maintained, the gaps (overlays or separations) will form and the solution may not converge. This aspect is discussed in the next section.

332

Finite Element Method with Applications in Engineering Desirable Requirements of the Shape Functions: Continuum has infinite DOF and it possesses minimum potential energy. Discretizing the domain by mesh of finite elements limits the DOF and renders the system more stiff, resulting in higher potential energy π. The displacement-based finite elements formulation yields an upper bound for π. The energy approaches the true value of π from above, as shown in Figure 10.3. The true minimum energy may never be reached even by refining the mesh, unless the shape functions are solution of the governing equations. The shape functions should satisfy some requirements to ensure that the numerical solution converges towards the true minimum, as the mesh is refined. Requirement 1: Interpolation function of an element must be able to represent the rigid body displacement. Rigid body displacements are those displacement modes that an element is able to undergo as a rigid body without stresses being developed. Constant terms in the shape functions will ensure this requirement. Requirement 2: Displacement functions must be capable of representing the constant generalized strain states within elements as elements get smaller. Generalized strains mean strains and/or curvatures. An element is said to be complete if displacement functions satisfy these two requirements. Linear terms in the polynomial satisfy this requirement for strains while second-order terms satisfy the requirement for curvatures. Reasoning for this requirement can easily be explained.

Figure 10.3 Convergence of potential energy

Figure 10.4 Evaluation of potential energy

333

Finite Element Method with Applications in Engineering Area under a curve y(x) is evaluated by dividing the curve into small intervals ∆xe, as shown in Figure 10.4 and integrating as

Now the potential energy π is given by

which, for one-dimensional problem of axial force can be written as

Similar reasoning can be made for higher dimensions. The state of strain in each element approaches constant value, when elements of the assemblage approach small size. The energy can then be found by the summation, as shown in one dimension. Requirement 3: Elements must be compatible. This requirement means that displacement within elements and across element boundaries must be continuous. When only the translational DOF are defined at the element nodes, only continuity of displacements must be preserved. However, when the rotational DOF are also defined in addition to the displacement DOF, as in the case of plate bending element, it is also necessary to satisfy the zero and first-order derivatives of the displacements. If this condition is not satisfied, displacements across thickness of plate will not be continuous at edges. Compatibility will also imply that elements have the same number of nodes, co-ordinates and interpolation functions at the inter-element boundaries. Elements that satisfy the above three requirements are called compatible or conforming elements. If these requirements are violated, the element is referred to as incompatible or non-confirming elements. Physically, compatibility ensures that no gaps or overlaps occur between elements. Requirement 4: An element should have no preferred direction. In other words, an element should be geometrically invariant (geometrically isotropic or spatially isotropic).

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Figure 10.5 The pascal triangle Geometric invariance is achieved if polynomial in shape function is complete, that is, it includes all the terms as shown in Pascal triangle of Figure 10.5. However, invariance can still be achieved if the corresponding orders of terms are selected on either side of the axis of symmetry. This requirement ensures that the polynomial expansion for element remains unchanged under a linear transformation from one Cartesian co-ordinates to another (Dunne, 1968). For bending of thin plates, difficulties of finding compatible displacement functions have led to use of non-confirming elements. Usually, a complete slope continuity is ignored for such elements, which still satisfy other requirements. It is essential that the nonconforming elements pass the patch test. It is a simple numerical test of element validity. A numerical test (Irons and Razzaque, 1977) is performed by imposing nodal displacements on an arbitrary patch of element, corresponding to any state of constant strain. If nodal equilibrium is simultaneously achieved without any external nodal forces and if constant stress field is obtained, the convergence is assured as mesh is refined. For further details on the patch test, refer (Zienkiewicz and Taylor, 1989). In the following section, two plate bending elements for thin plates, one non-conforming and the other for conforming elements, will be presented. Shape Functions for Plate Bending Elements: Equations necessary to derive plate bending elements are derived in the preceding sections. The pertinent equations to form element stiffness matrices are Numerous finite elements for plate bending have been developed over the years (Hrabok and Hrudey, 1984). In this section only two shape functions, which can be used to evaluate the stiffness matrix using Eq. (10.40), are presented.

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Finite Element Method with Applications in Engineering

Figure 10.6 Plate element (a) Rectangular plate geometry (b) Nodal DOF Non-conformal Shape Function: One of the popular rectangular plate bending element (Melosh, 1963) with 12-DOF is considered here. Figure 10.6 shows a rectangular plate bending element with nodal displacements wi and the rotations θxi and θyi (i = 1,2,3,4) where θx = ∂w /∂y and θy = –∂w / ∂x. The origin has been taken at (0, 0). Since the element has 12-DOF, a 12-term polynomial is selected as

The polynomial is complete up to the third-order terms (see Figure 10.5) and is also geometrically isotropic. Differentiating Eq. (10.41) with respect to x and y, vector {u} can be written as

336

Finite Element Method with Applications in Engineering From Eq. (10.42), curvatures and twist can be determined as

From Eqs. (10.41) and (10.43), it is observed that the interpolation function satisfies requirement 1 for rigid body displacement and requirement 2 for constant curvature as the element becomes smaller. Thus, this element is complete. Consider the compatibility of this element. Along edge 1–2 (y = 0)

In Eq. (10.44), it is noted that w(x, y) is cubic in x and the slope ∂w / ∂x is the same as in beam element. Constants a1, a2, a4, and a7 can be evaluated from the nodal displacements and rotations w1,θy1, w2 and θy2 as was done in the derivation of beam functions (Hermite polynomial). The resulting beam functions are given in Chapter 6, in Eq. (6.172) and are also listed later on in Eq. (10.48). Thus w and ∂w / ∂x are continuous along the edge 1 – 2. The slope ∂w/∂y is cubic in x involving four constants a3, a5, a8 and a11. However, only two DOF (θx1 and θx2) remain to determine these constants. The slope ∂w/∂y is not uniquely defined and the slope discontinuity occurs. Therefore, this element is non-conforming element and the solution may not render minimum potential energy. Evaluating Eq. (10.42) at each of the four nodes of the element, the following equation is obtained. { qe } = [G]{a} where

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Finite Element Method with Applications in Engineering

From Eq. (10.45),

is obtained. Substitution of Eq. (10.46) in Eq. (10.41) yields w(x, y) = [P]{a} = [P][G]-1 {qe} = [N]{qe}, thus the shape function matrix is given by

The stiffness matrix is then obtained from Eq. (10.40). Conforming Shape Function: It is possible to construct polynomials for a confirming 16DOF rectangular element by adding twist term θxy = ∂2w / ∂x∂y at each node of the four node element, considered in non-conforming element described above. However, some polynomials may not produce an invertible matrix [G]-1 as done in Eq. (10.46). An alternate way of forming the shape functions ensuring continuity of w and its normal derivatives ∂w / ∂x and ∂w / ∂y along element boundary is to employ product of Hermite polynomials (Bogner et al., 1968). The Hermite polynomials given in Eq. (6.177) in Chapter 6are listed here for completeness.

where Ls is the length of the element in s direction. The appropriate cubic interpolation polynomial in the expansion of w is

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Finite Element Method with Applications in Engineering

Eq. (10.49) can be written in matrix form as

The stiffness matrix is then obtained from Eq. (10.40).

10.3 DYNAMICS WITH FINITE ELEMENT METHOD 10.3.1 Introduction In the earlier chapters, it was assumed that the loads on the structure are static and do not vary with time in the finite element equations [K]{q} = {F}. These equations are accurate enough if the load {F} has a frequency less than one-quarter of the lowest natural frequency of the structure. However, loads due to moving vehicles, machinery, earthquake, blasts and so on have fairly high frequency content. For a structure subjected to any of these loads, it is essential to perform a dynamic analysis and include inertial effects. There is little difficulty in extending the static equilibrium equations to the dynamic case by including mass and damping matrices in the analysis. In the following sections, governing equations and solution techniques are presented. 10.3.2 Governing Equations The direct approach to formulate the finite element equations using the linear spring system described in Chapter 6 is presented first. Then, the energy approach for the general case is discussed. 10.3.2.1 Linear Spring Model Consider a two-DOF oscillator model of Figure 10.7 (a) and its free-body diagram shown in Figure 10.7 (b). Equations of motions for the system are merely expressions of the equilibrium of the individual masses, as follows (Clough and Penzien, 1975).

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Finite Element Method with Applications in Engineering

Thus, Eq. (10.51) can be written as

The above equation can be expressed in matrix form as

where {q}T = [u1 u2], [M] is the diagonal mass matrix, [C] is the damping matrix, and [K] is the stiffness matrix. Using the principle of virtual work, general finite element equations are formed.

Figure 10.7 Two-DOF oscillator: (a) Model; (b) Free-body diagram 10.3.2.2 Finite Element Formulation In Chapter 7, for an element the displacement vector {U} and the force vector {Fe} were given in Eqs. (7.57) and (7.68) as

where {U} is the displacement, {N} represents interpolation function matrix, {X} is the body force vector, {T} represents the surface traction vector, {P} contains external nodal forces, and {qe} contains the nodal displacement values. 340

Finite Element Method with Applications in Engineering The body force vector has to be modified to include the inertia and damping forces. The element force vector {

e

} is now written as

In Eq. (10.55), m is the mass per unit volume and c is the damping per unit volume. Thus, [Ke]{qe} = {Fe} of equation (7.69) modifies to

Rewriting this equation, the dynamic equilibrium equation takes the form

[M e] is the element consistent mass matrix, [C e] is the element consistent damping matrix and [K e] is the consistent stiffness matrix. [K e], [B] and [D], were defined in earlier chapters. Assembling the contributions from all elements of the discretized domain, the global dynamic equilibrium equation can be written as In the sequel, it will be assumed that the matrices [M], [C], and [K] are symmetric of size n × n, and the vectors {q} and {F(t)} are of size n × 1 In practice, determination of the damping matrix [C] is difficult and it is seldom known. Only the following case will be considered.

where α and β are determined experimentally (Clough and Penzein, 1975). Equation (10.56) can be written as

Equation (10.58) is of the same form as equation (10.56). In the sequel, the hat is suppressed and consider the form as

Equation (10.59) can be numerically solved by using either mode superposition method or direct time integration method. The mode superposition method is presented first followed

341

Finite Element Method with Applications in Engineering by a time integration method [Bathe (2001), Huebner et al. (1995), Clough and Penzien, (1975)]. 10.3.3 Mode Superposition Method The method is applied in two steps. In the first step, the eigenvalues (frequencies) and the eigenvectors (mode shapes) of the homogeneous Eq. (10.59) ({F(t)} = 0) are found. These modes are then superposed to solve for the forced vibration problems. 10.3.3.1 Free Vibrations For the case when {F(t)} = 0, Eq. (10.59) is the homogeneous differential equation given by To find the natural frequencies (eigenvalues) and the natural or principal modes (eigenvectors) of Eq. (10.60), the nodal vector {q(t)} is expressed as

where is the unknown vector of amplitudes and ω (rad/sec) is the natural frequency. Substituting Eq. (10.61) in Eq. (10.60) results in an eigenvalue problem

Eq. (10.62) has a non-trivial solution only when the determinant of Eq. (10.62) is zero, that is,

Since, [K] and [M] are assumed to be of order n × n, Eq. (10.63) yields n eigenvalues ω2r (r = 1,2,…, n). There will be a corresponding mode shape {(φ}r (r = 1,2,…, n) associated with each frequency ωr. Frequencies ωr will be positive if [K] and [M] are positive definite, which is generally the case. It is seldom practical to write the polynomial expansion for the determinant of Eq. (10.63) for large matrices. A number of standard computer programs are available in open literature as library subroutines for the purpose. In many cases, only a few modes {(φ}r (r = 1,2,…, p, p ≤ n) are required. There are several iterative solution procedures to solve for these modes (Clough and Penzien 1975; Bathe 2001). An important property of the mode shapes, which is very useful for solving forced vibration problems, is their orthogonality with respect to the weight [M]. The orthogonality condition can be stated as

where {φ}r and {φ}s are mode shapes corresponding to frequencies ωr and ωs, respectively. The orthogonality relation for the stiffness matrix [K] can be established from Eq. (10.62) as

342

Finite Element Method with Applications in Engineering Mr and Kr are called generalized mass and stiffness, respectively and ωr is the natural frequency of the r-th mode. Sometimes the mode shapes are normalized as generalized mass

which makes the

and the generalized stiffness

In the next section, only the mode shapes {φ}r will be employed to solve transient problems. 10.3.3.2 Transient Response The orthogonality properties [Eqs. (10.64) and (10.65)] of the mode shapes now may be used to uncouple the non-homogeneous equation of motion, Eq. (10.59). The solution vector {q(t)} is expressed by superposing the modes as

and {Y(t)} is the vector of unknown modal amplitudes. Note that [Ф] transforms the generalized coordinates {Y(t)} to the physical coordinates {Q(t)}. By substituting Eq. (10.66) in Eq. (10.59) and then pre-multiplying the resulting Eq. by [Ф]T, the following equation set is obtained.

Using the orthogonality Eqs. (10.64) and (10.65), Eq. (10.67) reduces to a set of uncoupled ordinary differential equations shown below.

where the generalized force Fr(t) = {(φ}Tr{F(t)}. Thus, the problem statement can be stated as

Initial conditions are

It can be noted that Eq. (10.68) is a governing differential equation for a single-DOF system. Solution can be obtained using the Duhamel integral as

where the constant C1 and C2 are to be determined from the initial conditions. The integral in the above equation may have to be evaluated numerically. Once all p solutions of Eq. (10.68) are determined, the complete response is obtained by superposing the modes as follows. 343

Finite Element Method with Applications in Engineering

The mode superposition method is favored if only a few modes are needed to describe the response. For example, an earthquake usually has only a few dominant frequencies in its frequency spectra. However, the method has also been used for a concentrated force (Green's function) in wave propagation problems (Datta and Shah, 2008), employing few modes p(p < n). 10.3.4 Direct Time Integration Method When a response for a short duration is sought, direct integration is an effective method of choice. It is easy in this procedure to incorporate the full damping matrix [C] and the conditions of Eq. (10.57) are not necessary. Of course, the computation time will increase. A numerical step-by-step procedure is used in this method to integrate Eq. [10.56(b)] in time directly without using any transformation matrix. It is based on deriving recursion relations that relate the values of {q}, and at time t to those at time t + ∆t, where ∆t is a small time interval. Starting at time t = 0 and incorporating the initial conditions, the scheme marches out at the time step ∆t. The algorithm continues to determine the solution at the discrete time t + ∆t, t + 2∆t,t+3∆t, …. etc., until a desired time length is reached. It may be noted that the method includes contribution of all n modes discussed in the previous section. Direct integration techniques are either explicit or implicit. If the recursion relations are substituted in the dynamic equations of equilibrium, Eq. [10.56(b)], at time t, the method is explicit. On the other hand, if the recursion relations are substituted at time t + ∆t, the method is implicit. Detailed discussion of these methods is beyond scope of this book. They can be found in advanced finite element textbooks (Bathe, 2001). In the following, both techniques are presented, the central difference method for the explicit case followed by the Newmark Beta method for the implicit case. 10.3.4.1 Central Difference Method Dynamic equilibrium relations in Eq. [10.56(b)] are a system of ordinary differential equations. Any convenient finite difference expressions for kinematic variables can be used to construct the algorithm. The central difference algorithm uses finite difference expressions for velocity and acceleration at time t as

The dynamic equilibrium equations, Eq. [10.56(b)], at time t are

where subscript indicates values at time t. Substituting relations for t and algorithm can be written of the form

t

of Eq. (10.72) into Eq. (10.73), the recursive

344

Finite Element Method with Applications in Engineering Eq. (10.74) can be expressed in a compact form shown below.

where the effective stiffness matrix [ ] and the effective force { (t)} are given by

The algorithm starts at t = 0 incorporating the initial conditions {q}0 and 0. The initial acceleration vector can be found from Eq. [10.56(b)] using these conditions. The 0 displacement {q}−∆t needed to start the algorithm is evaluated from Eq. (10.72) as

Computation then continues using Eq. (10.74) and Eq. (10.75). If the size of ∆t in Eq. (10.74) is selected larger than ∆tcr, the response may become unstable and it may grow without bound as the time increases. To obtain a stable solution the step size must be

where ωn is the largest frequency and Tn the shortest period of the equation (10.63) having matrices [M] and [K] of size n × n. The central difference method is said to be conditionally stable since it requires the use of a time step ∆t smaller than a critical time step ∆tcr. 10.3.4.2 Newmark Beta Method The Newmark Beta method is based on the following assumptions on displacement and velocity.

Eq. (10.78) is solved for resulting equations are

t+∆t

, which is then substituted in Eq. (10.79) to obtain

Dynamic equilibrium equations, Eq. [10.56(b)], at time t + ∆t are where the subscript indicates values at time t∆t. 345

t+∆t

. The

Finite Element Method with Applications in Engineering Substitution of relations for t and in the following recursive algorithm.

t

of Eqs. (10.80) and (10.81) into Eq. (10.82) results

Eq. (10.83) can be expressed in a compact form as

where the effective stiffness

and the effective force

are

The algorithm starts at t = 0 incorporating the initial conditions {q} and 0. 0 can be found from Eq. [10.56(b)] using these conditions. Then, the algorithm can march forward from Eq. (10.84). Performance of the Newmark Beta method has been studied extensively for stability (Bathe 2001; Zienkiewicz and Taylor 1989) and is not discussed here further. A good choice of parameters for an implicit method that is conditionally stable is α = 0.5 and β = 0.25. This is called the constant average acceleration method, as seen from Eqs. (10.78) and (10.79). For this choice of α and β there are no amplitude errors in any sine-wave motion, regardless of the frequency. However, the periods of the sine-wave motion are overestimated. The error magnifies as ∆t /T increases (Cook, 1981). The time integration method is favored for short duration loads and for impulsive loads. For example, blast loading has a wide frequency band and impact load has white noise as its frequency spectra. 10.4 NON-LINEAR ANALYSIS Final assembled equations arising from the application of FEM can be broadly represented in one of the following forms.

Eq. set (10.85) arises from application of FEM to static (time independent) or steady-state problems. Eq. sets (10.86) and (10.87), on the other hand, arise from the time-dependent (transient) situations. However, these time-dependent equation sets can be transformed to 346

Finite Element Method with Applications in Engineering an equivalent ‘pseudo static’ set by using appropriate numerical techniques (e.g. Section 10.3.4). Thus, the final equation set considered in an analysis is generally of the form described by Eq. set (10.85). If [K] is a constant coefficient matrix, the ensuing linear simultaneous equations can be solved by using methods such as those described in Appendix A. However, in certain cases, [K] = [K (q)]. For example, consider case of stress analysis of structural system/continuum. When the constitute model itself is non-linear (due to non-linear material behavior), or when original geometry deforms considerably with respect to the original geometry (i.e., geometric non-linearity) under application of loads, the assembled matrix will no longer be a constant coefficient matrix. In such cases of [K] = [K (q)], and the resulting equations are non-linear. Due to presence of non-linearity, principle of superposition cannot be used. The ensuing equations are required to be solved incrementally, in an iterative manner. Moreover, distinction has to be made between the deformed and the original undeformed state. Final (equilibrium) equations are written with respect to the deformed geometry by using reference system either as the original undeformed configuration (Lagrangian system) or the current deformed configuration (updated Lagrangian or Eulerian system). Detailed discussion of non-linear analyses is beyond the scope of this book. They can be found in advanced finite element textbook like (Zienkiewicz and Taylor, 1989; Bathe, 2001). Typical aspects of finite element formulation for non-linear stress analysis are discussed next. 10.4.1 Finite Element Formulation for Non-Linear Analysis Equilibrium conditions between internal and external forces must be satisfied irrespective of geometric or material non-linearity. If continuous displacement field in a system is represented by the nodal displacement vector stemming from the application of FEM, equilibrium equations can be obtained from the virtual work principle as

where is the sum of internal and external forces and is equivalent external nodal force vector (vector in the context of usual linear finite element analysis). On the other hand, is strain-displacement relationship matrix and is stress vector, which depend on the type of finite element used and dimensionality of the problem. In general, , and depend on displacements and, thus, the set of Eq. (10.88) is non-linear. Residuals can be visualized as nodal forces required to bring the assumed displacement pattern into equilibrium and, thus, the residuals should be reduced to zero as much as possible. Eq. set (10.88) is generally cast in the incremental form for ease in formulating iterative process, as

where is known as the tangent stiffness matrix, and is a combination of usual linear stiffness matrix and non-linear stiffness matrices arising from large displacements and/or stress levels. Solution of the ensuing non-linear equation set is discussed next. 10.4.2 Solution of Non-Linear Equations 347

Finite Element Method with Applications in Engineering Typical methods available in the literature to solve simultaneous non-linear equations are listed below. Newton–Raphson method Modified Newton–Raphson method Initial Stiffness method Secant method Arc-line method Arc-line method with line search Pre-conditioned conjugate gradient techniques All these methods have their relative merits and demerits. For the purpose of illustration, the Newton–Raphson method has been briefly discussed here. When the Newton–Raphson method is used, system of non-linear equations is represented as a set of functions for which roots are desired, that is,

These functions are expanded in a Taylor series about some arbitrary point, which represents an initial estimate of solution. Linear solution of the problem is generally considered as the initial estimate . Subsequently, the estimate is improved iteratively as

where

From Eq. (10.89),

and, thus, the incremental nodal displacement vector can be written as

Here,

is the tangential stiffness matrix evaluated on the basis of i.

The iterative process is repeated until the required convergence criterion is satisfied. Application of full load in a single step may cause solution procedure to diverge. Hence, load is generally applied in increments. Such incremental process is sometimes computationally cheaper and it also provides a complete load-displacement history. The Newton–Raphson method is illustrated in Figure 10.8 with a single load increment, where the applied load is ψ0 and the equilibrium solution is qEQN. On the other hand, K is usual linear stiffness term. Concepts in geometric and material non-linear analyses are illustrated next with the help of simple illustrative examples.

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Finite Element Method with Applications in Engineering

Figure 10.8 The Newton–Raphson procedure 10.4.3 Illustrative Examples Example 1 Consider the geometrically non-linear spring shown in the figure. If K (u) = 200 + 20u2N/mm and P = 1,140 N, find the displacement u.

Solution: For the single-DOF system depicted above, the restoring force is k.u = 200 u + 20 u3. Thus, the equivalent of Eq. set (10.88) can be shown to be

Taking differentiation with respect to u, the following equation is obtained. By comparing the above Eq. (2) with Eq. set (10.89), [KT] = 200 + 60u2. Eqs. (10.91) through (10.94) can be rewritten for this example as

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Finite Element Method with Applications in Engineering

The initial estimate u0 is obtained from the linear solution, which is obtained by setting nonlinear stiffness term to zero, as u0 = 1140/200 = 5.7 mm. Repeated application of Eq. (6) and then Eq. (3) for i = 0,1,2, … yields following results.

Similarly, for

Thus, it can be seen that the displacement is converging to 3 mm. It can be confirmed that the restoring force for 3 mm displacement is 200× 3 + 20× 33 = 1,140 N, which is the magnitude of externally applied force P. Example 2 A single-DOF system was considered in the previous example to illustrate geometric nonlinear analysis. In this example, a two-DOF geometrically non-linear spring system is considered to illustrate solution process for a set of non-linear simultaneous equations. For the two spring assemblage shown below, stiffness of each geometrically non-linear spring is as described in the previous example. Thus, k1 = 200 + 20uB2 and k2 = 200 + 20(uc − uB)2. Consider PB = 580 N and PC = 560 N. Compute the displacements at nodes B and C.

Solution: By following the formulation given in Chapter 6, the assembled matrix can be shown (after the application of boundary condition uA = 0) to be

From the above equation set, an equivalent form of Eq. (10.88) can be shown to be

350

Finite Element Method with Applications in Engineering By substituting expressions for K1 and K2 in the above equation and by using Eq. (10.89), it can be shown that

where

The initial estimate is obtained from the linear equation set obtained from Eq. (1) by considering only linear terms in K1 and K2 as

The residual force vector ψ (qi) required in the i-th iteration can be found from Eq. (2) by substituting uBi and uCi as values of uB and uC, respectively. Progress of numerical computations has been shown in the following table.

It can be seen from the table that as the iterative process progresses, (i) the residual forces reduce [column (2)], (ii) elements of incremental solution vector become smaller (column (4)) and (iii) solution converges to values uB= 3 mm and uC = 5 mm. Example 3 The bar assemblage shown in Figure (a) has linear geometric properties and idealized bilinear stress strain properties, as shown in Figure (b). Consider cross sectional area as A1= 500 mm2 and A2 = A3 = 300 mm2. Analyze the assemblage by using Newton–Raphson method. 351

Finite Element Method with Applications in Engineering

Solution: Consider the discretization shown in the Figure (c).

Here, the direction cosines are cos θ1= 0; sin θ1 = 1; cos θ2 = –0.6; sin θ2= 0.8; cos θ3 = 0.6; sin θ3 = 0.8 Due to bi-linear stress–strain curve and unequal force distribution in elements, it would be useful to find effective stiffnesses of three bar elements in terms of appropriate elasticity constants. Effective modulus of elasticity will be identical for elements 2 and 3, as they have identical area and are placed symmetrically with respect to the vertical axis (which is also the direction of load). However, element 1 has different stiffness and orientation as compared to elements 2 and 3. Thus, it can have different modulus of elasticity, depending on the induced strain levels. Thus, the stiffness coefficients for all the three elements are computed as follows. 352

Finite Element Method with Applications in Engineering

Element matrices in the local co-ordinates are

On the other hand, the global element matrices are obtained from formulation presented in Chapter 6 as

Note that U2 = V2 = U3 = V3 = U4 = V4 = 0. After application of these boundary conditions, the ensuing 2 × 2 assembled matrix is obtained as shown below.

Here, F1Y is the vertical load applied at node 1. From the above equation (and also from the vertical plane of symmetry), it can be concluded that U1 = 0. Thus, the only non-trivial equation is

353

Finite Element Method with Applications in Engineering Note that the numerical values of E1 and E2 depend on the strain levels. Residual force can be computed by considering the nodal equilibrium of node 1, as shown in the following figure.

Fe (1), Fe (2) and Fe (3) are magnitudes of forces in elements 1, 2 and 3, respectively, induced due to application of vertical downward force F at node 1. By applying equilibrium condition, following equations are obtained.

Here,

Eqs. (7) have been obtained from the given bi-linear stress–strain relations. Strain terms can be obtained from the formulation presented in Chapter 6 as

Thus, an equivalent of Eq. (10.88) for this example can be written as

where the stress components are computed from Eqs. (7). The initial estimate of the displacement can be obtained from Eq. (5) by setting E1 = E2 = 200 kN/mm2 and F1Y = –516 kN as

In the next iteration, the residual is estimated by computing stress components from Eqs. (7) and substituting the same in Eq. (9). The tangent stiffness matrix is simply considered to be K, defined in Eq. (5). Then, the incremental and the total updated displacements are 354

Finite Element Method with Applications in Engineering computed, respectively, by employing Eqs. (10.94) and (10.91), respectively. For example, consider i = 0. From Eq. (8),

Then, stresses are computed from Eqs. (7) to be

The tangent stiffness is obtained from Eq. (5) as

On the other hand, the residual force computed from Eq. (9) is obtained to be

Subsequently, the incremental displacement and the total displacement are computed, respectively, from Eqs. (10.94) and (10.91) as

Progress of the iterative process has been summarized below in the tabular form.

It can be observed from the table that the converged value of vertical displacement at node 1 isV1 = –25.0245 mm. It should be noted that a large load increment (like the one used in the current example) can be used only for elastic system, where loading and unloading paths are the same. For an inelastic system, however, smaller load increments must be used to trace load-displacement history.

10.4 NON-LINEAR ANALYSIS Final assembled equations arising from the application of FEM can be broadly represented in one of the following forms.

Eq. set (10.85) arises from application of FEM to static (time independent) or steady-state problems. Eq. sets (10.86) and (10.87), on the other hand, arise from the time-dependent (transient) situations. However, these time-dependent equation sets can be transformed to an equivalent ‘pseudo static’ set by using appropriate numerical techniques (e.g. Section 355

Finite Element Method with Applications in Engineering 10.3.4). Thus, the final equation set considered in an analysis is generally of the form described by Eq. set (10.85). If [K] is a constant coefficient matrix, the ensuing linear simultaneous equations can be solved by using methods such as those described in Appendix A. However, in certain cases, [K] = [K (q)]. For example, consider case of stress analysis of structural system/continuum. When the constitute model itself is non-linear (due to non-linear material behavior), or when original geometry deforms considerably with respect to the original geometry (i.e., geometric non-linearity) under application of loads, the assembled matrix will no longer be a constant coefficient matrix. In such cases of [K] = [K (q)], and the resulting equations are non-linear. Due to presence of non-linearity, principle of superposition cannot be used. The ensuing equations are required to be solved incrementally, in an iterative manner. Moreover, distinction has to be made between the deformed and the original undeformed state. Final (equilibrium) equations are written with respect to the deformed geometry by using reference system either as the original undeformed configuration (Lagrangian system) or the current deformed configuration (updated Lagrangian or Eulerian system). Detailed discussion of non-linear analyses is beyond the scope of this book. They can be found in advanced finite element textbook like (Zienkiewicz and Taylor, 1989; Bathe, 2001). Typical aspects of finite element formulation for non-linear stress analysis are discussed next. 10.4.1 Finite Element Formulation for Non-Linear Analysis Equilibrium conditions between internal and external forces must be satisfied irrespective of geometric or material non-linearity. If continuous displacement field in a system is represented by the nodal displacement vector stemming from the application of FEM, equilibrium equations can be obtained from the virtual work principle as

where is the sum of internal and external forces and is equivalent external nodal force vector (vector in the context of usual linear finite element analysis). On the other hand, is strain-displacement relationship matrix and is stress vector, which depend on the type of finite element used and dimensionality of the problem. In general, , and depend on displacements and, thus, the set of Eq. (10.88) is non-linear. Residuals can be visualized as nodal forces required to bring the assumed displacement pattern into equilibrium and, thus, the residuals should be reduced to zero as much as possible. Eq. set (10.88) is generally cast in the incremental form for ease in formulating iterative process, as

where is known as the tangent stiffness matrix, and is a combination of usual linear stiffness matrix and non-linear stiffness matrices arising from large displacements and/or stress levels. Solution of the ensuing non-linear equation set is discussed next.

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Finite Element Method with Applications in Engineering 10.4.2 Solution of Non-Linear Equations Typical methods available in the literature to solve simultaneous non-linear equations are listed below. Newton–Raphson method Modified Newton–Raphson method Initial Stiffness method Secant method Arc-line method Arc-line method with line search Pre-conditioned conjugate gradient techniques All these methods have their relative merits and demerits. For the purpose of illustration, the Newton–Raphson method has been briefly discussed here. When the Newton–Raphson method is used, system of non-linear equations is represented as a set of functions for which roots are desired, that is,

These functions are expanded in a Taylor series about some arbitrary point, which represents an initial estimate of solution. Linear solution of the problem is generally considered as the initial estimate . Subsequently, the estimate is improved iteratively as

where

From Eq. (10.89),

and, thus, the incremental nodal displacement vector can be written as

Here,

is the tangential stiffness matrix evaluated on the basis of i.

The iterative process is repeated until the required convergence criterion is satisfied. Application of full load in a single step may cause solution procedure to diverge. Hence, load is generally applied in increments. Such incremental process is sometimes computationally cheaper and it also provides a complete load-displacement history. The Newton–Raphson method is illustrated in Figure 10.8 with a single load increment, where the applied load is ψ0 and the equilibrium solution is qEQN. On the other hand, K is usual linear stiffness term. Concepts in geometric and material non-linear analyses are illustrated next with the help of simple illustrative examples.

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Figure 10.8 The Newton–Raphson procedure 10.4.3 Illustrative Examples Example 1 Consider the geometrically non-linear spring shown in the figure. If K (u) = 200 + 20u2N/mm and P = 1,140 N, find the displacement u.

Solution: For the single-DOF system depicted above, the restoring force is k.u = 200 u + 20 u3. Thus, the equivalent of Eq. set (10.88) can be shown to be

Taking differentiation with respect to u, the following equation is obtained.

By comparing the above Eq. (2) with Eq. set (10.89), [KT] = 200 + 60u2. Eqs. (10.91) through (10.94) can be rewritten for this example as

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Finite Element Method with Applications in Engineering

The initial estimate u0 is obtained from the linear solution, which is obtained by setting nonlinear stiffness term to zero, as u0 = 1140/200 = 5.7 mm. Repeated application of Eq. (6) and then Eq. (3) for i = 0,1,2, … yields following results.

Similarly, for

Thus, it can be seen that the displacement is converging to 3 mm. It can be confirmed that the restoring force for 3 mm displacement is 200× 3 + 20× 33 = 1,140 N, which is the magnitude of externally applied force P. Example 2 A single-DOF system was considered in the previous example to illustrate geometric nonlinear analysis. In this example, a two-DOF geometrically non-linear spring system is considered to illustrate solution process for a set of non-linear simultaneous equations. For the two spring assemblage shown below, stiffness of each geometrically non-linear spring is as described in the previous example. Thus, k1 = 200 + 20uB2 and k2 = 200 + 20(uc − uB)2. Consider PB = 580 N and PC = 560 N. Compute the displacements at nodes B and C.

Solution: By following the formulation given in Chapter 6, the assembled matrix can be shown (after the application of boundary condition uA = 0) to be

From the above equation set, an equivalent form of Eq. (10.88) can be shown to be

359

Finite Element Method with Applications in Engineering By substituting expressions for K1 and K2 in the above equation and by using Eq. (10.89), it can be shown that

where

The initial estimate is obtained from the linear equation set obtained from Eq. (1) by considering only linear terms in K1 and K2 as

The residual force vector ψ (qi) required in the i-th iteration can be found from Eq. (2) by substituting uBi and uCi as values of uB and uC, respectively. Progress of numerical computations has been shown in the following table.

It can be seen from the table that as the iterative process progresses, (i) the residual forces reduce [column (2)], (ii) elements of incremental solution vector become smaller (column (4)) and (iii) solution converges to values uB= 3 mm and uC = 5 mm. Example 3 The bar assemblage shown in Figure (a) has linear geometric properties and idealized bilinear stress strain properties, as shown in Figure (b). Consider cross sectional area as A1= 500 mm2 and A2 = A3 = 300 mm2. Analyze the assemblage by using Newton–Raphson method. 360

Finite Element Method with Applications in Engineering

Solution: Consider the discretization shown in the Figure (c).

Here, the direction cosines are cos θ1= 0; sin θ1 = 1; cos θ2 = –0.6; sin θ2= 0.8; cos θ3 = 0.6; sin θ3 = 0.8 Due to bi-linear stress–strain curve and unequal force distribution in elements, it would be useful to find effective stiffnesses of three bar elements in terms of appropriate elasticity constants. Effective modulus of elasticity will be identical for elements 2 and 3, as they have identical area and are placed symmetrically with respect to the vertical axis (which is also the direction of load). However, element 1 has different stiffness and orientation as compared to elements 2 and 3. Thus, it can have different modulus of elasticity, depending on the induced strain levels. Thus, the stiffness coefficients for all the three elements are computed as follows. 361

Finite Element Method with Applications in Engineering

Element matrices in the local co-ordinates are

On the other hand, the global element matrices are obtained from formulation presented in Chapter 6 as

Note that U2 = V2 = U3 = V3 = U4 = V4 = 0. After application of these boundary conditions, the ensuing 2 × 2 assembled matrix is obtained as shown below.

Here, F1Y is the vertical load applied at node 1. From the above equation (and also from the vertical plane of symmetry), it can be concluded that U1 = 0. Thus, the only non-trivial equation is

362

Finite Element Method with Applications in Engineering Note that the numerical values of E1 and E2 depend on the strain levels. Residual force can be computed by considering the nodal equilibrium of node 1, as shown in the following figure.

Fe (1), Fe (2) and Fe (3) are magnitudes of forces in elements 1, 2 and 3, respectively, induced due to application of vertical downward force F at node 1. By applying equilibrium condition, following equations are obtained.

Here,

Eqs. (7) have been obtained from the given bi-linear stress–strain relations. Strain terms can be obtained from the formulation presented in Chapter 6 as

Thus, an equivalent of Eq. (10.88) for this example can be written as

where the stress components are computed from Eqs. (7). The initial estimate of the displacement can be obtained from Eq. (5) by setting E1 = E2 = 200 kN/mm2 and F1Y = –516 kN as

In the next iteration, the residual is estimated by computing stress components from Eqs. (7) and substituting the same in Eq. (9). The tangent stiffness matrix is simply considered to be K, defined in Eq. (5). Then, the incremental and the total updated displacements are 363

Finite Element Method with Applications in Engineering computed, respectively, by employing Eqs. (10.94) and (10.91), respectively. For example, consider i = 0. From Eq. (8),

Then, stresses are computed from Eqs. (7) to be

The tangent stiffness is obtained from Eq. (5) as

On the other hand, the residual force computed from Eq. (9) is obtained to be

Subsequently, the incremental displacement and the total displacement are computed, respectively, from Eqs. (10.94) and (10.91) as

Progress of the iterative process has been summarized below in the tabular form.

It can be observed from the table that the converged value of vertical displacement at node 1 isV1 = –25.0245 mm. It should be noted that a large load increment (like the one used in the current example) can be used only for elastic system, where loading and unloading paths are the same. For an inelastic system, however, smaller load increments must be used to trace load-displacement history.

10.6 HYDRODYNAMICS SIMULATION OF SHALLOW WATER FLOW 10.6.1 Introduction Flow regime in rivers, channels, estuaries and coastal regions is a transient real world phenomenon. Solution to these problems can be obtained through physical models, analytical solutions or numerical models. Physical model is an attempt to reduce the complex problem into small observable scale in laboratory. Physical models are generally difficult to make, time consuming to prove (possibility of scale effects, etc.) and expensive. Analytical solutions refer to closed form solutions of governing equations. For rivers, channels or estuary problems, very few analytical solutions are available for simplified domain and boundary conditions. A numerical model is an attempt to represent system by solving governing equations that describe natural processes. Very complex geometry and 364

Finite Element Method with Applications in Engineering boundary conditions (peculiar to river or estuary problems) can be handled in numerical modelling (Vreugdenhil, 1994) using FEM. If numerical models are used for simulation related to river, estuary or coastal region, choice is between one-dimensional model and the one that incorporates two- or three-spatial dimensions (Hinwood and Wallis, 1975). One-dimensional models are easier to build, economical and fairly adequate for practical purposes. Due to complexity of estuary or river hydrodynamics, however, one-dimensional models may not give sufficiently accurate results. Hence, two-dimensional or threedimensional models are often required for a complete analysis of estuary problems. Problems related to numerical diffusion, dispersion and oscillation, demands of computer time and memory for simulation are some of the challenges in development of numerical models for two-dimensional and three-dimensional analyses (Vreugdenhil, 1994). Here, the two-dimensional finite element formulation for estuary or coastal hydrodynamics is presented with detailed formulation and application.

10.6.2 Governing Equations and Boundary Conditions For many coastal and estuarine flow problems, the vertical velocity component is relatively small in comparison with the horizontal velocity components U, V. Hence, the continuity equation and Navier–Stokes equations can be integrated over the depth and solved numerically to give the depth averaged velocity field (Martin and McCutcheon, 1998). Such flow field can generally be assumed to occur in unstratified or weakly stratified, estuaries, harbors, bays and coastal seas, where the wind shear is also relatively small. Shallow water equations in non-conservative form are used here for hydrodynamics of estuaries. Pertinent equations are described below (Gray and Lynch, 1979; Liggett, 1994). (1) Continuity equation

(2) Momentum equation in X direction

(3) Momentum equation in Y direction

Here, τ = g(U2 + V2)1/2/C2H; f = Coriolis parameter = 2 Ω sin θ; Ω = angular velocity of earth; a = depth mean eddy viscosity; θ = degree of latitude; g = acceleration due to gravity; z = H – h; H = total depth = h + ζ (see Figure 10.14); h = distance of bottom from reference plane; ζ =surface elevation from reference line; C = Chezy's Coefficient; W = wind shear component in the direction indicated by the suffix; t = time; U = depth averaged velocity in x direction; and V = depth averaged velocity in y direction. Initial and Boundary Conditions Governing equations are solved by using initial and boundary conditions. The simplest initial condition is that the fluid is at rest with a horizontal water level. This normally does not happen in reality. Influence of this wrong initial data gradually fades out due to wave damping by bottom friction.

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Finite Element Method with Applications in Engineering

Figure 10.14 Vertical section of a shallow water flow field Surface and bottom conditions come in two kinds. The kinematic conditions, for example, can be of the form that water particles do not cross either boundary. For the solid bottom, this means that the normal velocity component vanishes. Mathematically, the condition can be stated as

where Zb is the bottom level, measured from some horizontal reference level at the free surface and W is vertical velocity. Here, computations may be complicated as the surface may be moving by itself. Then, the relative normal velocity must vanish, as shown below.

Here h is the surface level measured from a horizontal reference level. At the bottom, one can assume that the viscous fluid ‘sticks’ to the bottom, that is, This is called the ‘no-slip’ condition. At the free surface, continuity of stresses is assumed, that is, the stresses in fluid just below the free surface are assumed to be the same as those in air just above. This means that the surface tension is not taken into account. For pressure,

where pa is the atmospheric pressure. The absolute pressure level is not important so it can be considered to be zero. 10.6.3 Finite Element Formulation In the present study, FEM based on Galerkin's formulation is used. Neglecting the eddy viscosity terms and spatial variations of atmospheric pressure and wind frictional effect, the shallow water (governing) equations can be written as (Gray, 1977; Lynch and Gray 1979)

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Finite Element Method with Applications in Engineering

Here, U = depth averaged velocity in x direction; V = depth averaged velocity in y direction; H = total depth = h + ζ (see Figure 10.14); h = distance of bottom from reference plane; C =Chezy's Coefficient; and f = Coriolis parameter. Unknown variables in shallow water equations are approximated in FEM as (Connors and Brebbia, 1976).

where n is the number of nodes (n = 3, for triangular element) per element. The Chezy's coefficient is taken as constant (but it can vary in some cases). Here, Ni are shape functions. In the present study, simple three node triangular elements are used to represent the domain as shown in Figure 10.9. The shape function details are given in Section 10.5.3.1. Variation of quantity can be expressed in matrix form as

Governing equation L (ɸ) is approximated in the context of weighted residual approach (Connor and Brebbia, 1976) as

If Wi = Ni, method weighted residual is called Galerkin's weighted residuals method, that is,

Galerkin's method is applied to the continuity Eq. (10.145) and momentum Eq. (10.146), Eq. (10.147) in the sequel. Continuity Equation

Momentum Equation in X Direction

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Finite Element Method with Applications in Engineering

where Momentum Equation in Y Direction

where

.

Dot

and

indicates

the

time

derivatives

.

Time Domain Discretization Eqs. (10.153), (10.155) and (10.157) can be written in the form

where R is function of variables U, V, H and spatial coordinate, ɸ is any unknown variable U, V, or H. Eq. (10.158) can be written in finite difference form as

Eqs. (10.153), (10.155) and (10.157) are rewritten below in form depicted by Eq. (10.159). Continuity Equation

Eq. (10.160) can be written in the following form

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Finite Element Method with Applications in Engineering

where

Momentum Equation in X Direction

Eq. (10.162) can be written as

where

Momentum Equation in Y Direction

Eq. (10.164) can be written as

where

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Finite Element Method with Applications in Engineering

Generally, explicit or implicit scheme is used for time integration. In explicit scheme, {R} is considered at time t. At time t, values of variable U, V and H are known. Thus, values of U, V and H can be computed explicitly. Time steps to be used in explicit method should be very small to get good accuracy in solution.

Figure 10.15 Rectangular channel domain Weighted average of {R} is considered at time t and time t + ∆t, when an implicit scheme is used.

Here, w0 = 1– w1. For implicit model, value of w1 can be taken as 0.7. For this value, system becomes unconditionally stable (Reddy, 1993). Larger time step can be taken with small decrease in accuracy of solution. Larger time step, in addition to decreasing accuracy of solution can also introduce some unwanted, numerically induced oscillations into solution. Numerical oscillation may occur, but they never become unstable for problems considered here. By superimposing element matrices for all elements in proper order, the global matrix for entire discretized domain can be obtained as follows. Here, {x} is a matrix or column vector consist of H, U or V at time instant t + ∆t. On the other hand, coefficient matrix [A] and column vector {R} are composed of values at time instant t. Thus, H, U and V can be obtained at t + ∆t time step by knowing values at t by using standard procedure such as Gauss elimination or iterative scheme discussed earlier. 10.6.3 Case Study Based on the above formulation, an implicit FEM based model has been developed and is applied to a straight simple channel (Figure 10.15) described by Gray and Lynch (1979). The channel has a length of 871.65 Km and width of 160 km. Here, the unknowns are horizontal velocities (U, V) and depth of water (H). Triangular elements have been used in discretization as shown in Figure 10.16. Grid has 176 elements and 115 nodes. For simulation, model is tested with constant bathymetry of 9.1435 meter. A sinusoidal tidal forcing function period of 12 hours and amplitude of 0.091435 m has been applied to the opened end of the channel. Cold start (U = V = 0, H = h) is taken as initial condition. 370

Finite Element Method with Applications in Engineering Normal flux to solid boundaries is taken as zero. Coriolis effect is neglected. Time step of ∆t = 350 sec is used. Solution obtained with models are shown in Figures 10.17 and 10.18 for surface elevation and velocities (U) after the steady-state has been reached. Small oscillations are present in solution when compared with analytical solution given by Gray and Lynch (1979). This is expected since the present model is based on numerical approximation.

Figure 10.16 Finite element grid used in computation of the rectangular channel

Figure 10.17 Surface elevation computed by FEM model compared with analytical solution

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Figure 10.18 Velocity computed by FEM model compared with analytical solution

10.7 FEM–SOFTWARE AND WEB RESOURCES 10.7.1 Introduction Due to tremendous development of FEM in various areas of Engineering and Sciences for the past 50 years or so, a large number of software and packages are available commercially and also as open sources. Even though few packages can solve problems in more than one area, most FEM packages are area specific and can solve only particular set of problems. Furthermore, while few packages can be employed to solve problems in three-, two- and one-dimensional domains, many packages can be employed to solve problems in one- or two-dimensional domains. Most of earlier packages did not have any graphical user interface (GUI) for discretization and data preparation and, thus, the preand post-processing had to be done separately. With recent advancements in computer sciences, most software packages presently available in market have GUI with various pre- and post-processing options to facilitate data preparation and interpretation of results. In this section, some of the commonly used FEM based software and packages in various areas of Engineering and Sciences are briefly described, with an aim to make reader aware of available software packages. Relative merits or demerits of competing software packages are not discussed here. Reader is referred to a comprehensive description on software packages at sitehttp://en.wikipedia.org/wiki/List_of_finite_element_software_packages. In addition, websites of the developers and vendors of specific software packages can be used to know about full capabilities and features to arrive at conclusion for selecting best possible software package(s) for the intended applications. 10.7.2 FEM in Structural Engineering As FEM was initially developed in the field of structural engineering, a large number of software packages have been developed during the past five decades in the area. Many software packages are available for structural design, planning and management. Exclusive design oriented software packages are available for structural steel design, reinforced cement concrete design, mechanical and aerospace structural design etc. In

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Finite Element Method with Applications in Engineering this section, some of the commonly used software packages in structural engineering have been briefly discussed. STAAD (http://www.bentley.com/en-US/Products/Structural+Analysis+and+Design) STAAD family of FEM analysis and design packages (STAAD.Pro/STAAD.beava) can be used for typical structures like multi-storied buildings, towers, culverts, plants, bridges, stadiums, marine structures and so on. It can easily accommodate design and loading requirements, in conformation of various international and national codes of practice. RAM Products/ Bentley Structural (http://www.bentley.com/en-US/Products/Structural+Analysis+and+Design) There are number of RAM modules (RAM Advanse / RAM Frame / RAM Concrete /RAM Concept / RAM Steel) for analysis and design of variety of structures, structural components: Structures such as trusses, continuous beams, frames of all types, retaining walls, masonry walls, tilt-up walls, shear walls, footings, etc. Static and dynamic analysis of two- and three-dimensional structures made up of common construction materials can also be performed for a variety of loading conditions Bentley Structural software can be used in structural design and construction documentation for structures in steel, concrete and timber in two- or three-dimensional. It gives the integrated analytical model with finite elements, nodes, boundary conditions and member releases, loads and load combinations. NISA Products (http://www.nisasoftware.com/Products) NISA is a FEM software for design and analysis to address the Automotive, Aerospace, Energy and Power, Civil, Electronics and Sporting Goods industries. NISA include integrated pre-/post- processing environment within a GUI and inter-operability with leading commercial CAD software. It can be used for solid and surface modelling, finite element mesh generation, various types of analyses such as that for stress, vibration, seismic, electromagnetic, fatigue, CFD/fluid structure interactions and so on. Some products (NISA/CIVIL, NISA DesignStudio) can be used for analysis, design and detailing of variety of RCC and steel structures. MSC Nastran (http://www.mscsoftware.com/products/msc_nastran.cfm) MSC Nastran is a general purpose finite element analysis solution for small to complex assemblies. It provides a wide range of modelling and analysis capabilities, including linear statics, displacement, strain, stress, vibration, heat transfer and so on. NEi Fatigue (http://www.nenastran.com/newnoran/fatigue) NEi Fatigue allows Engineers to calculate fatigue life of components under dynamic loading. Applications of NEi Fatigue include fatigue analysis for bodywork structures, axle components, crankshafts, rotary blades for wind power stations and so on. ANSYS (http://www.ansys.com/products/structural-mechanics) 373

Finite Element Method with Applications in Engineering ANSYS structural mechanics solutions offer a broad spectrum of capabilities covering a range of analysis types, elements, contact, materials, equation solvers and coupled physics. It has a CAD interface for the direct use of CAD models without any modifications or mediations. ABAQUS (http://www.simulia.com/) ABAQUS can be used in many different engineering fields such as civil, structural, automobile, mechanical, aerospace engineering and so on. It performs static and/or dynamic analysis and simulation on structures. It can deal with bodies with various loads, temperatures, contacts, impacts, and other environmental conditions. Pre- and postprocessing GUI interface available with the package can be used for visualization and animations. 10.7.3 FEM in Geotechnical Engineering Many software packages are available for geotechnical design, planning and management. A variety of software packages for foundation design, tunnel design, earthen dam and embankment design, retaining wall design and so on are available. In this section, some of the commonly used software packages in geotechnical engineering have been discussed. STAAD.foundation / RAM Foundation (http://www.bentley.com/en-US/Products/Structural+Analysis+and+Design) STAAD.foundation can automatically absorb the geometry, loads and results from a STAAD.Pro model to design foundations. It keeps track of changes made in the STAAD.Pro model and can merge reaction output to the existing foundation model. RAM Foundation is an integrated module within the RAM Structural System that performs design, evaluation and analysis of spread, continuous and pile cap foundations and so on. Midas GTS (http://www.midas-diana.com/products/product_gts.asp) Midas GTS is a Geotechnical and Tunnel Analysis System. It can be used to analyze tunnel intersections, deep foundations, excavations, embankments and slope stability, groundwater flow, effective stress analysis and vibration analysis for earthquake, blasting and so on. CRISP (http://www.mycrisp.com/products/crisp2d.html) CRISP can be used for a variety of geotechnical problems, including retaining structures, embankments, tunnels, foundations, geo-textile reinforcement, soil nailing, slope stability, borehole stability, construction sequence studies and so on. PLAXIS (http://www.plaxis.nl) Plaxis is a range of finite element packages intended for two- and three-dimensional analysis of foundation, deformation, stability, groundwater flow and so on in geotechnical engineering. 374

Finite Element Method with Applications in Engineering 10.7.4 FEM in Fluid Mechanics Many software packages are available for fluid flow analysis of pipes, open channels, rivers, estuaries, mechanical and aero systems and so on. Software packages are also available for computational fluid dynamics analysis, groundwater analysis, surface flow analysis, contaminant transport, etc. In this section, some of the commonly used software packages in the field of Fluid Mechanics are briefly discussed. ANSYS CFX (http://www.ansys.com/products) ANSYS CFX is based on computational fluid dynamics (CFD) technology for simulations and solutions of steady-state/transient analysis; laminar/turbulent flow analysis for a variety of problems. NEi Fluid Dynamics (http://www.nenastran.com/newnoran/fluidDynamics) NEi Fluid Dynamics provides CFD analysis solutions and models of three-dimensional fluid velocity, temperature and pressure for steady-state and transient applications. Various simulations like laminar and turbulent fluid flow, low speed and high speed compressible flow, forced-convection, natural convection and mixed flows and so on can be performed by using the software. NISA/3D-FLUID (http://www.nisasoftware.com/Portals) NISA/3D-FLUID can be used for gas flow from the low Mach number up to hypersonic speed. It allows specification of flow specific boundary conditions allows specification of flow variables such as density, temperature and its derivative, species mass fractions, and x, y, z momentums at any location in the computational domain. FEFLOW (http://www.feflow.info/about+M520d624e0eb) FEFLOW is a software package for modelling fluid flow and transport of dissolved constituents and/or heat transport processes in the subsurface. It can be used for flow and transport processes simulation in porous media, ranging from lab scale to continental scale and geotechnical applications for tunnel construction, construction site dewatering, dam seepage and so on. FEFLOW-F3 modeling/feflow-f3)

(http://www.swstechnology.com/groundwater-software/groundwater-

FEFLOW-F3 (Finite Element subsurface FLOW system) is a ground water modelling software and can be used for modelling of flow and transport processes in porous media under saturated and unsaturated conditions. FEMWATER (http://www.epa.gov/ceampubl/gwater/femwater/index.html) It is a software package of groundwater modelling system (GMS) for different groundwater models, including a three-dimensional finite element model of density-dependent flow and transport through saturated–unsaturated porous media. 10.7.5 FEM in Thermal and Automobile Engineering Many software packages are available for heat flow analysis and automobile design dealing with wide spectrum of problems encountered in heat transfer, complex real world automobile problems and so on. Some of the commonly used software packages are briefly discussed here. NEI Thermal Basic (http://www.nenastran.com/newnoran/thermalBasic)

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Finite Element Method with Applications in Engineering NEI Thermal Basic (TMG) provides solutions for linear and non-linear, steady-state and transient heat transfer processes including conduction, radiation, and free convection. It includes modelling of conduction, convection, radiation and phase change and it provides a range of thermal boundary conditions and solver controls. NISA/ HEAT (http://www.nisasoftware.com/Products/MechanicalApplications) NISA/Heat is a general purpose finite element program to analyze a wide spectrum of problems encountered in heat transfer dealing with concentrated heat flows, distributed heat flux, internal heat generation, radiation effects and so on. It can analyze non-linearity due to time and temperature-dependent materials and boundary conditions, such as temperature-dependent thermal conductivity and phase change (melting and freezing) and so on. LS-DYNA (http://www.easi.com/software-products-ls-dyna.aspx) LS-DYNA software is a general purpose transient dynamic finite element program used by automotive industries to analyze vehicle designs, especially for crash worthiness. It can be used for non-linear analysis; rigid body dynamics; quasi-static simulations; FEM-rigid multi-body dynamics coupling (MADYMO, CAL3D); real-time acoustics; design optimization; multi-physics coupling; structural-thermal coupling, adaptive re-meshing and so on. 10.7.6 FEM in Physics Some of the commonly used software packages for applications in physics such as acoustics, heat flow analysis and material studies and so on are briefly discussed below. COMSOL Multi-physics (http://www.comsol.com/products/) COMSOL Multi-physics (formerly FEMLAB) is a finite element analysis and solver software package for various physics and engineering applications, especially coupled phenomena, or multi-physics. The software has several modules like AC/DC Module, Acoustics Module, CAD Import Module, Chemical Engineering, Earth Science Module, Heat Transfer Module, Material Library, MEMS Module, RF Module, Structural Mechanics Module. Xenos (http://www.feildp.com/index.html) Xenos (X-ray/electron numerical optimization suite) is a suite of two- and threedimensional finite element programs to handle X-rays and electrons and has a focus on coupled or stand-alone applications for electric and magnetic field calculations, electron beam design, Monte Carlo modelling of radiation transport and thermal analysis and so on. 10.7.7 Multi-Field FEM Software Some of the commonly used software packages for multi-field applications are discussed below. NISA (http://www.nisasoftware.com) NISA is available to address the needs of the automotive, aerospace, energy and power, oil and gas, electronics packaging and civil engineering industries (two- and threedimensional rigid frames and trusses). NISA provides solutions in the areas of stress analysis, seismic analysis, vibration analysis, composite material analysis, fatigue analysis, thermal analysis, PCB analysis, Computational Fluid Dynamics, electromagnetic analysis, Civil structure analysis and so on. 376

Finite Element Method with Applications in Engineering Elmer (http://www.csc.fi/english/pages/elmer) Elmer is an open source multi-physical simulation software. Elmer includes physical models of fluid dynamics, structural mechanics, electromagnetics, heat transfer and acoustics, for example. These are described by partial differential equations; which Elmer solves by FEM. One can use basic element shapes in one-, two- and three-dimensional domains with the Lagrange shape functions of k ≤ 2 and FEM solvers (without multiple acceleration). Assembly and iterative solution can also be done in parallel. FlexPDE6 (http://www.pdesolutions.com/) FlexPDE6 is Multi-Physics Finite Element Solution Environment for Partial Differential Equations. It can be used for one-, two- or three-dimensional Multi-Physics PDE problems of heat flow, stress analysis, fluid mechanics, chemical reactions, electromagnetics, diffusion, etc. It has a self-contained processing system that analyses problem description, symbolically from Galerkin's finite element integrals, derivatives and dependencies. It builds a coupling matrix and solves it and plots the results. 10.7.8 FEM in Other Fields Other than the various fields mentioned above, FEM has been used in many other fields. Many software packages are available for various areas like electrical engineering, electronics, bio-medical engineering and so on. In this section, some of the commonly used software packages in such fields are discussed briefly. FEAP (http://www.nisasoftware.com) FEAP has been developed for analyzing stress, random vibration, fatigue life, threedimensional convective fluid flow and thermal analysis of printed circuit boards (PCBs), mounted components and electronic systems. NEi Nastran (www.waveaxis.com/nei-nastran.html) NEi Nastran provides solutions for bio-medical engineering to evaluate and optimize many performance and reliability aspects of designs and manufacturing processes encountered in the field. Design and deployment of cardio-vascular related systems such as vascular stents, catheters, drug delivery systems, pacemakers, defibrillators and heart valve replacements can be done using this software. Design and deployment of orthopedic related devices such as knee replacement, hip implants, spinal implants, cartilage and joint replacement, human modelling, virtual biomechanics and biocompatible materials analysis, can also be performed.

10.8 CONCLUDING REMARKS Many approximate numerical analysis methods have evolved in the last six decades for solution of complex engineering problems due to advent of high speed computers. Out of various available numerical techniques, finite difference method (FDM), finite element method (FEM), boundary element method (BEM), finite volume method (FVM) and meshless method have become popular among scientists and engineers. With advancements in computational science and development of modern computers and high speed processors, application of these numerical techniques has become much easier. Out of these numerical techniques, one of the most important advances in the field of numerical methods was the development of FEM in the 1950s. FEM has become a very popular and practical tool for all type of computing in engineering and science. As demonstrated in this book, FEM is robust and thoroughly developed method. It is widely used in all engineering and science fields due to its versatility and 377

Finite Element Method with Applications in Engineering applicability to complex geometry and flexibility for many types of linear and non-linear problems. Most practical engineering problems related to solids, structures, fluids and materials are currently solved using well developed FEM packages. As a numerical method, it appears that FEM has reached a point where no significant break though may be expected in the near future. However, future growth of FEM will involve broader applications to further engineering problems, further refinement in the basic theorems and formulations and further efficient pre-processing and post-processing tools. Various sophisticated tools are available for pre-processing of data and postprocessing of FEM simulation results. However, further efforts are required to efficiently automate pre- and post-processing tasks for more user friendly and error free operations. Tremendous growth in computer technology will undoubtedly continue to have significant impact on further evolution of FEM. Improved efficiency achieved in computer hardware and software tools such as micro/nano processors and large-scale parallel processing will need significant improvements in finite element model development and applications. From Engineer's view point, FEM can always be made more efficient and easier to use with sophisticated pre- and post-processing tools. As FEM is applied to very large and complex problems, it is very important that the solution process remains efficient and economical. With continuing demand and economic pressures to further improve innovations, efficiency and engineering productivity, this century will see further growth of FEM in engineering design and analysis. FEM will remain more exciting and will be an important part of Engineer's design and analysis toolkit.

REFERENCES AND FURTHER READING Arlett, P. L., Bahrani, A. K., and Zienkiewicz, O. C. (1968). Application of Finite Element Method to the Solution of Helmholtz's Equation, Proc. IEE, 115(12):1762–1766. Arpita, M. (2008). FEM-GA based simulation-optimization models for groundwater remediation, M.Tech. Thesis, Department of Civil Engineering, Indian Institute of Technology, Bombay. Backstrom, G. (1999). Fluid Dynamics by Finite Element Analysis, Studentlitteratur, Lund. Baker, A. J. (1983). Finite Element Computational Fluid Mechanics, McGraw Hill, Hemisphere, New York. Bathe, K. J. (2001). Finite Element Procedure, Prentice Hall of India, New Delhi. Bear, J. (1972). Dynamics of Fluids in Porous Media. Elsevier, New York. Bear, J. (1979). Hydraulics of Groundwater. McGraw Hill, New York. Bogner, F. K., Fox, R. L., and Schmidt, L. A. (1968). The generation of inter-elementcompatible stiffness and mass matrices by the use of interpolation formulae, Proceedings of 1st conference on matrix methods in structural mechanics, Wright-Patterson Air Force Base, Dayton, OH, pp. 397–443. Brebbia, C. A., and Walker, S. (1979). Dynamic Analysis of Offshore Structures, NewnesButterworths, London. Chapra, S. C., and Canale, R. P. (2002). Numerical Methods for Engineers, 4th ed., Tata McGraw Hill, New Delhi. Chung, T. J. (1978). Finite Element Analysis in Fluid Dynamics, McGraw Hill, New York. Clough, R. W., and Penzien, J. (1975). Dynamics of Structures, McGraw Hill. 378

Finite Element Method with Applications in Engineering Connors, J. J., and Brebbia, C. A. (1976). Finite Element Techniques for Fluid Flow, Butterworth, London. Cook, R. D. (1981). Concepts and Applications of Finite Element Analysis, John Wiley and Sons. Datta, S. K., and Shah, A. H. (2008). Elastic Waves in composite Media and Structures, CRC Press, Boca Raton. Desai, C. S., and Abel J. F. (1987). Introduction to the Finite Element Method — A Numerical Method for Engineering Analysis, Wads Worth, California. Dunne, P. (1968). Complete polynomial displacement fields for the finite element method, Aeronaut. J., 72: 246–247. Freeze, R. A., and Cherry, J. A. (1979). Groundwater, Prentice Hall, Englewood Cliffs, NJ. Gray, W. G. (1977). An efficient finite element scheme for two-dimensional surface water computation, Finite Element in Water Resources, Vol. I, Brebbia, C. A., Gray, W. G. and Pinder, G. F., (eds.), Pentech Press, London, pp. 4.33–4.49. Gray, W. G., and Lynch, D. R. (1979). On the control of noise in finite element tidal computations: a semi-implicit approach, Computer and Fluids,7:47–67. Hinwood, J. B., and Wallis, I. G. (1975). Review of Models of Tidal Waters, J. Hydra. Eng., ASCE, 101(10). Hrabok, M. M, and Hrudey, T. M. (1984). A review and catalogue of plate bending finite elements, Computers and Structures, 19(3):479–495. Huebner, K. H., Thornton, E. A., and Byron, T. G. (1995). The Finite Element Method for Engineers, John Wiley and Sons. Huyakorn, P. S., and Pinder, G. F. (1983). Computational Methods in Subsurface Flow. Academic Press, New York, pp. 473. Irons, B. M., and Razzaque, A. (1977). Experience with the patch test for convergence of finite element method, in Mathematical Foundation of the Finite Element Method, Aziz, A. K. (ed.). Academy Press, pp. 557–587. Istok, J. (1989). Groundwater Modeling by the Finite Element Method, American Geophysical Union, Washington. Kinsler, L. E., and Frey, A. R. (1950). Fundamentals of Acoustics, John Wiley and Sons, New York. Krishnamoorthy, C. S. (1994). Finite Element Analysis: Theory and Programming, Tata McGraw Hill, New Delhi. Liggett, J. A. (1994). Fluid Mechanics, McGraw Hill, Singapore. Lynch, D. R., and Gray, W. G. (1979). A wave equation model for finite element tidal computations, Computer and Fluids, 7:207–228. MacCamy, R. C., and Fuchs, R. A. (1945). Wave Forces on Piles: A Diffraction Theory, US Army Coastal Engineering Center, Tech. Mem. No. 69. Martin, J. L., and McCutcheon, S. C. (1998). Hydrodynamics and Transport for Water Quality Modelling, Lewis Publishers, London.

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Finite Element Method with Applications in Engineering Melosh, R. J. (1963). Basis for derivation of matrices for the direct stiffness method, J. Am. Inst. Aeronaut. Astron. 1(7):1631–1637. Mindlin, R. D. (1951). Influence of rotatory inertia and shear on flexural motion of isotropic, elastic plates, J. Appl. Mech., 31–38. Mohr, G. A. (1992). Finite Elements for Solids, Fluids and Optimization, Oxford University Press, New York. Pinder, G. F., and Gray, W. G. (1977). Finite Element Simulation in Surface and Subsurface Hydrology, Academic Press, New York. Pironneau, O. (1989). Finite Element Methods for Fluids, John Wiley and Sons, Chichester. Oden, J. T. (1969). A general theory of finite elements: II. Applications, Int. J. Numer. Methods in Eng., 1(3). Reddy, J. N. (1993). An Introduction to the Finite Element Method, McGraw Hill, New York. Reddy, J. N. (1999). Theory and Analysis of Elastic Plates, Taylor and Francis, PA, USA. Reddy, J. N., and Gartling, D. K. (1994). The Finite Element Method in Heat Transfer and Fluid Dynamics, CRC Press, London. Rice, J. (1993). Numerical Methods, Software, and Analysis, Academic Press, New York. Segerlind, L. J. (1985). Applied Finite Element Analysis, John Wiley and Sons, New York. Stasa, F. L. (1985). Applied Finite Element Analysis for Engineers, CBS Publ. Japan, New York. Streeter, V. L., Wylie, E. B., and Bedford, K. W. (1998). Fluid Mechanics, WCB/ McGraw Hill, Boston. Sun, N. (1996). Mathematical Modeling of Groundwater Pollution, Springer Publishing House, New York. Taylor, C., Patil, B. S., and Zienkiewicz, O. C. (1969). Harbor Oscillation: A Numerical Treatment for Undamped Natural Modes, Proc. Inst. Civil Eng., 43:141–155. Timoshenko, S. P., and Woinowsky-Krieger, S. (1959). Theory of plates and shells, Second ed. McGraw Hill International ed., Singapore. Vreugdenhil, C. B. (1994). Numerical Methods for Shallow-Water Flow, Kluwer Academic Publishers, London. Wang, H., and Anderson, M. P. (1982). Introduction to Groundwater Modeling Finite Difference and Finite Element Methods, W. H. Freeman and Company, New York. White, R. E. (1985). An Introduction to the Finite Element Method with Applications to Nonlinear Problems, John Wiley and Sons, New York. Zheng, C., and Bennett, G. D. (1995). Applied Contaminant Transport Modeling-Theory and Practice, Van Nostrand Reinhold, New York, NY. Zienkiewicz, O. C. (1980). Finite Element Method, 3rd ed., McGraw Hill Book, New York. Zienkiewicz, O. C., and Taylor, R. L. (1989). The Finite Element Methods, 4th ed., Vol. 1: Basic Formulation and Linear Problems, Vol. 2: Solid and Fluid Mechanics; Dynamics and Non-linearity McGraw Hill Company Limited, London.

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Appendix A

REVIEW OF MATRIX ALGEBRA AND MATRIX CALCULUS A.1 REVIEW OF MATRIX ALGEBRA Finite element method can be viewed as a means of forming matrix equation that approximates behavior of a physical system. Hence, any finite element application results in forming, manipulating and solving matrix equations. Therefore, matrix algebra pertaining to finite element analysis is reviewed here. A.1.1 Definition of a Matrix A matrix is an m × n array of numbers arranged in m rows and n columns.

Size of [a] = m × n ai j → element of [a] corresponding to i th row and j th column. A.1.2 Types of Matrices 1. Column matrix (vector) – n = 1:

2. Row matrix (transposed vector) – m = 1:

3. Rectangular matrix (m ≠ n):

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Finite Element Method with Applications in Engineering 4. Square matrix (m = n):

5. Symmetric matrix:

Symmetric matrix is always a square matrix. 6. Diagonal matrix: It is a square matrix, where only the diagonal elements are non-zero, i.e., aij = 0, if i ≠ j.

7. Identity matrix (unit matrix): A diagonal matrix with ones on the main diagonal is called an identity matrix.

8. Transposed matrix: It is obtained by interchanging rows and columns.

Transpose of [a] = [a] = [b] = T

bij = aji Note: For a symmetric matrix [a] = [a]. T

9. Null matrix: All elements are zero in the null matrix.

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Finite Element Method with Applications in Engineering 10. Triangular matrix: It can be either an upper or lower triangular matrix. Upper triangular matrix:

Lower triangular matrix:

11. Banded matrix:

A.1.3 Equality of Matrices

Two matrices are said to be equal only when they are equal element by element. A.1.4 Addition or Subtraction of Matrices Matrices must be of the same size and elements must be added or subtracted term by term. or, in subscript (index) notation, it can be shown that cij = aij + bij = bij + aij

Let

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A.1.5 Scalar Multiplication

Let

Then A.1.6 Matrix Multiplication Two matrices [a] and [b] can be multiplied only if the number of columns in matrix [a] equals the number of rows in matrix [b].

c11 = 4(–2) + 3(0) + (–1)(1) = –9 c12 = 4(0) + 3(1) + (–1)(1) = 2 c21 = 2(–2) + (0) + 1(1) = –3 c22 = 2(0) + 0(1) + 1(1) = 1

Hence, [c] = In general, matrix multiplication is not commutative. i.e., [a][b] ≠ [b][a] A.1.7 Inverse of a Square Matrix

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There are several ways of computing an inverse of a matrix. When the size of matrix is small (m = n ≤ 4), inverse can be computed easily by employing cofactor method. Co-factor or adjoint method:

where [c] = cofactor matrix.

Mij is the minor of aij and is defined as the determinant of the matrix obtained by deleting the row and column belonging to aij from [a].

Inverse exists only if |a| ≠ 0 (non-singular matrix).

Then: c11= (-1) (1 ×1 – 1 × 0) = 1 2

c12 = (-1) (2 × 1 – 1 × 0) = –2 3

c13 = (-1) (2 ×1 – 1 × 1) = 1 4

c21 = (-1) (2 × 1 – 1 × 1) = –1 3

c22= (-1) (2 ×1 – 1 × 1) = 1 4

c23 = (-1) (2 ×1 – 1 × 2) = 0 5

c31 = (-1) (2 ×0 – 2 ×1) = –1 5

c32 = (-1) (2× 0 – 1 × 1) = 2 5

c33 = (-1) (2 ×1 – 2 × 2) = –2 6

Hence

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A.1.8 Some Useful Notes on Matrices 1. 2. 3. 4. 5. if λ is a scalar λ = λ 6. T

7. 8. If [k] is symmetric then [A] [k][A] is symmetric. 9. Proof: T

10. 11. 12. 13. If [a] is symmetric, [a] is also symmetric. 14. Proof: –1

15.

A.2 MATRIX CALCULUS Let the elements of [a] be functions of a parameter t. Then, [b] =

= [a],t implies bij=

and [b] =∫ [a]dt implies bij =∫aijdt

Example:

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A.3 SOLUTION OF LINEAR SIMULTANIOUS EQUATIONS An application of finite element method results in a set of simultaneous equations as shown below.

In matrix notation,

It is assumed that |K| ≠ 0 and { f } ≠ 0. Under these conditions, the above system is a nonhomogeneous equation system. A non-homogeneous equation system can be solved by using any of the following methods: 1. 2. 3. 4. 5. 6. 7.

Inversion of co-efficient matrix Gauss–elimination method Cramer's rule Gauss–Siedel method Gauss–Jordan method LU decomposition method Jacobi method

The first two methods will be discussed here. (a) Inversion of co-efficient matrix By pre-multiplying both sides of the equation [K]{ q} = {f } with the inverse of the coefficient matrix [K], following equation is obtained.

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The method of inversion of coefficient matrix can be used for hand calculations of 2 × 2 and 3×3 matrices. (b) Gauss-elimination method This method is widely used in many commercially available finite element packages. It is implemented in the teaching purpose finite element program SFEAP too. The method involves triangularization of the coefficient matrix [K] and evaluation of the unknowns {q} by back-substitution starting from the last equation. Steps 1. Augment the n × n coefficient matrix [Kij] with the right hand side vector {f} to form n × (n +1) matrix. 2. Inter-change rows if necessary to make the value of K11 the largest magnitude of any coefficient in the first column. 3. Create zeros in the second through nth rows in the first column by subtracting Ki1/K11 times the first row from the i-th row (i = 2, ………,n). 4. Repeat for each i = 2 to n − 1: 1. Inter-change rows if necessary to make Kii the largest magnitude in the i-th column by inter-changing rows (consider rows only i to n). 2. After completion of step 4.a, for each row j = i +1, … n,

At the completion of Step 4, the co-efficient matrix [Kij]n×n is transformed into an upper triangular matrix. Solve for qn from the nth equation as qn = Kn.n+1/knn. Solve for qi (i = n − 1, n – 2, .., 1) from n – 1 through the first equation in turn by using equation

For computer implementation of Gauss-elimination method, the above steps can be summarized as

Note: when i > n, summation is interpreted as zero. Example: Solve the equations given below by employing Gauss elimination method. 388

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Step 1: n = 3

Step 2: Not necessary to interchange rows. Steps 3 and 4:

Steps 5 and 6:

Summary of equations: Step 1: Choose ranges of i, j, l.

Step 2: Compute updated Kij : 389

Finite Element Method with Applications in Engineering For l = 1, i = 2, j = 1, 2, 3, 4:

For l = 1, i = 3, j = 1, 2, 3, 4:

For l = 2, i = 3, j = 2, 3, 4:

Step 3: Compute qi's i = n = 3:

i = n – 1 = 2:

390

Finite Element Method with Applications in Engineering i = n – 2= 1:

A FORTRAN–77 code for this method is given next. C

ROUTINE TO SOLVE [A]{X} = {C} BY USING GAUSS-ELIMINATION DIMENSION A(10,10), C(10) READ (5,*)N READ (5,*) ((A(I,J),J = 1,N),I = 1.N) READ (5,*)(C(I), I = 1.N) WRITE(6,5)

5

FORMAT(/1X,’MATRIX-A’/1X,10(‘=’)/) DO 10 I = 1.N

10

WRITE(6,15)(A(I,J), J = 1, N)

15

FORMAT(10F12.5) WRITE(6,25)(C(I),I = 1,N)

25

FORMAT(//1X,’VECTOR-C’/1X,10(‘=’)//(10F12.5))

C C

FORWARD ELIMINATION

C

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Finite Element Method with Applications in Engineering

DO 20 L= 1, N − 1 DO 20 I = L + 1,N TEMP = A(I,L)/A(L,L) DO 30 J = L+ 1,N 30

A(I,J) = A(I,J)- TEMP*A(L,J)

20

C(I) = C(I)-TEMP*C(L)

C C

BACK – SUBSTITUTION…

C C(N) = C(N)/A(N,N) DO 40 II = 1, N–1 I = N-II SUM = 0.0 DO 50 J = I + 1, N 50

SUM = SUM+A(I,J)*C(J)

40

C(I) = (C(I)-SUM)/A(I,I) WRITE(6,35)(C(I),I = 1,N)

35

FORMAT(//1X,’SOLUTION VECTOR’ /1X,15(‘=’)//(10F12.5)/) STOP

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END

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Finite Element Method with Applications in Engineering

Appendix B

ELEMENTS VARIATIONS

OF

CALCULUS

OF

B.1 GENERAL Governing equations in engineering are generally second (or fourth) order ordinary or partial differential equations. This requires the function to be continuous having derivatives up to order two (or four). Description of a problem is not complete until correct associated boundary conditions are specified. If there is a function of the functions (functional) for the governing equation, the governing equations as well as the appropriate boundary conditions can be derived through the technique of Calculus of Variations (CV). The functional requires the functions to be continuous having lower order derivatives. Rich history of the CV of the last 300 years can be found in the book by Goldstine (1980). One of the major advantages of CV is that it reduces the requirement of the order of derivatives. In the Finite Element Method, this advantage is exploited to derive a system of linear algebraic equations (designated as equilibrium equations) for an approximate solution. The stiffness matrix for a differential equation was derived using the variational approach in Example 4.1 of Chapter 4. However, it may be mentioned that if the differential equation is dispersive (dissipative) as in Example 4.2, no functional can be constructed. In such a case, resort is taken to one of the weighted residual approach. Galerkin's method is one of the widely used approaches, which is described in the Appendix C. The variational technique is briefly described here.

B.2 THE EULER–LAGRANGE EQUATION For a one-dimensional problem, consider functions whose inputs are functions and whose outputs are real number as:

where x is an independent variable, u(x) a dependent variable and a prime denotes differentiation with respect to x. The function u(x) is continuous having derivatives up to order two and satisfies the boundary conditions

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Finite Element Method with Applications in Engineering

Figure B.1 Admissible comparison paths: (a) Single path, and (b) Family of paths

A function like I [u(x)] is actually called a functional, this name is used to distinguish I [u(x)] from ordinary functions whose domain consist of ordinary variables. Function F(x; u, u′, u″) in the integral is to be viewed as an ordinary function of the variables x, u, u′ and u″. It is required to find u(x) such that I [u(x)] is stationary (minimum or maximum). Let u(x) be the true solution which makes I [u(x)] stationary and U(x) be a comparison or competing path as shown in Figure B.1(a). Then U(x) = u(x) + η(x) where the neighboring curve η(x) is continuous having a derivative up to order two and satisfies the boundary conditions

However, there are infinitely many comparison paths U1(x), U2(x), U3(x),…., etc. as shown in Figure B.1(b). This dilemma can be circumvented by introducing a real parameter α, such that

It can be noticed that Eq. (B.4) represents a family of curves, which includes u(x) when α = 0. Thus, a new problem can be posed as

Thus, the problem reduces to finding minimum or maximum (stationary) for a function of a real variable α. A necessary condition from the calculus of single variable to find maxima/minima is

After differentiating I(α) and setting α = 0 the sought maxima or minima can be recovered. Using the chain rule the differentiation of Eq. (B.5) is evaluated as

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Finite Element Method with Applications in Engineering

Noting that

and setting α = 0

The governing differential equation and consistent boundary conditions can be recovered by integrating by parts Eq. (B.8). Integrating by parts gives

Substituting Eq. (B.9) in Eq. (B.7), the resulting equation can be written as

1. Euler−Lagrange Equation:

Since η(x) is arbitrary, the integral in Eq. (B.10) yields the governing differential equation as

2. Boundary Conditions:

The boundary terms in Eq. (B.10) yield the consistent boundary conditions

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Finite Element Method with Applications in Engineering

Note that one term from each row has to be specified in the above boundary conditions. If F(x; u, u′) is not a function of u″, Euler−Lagrange equation is

The boundary conditions are

EXAMPLE B.1: In pure bending of a prismatic beam having bending rigidity EI the functional for the bending energy can be written as

where w is the lateral displacement of q is the distributed load. Find the differential equations and the boundary conditions.

Now written as

where w is the dependent variable. Eq. (B.11) can be

Noting

For boundary conditions it is necessary to specify,

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Finite Element Method with Applications in Engineering

B.3 THE VARIATIONAL OPERATOR The Euler−Lagrange equations were obtained by taking a competing path in the previous section. In this section, variational calculus is introduced, which is very useful in deriving formally the governing equations in the finite element analysis. In the rigid body dynamics, one encounters the variational operator, δ, when the work is described as δW = F. δr, where δW is the virtual work, F is the force vector and δr is the virtual displacement. The adjective virtual means not being so in actual fact but being so in essence or effect. Here, the variational operator will be used in a broader sense or generalized sense. The quantity u(x), u(x, y), or u(x, y, z) can be temperature, electric potential, potential flow distribution, displacement etc., and apply the δ operator to it. It may be noted that the δ variation can be applied only to the dependent variables and not to the independent variables. For example, the total variation for u(x, y) can be shown to be

Figure B.2 Differences between ∆h, dh and δh

In order to contrast the differential and variation, consider a one-dimension case of the function h(x) shown in Figure B.2. At point a the change in h(x), ∆h, resulting from a change in the independent variable x, ∆x = dx, the differential dh = h′(a)dx, and δh are shown in the figure. It can be seen that ∆h and δh are a very close approximation but not the same, δh stands by itself and does not depend on the independent variable x. The true path, u(x), is now varied by giving a small variation δu (refer to Figure B.1 (a) in which η(x) is replaced by δu(x)). From Eq. (B.4), U(x) = u(x) + αη(x) = u(x) + δu, which implies αη(x) = δu. In the following, a commutative property with respect to differentiation, of the δ operator is established.

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Finite Element Method with Applications in Engineering

The variational principle is now applied to functional I(u) where u is considered as being a function of two independent variables x and y.

Noting δx = 0 and δy = 0,

and using Green's identity the last term can be written as

where C is the contour of integration. For stationary value of I, δI = 0, which yields the equilibrium equation and boundary conditions as

Boundary conditions:

.

EXAMPLE B.2 Find the governing differential equation and boundary condition for the functional

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Finite Element Method with Applications in Engineering where

(a) Using Eq. (B.15)

(b) Solving formally:

Note f (x,y) is a function of independent variables only and, therefore, δf = 0.

But

The first integral yields the differential equation: applying Green's identity, yields:

and the second integral after .

REFERENCES AND FURTHER READING Finlayson, B. A. (1972). The Method of Weighted Residuals and Variational Principles, Academic Press, New York. Mikhlin, S. G. (1964). Variation Methods in Mathematical Physics, Macmillan Co., New York. Goldstine, H. H. (1980). The History of the Calculus of Variations from 17th through 19th Century, Springer−Verlag, New York.

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Finite Element Method with Applications in Engineering

Appendix C

EXAMPLE ILLUSTRATING GALERKIN'S METHOD

USE

OF

C.1 GENERAL Methods of weighted residual are used when differential equations (that describe the behavior of physical system) are known. Of all the weighted residual methods, Galerkin's method is employed extensively in the context of finite element analysis. The use of the method is demonstrated by employing an illustrative example.

C.2 EXAMPLE A physical phenomenon is described by the differential equation

such that 1. Derive a 2-node line element having length L, by employing Galerkin's method. Assume approximate solution of the form

where 2. Solve the differential equation by employing (i) two; (ii) three and (iii) four such elements in each case. Use equal length elements in each case. Solution: 1. For an i-th generic element having length L, the Galerkin's equations are given from Eq. (2.9) as

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Finite Element Method with Applications in Engineering

Therefore, from Eq. (C.1),

can be obtained. By substituting

and the expression for u into Eq. (C.2), following equations can be derived.

Then for m = 1,

Note that N1 = 1 at x = 0, N1 = 0 at x = L, and u′ = u′1 at x = 0. Hence, from the above equation,

can be formulated. Similarly, for m = 2, following equation can be derived.

Noting N2 = 0 at x = 0, N2 = 1 at x = L and u′ = u′2 at x = L, equation

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Finite Element Method with Applications in Engineering

can be obtained from Eq. (C.7). Therefore, from Eqs. (C.6) and (C.8),

2. (i) Solution of the differential equation by using two elements:

By assembling the element equations, the global equations take form

from which, 4.3334u2 = 0.25, or u2 = 0.0577. 403

Finite Element Method with Applications in Engineering (ii) Solution of the differential equation by using three elements:

By assembling element contributions and enforcing boundary conditions, the global equations

(iii) Solution of the differential equation by using four elements

404

Finite Element Method with Applications in Engineering

By assembling element contributions and enforcing boundary conditions, the global equation set takes form

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Finite Element Method with Applications in Engineering

Appendix D

REVIEW OF GAUSS QUADRATURE PROCEDURE FOR NUMERICAL INTEGRATION D.1 GENERAL It was shown in Chapter 7 that integrals, which may need to be evaluated numerically, are of the form

In finite element analysis, ‘Gauss Quadrature’ procedure is used extensively for numerical integration.

D.2 FUNDAMENTAL CONCEPT Consider a simple curve shown in Figure D.1. Application of trapezoidal rule is shown in Part (a) of the figure to evaluate area under the curve between the limits of a and b. The area of the trapezoid is given by

where w = b – a = width of the trapezoid. the above equation can be alternatively written as

By employing trapezoidal rule, the area under the curve between the limits a and b is approximated as the area of the trapezoid. However, this approximation may give rise to significant errors, especially when large trapezoids are considered. By using Gauss Quadrature, on the other hand, instead of choosing A and B at the end of interval of interest, two intermediate points C and D are chosen, as shown in Part (b) of Figure D.1. A trapezoid is then constructed by drawing a straight line through these points and extending it to the end of the interval. Thus, a part of the trapezoid lies outside the curve (upper corners in the figure), while part of the curve lies outside the trapezoid. By appropriately choosing points C and D, it is possible to balance the two areas so that the area in the trapezoid equals the area under the curve. The resulting approximation can then give exact integration. The Gauss Quadrature rule essentially consists of a simple way of choosing C and D to get accurate integration. The C and D are termed as Gauss points, and the integration is sought of the form given by Eq. (D.3). The W1 and W2 in the equation are commonly referred to as weights associated with Gauss points.

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Figure D.1 Two-point Gauss rule as derived from trapezoidal rule In a more general form, if integration I is required by employing m Gauss points, it can be shown that

where m is the number of Gauss points and Wk indicates weight with the k th Gauss point. A general m point Gauss Quadrature procedure is sketched in Figure D.2. The locations of Gauss points and associated weights, for different Gauss Quadrature, on the other hand are presented on Table D.1. Note that generally, Gauss Quadrature, which employs n Gauss points, is exact if the integration is polynomial of degree 2n – 1 or less.

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Finite Element Method with Applications in Engineering

Figure D.2 m point Gauss quadrature Table D.1 Table of Gauss points for integration from –1 to +1 Number of Gauss Points

Locations of Gauss Points, ξ

Associated Weights, W

1.

ξ1 = 0

2.00000 00000 00000

2.

ξ1,ξ2 = ±0.577350269189626

1.000

3.

ξ1,ξ3 = ±0.77459666924148

5/9 = 0.5555555555556

ξ2 = 0.000000000000000

8/9 = 0.8888888888889

ξ1,ξ4 = ±0.86113 63115 94053

0.34785 48451 37454

ξ2,ξ3 = ±0.33998 10435 84856

0.65214 51548 62546

ξ1,ξ5 = ±0.90617 98459 38664

0.23692 68550 56189

ξ2,ξ4 = ±0.53846 93101 05683

0.47862 86704 99366

ξ3 = ±0.00000 00000 00000

056888 88888 88889

ξ1,ξ6 = ±0.93246 95142 03152

0.17132 44923 79170

ξ2,ξ5 = ±0.66120 93864 66265

0.36076 15730 48139

4.

5.

6.

i

408

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Finite Element Method with Applications in Engineering

7.

8.

9.

10.

ξ3,ξ4 = ±0.23861 91860 83197

0.46791 39345 72691

ξ1,ξ7 = ±0.94910 79123 42759

0.12948 49661 68870

ξ2,ξ6 = ±0.74153 11855 99394

0.27970 53914 89277

ξ3,ξ5 = ±0.40584 51513 77397

0.38183 00505 05119

ξ4 = ±0.00000 00000 00000

0.41795 91836 73469

ξ1,ξ8 = ±0.96028 98564 97536

0.10122 85362 90376

ξ2,ξ7 = ±0.79666 64774 13627

0.22238 10344 53374

ξ3,ξ6 = ±0.52553 24099 16329

0.31370 66458 77887

ξ4,ξ5 = ±0.18343 46424 95650

0.36268 37833 78362

ξ1,ξ9 = ±0.96816 02395 07626

0.08127 43883 61574

ξ2,ξ8 = ±0.83603 11073 26636

0.18064 81606 94857

ξ3,ξ7 = ±0.61337 14327 00590

0.26061 06964 02935

ξ4,ξ6 = ±0.32425 34234 03809

0.31234 70770 40003

ξ5 = ±0.00000 00000 00000

0.33023 93550 01260

ξ1,ξ10 = ±0.97390 65285 17172

0.06667 13443 08688

ξ2,ξ9 = ±0.86506 33666 88985

0.14945 13491 50581

ξ3,ξ8 = ±0.67940 95682 99024

0.21908 63625 15982

ξ4,ξ7 = ±0.43339 53941 29247

0.26926 67193 09996

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Finite Element Method with Applications in Engineering

ξ5,ξ6 = ±0.14887 43389 81631

0.29552 42247 14753

Example Evaluate by using 1, 2 and 3 Gauss points. Solution: Exact value of I is

In order to use numerical integration, change of variable is required.

Therefore, for the given data,

Thus,

and

(i) 1– Gauss point (m = 1)

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Finite Element Method with Applications in Engineering

(ii) 2 – Gauss points (m = 2)

(iii) 3 – Gauss points (m = 3)

D.3 GAUSS QUADRATURE FOR TWO DIMENSION Gauss quadrature rule was summarized in the previous section for line integration. It is extended for area integration in this section. Consider

By applying Gauss Quadrature for line integration, it can be shown that

Here, m and n are, respectively, the number of Gauss points in the ξ and η directions; (ξk,η1) are the co-ordinates of Gauss points in the ξ–η plane; and Wk and Wl are Gauss weights in ξ and η directions, respectively. Depending upon the number of Gauss points in 411

Finite Element Method with Applications in Engineering the ξ and η directions, various Gauss quadrature rules can be established (refer to Figure D.3). (i) 1 × 1 Gauss Quadrature: From Eq. (D.6), for 1×1 Gauss Quadrature,

Figure D.3 Different Gauss quadrature rules for area integration (ii) 2 × 2 Gauss Quadrature: From Eq. (D.6) and Table D.1, the 2 × 2 Gauss Quadrature can be written as

(iii) 3 × 3 Gauss Quadrature: From Eq. (D.6) and Table D.1, the 3×3 Gauss Quadrature can be written as

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Appendix E

USER'S MANUAL FOR THE SIMPLIFIED FINITE ELEMENT ANALYSIS PROGRAM (SFEAP) E.1 PREAMBLE The simplified finite element analysis program (sfeap) can be used for static linear-elastic analysis of plane and space trusses, beam and plane frames, two- and three-dimensional problems and axisymmetric solids. There are eight element subroutines, which are listed in Table E.1. They are numbered as ELMT1 to ELMT8 for referencing. The program was coded using the theory presented in the text. A copy of the program in PDF format is given at the end of this appendix and the executable and the FORTRAN versions of the program are included in the CD. The first three subroutines employ one-dimensional element; ELMT1 and ELMT2 use truss elements while ELMT3 employs beam elements. The next two subroutines use twodimensional elements in rectangular Cartesian co-ordinates; ELMT4 uses 3-node constant strain triangles and ELMT5 employs three to nine nodes isoparametric elements. ELMT6 and ELMT7 use two-dimensional, three to nine node isoparametric elements in cylindrical (r, z) co-ordinates. The subroutine ELMT8 employs 8-noded, three-dimensional brick elements. Table E.1 List of Programs Program Number

Program

ELMT1

Plane truss element

ELMT2

3 - D truss element

ELMT3

Plane beam/frame element

ELMT4

CST element for plane stress/strain problem

ELMT5

3 - 9 variable nodes isoparametric plane element

ELMT6

3 - 9 variable nodes isoparametric element for axisymmetric analysis m=0

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Finite Element Method with Applications in Engineering

ELMT7

3 - 9 variable nodes isoparametric element for axisymmetric analysis m>0

ELMT8

8 node brick element for 3-D stress analysis

ELMT9

′Dummy subroutine′ for the user to write

ELMT10

′Dummy subroutine′ for the user to write

E.2 PROGRAM LAYOUT Program sfeap is organized as follows:

* SUBROUTINE DISTRL

(ICN=2) IS ACTIVE ONLY IN ELMT3, ELMT5, ELMT6 AND ELMT7.

** SUBROUTINE STRESS

(ICN=4) IS CALLED BY EVERY ELEMENT MODULE. AT PRESENT WE HAVE TO INPUT LS IN ELMT5, ELMT6 AND ELMT7.

***SUBROUINE RLOAD

IS CALLED BY EVERY ELEMENT MODULE TO CALCULATE EQUIVALENT NODAL FORCES DUE TO SINKING SUPPORTS. 415

Finite Element Method with Applications in Engineering

MAIN ROUTINE FUNCTION MAIN PROGRAM

Reads control information and evaluates the space for arrays.

PROCESS

Allocates Dynamic Memories to the arrays.

BANDW

Computes the half bandwidth.

LOAD

Read and write the concentrated nodal load data and compute the global load vector stemming from the concentrated loads.

DISTRL

Compute and assemble the equivalent, elemental load vectors due to the distributed loads (if any).

ASSEMB

Assemble element matrices.

BANSOL

Solve the global equation by using Gauss elimination method.

DISP OR DISP1

Print the computed nodal displacements (post-processing).

STRESS

Compute and print the ‘secondary’ unknowns i.e. compute element stresses/forces (post-processing).

E.2.1 Global Variables and Arrays Following FORTRAN variables and arrays are declared in the main program and process. INTEGER VARIABLES NUMNP

Total no. of nodes

NUMEL

Total no. of elements

NDF

No. of nodal DOF (degrees of freedom 1 / 2 / 3)

NDM

No. of dimension (1 / 2 / 3)

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Finite Element Method with Applications in Engineering

MNEL

Maximum no. of nodes per element

NUMAT

No. of material sets

NSN

No. of supported nodes

NPROP

No. of material data

ARRAYS

The FORTRAN program uses dynamic memory allocation. To achieve this feature following integers are defined in the Main Program and passed on to the program Process.

MST=MNEL*NDF NST=MST NEQ2=NDF*NUMNP N1=1 I2=1 N4=N1 N5=N1+NDM*NUMNP I3=I2+NUMEL I4=I3+NUMEL I6=I4+MNEL*NUMEL N8=N5+NPROP*NUMAT I7=I6+NDF*NUMNP N9=N8+NDF*NUMNP N10=N9+MST*MST N11=N10+NDM*MNEL 417

Finite Element Method with Applications in Engineering

N12=N11+NDF*MNEL N13=N12+MST N14=N13+NEQ2 I14=I7+NUMAT N15=N14+NEQ2*NEQ2 N16=N15+3*NUMEL NTOT=N16+NEQ2+1000 IATOT= I14

Integer Arrays

IA(I2)

= NELMAT(NUMEL)

Stores material set numbers of each element

IA(I3)

= NELND(NUMEL)

Stores no. of nodes per each element

IA(I4)

= NP(MNEL,NUMEL)

Stores element nodal connectivity data (i.e. global node numbers corresponding to the local node numbers of each element.)

IA(I6)

= ID(NDF,NUMNP)

Stores ID number or location number of each nodal DOF

IA(I7)

= MEL(NUMAT)

Stores element type number corresponding to each material set

IA(I14)

= NLD(NUMEL)

Stores the element numbers which have distributed loads

Real Arrays (in double precision) A(N1)

= X(NDM,NUMNP)

Stores global co-ordinates of all the nodes

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Finite Element Method with Applications in Engineering

A(N5)

= D(NPROP,NUMAT)

Stores material data for each material set

A(N8)

= U(NDF,NUMNP)

Stores global displacements of each node

A(N9)

= SL(MST*MST)

Stores element stiffness matrix [k ]

A(N10)

= XL(NDM,MNEL)

Stores global co-ordinates of nodes of an element

A(N11)

= UL(NDF,MNEL)

Stores elemental global displacement component {q }

A(N12)

= RL(MST)

Stores elemental level global load components {f }

A(N13)

= R(NEQ2)

Stores global load components {F}

A(N14)

= S(NEQ2,NEQ2)

Stores global stiffness matrix, [K], in the banded form

A(N15)

= STR(3,NUMEL)

Stores the intensities of the distributed loads at the nodes

e

e

e

E.2.2 Notes on Element Subroutines ELMTn All the element subroutines have following common structures:

SUBROUTINE ELMTn(DL,NEL,XL,SL,UL,RL,ICN) IMPLICIT REAL*8 (A-H, O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,N PROP COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST DIMENSION XL(NDM,1),SL(NST,1),DL(1),A(4)UL(1),RL(1) GO TO (1, 2, 3, 4), ICN 1. This section (ICN=1) is activated by the subroutine READG to read all material characteristics data that would be used at a later stage to compute stiffness matrix, stress etc. Data generated in this section may be stored in the array DL(NPROP,NUMAT). 2. This section (ICN=2) is activated by the subroutine DISTRL to calculate the consistent nodal load vector RL(MST) due to traction forces, if any, acting on the

419

Finite Element Method with Applications in Engineering surface of an element. The intensity of the distributed load is brought in by the array UL. 3. In this section (ICN=3), the element stiffness matrix SL(MST,MST) is generated by using the data available in the array DL and XL. 4. The stresses (or stress resultants) are calculated in this section (ICN=4) by using the data available in DL, XL, and UL. NOTES: Only the arrays DL, SL and RL are variables to be generated by this element subroutine. Other arrays and parameters in the argument list are already specified and must not be redefined by this subroutine. Array DL should also not be redefined for ICN > 1. 1. If a section is not required, it can be omitted by putting a RETURN statement in that section. Variables and arrays in the ELMTn subroutine Following FORTRAN variables and arrays are used in an element subroutine ELMT n.

NUMNP

No. of nodal points

NUMEL

No. of elements

NDF

No. of degrees of freedom per node

NDM

No. of spatial dimension

MNEL

Maximum no. of nodes per element

NUMAT

No. of material characteristic sets

NSN

No. of supported nodes

NEQ

Total no. of equations for the entire structure

MBAND

Half band width

NPROP

Maximum number of material characteristics data associated with the element type

MA

Material characteristic number

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Finite Element Method with Applications in Engineering

IELB

Element type number (in the LIBRARY)

IELG

Current element number (in the entire structure)

NST

Total number of DOF's in an element NST =NDF*NEL

NXSJ

This is a switch. Its value is set to one if negative Jacobian is encountered. Otherwise it is zero.

NEL

No. of nodes actually connected to the current element

ISWICH

This is a logical switch. If it is set. TRUE. in the main program, the Jacobian of each element will be printed (can be used to debug data.)

DL

Local array of material data. Information stored is used later to compute stiffness, stresses etc.

XL

Local array of global co-ordinates of element nodes

SL

Local array to store element stiffness matrix. This array is initialized to zero before the element subroutine is called.

UL

Local array of displacements of element nodes in global axes. This array is also used to pass on the distributed load data to the element subroutine.

RL

Local array of equivalent nodal load vector. This array is initialized to zero before the element subroutine is called.

ICN

Switch used to direct control to appropriate part of the element subroutine. Its value is automatically fixed depending upon the various level of the program.

E.2.3 Input Instructions Sample examples for each element along with the input data are given in the Example section followed by the listing of the FORTRAN source code. Following input data are required for the simplified finite element analysis (sfeap). Format to input the data are given in the parenthesis for each line. All the ‘integer’ input data are in format (I5) whereas data for all the ‘real’ variables are in format (E10.0). A.

TITLE (15A4)

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Finite Element Method with Applications in Engineering

Columns 1 − 60

B. CONTROL INFORMATION

Title to be printed with the output.

(815)

Column Variables 1–5 NUMNP

No. of nodes

6–10 NUMEL

No. of elements

11–15 NDF

No. of degrees of freedom per node

16–20 NDM

No. of dimensions

21–25 MNEL

Maximum no. of nodes per element

26–30 NUMAT

No. of material sets

31–35 NSN

No. of supported nodes (no. of nodes at which at least one DOF is prescribed.)

36–40 NPROP

No. of material data

C. CO-ORDINATES (one line per node) (2I5, 3E10.0)

(2I5, 3E10.0)

Columns 1–5

Node number

6–10

Generation increment

11–20

X1* coordinate

21–30

X2* coordinate 422

Finite Element Method with Applications in Engineering

31–40

X3* coordinate

*(X1, X2) = (X, Y) for programs ELMT1, ELMT3, ELMT4, ELMT5; (X1, X2, X3) = (X, Y, Z) for ELMT2 and ELMT8 and (X1, X2) = (R, Z) for ELMT6 and ELMT7. D. ELEMENT DATA (one line per node)

(13I5)

Columns 1–5

Element number

6–10

Generation increments for nodes (Element numbers will be increased by one.)

11–15

Material set number

16–20

No. of nodes in the element

21–25

Node – 1 number

26–30

Node – 2 number

31–35

Node – 3 number

36–40

Node – 4 number

41–45

Node – 5 number

46–50

Node – 6 number

51–55

Node – 7 number

56–60

Node – 8 number

61–65

Node – 9 number

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Finite Element Method with Applications in Engineering

E. BOUNDRY CONDITIONS (one line per restrained node)

(4I5)

Columns 1–5

Node number

6–10

DOF – 1 boundary code

11–15

DOF – 2 boundary code

16–20

DOF – 3 boundary code

Boundary code: 1 – restrained 0 – free F. NODAL DISPLACEMENT DATA (one line per node)

(I5,5x,3E10.0)

Columns 1–5

Node number

11–20

Displacement – 1

21–30

Displacement – 2

31–40

Displacement – 3

Terminate with a blank line Note that the input displacement is ignored if the corresponding boundary code is zero. G. MATERIAL PROPERTY (CHARACTERISTIC) DATA

(2I5)

Columns 1–5

Material set number

5–10

Element type number

424

Finite Element Method with Applications in Engineering Each material set input line must be followed immediately by the material property data required for the element type. Refer to the Input/output information for elements section. Any change in the material property data will require a new material set number. Thus, the total number of material sets, NUMAT, has to be properly estimated. H. NODAL LOAD DATA (one line per loaded node)

(I5,5x,3E10.0)

Columns 1–5

Node number

11–20

Force – 1

21–30

Force – 2

31–40

Force – 3

Terminate with a blank line. I. DISTRIBUTED LOAD DATA (one line per loaded element)

(I5,5x,3E10.0)

Columns 1–5

Node number

11–20

Intensity – 1

21–30

Intensity – 2

31–40

Intensity – 3

Terminate with a blank line.

At present, the program can handle only normal distributed load for the beam element of ELMT3, four node isoparametric elements of ELMT5, and four or higher than five nodes elements for ELMT6 and ELMT7. Variation of the load for ELMT5 is assumed to be linear acting on the edge ξ, of (ξ, η) or s = +1 of (s,t). For ELMT6 and ELMT7, intensities of the normal stress at the local nodes 2 and 3 (and also node 6 if the element is quadratic) are input. E.2.3.1 Values of the variables in the ELMT n subroutine: 425

Finite Element Method with Applications in Engineering

E.2.3.2 Input/output information for elements: ELMT1: Plane Truss Element MATERIAL DATA (one for each material set)

(2E10.0)

Columns 1–10E

Young's modulus

11–20 AR

Area of cross-section

OUTPUT

1. Displacements in global axes. 2. Axial load P (stress resultant) - tension if the sign is positive - compression if the sign is negative.

ELMT2: 3-D Truss Element MATERIAL DATA (one for each material set)

(2E10.0)

Columns 1–10 E

Young's modulus

426

Finite Element Method with Applications in Engineering

11–20 AR

OUTPUT

Area of cross-section

1. Displacements in global axes. 2. Axial load P (stress resultant) - tension if the sign is positive - compression if the sign is negative.

ELMT3: Plane Beam/Frame Element MATERIAL DATA (one for each material set)

(3E10.0,2I5)

Columns 1–10 E

Young's modulus

11–20 AR

Area of cross-section

21–30 AI

Second moment of area (moment of inertia)

31–35 IH

Release code* of hinge at end – 1

36–40 JH

Release code of hinge at end – 2

* code = 1 : Release; code = 0: Fixed 1. Deflections in global axes. [U1 V1 θ1 U2 V2 θ2]

T

2. Stress resultants in local axes [P1 V1 M1 P2 V2 M2]

T

427

Finite Element Method with Applications in Engineering

ELMT4: CST Element for Plane Stress/Strain Problem MATERIAL DATA material set)

(one

for

each

(3E10.0, I5)

Columns 1–10 E

Young's modulus

11–20 ANU

Poisson's ratio

21–30 TH

Thickness

31–35 KODE

KODE equal to zero for plane strain, 1 for plane stress

OUTPUT:

1. Displacements in the global axes. 2. Stresses at the center. Note that the stresses are constant throughout an element.

ELMT5: 3 – 9 Variable Nodes Isoparametric Plane Element MATERIAL DATA material set)

(one

for

each

(3E10.0,4 I5)

Columns 1–10 E

Young's modulus

11–20 ANU

Poisson's ratio

21–30 TH

Thickness

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Finite Element Method with Applications in Engineering

31–35 KODE

KODE equal to zero for plane strain, 1 for plane stress

36–40 L

Order of quadrature for stiffness

41–45 LS

No. of stress points in each direction

46–50 LLD

Order of quadrature for consistent loads

OUTPUT:

1. Displacements in the global axes. 2. Stresses at the Gauss points.

ELMT6: 3 − 9 Variable Nodes Isoparametric Element for Axisymmetric Analysis m = 0 MATERIAL DATA (one for each material set)

(2E10.0,3 I5)

Columns 1–10 E

Young's modulus

11–20 ANU

Poisson's ratio

21–25 L

Order of quadrature for stiffness

26–30 LS

No. of stress points in each direction

31–35 LLD

Order of quadrature for consistent loads

OUTPUT:

1. Displacements in the global axes. 2. Stresses at the Gauss points.

ELMT7: 3 − 9 Variable Nodes Isoparametric Element for Axisymmetric Analysis m > 0 429

Finite Element Method with Applications in Engineering

MATERIAL DATA (one for each material set)

(2E10.0,4 I5)

Columns 1–10 E

Young's modulus

11–20 ANU

Poisson's ratio

21–25 L

Order of quadrature for stiffness

26–30 LS

No. of stress points in each direction

31–35 LLD

Order of quadrature for consistent loads

36–40 KODE*

KODE is 1 for ‘core’ element, and zero otherwise.

* When KODE is 1, the program assumes that the nodes of a core element are numbered in such a way that the local nodes 1 and 4 (and also node 8 if applicable) are located on the z– axis. OUTPUT:

1. Displacements in the global axes. 2. Stresses at the Gauss points.

ELMT8: 8 Node Brick Element for 3 –H D Stress Analysis MATERIAL DATA (one for each material set)

(2E10.0)

Columns 1–10 E

Young's modulus

11–20 ANU

Poisson's ratio

430

Finite Element Method with Applications in Engineering

OUTPUT:

1. Displacements in the global axes. 2. Stresses at the Gauss points.

E.3.3 EXAMPLES Example 1

Analyze the plane truss shown in the figure above. The number in parentheses beside each member indicates the cross-sectional area in square centimeters (cm2). The truss is subjected to concentrated loads at joints E and F as shown. Moreover, the support at C sinks by 0.36 cm. Assume Young's modulus, E, for all members as 20,000 kN/cm2. The finite element representation of the problem is shown in the figure above. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. Input Data

431

Finite Element Method with Applications in Engineering

Example 2 Analyze the space truss shown in the figure below. The truss is subjected to a concentrated load at joint A as shown. Assume for all members, Young's modulus, E, as 200 GPa 5 200*106 kN / m2 and area, A, as 0.03 m2.

432

Finite Element Method with Applications in Engineering

The finite element representation of the problem is also shown in the figure. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. Input Data

Example 3(a) Analyze the plane frame shown in the figure below. The frame has simple supports at nodes 1 and 6. A load of P = 100 kN is acting at node 4. For each element (member) assume Young's modulus, E = 200 GPa, Area, A = 0.01 m2, and Moment of Inertia, I = 2 × 10–4 m4.

433

Finite Element Method with Applications in Engineering

The finite element representation of the problem is shown in the figure above. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. Input Data

Example 3(b) Analyze the plane frame shown in the figure below. The finite element representation of the problem is shown in the figure. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. The frame is fixed at nodes 1 and 3.

434

Finite Element Method with Applications in Engineering A load of 5 kN/m is acting on member 1. For each element (member) assume Young's modulus, E = 200 GPa, Area, A = 0.01 m2, and Moment of Inertia, I = 2.0×10–4 m4.

Input Data

Example 4: The figure below shows a 16 element model of a rectangular steel plate with one side fixed and the opposite side under uniformly distributed tensile load of 1,000 N/m. Assume Young's modulus, E as 20.0E + 06 N/m2, Poisson's ratio, v as 0.3 and thickness as 0.1 m. Perform plane stress analysis.

435

Finite Element Method with Applications in Engineering

The finite element representation of the problem is shown in the figure above. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. Input Data

436

Finite Element Method with Applications in Engineering Example 5: A cantilever beam of span 9 m is subjected to a tip load of 4 kN. Analyze the beam by employing 4-node elements under plane stress condition. Assume Young's modulus, E as 2,000 kN/m2, Poisson's ratio, v as 0.2 and width as 1 m.

The finite element representation of the problem is shown in the figure above. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. Input Data

437

Finite Element Method with Applications in Engineering Example 6: A cylinder, shown in the figure, is subjected to uniform external pressure σrr = 1 GPa. Assume Young's modulus E = 200 GPa, Poisson's ratio v = 0.2, and radii a = 0.5 m and b = 1.0 m. The finite element representation of the problem is also shown in the figure above. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers.

The problem is purely axisymmetric and involves only one harmonic, m = 0. The rigid body displacement in the z-direction is restrained at the nodes on the face z = 2. Input Data

438

Finite Element Method with Applications in Engineering

Example 7: A solid cylinder, shown in the figure, is subjected to external pressure of form where p0 = 1 GPa. Assume Young's modulus E =20 GPa, Poisson's ratio v = 0.2, and radii a = 0.0 m and b = 1.0 m.

439

Finite Element Method with Applications in Engineering The finite element representation of the problem is also shown in the figure above. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. The problem involves one harmonic, m = 1. The rigid body displacement in the z − direction is restrained at the nodes on the face z = 2. Note that one side of elements 1 and 3 are on z − axis. Input Data

Example 8: A cantilever beam of span 120 cm is subjected to an end load of 40 kN as shown in the figure below. Analyze the beam by employing 8 − node brick elements. The beam is 10 cm deep and 2 cm wide. Assume for all members, Young's modulus, E, as 20,000 kN / cm2 and Poisson's ratio v = 0.3.

440

Finite Element Method with Applications in Engineering

The finite element representation of the problem is also shown in the figure. Node numbers are indicated by Arabic numbers, while element numbers are denoted by encircled Arabic numbers. Input Data

441

Finite Element Method with Applications in Engineering

E.3.4 Listing of SFEAP C PROGRAM SFEAP C Simplified Finite Element Analysis Program (SFEAP) C C Program SFEAP has been developed jointly by C Drs. R. Paskaramoorthy, W. M. Karunasena, Yogesh Desai & C Prof. Arvind Shah, Dept. of Civil Eng. C corrected version January 8, 2000 C C Version02 completed on October 15, 2008. C Version02: In this version three variables, NNDS, NLN & NELDL 442

Finite Element Method with Applications in Engineering C are added to simplify making frames for Visual Basic.dotnet. C Also the output is split in two - output and results. C Trying to make ONE D arrays and no prior dimensioning. C Revised this program on October 27, 2008 in working condition. CC Version03: Added 8node brick element as ELMT08 Q

**************************************

C

*

*

C

*

SFEP: SIMPLIFIED FINITE ELEMENT PROGRAM FOR STATIC

C

*

ANALYSIS

C

*

CONTROL INFORMATION *

C

*

NUMNP = NO. OF NODES *

C

*

NUMEL = NO. OF ELEMENTS

C

*

NDF = NO. OF DEGREES OF FREEDOM *

C

*

NDM = NO. OF DIMENSIONS

C

*

MNEL = MAXM. NO. OF NODES/ELEMENT

C

*

NUMAT = NO. OF MATERIAL SETS

*

C

*

NSN = NO. OF SUPPORTED NODES

*

C

*

NPROP = NO .OF MATERIAL DATA

*

C

*

MST = MNEL*NDF

C

*

NEQ2 = NDF*NUMNP

C

*

*

C

*

ALL ARRAYS RESIDE IN THE ONE DIMENSIONAL ARRAY 'A' *

C

*

THEIR LOCATIONS ARE GIVEN BELOW *

C

*

A(NI) = X(NDM,NUMNP)

C

*

IA(I2) = NELMAT(NUMEL) *

C

*

IA(I3) = NELND(NUMEL)

C

*

IA(I4) = NP(MNEL,NUMEL) *

C

*

A(N5) = D(NPROP,NUMAT) *

C

*

IA(I6) = ID(NDF,NUMNP)

*

C

*

IA(I7) = MEL(NUMAT)

*

C

*

A(N8) = U(NDF,NUMNP)

*

C

*

A(N9) = SL(MST*MST)

*

C

*

A(N10) = XL(NDM,MNEL)

*

C

*

A(N11) = UL(NDF,MNEL)

*

*

* * *

* *

* *

443

*

Finite Element Method with Applications in Engineering C

*

A(N12) = RL(MST)

*

C

*

A(N13) = R(NEQ2) *

C

*

IA(I14) = NLD(NUMEL)

*

C

*

A(N14) = S(NEQ2,NEQ2)

*

C

*

A(N15) = STR(3,NUMEL)

*

C

*

ISWICH =.TRUE. THEN

*

C

*

ACTIVATES,PRINTING OF JACOBIAN OF EACH ELEMENT

C

*

corrected versión January 8, 2000 *

Q

*************************************************

*

IMPLICIT REAL*8 (A–H,O–Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK2/ISWICH COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST COMMON/BLK5/OM,NH COMMON/BLK6/NNDS, NLN, NELDL COMMON/BLK7/ N1,I2,N4,N5,I3,I4,I6,N8,I7,N9,N10, *N11,N12,N13,N14,I14,N15,N16,N17,NTOT,IATOT LOGICAL ISWICH CHARACTER*4 HED(15) ISWICH=.FALSE. NH=0 C C….READ & WRITE CONTROL INFORMATION C OPEN(5,FILE= ‘input.dat’) OPEN(6,FILE=‘output.dat’) OPEN(11,FILE=‘result.dat’) READ(5,2) HED,NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NPROP 2

FORMAT(15A4/9I5) WRITE(6,3) HED,NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NPROP

3 FORMAT(′′,15A4// 1 ‘ NO. OF NODES………………………………=’,I4/ 2 ‘ NO. OF ELEMENTS…………………………=’,I4/ 3 ‘ NO. OF DEGREES OF FREEDOM PER NODE…………………………=’,I4/ 444

Finite Element Method with Applications in Engineering 4 ‘ NO. OF DIMENSIONS…………………………=’,I4/ 5 ‘ MAXM. NO. OF NODES/ELEMENT………………………………..=’,I4/ 6 ‘ NO. OF MATERIAL SETS…………………………=’,I4/ 7 ‘ NO. OF SUPPORTED NODES…………………………=’,I4/ 8 ‘ NO. OF MATERIAL DATA…………………………=’,I4/ C C…. SET POINTERS FOR ALLOCATION OF DATA ARRAYS C MST=MNEL*NDF NST=MST C NEQ2=NDF*NUMNP C N1=1 I2=1 N4=N1 N5=N1+NDM*NUMNP I3=I2+NUMEL I4=I3+NUMEL I6=I4+MNEL*NUMEL N8=N5+NPROP*NUMAT I7=I6+NDF*NUMNP N9=N8+NDF*NUMNP N10=N9+MST*MST N11=N10+NDM*MNEL N12=N11+NDF*MNEL N13=N12+MST N14=N13+NEQ2 I14=I7+NUMAT N15=N14+NEQ2*NEQ2 N16=N15+3*NUMEL NTOT=N16+NEQ2+1000 IATOT= I14 CALL PROCESS CLOSE (5) 445

Finite Element Method with Applications in Engineering CLOSE (6) CLOSE(11) END C**************************************************************** SUBROUTINE PROCESS IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK2/ISWICH COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST COMMON/BLK5/OM,NH COMMON/BLK6/NNDS, NLN, NELDL COMMON/BLK7/ N1,I2,N4,N5,I3,I4,I6,N8,I7,N9,N10, *N11,N12,N13,N14,I14,N15,N16,N17,NTOT,IATOT LOGICAL ISWICH REAL*8, POINTER :: A(:) INTEGER*4, POINTER :: IA(:) INTEGER*4 ERROR ALLOCATE(A(NTOT),STAT=ERROR) IF (ERROR.NE.0) THEN PRINT*, “ERROR, A NOT ALLOCATED. ABORTING.” PRINT*, “ERROR=”,ERROR STOP END IF ALLOCATE(IA(IATOT),STAT=ERROR) IF (ERROR.NE.0) THEN PRINT*, “ERROR, IA NOT ALLOCATED. ABORTING.” PRINT*, “ERROR=”,ERROR STOP END IF ISWICH=.FALSE. CALL READG(A(N1),IA(I2),IA(I3),IA(I4),A(N5),IA(I6),IA(I7),A(N8), *A(N10),A(N9),A(N11),A(N12)) C

SUBROUTINE READG(X,NELMAT,NELND,NP,D,ID,MEL,U,XL,SL,UL,RL) CALL BANDW(IA(I3),IA(I4),MBAND) 446

Finite Element Method with Applications in Engineering C

SUBROUTINE BANDW(NELND,NP,MB) N14=N13+NEQ CALL LOAD(A(N13),IA(I6),A(N12))

C

SUBROUTINE LOAD(R,ID,RL) I14=I7+NUMAT N15=N14+NEQ*MBAND N16=N15+3*NUMEL CALL DISTRL(A(N1),IA(I2),IA(I3),IA(I4),A(N5),A(N9), *A(N10),A(N12),A(N13),IA(I6),IA(I7),IA(I14),A(N15))

C

SUBROUTINE DISTRL(X,NELMAT,NELND,NP,D,SL,XL,RL,R,ID,MEL,NLD,STR) N17=N14+NEQ*MBAND CALL ASSEMB(A(N1),IA(I2),IA(I3),IA(I4),A(N5),A(N9), *A(N10),A(N12),A(N14),A(N13),IA(I6),IA(I7),A(N11),A(N8))

C

SUBROUTINE ASSEMB(X,NELMAT,NELND,NP,D,SL,XL,RL,S,R,ID,MEL,UL,U) CALL BANSOL(A(N14),A(N13),NEQ,MBAND)

C

SUBROUTINE BANSOL(S,R,NEQ,MBAND) IF(NH.NE.1) CALL DISP(A(N13),A(N8),IA(I6))

C

SUBROUTINE DISP(R,U,ID)

C IF(NH.EQ.1) CALL DISP1(A(N13),A(N8),IA(I6),A(N1)) C

SUBROUTINE DISP1(R,U,ID,X)

C CALL STRESS(A(N1),IA(I2),IA(I3),IA(I4),A(N5),A(N8),A(N10), A(N11),IA(I7),A(N12),A(N9))

1

C SUBROUTINE STRESS(X,NELMAT,NELND,NP,D,U,XL,UL,MEL,RL,SL) END C****************************************************************** SUBROUTINE READG(X,NELMAT,NELND,NP,D,ID,MEL,U,XL,SL,UL,RL) C C…. SUBROUTINE TO GENERATE THE MISSING DATA C IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP DIMENSION X(NDM,1),NELMAT(1),NELND(1),NP(MNEL,1),XL(1),U(NDF,1), D(NPROP,1),ID(NDF,1),MEL(1),SL(1),UL(1),RL(1) 447

1

Finite Element Method with Applications in Engineering C C…. READ & WRITE NODE DATA C WRITE(6,280) 280 FORMAT(//10H NODE DATA// 1

5H NODE,11X,5H1-ORD,11X,5H2-ORD,11X,5H3-ORD)

C NOLD=0 NXOLD=0 10 READ (5,20) N,NX,(X(I,N),I=1,NDM) IF (NXOLD.EQ.0) GO TO 40 NUMINC=(N-NOLD)/NXOLD IF(NUMINC.LE.1) GO TO 40 XNUM=NUMINC NUMINC=NUMINC-1 DO 30 I=1,NDM NN=NOLD ST=(X(I,N)-X(I,NOLD))/XNUM DO 30 K=1,NUMINC NN=NN+NXOLD 30 X(I,NN)=X(I,NOLD)+K*ST 40 NOLD=N NXOLD=NX IF(N.EQ.NUMNP) GO TO 300 GO TO 10 20 FORMAT(2I5,3E10.0) C 300 DO 200 J=1,NUMNP WRITE(6,210) J,(X(I,J),I=1,NDM) 200 CONTINUE 210 FORMAT(I5,3F16.3) C C…. READ & WRITE ELEMENT DATA C WRITE(6,240) 448

Finite Element Method with Applications in Engineering MOLD=0 MXOLD=0 50 READ(5,60) M,MX,NELMAT(M),NEL,(NP(I,M),I=1,MNEL) NELND(M)=NEL IF(MXOLD.EQ.0) GO TO 70 NUMINC=(M-MOLD) IF(NUMINC.LE.1) GO TO 70 NUMINC=NUMINC-1 DO 80 K=1,NUMINC KK=K+MOLD NELMAT(KK)=NELMAT(MOLD) NELND(KK)=NELND(MOLD) DO 80 I=1,NEL 80 NP(I,KK)=NP(I,MOLD)+MXOLD*K 70 MOLD=M MXOLD=MX IF(M.EQ.NUMEL) GO TO 310 GO TO 50 60 FORMAT(13I5) C 310 DO 220 J=1,NUMEL NEL=NELND(J) WRITE(6,230) J,NELMAT(J),NELND(J),(NP(I,J),I=1,NEL) 220 CONTINUE 230 FORMAT(I5,I9,I5,1X,9(1X,I5)) 240 FORMAT(///' ELEMENT DATA'//' ELMT MATERIAL NEL NOD-1 NOD-2', *1X,'NOD-3 NOD-4 NOD-5 NOD-6 NOD-7 NOD-8 NOD-9') C C…. READ & WRITE BOUNDARY CONDITIONS C DO 90 I=1,NDF DO 90 J=1,NUMNP U(I,J)=0.0 90 ID(I,J)=0 WRITE(6,120) 449

Finite Element Method with Applications in Engineering DO 130 J=1,NSN READ(5,100) N,(ID(I,N),I=1,NDF) WRITE(6,110) N,(ID(I,N),I=1,NDF) 130 CONTINUE 110 FORMAT(I5,6I8) 100 FORMAT(7I5) 120 FORMAT(//' BOUNDARY CONDITION CODES'// 1 1X,'NODE',3X,'1-STF',3X,'2STF',3X,'3-STF') C C…. GENERATE ID NO FOR THE ENTIRE STRUCTURE C NEQ=0 DO 161 J=1,NUMNP DO 163 I=1,NDF IF(ID(I,J).EQ.0) GO TO 162 ID(I,J)=0 GO TO 163 162 NEQ=NEQ+1 ID(I,J)=NEQ 163 CONTINUE 161 CONTINUE C C…. READ & WRITE THE NODAL DISPLACEMENTS IF SPECIFIED C NN=0 18 READ(5,15) N,(UL(I),I=1,NDF) IF(N.EQ.0) GO TO 16 NN=NN+1 IF(NN.EQ.1) WRITE(6,21) WRITE(6,22) N,(UL(I),I=1,NDF) DO 17 I=1,NDF 17 U(I,N)=UL(I) GO TO 18 15 FORMAT(I5,5X,3E10.0) 22 FORMAT(I5,3E15.4) 450

Finite Element Method with Applications in Engineering 21 FORMAT(//' NODAL DISPLACEMENT DATA'// * 1X,'NODE',9X,'1-DISP',9X,'2-DISP',9X,'3-DISP') C C….READ & WRITE MATERIAL DATA C 16 DO 11 I=1,NPROP DO 11 J=1,NUMAT 11 D(I,J)=0.0 C DO 12 J=1,NUMAT READ(5,13) MA,IELB 13 FORMAT(2I5) MEL(MA)=IELB WRITE(6,14) MA,IELB 14 FORMAT(///' MATERIAL SET NUMBER',I5,9X,' ELMT LIB NUMBER',I5/) CALL ELMLIB(D(1,MA),NEL,XL,SL,UL,RL,1,IELB) 12 CONTINUE RETURN END C******************************************************************* SUBROUTINE BANDW(NELND,NP,MB) C C…. THIS SUBROUTINE CALCULATES THE HALF BAND WIDTH C IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP DIMENSION NELND(1),NP(MNEL,1) C MB=0 DO 10 J=1,NUMEL NEL=NELND(J) NLAR=NP(1,J) NSMA=NP(1,J) DO 20 I=1,NEL 451

Finite Element Method with Applications in Engineering IF(NSMA.GT.NP(I,J)) NSMA=NP(I,J) IF(NLAR.LT.NP(I,J)) NLAR=NP(I,J) 20 CONTINUE MM=(NLAR−NSMA+1)*NDF IF(MB.GT.MM) GO TO 10 MB=MM MEL=J 10 CONTINUE WRITE(6, 31) NEQ 31 FORMAT(/'NUMBER OF EQUATION, NEQ =' I5) WRITE(6,30) MB,MEL 30 FORMAT(/' HALF BAND WIDTH =',I5,5X,' ELMT NUMBER =',I5/) RETURN END C****************************************************************** SUBROUTINE LOAD(R,ID,RL) IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP DIMENSION R(1),RL(1),ID(NDF,1) C C…. INITIALIZE GLOBAL LOAD VECTOR C DO 10 I=1,NEQ 10 R(I)=0.0 C C…. READ & WRITE NODAL LOAD DATA AND FORM GLOBAL LOAD VECTOR C NN=0 40 READ(5,50) N,(RL(I),I=1,NDF) IF(N.EQ.0) GO TO 70 NN=NN+1 IF(NN.EQ.1) WRITE(6,20) WRITE(6,60) N,(RL(I),I=1,NDF) DO 80 I=1,NDF 452

Finite Element Method with Applications in Engineering II=ID(I,N) IF(II.EQ.0) GO TO 80 R(II)=R(II)+RL(I) 80 CONTINUE GO TO 40 50 FORMAT(I5,5X,3E10.0) 60 FORMAT(I5,6X,3E16.4) 20 FORMAT(//' NODAL LOAD DATA'// 1 5H NODE,16X,6H1-LOAD,10X,6H2-LOAD,10X,6H3-LOAD) C 70 RETURN END C**************************************************************** SUBROUTINE DISTRL(X,NELMAT,NELND,NP,D,SL,XL,RL,R,ID,MEL,NLD,STR) IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST COMMON/BLK4/ICOUNT DIMENSION X(NDM,1),NELMAT(1),NELND(1),NP(MNEL,1),D(NPROP,1), *SL(1),XL(NDM,1),ID(NDF,1),LM(27),RL(1),R(1),MEL(1),NLD(1),STR(3,1) C NN=0 200 READ(5,10) IELG,P1,P2,P3 10 FORMAT(I5,5X,3E10.0) IF(IELG.EQ.0) GO TO 100 NN=NN+1 IF(NN.EQ.1) WRITE(6,15) 15 FORMAT(/1X,'DISTRIBUTED LOAD'/ *1X,'ELMT',17X,'UDL-1',11X,'UDL-2',11X,'UDL-3') NLD(NN)=IELG STR(1,NN)=P1 STR(2,NN)=P2 STR(3,NN)=P3 WRITE(6,25) IELG,P1,P2,P3 453

Finite Element Method with Applications in Engineering 25 FORMAT(I5,6X,3E16.4) GO TO 200 C C…. EVALUATE AND ASSEMBLE THE CONSISTENT LOAD VECTOR C 100 IF(NN.EQ.0) RETURN DO 30 L=1,NN IELG=NLD(L) MA=NELMAT(IELG) IELB=MEL(MA) NEL=NELND(IELG) ICOUNT=L DO 40 J=1,NEL NG=NP(J,IELG) DO 40 I=1,NDM 40 XL(I,J)=X(I,NG) NST=NDF*NEL C C…. INITIALIZE RL C DO 50 I=1,NST 50 RL(I)=0.0 CALL ELMLIB(D(1,MA),NEL,XL,SL,STR(1,L),RL,2,IELB) C C…. FORM LOCATION MATRIX C DO 60 J=1,NEL N=NP(J,IELG) NL=NDF*(J-1) DO 60 K=1,NDF 60 LM(NL+K)=ID(K,N) C C…. ASSEMBLE C DO 70 I=1,NST 454

Finite Element Method with Applications in Engineering IF(LM(I).EQ.0) GO TO 70 II=LM(I) R(II)=R(II)+RL(I) 70 CONTINUE 30 CONTINUE RETURN END C******************************************************************* SUBROUTINE ASSEMB(X,NELMAT,NELND,NP,D,SL,XL,RL,S,R,ID,MEL,UL,U) IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST DIMENSION X(NDM,1),NELMAT(1),NELND(1),NP(MNEL,1),U(NDF,1), 1 D(NPROP,1),SL(1),XL(NDM,1),ID(NDF,1), 2 LM(54),S(NEQ,1),RL(1),R(1),MEL(1),UL(NDF,1) C C… INITIALIZE GLOBAL STIFFNESS MATRIX C DO 10 I=1,NEQ DO 10 J=1,MBAND 10 S(I,J)=0.0 NXSJ=0.0 C C DO 20 L=1,NUMEL MA=NELMAT(L) IELB=MEL(MA) IELG=L NEL =NELND(L) DO 70 J=1,NEL NG=NP(J,L) DO 71 I=1,NDM 71 XL(I,J)=X(I,NG) DO 72 I=1,NDF 72 UL(I,J)=U(I,NG) 455

Finite Element Method with Applications in Engineering 70 CONTINUE NST=NDF*NEL NST2=NST*NST C C….INITIALIZE RL & SL C DO 90 I=1,NST 90 RL(I)=0.0 DO 40 I=1,NST2 40 SL(I)=0.0 CALL ELMLIB(D(1,MA),NEL,XL,SL,UL,RL,3,IELB) C C…. FORM LOCATION MATRIX C DO 30 J=1,NEL N=NP(J,L) NL=NDF*(J-1) DO 30 K=1,NDF 30 LM(NL+K)=ID(K,N) C C…. ASSEMBLE C DO 50 I=1,NST IF(LM(I).EQ.0) GO TO 50 II=LM(I) R(II)=R(II)+RL(I) DO 60 J=1,NST JJ=LM(J)-II+1 IF(JJ.LE.0) GO TO 60 K=NST*(J-1)+I S(II,JJ)=S(II,JJ)+SL(K) 60 CONTINUE 50 CONTINUE 20 CONTINUE IF(NXSJ.EQ.0) GO TO 45 456

Finite Element Method with Applications in Engineering WRITE(6,55) 55 FORMAT(//' ****PROG TERMINATED: NEG JACOBIAN ENCOUNTERED****'//) STOP 45 RETURN END C******************************************************************* SUBROUTINE BANSOL(S,R,NEQ,MBAND) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION S(NEQ,1),R(NEQ) C C…. REDUCE STIFFNESS MATRIX C DO 60 N=1,NEQ IF (S(N,1).GT.1.0D-40) GO TO 30 WRITE (6,20) N 20 FORMAT(///30H ZERO ON DIAGONAL FOR EQUATION,I4) STOP 30 DO 50 M=2,MBAND FACT=S(N,M)/S(N,1) I=N+M-1 IF (I.GT.NEQ) GO TO 50 J=0 DO 40 K=M,MBAND J=J+1 40 S(I,J)=S(I,J)-FACT*S(N,K) 50 S(N,M)=FACT 60 CONTINUE C C…. REDUCE LOAD VECTOR C 70 DO 90 N=1,NEQ DO 80 M=2,MBAND I=N+M-1 IF (I.GT.NEQ) GO TO 90 80 R(I)=R(I)-S(N,M)*R(N) 457

Finite Element Method with Applications in Engineering 90 R(N)=R(N)/S(N,1) C C…. BACK SUBSTITUTE C N=NEQ 100 N=N-1 IF (N.EQ.0) GO TO 120 DO 110 M=2,MBAND I=N+M-1 IF (I.GT.NEQ) GO TO 110 R(N)=R(N)-S(N,M)*R(I) 110 CONTINUE GO TO 100 C 120 RETURN END C*****7*********************************************************** SUBROUTINE DISP(R,U,ID) IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK2/ISWICH COMMON/BLK5/OM,NH LOGICAL ISWICH DIMENSION R(1),U(NDF,1),ID(NDF,1) ISWICH=.FALSE. C C…. WRITE NODE DISPLACEMENTS C WRITE (11,20) 20 FORMAT(//19H NODE DISPLACEMENTS// 1

5H NODE,16X,6H1-DISP,10X,6H2-DISP,10X,6H3-DISP)

C DO 10 J=1,NUMNP DO 30 I=1,NDF 458

Finite Element Method with Applications in Engineering II=ID(I,J) IF(II.NE.0) U(I,J)=R(II) 30 CONTINUE WRITE(11,40) J,(U(I,J),I=1,NDF) 10

CONTINUE

40

FORMAT(I5,6X,3E16.4) RETURN END

C******************************************************************* SUBROUTINE DISP1(R,U,ID,X) IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK5/OM,NH DIMENSION R(1),U(NDF,1),ID(NDF,1),X(NDM,1) C C…. WRITE NODE DISPLACEMENTS C WRITE (11,20) NH 20 FORMAT(//19H NODE DISPLACEMENTS,5X,5X,'HARMONIC NUMBER =',I3// 1

5H NODE,16X,6HR-DISP,10X,6HZ-DISP,10X,6HT-DISP)

FACT=DSQRT(2.0D00) DO 10 J=1,NUMNP DO 30 I=1,NDF II=ID(I,J) IF(II.EQ.0) GO TO 50 U(I,J)=R(II) GO TO 30 50 IF(DABS(X(1,J)).GT.1.0D-03) GO TO 30 IF(I.NE.3) GO TO 30 U(1,J)=U(1,J)/FACT U(3,J)=U(1,J) 30 CONTINUE WRITE(11,40) J,(U(I,J),I=1,NDF) 10 CONTINUE 459

Finite Element Method with Applications in Engineering 40 FORMAT(I5,6X,3E16.4) RETURN END C******************************************************************* SUBROUTINE STRESS(X,NELMAT,NELND,NP,D,U,XL,UL,MEL,RL,SL) IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST DIMENSION X(NDM,1),NELMAT(1),NELND(1),NP(MNEL,1), 1 D(NPROP,1),U(NDF,1),XL(NDM,1),UL(NDF,1),MEL(1),RL(1),SL(1) C DO 20 L=1,NUMEL MA=NELMAT(L) IELB=MEL(MA) IELG=L NEL=NELND(L) DO 70 J=1,NEL NG=NP(J,L) DO 90 I=1,NDM 90 XL(I,J)=X(I,NG) DO 70 K=1,NDF 70 UL(K,J)=U(K,NG) C CALL ELMLIB(D(1,MA),NEL,XL,SL,UL,RL,4,IELB) 20 CONTINUE RETURN END C******************************************************************* SUBROUTINE ELMLIB(DL,NEL,XL,SL,UL,RL,ICN,IELB) C C…. THIS SUBROUTINE CALLS THE APPROPRIATE ELEMENT C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION DL(1),XL(1),SL(1),UL(1),RL(1) 460

Finite Element Method with Applications in Engineering C GO TO (1,2,3,4,5,6,7,8),IELB 1 CALL ELMT1(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 2 CALL ELMT2(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 3 CALL ELMT3(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 4 CALL ELMT4(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 5 CALL ELMT5(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 6 CALL ELMT6(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 7 CALL ELMT7(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 8 CALL ELMT8(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 9 CALL ELMT9(DL,NEL,XL,SL,UL,RL,ICN) GO TO 100 10 CALL ELMT10(DL,NEL,XL,SL,UL,RL,ICN) 100 RETURN END C******************************************************************* SUBROUTINE ELMT1(DL,NEL,XL,SL,UL,RL,ICN) C C…. PLANE TRUSS ELEMENT C IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST DIMENSION XL(NDM,1),SL(NST,1),DL(1),A(4),UL(1),RL(1) C GO TO (1,2,3,4),ICN 461

Finite Element Method with Applications in Engineering C C…. READ MATERIAL PROPERTIES C 1

READ(5,10) E,AR WRITE(6,15)E,AR DL(1)=E DL(2)=AR

10

FORMAT(2E10.0)

15

FORMAT(′ YOUNGS MODULAS ……… =′,E13.4/

*

′ AREA OF CROSS SECTION .. =′,E13.4) RETURN

2

RETURN

C C…. FORM ELEMENT STIFFNESS MATRIX C 3

DX=XL(1,2)-XL(1,1) DY=XL(2,2)-XL(2,1) XXL=DSQRT(DX**2+DY**2) A(1)=-DX/XXL A(2)=-DY/XXL A(3)=-A(1) A(4)=-A(2) EAL=DL(1)*DL(2)/XXL

C DO 30 I=1,NST DO 30 J=I,NST SL(I,J)=A(I)*A(J)*EAL 30

SL(J,I)=SL(I,J) CALL RLOAD(SL,UL,NST,RL) RETURN

C C…. EVALUATE STRESS RESULTANTS C 4

IF(IELG.EQ.1) WRITE(11,40) DX=XL(1,2)-XL(1,1) 462

Finite Element Method with Applications in Engineering DY=XL(2,2)-XL(2,1) XXL=DSQRT(DX*DX+DY*DY) COS=DX/XXL SIN=DY/XXL DU=UL(3)-UL(1) DV=UL(4)-UL(2) EAL=DL(1)*DL(2)/XXL P=EAL*(COS*DU+SIN*DV) WRITE(11,50) IELG,P 50 FORMAT(I5,6X,E16.4) 40 FORMAT(/// ′ ELEMENT FORCES′// *1X, ′ELMT′,17X, ′FORCE′) RETURN END C******************************************************************* SUBROUTINE RLOAD(SL,UL,NST,RL) C C…. FORM RHS VECTOR FOR SUPPORT SETTLEMENTS C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION SL(NST,1),UL(1),RL(1) C DO 110 J=1,NST IF(DABS(UL(J)).LT.1.0D-10) GO TO 110 DO 115 I=1,NST 115 RL(I)=RL(I)-SL(I,J)*UL(J) 110 CONTINUE RETURN END C*****7************************************************************* SUBROUTINE ELMT2(DL,NEL,XL,SL,UL,RL,ICN) C C…. 3–D TRUSS ELEMENT C IMPLICIT REAL*8 (A−H,O−Z) 463

Finite Element Method with Applications in Engineering

COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST DIMENSION XL(NDM,1),SL(NST,1),DL(1),A(6),UL(1),RL(1) C GO TO (1,2,3,4),ICN C C…. READ MATERIAL PROPERTIES C 1 READ(5,10) E,AR WRITE(6,15)E,AR DL(1)=E DL(2)=AR 10 FORMAT(2E10.0) 15 FORMAT (′YOUNGS MODULAS ……… =′,E13.4/ *

′ AREA OF CROSS .. =′,E13.4/ RETURN

2 RETURN C C…. FORM ELEMENT STIFFNESS MATRIX C 3 DX=XL(1,2)-XL(1,1) DY=XL(2,2)-XL(2,1) DZ=XL(3,2)-XL(3,1) XXL=DSQRT(DX**2+DY**2+DZ**2) A(1)=-DX/XXL A(2)=-DY/XXL A(3)=-DZ/XXL A(4)=-A(1) A(5)=-A(2) A(6)=-A(3) EAL=DL(1)*DL(2)/XXL C DO 30 I=1,NST DO 30 J=I,NST 464

Finite Element Method with Applications in Engineering SL(I,J)=A(I)*A(J)*EAL 30 SL(J,I)=SL(I,J) CALL RLOAD(SL,UL,NST,RL) RETURN C C…. EVALUATE STRESS RESULTANTS C 4 IF(IELG.EQ.1) WRITE(11,40) DX=XL(1,2)-XL(1,1) DY=XL(2,2)-XL(2,1) DZ=XL(3,2)-XL(3,1) XXL=DSQRT(DX**2+DY**2+DZ**2) C1=DX/XXL C2=DY/XXL C3=DZ/xxl DU=UL(4)-UL(1) DV=UL(5)-UL(2) DW=UL(6)-UL(3) EAL=DL(1)*DL(2)/XXL P=EAL*(C1*DU+C2*DV+C3*DW) WRITE(11,50) IELG,P 50 FORMAT(I5,6X,E16.4) 40 FORMAT(/// ′ ELEMENT RETURN END C******************************************************************* SUBROUTINE ELMT3(DL,NEL,XL,SL,UL,RL,ICN) C C…. PLANE BEAM ELEMENT C IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST COMMON/BLK4/ICOUNT 465

Finite Element Method with Applications in Engineering DIMENSION XL(NDM,1),SL(NST,1),DL(1),UL(1),RL(1),A(3,6),EK(3,3), 1 EKAA(3,6),FEF(4) C GO TO (1,2,3,4),ICN C C…. READ & WRITE MATERIAL PROPERTIES C 1 READ(5,10) E,AR,AI,IH,JH WRITE(6,15) E,AR,AI,IH,JH DL(1)=E*AR DL(2)=E*AI DL(3)=IH DL(4)=JH 10 FORMAT(3E10.0,2I5) 15 FORMAT (′ YOUNGS MODULAS………. = ′,E13.4/ *

′ AREA OF CROSS SECTION … =′,E13.4/

*

′ SECOND MOMENT OF AREA … =′,E13.4/

*

′ HINGE AT I-END ………. =′,I13/

*

′ HINGE AT J-END ………. =′,I13)

RETURN C C…. EVALUATE THE CONSISTENT LOAD VECTOR C 2 IF(ICOUNT.EQ.1) WRITE(6,105) 105 FORMAT(///1X, ′FIXED END FORCES′// *1X, ′ELMT′,9X, ′SF-1′,9X, ′BM-1′,9X, ′SF-2′,9X, ′BM-2′) P1=UL(1) P2=UL(2) DX=XL(1,2)-XL(1,1) XXL=DABS(DX) RL(2)= (7.0*P1+3.0*P2)*XXL/20.0 RL(5)= (3.0*P1+7.0*P2)*XXL/20.0 RL(3)= (P1/20.0+P2/30.0)*XXL*XXL RL(6)=-(P1/30.0+P2/20.0)*XXL*XXL FEF(1)=-RL(2) 466

Finite Element Method with Applications in Engineering FEF(2)=-RL(3) FEF(3)=-RL(5) FEF(4)=-RL(6) WRITE(6,115) IELG,(FEF(I),I=1,4) 115 FORMAT(I5,4E13.4) cosA = DX/XXL sinA = DY/XXL RL(1) = -RL(2)*sinA RL(2) = RL(2)*cosA RL(4) = -RL(5)*sinA RL(5) = RL(5)*cosA RETURN C C…. FORM ELEMENT STIFFNESS MATRIX C 3 DX=XL(1,2)-XL(1,1) DY=XL(2,2)-XL(2,1) XXL=DSQRT(DX*DX+DY*DY) COSA=DX/XXL SINA=DY/XXL EAL=DL(1)/XXL EIL=DL(2)/XXL EK(1,1)=EAL EK(1,2)=0.0 EK(1,3)=0.0 EK(2,1)=0.0 EK(2,2)=4.0*EIL EK(2,3)=2.0*EIL EK(3,1)=0.0 EK(3,2)=EK(2,3) EK(3,3)=EK(2,2) C IH=DL(3) JH=DL(4) CALL TRAN(A,IH,JH,COSA,SINA,XXL) 467

Finite Element Method with Applications in Engineering DO 30 I=1,3 DO 30 J=1,6 TEMP=0.0 DO 35 K=1,3 35 TEMP=TEMP+EK(I,K)*A(K,J) 30 EKAA(I,J)=TEMP IF(ICN.EQ.4) GO TO 60 C DO 50 I=1,6 DO 50 J=1,6 DO 50 K=1,3 50 SL(I,J)=SL(I,J)+A(K,I)*EKAA(K,J) CALL RLOAD(SL,UL,NST,RL) RETURN C C…. EVALUATE STRESSES C 4 IF(IELG.EQ.1) WRITE(11,100) GO TO 3 60 CALL TRAN(A,IH,JH,1.0D00,0.0D00,XXL) DO 70 I=1,6 DO 70 J=1,6 SL(I,J)=0.0 DO 70 K=1,3 70 SL(I,J)=SL(I,J)+A(K,I)*EKAA(K,J) C DO 80 I=1,6 RL(I)=0.0 DO 80 J=1,6 80 RL(I)=RL(I)+SL(I,J)*UL(J) WRITE(11,90) IELG,(RL(I),I=1,6) 90 FORMAT(I5,5X, ′1′,3E16.4/10X, ′2′,3E16.4) 100 FORMAT(/// ′ ELEMENT FORCES ′// *1X, ′ELMT′,2X, ′NODE ′,11X, ′AXIAL′,11X, ′SHEAR′,9X, ′BENDING′/ *

22X, ′FORCE′,11X, ′FORCE′,9X,′ MOMENT′/) 468

Finite Element Method with Applications in Engineering RETURN END C**************************************************************** SUBROUTINE TRAN(A,IHI,IHJ,COSA,SINA,XL) C C…. FORM ELEMENT TRANSFORMATION MATRIX C IMPLICIT REAL*8(A-H,O-Z) DIMENSION A(3,6) C C…. NO HINGES C A(1,1)= −COSA A(1,2)= −SINA A(1,3)=0.0 A(1,4)= COSA A(1,5)= SINA A(1,6)= 0.0 A(2,1)= −SINA/XL A(2,2)= COSA/XL A(2,3)= 1.0 A(2,4)= SINA/XL A(2,5)= −COSA/XL A(2,6)= 0.0 A(3,1)= −SINA/XL A(3,2)= COSA/XL A(3,3)= 0.0 A(3,4)= SINA/XL A(3,5)= −COSA/XL A(3,6)= 1.0 C IF (IHI.EQ.0.AND.IHJ.EQ.0) GO TO 60 IF (IHI.NE.0.AND.IHJ.EQ.0) GO TO 20 IF (IHI.EQ.0.AND.IHJ.NE.0) GO TO 40 C 469

Finite Element Method with Applications in Engineering C…. HINGES AT I AND J ENDS C DO 10 I=2,3 DO 10 J=1,6 10 A(I,J) =0.0 GO TO 60 C C…. HINGE AT I END ONLY C 20 DO 30 J=1,6 30 A(2,J) =-A(3,J)/2 GO TO 60 C C…. HINGE AT J END ONLY C 40 DO 50 J=1,6 50 A(3,J) =-A(2,J)/2 60 RETURN END C*****7************************************************************* SUBROUTINE ELMT4(DL,NEL,XL,SL,UL,RL,ICN) C C…. CST ELEMENT FOR PLANE STRESS/STRAIN PROBLEM C IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK2/ISWICH COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST LOGICAL ISWICH DIMENSION XL(NDM,1),SL(NST,1),DL(1),RL(1),UL(1),EPS(3),SIG(3), 1

A(3),B(3)

ISWICH=.FALSE. C GO TO (1,2,3,4),ICN 470

Finite Element Method with Applications in Engineering C C…. READ & WRITE MATERIAL PROPERTIES C 1 READ(5,10) E,ANU,TH,KODE 10 FORMAT(3E10.0,I5) WRITE(6,11) E,ANU,TH,KODE 11 FORMAT (′ YOUNGS MODULAS ………………………. = ′,E13.4/ *

′ POISSONS RATIO ………………………. =′,E13.4/

*

′ THICKNESS …………………………… =′,E13.4/

*

′ KODE (1 FOR PL STRESS, ZERO FOR PL STRAIN) =′,I13/)

DL(1)=E DL(2)=ANU DL(3)=TH DL(4)=KODE RETURN 2 RETURN C C…. FORM THE ELEMENT STIFFNESS MATRIX C 3 IF(ISWICH) WRITE(6,210) IELG 210 FORMAT(// ′ ELEMENT NUMBER =′,I3) C C…. IF KODE=0, THEN PLANE STRAIN. OTHERWISE PLANE STRESS. C KODE=DL(4) I=KODE IF(I.NE.0) I=1 IF(I.EQ.0) I=2 D1=DL(1)*(1.+(1-I)*DL(2))/(1.0+DL(2))/(1.0-I*DL(2)) D2=D1*DL(2)/(1.+(1-I)*DL(2)) D3=DL(1)*0.5/(1.+DL(2)) C CALL AB(XL,A,B,AR) IF(ISWICH) WRITE(6,420) AR 420 FORMAT(′ AREA = ′,F10.5) 471

Finite Element Method with Applications in Engineering IF(AR.GT.1.0D-05) GO TO 25 NXSJ=1 WRITE(6,26) IELG 26 FORMAT(′ NEGATIVE AREA ENCOUNTERED FOR ELMT ′,I4) 25 DV=DL(3)*0.25/AR D11=D1*DV D12=D2*DV D33=D3*DV C C…. FOR EACH J NODE COMPUTE DB=D*BJ C DO 50 J=1,NEL DB11=D11*B(J) DB12=D12*A(J) DB21=D12*B(J) DB22=D11*A(J) DB31=D33*A(J) DB32=D33*B(J) C C…. FOR EACH I NODE COMPUTE S=BI*DB C DO 60 I=1,J SL(I+I-1,J+J-1)=SL(I+I-1,J+J-1)+DB11*B(I)+DB31*A(I) SL(I+I-1,J+J )=SL(I+I-1,J+J )+DB12*B(I)+DB32*A(I) SL(I+I ,J+J-1 )=SL(I+I ,J+J-1)+DB21*A(I)+DB31*B(I) SL(I+I ,J+J )=SL(I+I ,J+J )+DB22*A(I)+DB32*B(I) 60 CONTINUE 50 CONTINUE C C…. COMPUTE LOWER TRIANGULAR PART BY SYMMETRY C DO 70 I=2,NST DO 70 J=1,I 70 SL(I,J)=SL(J,I) CALL RLOAD(SL,UL,NST,RL) 472

Finite Element Method with Applications in Engineering RETURN C C…. EVALUATE THE STRESSES C 4 IF(IELG.EQ.1) WRITE(11,100) I=DL(4) IF(I.NE.0) I=1 IF(I.EQ.0) I=2 D1=DL(1)*(1.+(1-I)*DL(2))/(1.0+DL(2))/(1.0-I*DL(2)) D2=D1*DL(2)/(1.+(1-I)*DL(2)) D3=DL(1)*0.5/(1.+DL(2)) CALL AB(XL,A,B,AR) C DO 13 I=1,3 13 EPS(I)=0.0 XX=(XL(1,1)+XL(1,2)+XL(1,3))/3.0 YY=(XL(2,1)+XL(2,2)+XL(2,3))/3.0 C DO 14 J=1,NEL EPS(1)=EPS(1)+B(J)*UL(J+J-1) EPS(2)=EPS(2)+A(J)*UL(J+J) 14 EPS(3)=EPS(3)+A(J)*UL(J+J-1)+B(J)*UL(J+J) C SIG(1)=(D1*EPS(1)+D2*EPS(2))/(2.0*AR) SIG(2)=(D2*EPS(1)+D1*EPS(2))/(2.0*AR) SIG(3)=D3*EPS(3)/(2.0*AR) WRITE(11,15) IELG,XX,YY,(SIG(I),I=1,3) 12 CONTINUE 15 FORMAT(I5,5E12.4) 100 FORMAT(/// ′ S T R E S S ′// 1 1X, ′ELMT ′,7X, ′1-ORD ′,7X, ′2-ORD ′,3X, ′XX-STRESS ′,3X, 2 ′YY-STRESS,3X, ′XY-STRESS ′) RETURN END C******************************************************************* 473

Finite Element Method with Applications in Engineering SUBROUTINE AB(XL,A,B,AR) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION A(3),B(3),XL(2,3) C A(1)=XL(1,3)-XL(1,2) A(2)=XL(1,1)-XL(1,3) A(3)=XL(1,2)-XL(1,1) C B(1)=XL(2,2)-XL(2,3) B(2)=XL(2,3)-XL(2,1) B(3)=XL(2,1)-XL(2,2) AR=0.5*(A(1)*B(3)-A(3)*B(1)) RETURN END C*****7************************************************************* SUBROUTINE ELMT5(DL,NEL,XL,SL,UL,RL,ICN) C C…. 3-9 VARIABLE NODED ISOPARAMETRIC PLANE ELEMENT C IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK2/ISWICH COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST LOGICAL ISWICH DIMENSION XL(NDM,1),SL(NST,1),DL(1),ZI(9),ET(9),WT(9),SHP(3,9), 1

XS(2,2),RL(1),UL(1),EPS(3),SIG(3)

C GO TO (1,2,3,4),ICN C C…. READ & WRITE MATERIAL PROPERTIES C 1 READ(5,10) E,ANU,TH,KODE,L,LS,LLD 10 FORMAT(3E10.0,4I5) WRITE(6,11) E,ANU,TH,KODE,L,LS,LLD 474

Finite Element Method with Applications in Engineering 11 FORMAT(′ YOUNGS MODULAS ………………………. = ′,E13.4/ *

′ POISSONS RATIO ………………………. = ′,E13.4/

*

′ THICKNESS …………………………… = ′,E13.4/

*

′ KODE (1 FOR PL STRESS, ZERO FOR PL STRAIN) = ′,I13/

*

′ ORDER OF QUADRATURE FOR STIFFNESS ……… = ′,I13/

*

′ NO. OF STRESS POINTS IN EACH DIRECTION …. = ′,I13/

*

′ ORDER OF QUAD FOR CONSISTENT LOAD ……… = ′,I13)

DL(1)=E DL(2)=ANU DL(3)=TH DL(4)=KODE DL(5)=L DL(6)=LS DL(7)=LLD RETURN C C…. EVALUATE THE CONSISTENT LOAD VECTOR C 2 P1=UL(1) P2=UL(2) LINT=DL(7) CALL PGAUSL(LINT,ZI,ET,WT) DO 105 K=1,LINT CALL SHAPE(ZI(K),ET(K),XL,SHP,XS,XSJ,NEL,1) SIGMA=SHP(3,2)*P1+SHP(3,3)*P2 DV=SIGMA*WT(K)*DL(3) DO 120 I=1,NEL RL(I+I-1)=RL(I+I-1)+SHP(3,I)*XS(2,2)*DV 120 RL(I+I)=RL(I+I)-SHP(3,I)*XS(2,1)*DV 105 CONTINUE RETURN C C…. FORM THE ELEMENT STIFFNESS MATRIX C 3 IF(ISWICH) WRITE(6,210) IELG 475

Finite Element Method with Applications in Engineering 210 FORMAT(// ′ ELEMENT NUMBER = ′,I3) C C…. IF KODE=0, THEN PLANE STRAIN. OTHERWISE PLANE STRESS. C KODE=DL(4) I=KODE IF(I.NE.0) I=1 IF(I.EQ.0) I=2 D1=DL(1)*(1.+(1-I)*DL(2))/(1.0+DL(2))/(1.0-I*DL(2)) D2=D1*DL(2)/(1.+(1-I)*DL(2)) D3=DL(1)*0.5/(1.+DL(2)) C IF(IELG.EQ.1) LINT=0 L=DL(5) IF(L*L.NE.LINT) CALL PGAUSS(L,LINT,ZI,ET,WT) C DO 30 K=1,LINT CALL SHAPE(ZI(K),ET(K),XL,SHP,XS,XSJ,NEL,2) IF(ISWICH) WRITE(6,420) ZI(K),ET(K),XSJ 420 FORMAT(′ FORMAT(′ ZI= ′,F10.5,5X, ′ET= ′,F10.5,5X, ′XSJ= ′,F10.5) IF(XSJ.GT.1.0D-05) GO TO 25 NXSJ=1 WRITE(6,26) IELG 26 FORMAT(′ NEGATIVE JACOBIAN ENCOUNTERED FOR ELMT ′,I4) 25 DV=XSJ*WT(K)*DL(3) D11=D1*DV D12=D2*DV D33=D3*DV C C…. FOR EACH J NODE COMPUTE DB=D*BJ C DO 50 J=1,NEL DB11=D11*SHP(1,J) DB12=D12*SHP(2,J) DB21=D12*SHP(1,J) 476

Finite Element Method with Applications in Engineering DB22=D11*SHP(2,J) DB31=D33*SHP(2,J) DB32=D33*SHP(1,J) C C…. FOR EACH I NODE COMPUTE S=BI*DB C DO 60 I=1,J SL(I+I-1,J+J-1)=SL(I+I-1,J+J-1)+SHP(1,I)*DB11+SHP(2,I)*DB31 SL(I+I-1,J+J )=SL(I+I-1,J+J )+SHP(1,I)*DB12+SHP(2,I)*DB32 SL(I+I ,J+J-1)=SL(I+I ,J+J-1)+SHP(2,I)*DB21+SHP(1,I)*DB31 SL(I+I ,J+J )=SL(I+I ,J+J )+SHP(2,I)*DB22+SHP(1,I)*DB32 60 CONTINUE 50 CONTINUE 30 CONTINUE C C…. COMPUTE LOWER TRIANGULAR PART BY SYMMETRY C DO 70 I=2,NST DO 70 J=1,I 70 SL(I,J)=SL(J,I) CALL RLOAD(SL,UL,NST,RL) RETURN C C…. EVALUATE THE STRESSES C 4 IF(IELG.EQ.1) WRITE(11,100) L=DL(6) I=DL(4) IF(I.NE.0) I=1 IF(I.EQ.0) I=2 D1=DL(1)*(1.+(1-I)*DL(2))/(1.0+DL(2))/(1.0-I*DL(2)) D2=D1*DL(2)/(1.+(1-I)*DL(2)) D3=DL(1)*0.5/(1.+DL(2)) IF(L*L.NE.LINT) CALL PGAUSS(L,LINT,ZI,ET,WT) C 477

Finite Element Method with Applications in Engineering DO 12 L=1,LINT CALL SHAPE(ZI(L),ET(L),XL,SHP,XS,XSJ,NEL,2) DO 13 I=1,3 13 EPS(I)=0.0 XX=0.0 YY=0.0 C DO 14 J=1,NEL XX=XX+SHP(3,J)*XL(1,J) YY=YY+SHP(3,J)*XL(2,J) EPS(1)=EPS(1)+SHP(1,J)*UL(J+J-1) EPS(2)=EPS(2)+SHP(2,J)*UL(J+J) 14 EPS(3)=EPS(3)+SHP(1,J)*UL(J+J)+SHP(2,J)*UL(J+J-1) C SIG(1)=D1*EPS(1)+D2*EPS(2) SIG(2)=D2*EPS(1)+D1*EPS(2) SIG(3)=D3*EPS(3) WRITE(11,15) IELG,XX,YY,(SIG(I),I=1,3) 12 CONTINUE 15 FORMAT(I5,5E12.4) 100 FORMAT(/// ′ S T R E S S ′// 1 1X, ′ELMT ′,7X, ′1-ORD ′,7X, ′2-ORD ′,3X, ′XX-STRESS ′,3X, 2 ′YY-STRESS ′,3X, ′XY-STRESS ′) RETURN END C******************************************************************* SUBROUTINE SHAPE(SS,TT,X,SHP,XS,XSJ,NEL,KEY) C C… SHAPE FUNCTIONS FOR ISOPARAMETRIC ELEMENTS C… KEY=1 DERIVATIVES ARE WRT NATURAL COORDINATES C…

2 DERIVATIVEA ARE WRT GLOBAL COORDINATES

C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION SHP(3,1),X(2,1),S(4),T(4),XS(2,2),SX(2,2) DATA S/-0.5D00,0.5D00,0.5D00,-0.5D00/ 478

Finite Element Method with Applications in Engineering DATA T/-0.5D00,-0.5D00,0.5D00,0.5D00/ C C… FORM 4-NODE QUADRILATERAL SHAPE FUNCTIONS C DO 100 I=1,4 SHP(3,I)=(0.5+S(I)*SS)*(0.5+T(I)*TT) SHP(1,I)=S(I)*(0.5+T(I)*TT) 100 SHP(2,I)=T(I)*(0.5+S(I)*SS) IF(NEL.GT.3) GO TO 20 DO 10 I=1,3 10 SHP(I,3)=SHP(I,3)+SHP(I,4) 20 IF(NEL.GT.4) CALL SHAP2(SS,TT,SHP,NEL) C C…. CONSTRUCT JACOBIAN AND ITS INVERSE C DO 130 I=1,2 DO 130 J=1,2 XS(I,J)=0.0 DO 130 K=1,NEL 130 XS(I,J)=XS(I,J)+SHP(I,K)*X(J,K) XSJ=XS(1,1)*XS(2,2)-XS(1,2)*XS(2,1) IF(KEY.EQ.1) RETURN C SX(1,1)= XS(2,2)/XSJ SX(1,2)=-XS(1,2)/XSJ SX(2,1)=-XS(2,1)/XSJ SX(2,2)= XS(1,1)/XSJ C C… FORM GLOBAL DERIVATIVES C DO 140 I=1,NEL TEMP =SHP(1,I)*SX(1,1)+SHP(2,I)*SX(1,2) SHP(2,I)=SHP(1,I)*SX(2,1)+SHP(2,I)*SX(2,2) 140 SHP(1,I)=TEMP RETURN 479

Finite Element Method with Applications in Engineering END C**************************************************************** SUBROUTINE SHAP2(S,T,SHP,NEL) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION SHP(3,1) S2=(1.-S*S)/2 T2=(1.-T*T)/2 DO 100 I=5,8 DO 100 J=1,3 100 SHP(J,I)=0.0 C SHP(1,5)=-S*(1.-T) SHP(2,5)=-S2 SHP(3,5)= S2*(1.-T) IF(NEL.LT.6) GO TO 107 C SHP(1,6)= T2 SHP(2,6)=-T*(1.+S) SHP(3,6)= T2*(1.+S) IF(NEL.LT.7) GO TO 107 C SHP(1,7)= -S*(1.+T) SHP(2,7)= S2 SHP(3,7)= S2*(1.+T) IF(NEL.LT.8) GO TO 107 C SHP(1,8)= -T2 SHP(2,8)= -T*(1.-S) SHP(3,8)= T2*(1.-S) IF(NEL.LT.9) GO TO 107 C SHP(1,9)= -S*T2*4.0 SHP(2,9)= -T*S2*4.0 SHP(3,9)= 4.*S2*T2 C 480

Finite Element Method with Applications in Engineering C…. CORRECT CORNER NODES FOR INTERIOR NODE (LAGRANGIAN) C DO 106 J=1,3 DO 105 I=1,4 105 SHP(J,I)=SHP(J,I)-0.25*SHP(J,9) C…. CORRECT MID SIDE NODES FOR INTERIOR NODE DO 106 I=5,8 106 SHP(J,I)=SHP(J,I)-0.5*SHP(J,9) C C…. CORRECT CORNER NODES FOR PRESENCE OF MIDSIDE NODES C 107 DO 108 J=1,3 SHP(J,1)=SHP(J,1)-0.5*(SHP(J,5)+SHP(J,8)) SHP(J,2)=SHP(J,2)-0.5*(SHP(J,5)+SHP(J,6)) SHP(J,3)=SHP(J,3)-0.5*(SHP(J,6)+SHP(J,7)) 108 SHP(J,4)=SHP(J,4)-0.5*(SHP(J,7)+SHP(J,8)) RETURN END C******************************************************************* SUBROUTINE PGAUSL(LINT,ZI,ET,WT) C C… GAUSS POINTS AND WEIGHTS TO CALCULATE THE EQUIVALENT NODAL C… LOAD VECTOR DUE TO SURFACE TRACTION AT THE FACE ZI=+1 C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION ZI(1),ET(1),WT(1) GO TO (1,2,3),LINT C C… 1 PT INTEGRATION C 1 ZI(1)=1. ET(1)=0. WT(1)=2. RETURN C 481

Finite Element Method with Applications in Engineering C… 2 PT INTEGRATION C 2 G = 1.0D00/DSQRT(3.0D00) ZI(1)=1.0 ZI(2)=1.0 ET(1)= G ET(2)=-G WT(1)=1.0D00 WT(2)=1.0D00 RETURN C C… 3 PT INTEGRATION C 3 G = DSQRT(0.6D00) ZI(1)=1.0 ZI(2)=1.0 ZI(3)=1.0 ET(1)=-G ET(2)= 0.0 ET(3)= G WT(1)= 5./9. WT(2)= 8./9. WT(3)= 5./9. RETURN END C******************************************************************* SUBROUTINE PGAUSS(L,LINT,ZI,ET,WT) C C… GAUSS POINTS AND WEIGHTS FOR TWO DIMENSIONS C IMPLICIT REAL*8 (A-H,O-Z) DIMENSION LR(9),LZ(9),LW(9),ZI(1),ET(1),WT(1) DATA LR/-1,1,1,-1,0,1,0,-1,0/,LZ/-1,-1,1,1,-1,0,1,0,0/ DATA LW/4*25,4*40,64/ LINT=L*L 482

Finite Element Method with Applications in Engineering GO TO (1,2,3),L C C… 1*1 INTEGRATION C 1 ZI(1)=0. ET(1)=0. WT(1)=4. RETURN C C… 2*2 INTEGRATION C 2 G = 1.0D00/DSQRT(3.0D00) DO 21 I=1,4 ZI(I)=G*LR(I) ET(I)=G*LZ(I) 21 WT(I)=1.0D00 RETURN C C… 3*3 INTEGRATION C 3 G = DSQRT(0.6D00) H = 1.0D00/81.0D00 DO 31 I=1,9 ZI(I)=G*LR(I) ET(I)=G*LZ(I) 31 WT(I)=H*LW(I) RETURN END C*****7************************************************************* SUBROUTINE ELMT6(DL,NEL,XL,SL,UL,RL,ICN) C C…. 3-9 VARIABLE NODED ISOPARAMETRIC ELEMENT FOR AXISYMMETRIC C…. ANALYSIS M=0 C IMPLICIT REAL*8 (A-H,O-Z) 483

Finite Element Method with Applications in Engineering

COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK2/ISWITCH COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST COMMON/BLK5/OM,NH LOGICAL ISWITCH DIMENSION XL(NDM,1),SL(NST,1),DL(1),ZI(9),ET(9),WT(9),SHP(3,9), *XS(2,2),RL(1),UL(1),EPS(4),SIG(4),DS(3,3),BJ(4,2),BI(4,2),DB(4,2) C GO TO (1,2,3,4),ICN C C…. READ & WRITE MATERIAL PROPERTIES C OM = 0.0 1 READ(5,10) E,ANU,L,LS,LLD 10 FORMAT(2E10.0,4I5) WRITE(6,11) E,ANU,L,LS,LLD 11 FORMAT(′ YOUNGS MODULAS ……………………… = ′,E13.4/ *

′ POISSONS RATIO ………………………. = ′,E13.4/

*

′ ORDER OF QUADRATURE FOR STIFFNESS ……… = ′,I13/

*

′ NO. OF STRESS POINTS IN EACH DIRECTION …. = ′,I13/

*

′ ORDER OF QUAD FOR CONSISTENT LOAD ……… = ′,I13/)

RHO = 1.0 DL(1)=E DL(2)=ANU DL(3)=RHO DL(4)=L DL(5)=LS DL(6)=LLD PI=4.0*ATAN(1.0) RETURN C C…. EVALUATE THE CONSISTENT LOAD VECTOR C 2 P1=UL(1) 484

Finite Element Method with Applications in Engineering P2=UL(2) LINT=DL(6) CALL PGAUSL(LINT,ZI,ET,WT) DO 105 K=1,LINT CALL SHAPE(ZI(K),ET(K),XL,SHP,XS,XSJ,NEL,1) SIGMA=SHP(3,2)*P1+SHP(3,3)*P2 IF(NEL.GT.4) SIGMA=SIGMA+SHP(3,6)*UL(3) RR=0.0 DO 125 J=1,NEL 125 RR=RR+SHP(3,J)*XL(1,J) DV=2.0*PI*SIGMA*RR*WT(K) DO 120 I=1,NEL RL(I+I-1)=RL(I+I-1)+SHP(3,I)*XS(2,2)*DV 120 RL(I+I)=RL(I+I)-SHP(3,I)*XS(2,1)*DV 105 CONTINUE RETURN C C…. FORM THE ELEMENT STIFFNESS MATRIX C 3 IF(ISWITCH) WRITE(6,210) IELG 210 FORMAT(// ′ ELEMENT NUMBER = ′,I3) C 6 E=DL(1) ANU=DL(2) FACT=E/(1.0+ANU)/(1.0-2.0*ANU) DS(1,1)=FACT*(1.0-ANU) DS(2,2)=DS(1,1) DS(3,3)=DS(1,1) DS(1,2)=FACT*ANU DS(1,3)=DS(1,2) DS(2,1)=DS(1,2) DS(2,3)=DS(1,2) DS(3,1)=DS(1,2) DS(3,2)=DS(1,2) DS44=FACT*(0.5-ANU) 485

Finite Element Method with Applications in Engineering IF(ICN.EQ.4) GO TO 5 C C…. INITIALIZE BI AND BJ MATRICES C DO 110 I=1,4 DO 110 J=1,2 BI(I,J)=0.0 110 BJ(I,J)=0.0 C IF(IELG.EQ.1) LINT=0 L=DL(4) IF(L*L.NE.LINT) CALL PGAUSS(L,LINT,ZI,ET,WT) C DO 30 K=1,LINT CALL SHAPE(ZI(K),ET(K),XL,SHP,XS,XSJ,NEL,2) IF(ISWITCH) WRITE(6,420) ZI(K),ET(K),XSJ 420 FORMAT (′ZI=′,F10.5,5X,′XSJ=′,F10.5) IF(XSJ.GT.1.0D-05) GO TO 25 NXSJ=1 WRITE(6,26) IELG 26 FORMAT (′ NEGATIVE JACOBIAN ENCOUNTERED FOR ELMT’,I4) C 25 RR=0.0 DO 35 I=1,NEL 35 RR=RR+SHP(3,I)*XL(1,I) DV=2.0*PI*RR*XSJ*WT(K) RHO=DL(3) DM=2.0*PI*RR*XSJ*WT(K)*RHO*OM*OM C C….FORM BJ MATRIX C JJ=1 DO 50 J=1,NEL BJ(1,1)=SHP(1,J) BJ(2,1)=SHP(3,J)/RR 486

Finite Element Method with Applications in Engineering BJ(4,1)=SHP(2,J) BJ(3,2)=SHP(2,J) BJ(4,2)=SHP(1,J) C C…. FOR EACH J NODE COMPUTE DB=D*BJ C DO 70 K2=1,2 DO 80 K1=1,3 DB(K1,K2)=0.0 DO 80 KKK=1,3 80 DB(K1,K2)=DB(K1,K2)+DS(K1,KKK)*BJ(KKK,K2) 70 DB(4,K2)=DS44*BJ(4,K2) C C…. FORM BI MATRIX C II=1 DO 60 I=1,J BI(1,1)=SHP(1,I) BI(2,1)=SHP(3,I)/RR BI(4,1)=SHP(2,I) BI(3,2)=SHP(2,I) BI(4,2)=SHP(1,I) C C…. FOR EACH I NODE, COMPUTE S=BI*DB C DO 90 K1=1,2 IR=II+K1-1 DO 90 KK=1,2 IC=JJ+KK-1 IF(K1.EQ.KK) SL(IR,IC)=SL(IR,IC)-DM*SHP(3,I)*SHP(3,J) DO 90 KKK=1,4 90 SL(IR,IC)=SL(IR,IC)+DV*BI(KKK,K1)*DB(KKK,KK) 60 II=II+NDF 50 JJ=JJ+NDF 30 CONTINUE 487

Finite Element Method with Applications in Engineering C C…. COMPUTE LOWER TRIANGULAR PART BY SYMMETRY C DO 75 I=2,NST DO 75 J=1,I 75 SL(I,J)=SL(J,I) CALL RLOAD(SL,UL,NST,RL) RETURN C C…. EVALUATE THE STRESSES C 4 IF(IELG.EQ.1) WRITE(11,100) LS=DL(5) GO TO 6 5 IF(LS*LS.NE.LINT) CALL PGAUSS(LS,LINT,ZI,ET,WT) C DO 12 L=1,LINT CALL SHAPE(ZI(L),ET(L),XL,SHP,XS,XSJ,NEL,2) DO 13 I=1,4 13 EPS(I)=0.0 RR=0.0 ZZ=0.0 C DO 14 J=1,NEL RR=RR+SHP(3,J)*XL(1,J) ZZ=ZZ+SHP(3,J)*XL(2,J) J1=J+J-1 J2=J+J EPS(1)=EPS(1)+SHP(1,J)*UL(J1) EPS(2)=EPS(2)+SHP(3,J)*UL(J1) EPS(3)=EPS(3)+SHP(2,J)*UL(J2) 14 EPS(4)=EPS(4)+SHP(2,J)*UL(J1)+SHP(1,J)*UL(J2) EPS(2)=EPS(2)/RR C SIG(1)=DS(1,1)*EPS(1)+DS(1,2)*EPS(2)+DS(1,3)*EPS(3) 488

Finite Element Method with Applications in Engineering SIG(2)=DS(2,1)*EPS(1)+DS(2,2)*EPS(2)+DS(2,3)*EPS(3) SIG(3)=DS(3,1)*EPS(1)+DS(3,2)*EPS(2)+DS(3,3)*EPS(3) SIG(4)=DS44*EPS(4) WRITE(11,15) IELG,RR,ZZ,(SIG(I),I=1,4) 12 CONTINUE 15 FORMAT(I5,2E12.4,4E11.3) 100 FORMAT(/// ′ S T R E S S′// 1 1X,′ELMT′,7X,′R-ORD′,7X,′Z-ORD′,3X,′RR-STRES′,3X, 2 ′ TT-STRES′,3X,′ZZ-STRES′,3X,′RZ-STRES′) RETURN END C*****7************************************************************* SUBROUTINE TRANS(SL,NST,NEL,NH) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION SL(NST,1),IEQ(3) DATA IEQ/1,10,22/ C IF(NH.GT.1) RETURN FACT=DSQRT(0.5D00) KK=3 IF(NEL.LT.8) KK=2 DO 30 K=1,KK IR=IEQ(K) C C…. MODIFY THE COLUMNS C DO 10 I=1,NST SK1=SL(I,IR) SK2=SL(I,IR+2) SL(I,IR)=(SK1+SK2)*FACT 10 SL(I,IR+2)=(SK2-SK1)*FACT C C…. MODIFY THE ROWS C DO 20 J=1,NST 489

Finite Element Method with Applications in Engineering SK1=SL(IR,J) SK2=SL(IR+2,J) SL(IR,J)=(SK1+SK2)*FACT 20 SL(IR+2,J)=(SK2/SK1)*FACT 30 CONTINUE RETURN END C******************************************************************* SUBROUTINE ELMT7(DL,NEL,XL,SL,UL,RL,ICN) C C…. 3-9 VARIABLE NODED ISOPARAMETRIC ELEMENT FOR AXISYMMETRIC C…. ANALYSIS M > 0… C IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP COMMON/BLK2/ISWITCH COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST COMMON/BLK5/OM,NH LOGICAL ISWITCH DIMENSION XL(NDM,1),SL(NST,1),DL(1),ZI(9),ET(9),WT(9),SHP(3,9), *XS(2,2),RL(1),UL(1),EPS(6),SIG(6),DS(3,3),BJ(6,3),BI(6,3),DB(6,3) C GO TO (1,2,3,4),ICN C C…. READ & WRITE MATERIAL PROPERTIES C 1 READ(5,10) E,ANU,L,LS,LLD,KODE, NH 10 FORMAT(2E10.0,5I5) OM = 0.0 RHO = 1.0 WRITE(6,11) E,ANU,L,LS,LLD,KODE,NH 11 FORMAT (′ YOUNGS MODULAS ………………………. =’,E13.4/ *

′ POISSONS RATIO ………………………. =’,E13.4/

*

′ ORDER OF QUADRATURE FOR STIFFNESS ……… =’,I13/ 490

Finite Element Method with Applications in Engineering *

′ NO. OF STRESS POINTS IN EACH DIRECTION …. =’,I13/

*

′ ORDER OF QUAD FOR CONSISTENT LOAD ……… =’,I13/

*

′ KODE (=1 FOR CORE ELEMENTS ……………. =’,I13/

*

′ HARMONIC NUMBER………………………..=’,I13) DL(1)=E DL(2)=ANU DL(3)=RHO DL(4)=L DL(5)=LS DL(6)=LLD DL(7)=KODE PI=4.0*ATAN(1.0) RETURN

C C…. EVALUATE THE CONSISTENT LOAD VECTOR C 2 P1=UL(1) P2=UL(2) LINT=DL(6) CALL PGAUSL(LINT,ZI,ET,WT) DO 105 K=1,LINT CALL SHAPE(ZI(K),ET(K),XL,SHP,XS,XSJ,NEL,1) SIGMA=SHP(3,2)*P1+SHP(3,3)*P2 IF(NEL.GT.4) SIGMA=SIGMA+SHP(3,6)*UL(3) RR=0.0 DO 125 J=1,NEL 125 RR=RR+SHP(3,J)*XL(1,J) DV=PI*SIGMA*RR*WT(K) II=1 DO 120 I=1,NEL RL(II) =RL(II )+SHP(3,I)*XS(2,2)*DV RL(II+1)=RL(II+1)-SHP(3,I)*XS(2,1)*DV RL(II+2)=RL(II+2)+0.0 120 II =II+NDF 105 CONTINUE 491

Finite Element Method with Applications in Engineering RETURN C C…. FORM THE ELEMENT STIFFNESS MATRIX C 3 IF(ISWITCH) WRITE(6,210) IELG 210 FORMAT(// ′ ELEMENT NUMBER = ’, I3) C 6 E=DL(1) ANU=DL(2) FACT=E/(1.0+ANU)/(1.0-2.0*ANU) DS(1,1)=FACT*(1.0-ANU) DS(2,2)=DS(1,1) DS(3,3)=DS(1,1) DS(1,2)=FACT*ANU DS(1,3)=DS(1,2) DS(2,1)=DS(1,2) DS(2,3)=DS(1,2) DS(3,1)=DS(1,2) DS(3,2)=DS(1,2) DS44=FACT*(0.5-ANU) DS55=DS44 DS66=DS44 IF(ICN.EQ.4) GO TO 5 C C…. INITIALIZE BI AND BJ MATRICES C DO 110 I=1,6 DO 110 J=1,3 BI(I,J)=0.0 110 BJ(I,J)=0.0 C IF(IELG.EQ.1) LINT=0 L=DL(4) IF(L*L.NE.LINT) CALL PGAUSS(L,LINT,ZI,ET,WT) C 492

Finite Element Method with Applications in Engineering DO 30 K=1,LINT CALL SHAPE(ZI(K),ET(K),XL,SHP,XS,XSJ,NEL,2) IF(ISWITCH) WRITE(6,420) ZI(K),ET(K),XSJ 420 FORMAT (′ ZI=’,F10.5,5X,’ET=’,F10.5,5X,’XSJ=’,F10.5) IF(XSJ.GT.1.0D-05) GO TO 25 NXSJ=1 WRITE(6,26) IELG 26 FORMAT (′ NEGATIVE JACOBIAN ENCOUNTERED FOR ELMT’,I4) C 25 RR=0.0 DO 35 I=1,NEL 35 RR=RR+SHP(3,I)*XL(1,I) DV=PI*RR*XSJ*WT(K) RHO=DL(3) DM=PI*RR*XSJ*WT(K)*RHO*OM*OM C C…. FORM BJ MATRIX C JJ=1 DO 50 J=1,NEL BJ(1,1)= SHP(1,J) BJ(3,1)= SHP(3,J)/RR BJ(4,1)= SHP(2,J) BJ(5,1)=-SHP(3,J)*NH/RR BJ(2,2)= SHP(2,J) BJ(4,2)= SHP(1,J) BJ(6,2)=-SHP(3,J)*NH/RR BJ(3,3)=-SHP(3,J)*NH/RR BJ(5,3)= SHP(3,J)/RR-SHP(1,J) BJ(6,3)=-SHP(2,J) C C…. FOR EACH J NODE, COMPUTE DB=D*BJ C DO 70 KK=1,NDF DO 80 K1=1,3 493

Finite Element Method with Applications in Engineering DB(K1,KK)=0.0 DO 80 KKK=1,3 80 DB(K1,KK)= DB(K1,KK)+DS(K1,KKK)*BJ(KKK,KK) DB(4,KK) = DS44*BJ(4,KK) DB(5,KK) = DS55*BJ(5,KK) 70 DB(6,KK) = DS66*BJ(6,KK) C C…. FORM BI MATRIX C II=1 DO 60 I=1,J BI(1,1)= SHP(1,I) BI(3,1)= SHP(3,I)/RR BI(4,1)= SHP(2,I) BI(5,1)=-SHP(3,I)*NH/RR BI(2,2)= SHP(2,I) BI(4,2)= SHP(1,I) BI(6,2)=-SHP(3,I)*NH/RR BI(3,3)=-SHP(3,I)*NH/RR BI(5,3)= SHP(3,I)/RR-SHP(1,I) BI(6,3)=-SHP(2,I) C C…. FOR EACH I NODE, COMPUTE S=BI*DB C DO 90 K1=1,3 IR=II+K1-1 DO 90 KK=1,3 IC=JJ+KK-1 IF(K1.EQ.KK) SL(IR,IC)=SL(IR,IC)-DM*SHP(3,I)*SHP(3,J) DO 90 KKK=1,6 90 SL(IR,IC)=SL(IR,IC)+DV*BI(KKK,K1)*DB(KKK,KK) 60 II=II+NDF 50 JJ=JJ+NDF 30 CONTINUE C 494

Finite Element Method with Applications in Engineering C…. FORM LOWER TRIANGULAR PART BY REFLECTION C DO 75 I=2,NST DO 75 J=1,I 75 SL(I,J)=SL(J,I) KODE=DL(7) IF(NH.EQ.1.AND.KODE.EQ.1) CALL TRANS(SL,NST,NEL,NH) CALL RLOAD(SL,UL,NST,RL) RETURN C C…. EVALUATE THE STRESSES C 4 IF(IELG.EQ.1) WRITE(11,100) LS=DL(5) GO TO 6 5 IF(LS*LS.NE.LINT) CALL PGAUSS(LS,LINT,ZI,ET,WT) C DO 12 L=1,LINT CALL SHAPE(ZI(L),ET(L),XL,SHP,XS,XSJ,NEL,2) DO 13 I=1,6 13 EPS(I)=0.0 RR=0.0 ZZ=0.0 DO 14 J=1,NEL RR=RR+SHP(3,J)*XL(1,J) 14 ZZ=ZZ+SHP(3,J)*XL(2,J) C JJ=0 DO 55 J=1,NEL BJ(1,1)= SHP(1,J) BJ(3,1)= SHP(3,J)/RR BJ(4,1)= SHP(2,J) BJ(5,1)=-SHP(3,J)*NH/RR BJ(2,2)= SHP(2,J) BJ(4,2)= SHP(1,J) 495

Finite Element Method with Applications in Engineering BJ(6,2)=-SHP(3,J)*NH/RR BJ(3,3)=-SHP(3,J)*NH/RR BJ(5,3)= SHP(3,J)/RR-SHP(1,J) BJ(6,3)=-SHP(2,J) C DO 65 I=1,6 DO 65 K=1,3 65 EPS(I)=EPS(I)+BJ(I,K)*UL(JJ+K) 55 JJ=JJ+NDF C SIG(1)=DS(1,1)*EPS(1)+DS(1,2)*EPS(2)+DS(1,3)*EPS(3) SIG(2)=DS(2,1)*EPS(1)+DS(2,2)*EPS(2)+DS(2,3)*EPS(3) SIG(3)=DS(3,1)*EPS(1)+DS(3,2)*EPS(2)+DS(3,3)*EPS(3) SIG(4)=DS44*EPS(4) SIG(5)=DS55*EPS(5) SIG(6)=DS66*EPS(6) WRITE(11,15) IELG,RR,ZZ,(SIG(I),I=1,6) 12 CONTINUE 15 FORMAT(I5,2E12.4,6E11.3) 100 FORMAT(/// ′ S T R E S S ′// 1 1X, ′ ELMT ′,7X, ′ R-ORD ′,7X, ′ Z-ORD ′,3X, ′RR-STRES′,3X, 2 ′ TT-STRES ′,3X,′ ZZ-STRES ′,3X,′ RZ-STRES ′,3X, ′RT-STRES ′,3X, 3 ′ ZT-STRES′) RETURN END C C****************************************************************** C SUBROUTINE ELMT8(DL,NEL,XL,SL,UL,RL,ICN) C C…. 8 NODE BRICK ELEMENT FOR 3-D STRESS ANALYSIS C IMPLICIT REAL*8 (A-H,O-Z) COMMON/BLK1/NUMNP,NUMEL,NDF,NDM,MNEL,NUMAT,NSN,NEQ,MBAND,NPROP 496

Finite Element Method with Applications in Engineering COMMON/BLK2/ISWICH COMMON/BLK3/MA,IELB,IELG,NST,NXSJ,MST LOGICAL ISWICH DIMENSION XL(NDM,1),SL(NST,1),DL(1),RL(1),UL(1),EPS(6),SIG(6) DIMENSION SHP(4,8),RG(8),SG(8),TG(8),WG(8) ISWICH=.FALSE. IFLAG=0 C GO TO (1,2,3,4),ICN C C…. READ & WRITE MATERIAL PROPERTIES C 1 CONTINUE READ(5,10) E,ANU 10 FORMAT(2E10.0) WRITE(6,11) E,ANU 11 FORMAT (′ YOUNGS MODULUS …………………………..=′, E13.4/ *

′ POISSONS RATIO …………………………..=′,E13.4/)

DL(1)=E DL(2)=ANU RETURN C 2 RETURN C C…. FORM THE ELEMENT STIFFNESS MATRIX C 3 CONTINUE E=DL(1) ANU=DL(2) DL1=E*(1.0D0-ANU)/(1.0D0+ANU)/(1.0D0-2.0D0*ANU) DL2=ANU*DL1/(1.0D0-ANU) DL3=E/2.0D0/(1.0D0+ANU) IF(ISWICH) WRITE(11,310) IELG 310 FORMAT(//′ ELEMENT NUMBER =′,I3) CALL PGAUSS3(RG,SG,TG,WG) 497

Finite Element Method with Applications in Engineering DO 320 L=1,NEL CALL SHAPE3(RG(L),SG(L),TG(L),XL,SHP,XSJ,NDM,NEL,.FALSE.) DO 330 I =1,NEL I3=3*I I1=I3-2 I2=I3-1 DO 330 J =I,NEL J3=3*J J1=J3-2 J2=J3-1 SL(I1,J1) = SL(I1,J1)+(DL1*SHP(1,I)*SHP(1,J) *

+DL3*(SHP(2,I)*SHP(2,J)+SHP(3,I)*SHP(3,J)))*XSJ

SL(I1,J2)= SL(I1,J2)+(DL2*SHP(1,I)*SHP(2,J) *

+DL3*SHP(2,I)*SHP(1,J))*XSJ

SL(I1,J3)= SL(I1,J3)+(DL2*SHP(1,I)*SHP(3,J) *

+DL3*SHP(3,I)*SHP(1,J))*XSJ

SL(I2,J1)= SL(I2,J1)+(DL2*SHP(2,I)*SHP(1,J) *

+DL3*SHP(1,I)*SHP(2,J))*XSJ

SL(I2,J2) = SL(I2,J2)+(DL1*SHP(2,I)*SHP(2,J) *

+DL3*(SHP(1,I)*SHP(1,J)+SHP(3,I)*SHP(3,J)))*XSJ

SL(I2,J3)= SL(I2,J3)+(DL2*SHP(2,I)*SHP(3,J) *

+DL3*SHP(3,I)*SHP(2,J))*XSJ

SL(I3,J1)= SL(I3,J1)+(DL2*SHP(3,I)*SHP(1,J) *

+DL3*SHP(1,I)*SHP(3,J))*XSJ

SL(I3,J2)= SL(I3,J2)+(DL2*SHP(3,I)*SHP(2,J) *

+DL3*SHP(2,I)*SHP(3,J))*XSJ

330 SL(I3,J3) = SL(I3,J3)+(DL1*SHP(3,I)*SHP(3,J)+DL3*(SHP(1,I)* *

SHP(1,J)+SHP(2,I)*SHP(2,J)))*XSJ

320 CONTINUE C C…. COMPUTE LOWER TRIANGULAR PART BY SYMMETRY C DO 340 I = 1,24 DO 340 J = I,24 340 SL(J,I) = SL(I,J) 498

Finite Element Method with Applications in Engineering CALL RLOAD(SL,UL,NST,RL) RETURN C 4 CONTINUE C C…. EVALUATE THE STRESSES C IF(IELG.EQ.1) WRITE(11,100) CALL PGAUSS3(RG,SG,TG,WG) DO 400 L=1,NEL CALL SHAPE3(RG(L),SG(L),TG(L),XL,SHP,XSJ,NDM,NEL,.FALSE.) DO 410 I=1,6 410 EPS(I)=0.0D0 XX=0.0D0 YY=0.0D0 ZZ=0.0D0 JJ=0 DO 420 J=1,NEL C XX=XX+SHP(4,J)*XL(1,J) YY=YY+SHP(4,J)*XL(2,J) ZZ=ZZ+SHP(4,J)*XL(3,J) C EPS(1)=EPS(1)+SHP(1,J)*UL(JJ+1) EPS(2)=EPS(2)+SHP(2,J)*UL(JJ+2) EPS(3)=EPS(3)+SHP(3,J)*UL(JJ+3) EPS(4)=EPS(4)+SHP(2,J)*UL(JJ+1)+SHP(1,J)*UL(JJ+2) EPS(5)=EPS(5)+SHP(3,J)*UL(JJ+2)+SHP(2,J)*UL(JJ+3) EPS(6)=EPS(6)+SHP(3,J)*UL(JJ+1)+SHP(1,J)*UL(JJ+3) C 420 JJ=JJ+NDF C SIG(1)=DL1*EPS(1)+DL2*EPS(2)+DL2*EPS(3) SIG(2)=DL2*EPS(1)+DL1*EPS(2)+DL2*EPS(3) SIG(3)=DL2*EPS(1)+DL2*EPS(2)+DL1*EPS(3) 499

Finite Element Method with Applications in Engineering SIG(4)=DL3*EPS(4) SIG(5)=DL3*EPS(5) SIG(6)=DL3*EPS(6) C WRITE(11,200) IELG,L,XX,YY,ZZ,SIG C 400 CONTINUE RETURN C 100 FORMAT(5X,′ELEMENT STRESSES′//4X,′ ELEM ′,4X,′ GP ′,2X, *

3X,′ 1-COORD ′,5X,′ 2-COORD ′,5X,′ 3-COORD ′,3X,′ 11-STRESS ′,

*

3X,′ 22-STRESS ′,3X,′ 33-STRESS ′,3X,′ 12-STRESS ′,3X,′ 23-STRESS ′,

*

3X,′ 31-STRESS ′,3X)

200 FORMAT(1X,I5,2X,I5,2X,9E12.4) END C C************************************************************* C SUBROUTINE PGAUSS3(R,S,T,W) C C…..2 POINT GAUSS INTEGRATION C IMPLICIT REAL*8(A-H,O-Z) DIMENSION LR(8),LS(8),LT(8),R(1),S(1),T(1),W(1) DATA LR/-1,1,1,-1,-1,1,1,-1/ DATA LS/-1,-1,1,1,-1,-1,1,1/ DATA LT/1,1,1,1,-1,-1,-1,-1/ G=1.0D0/SQRT(3.0D0) DO 10 I=1,8 R(I)=G*LR(I) S(I)=G*LS(I) T(I)=G*LT(I) 10 W(I)=1.0D0 RETURN END 500

Finite Element Method with Applications in Engineering C C************************************************************* C SUBROUTINE SHAPE3(RR,SS,TT,X,SHP,XSJ,NDM,NEL,IFLAG) C C…..SHAPE FUNCTIONS FOR 8 NODE 3-D BRICK ELEMENT C IMPLICIT REAL*8(A-H,O-Z) LOGICAL IFLAG DIMENSION SHP(4,1),X(NDM,1),R(8),S(8),T(8),SX(3,3),XS(3,3) DATA R/-1.0D0,1.0D0,1.0D0,-1.0D0,-1.0D0,1.0D0,1.0D0,-1.0D0/ DATA S/-1.0D0,-1.0D0,1.0D0,1.0D0,-1.0D0,-1.0D0,1.0D0,1.0D0/ DATA T/1.0D0,1.0D0,1.0D0,1.0D0,-1.0D0,-1.0D0,-1.0D0,-1.0D0/ C DO 10 I =1,8 SHP(4,I) =0.125D0*(1.0D0+RR*R(I))*(1.0D0+SS*S(I))*(1.0D0+TT*T(I)) SHP(1,I) =0.125D0*R(I)*(1.0D0+SS*S(I))*(1.0D0+TT*T(I)) SHP(2,I) =0.125D0*(1.0D0+RR*R(I))*S(I)*(1.0D0+TT*T(I)) SHP(3,I) =0.125D0*(1.0D0+RR*R(I))*(1.0D0+SS*S(I))*T(I) 10 CONTINUE C C…. CONTSTRUCTION OF JACOBIAN AND INVERSE C DO 30 I = 1,NDM DO 30 J = 1,3 XS(I,J) = 0.0D0 DO 30 K=1,NEL 30 XS(I,J) =XS(I,J)+X(I,K)*SHP(J,K) IF (NDM.LT.3) RETURN XSJ = XS(1,1)*(XS(2,2)*XS(3,3)-XS(2,3)*XS(3,2))-XS(1,2)*(XS(2,1) * *XS(3,3)-XS(2,3)*XS(3,1))+XS(1,3)*(XS(2,1)*XS(3,2)-XS(2,2)* * XS(3,1)) IF (IFLAG) RETURN SX(1,1) = (XS(2,2)*XS(3,3)-XS(2,3)*XS(3,2))/XSJ SX(2,1) =-(XS(2,1)*XS(3,3)-XS(2,3)*XS(3,1))/XSJ 501

Finite Element Method with Applications in Engineering SX(3,1) = (XS(2,1)*XS(3,2)-XS(2,2)*XS(3,1))/XSJ SX(1,2) =-(XS(1,2)*XS(3,3)-XS(1,3)*XS(3,2))/XSJ SX(2,2) = (XS(1,1)*XS(3,3)-XS(1,3)*XS(3,1))/XSJ SX(3,2) =-(XS(1,1)*XS(3,2)-XS(1,2)*XS(3,1))/XSJ SX(1,3) = (XS(1,2)*XS(2,3)-XS(1,3)*XS(2,2))/XSJ SX(2,3) =-(XS(1,1)*XS(2,3)-XS(1,3)*XS(2,1))/XSJ SX(3,3) = (XS(1,1)*XS(2,2)-XS(1,2)*XS(2,1))/XSJ C C…. FORM GLOBAL DERIVATIVES C DO 40 I=1,NEL TP1 = SHP(1,I)*SX(1,1)+SHP(2,I)*SX(2,1)+SHP(3,I)*SX(3,1) TP2 = SHP(1,I)*SX(1,2)+SHP(2,I)*SX(2,2)+SHP(3,I)*SX(3,2) SHP(3,I) = SHP(1,I)*SX(1,3)+SHP(2,I)*SX(2,3)+SHP(3,I)*SX(3,3) SHP(1,I) = TP1 40 SHP(2,I) = TP2 RETURN END C******************************************************************** SUBROUTINE ELMT9(DL,NEL,XL,SL,UL,RL,ICN) RETURN END SUBROUTINE ELMT10(DL,NEL,XL,SL,UL,RL,ICN) RETURN END C******************************************************************

502

Finite Element Method with Applications in Engineering

Appendix F

GRAPHICAL INTERFACE FOR THE SIMPLIFIED FINITE ELEMENT ANALYSIS PROGRAM (SFEAP) F.1 PROGRAMS The simplified finite element analysis program (sfeap) can be used for static linear-elastic analysis of plane and space trusses, beam and plane frames, two and three-dimensional problems, and axisymmetric solids. There are eight Finite Element programs contained on the accompanying CD-ROM. The programs are listed in Table F.1, and are numbered as P1 to P8 for referencing. All the programs were coded using the theory presented in the text. F.1.1 Preamble The first three programs use one dimensional element; P1 and P2 use truss elements while P3 employs beam elements. The next two use two-dimensional elements in rectangular Cartesian co-ordinates; P4 uses three nodes constant strain triangles and P5 employs three to nine nodes isoparametric elements. P6 and P7 use two-dimensional three to nine nodes isoparametric elements in cylindrical (r, z) co-ordinates. The last program P8 employs eight nodded three-dimensional brick elements. The executables resulting from the finite element code written in FORTRAN 77 have been “wrapped” in a graphical user interface (GUI) developed using Visual C# for ease of use. The GUI requires at least Microsoft .Net (“dot” Net) framework 2.0 which can be downloaded freely fromhttp://www.microsoft.com. FlexCell Grid Control is used for data visualization and import/export features. Moreover, the GUI software uses the Microsoft Windows GDI+ for drawing figures (more about drawing figures is discussed later on). The software has been tested using 32 bit and 64 bit versions of Microsoft Windows XP. It is recommended that the screen resolution be set to at least 1,024 × 768 pixels for optimal visual experience. ®

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All the programs are standard Microsoft Windows program. As such, it possesses the standard controls including minimize, restore, and exit controls, as well as horizontal and vertical scroll bars, if required. For each program data has to be entered in six separate screens (frames). The screens are Control Information, Co-ordinates, Element Data, Boundary Conditions and Nodal Displacement Data, Material Characteristic Data, and Nodal and Distributed Load Data. All the screens will be explained in the section Executing Programs. Note that all input (force, displacement, material properties, etc.) must use a consistent set of units. Moreover, all inputs that are floating point quantities require the use of a decimal point, and can, if desired, be entered in “E” format (e.g., 20,000.0 or 2.0E+04 or 2.0e+04). Each program has its own manual which can be viewed by Clicking the icon ®

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Table F.1 List of Programs Program Number

Program

503

Finite Element Method with Applications in Engineering

P1

Analysis of plane truss

P2

Analysis of three-dimensional truss

P3

Analysis of plane frames or beam

P4

Plane stress/strain analysis using CST element

P5

Plane stress/strain analysis using isoparametric element

P6

Axisymmetric analysis for circumferential harmonic = 0

P7

Axisymmetric analysis for circumferential harmonic > 0

P8

Three-dimensional stress analysis using brick elements

F.2 EXECUTING PROGRAMS From the Start Menu → All Programs → SFEAP → the SFEAP icon will open the Table of Contents screen as shown in Figure F.1. Manual buttons are next to the corresponding program buttons. Six input screens of every program are very similar and hence only the case of the first program P1 is discussed with comments for other programs. It may be noted that every screen has an Exit button which allows the user to exit the program. An electronic copy of Appendix F will open by the button Appendix F, at the top left corner of the Table of Contents screen. on the first button, Analysis of Plane Truss, will open a screen similar to that shown in Figure F.2. The manual for the program, in PDF*, can be seen by the corresponding manual button in the Table of Contents. The manual will guide the user through the entire process of executing the program. Each manual is short. It is recommended that the user should read the applicable manual and experiment with the sample data to get familiar with the programs. For practice, a problem with output is given at the end of the manual. 1. CONTROL INFORMATION Note that the only buttons present at this time are the Control Information on the top left region and the Load Sample Data, Browse Input and Exit buttons located on the bottom right region of the screen. It can be seen that there are five entries, which are highlighted yellow. The data in these are already filled in and they cannot be edited. All the programs will have some of the entries highlighted. It is possible to load the sample data by Load Sample Data button. Also it is possible to load previously stored input file by Browse Input button. New data has to be typed in this screen. The input in the Control Information screen dictates the number of rows and/or columns in the tables in all subsequent screens. It is essential that in this as well as all other screens the entries 504

Finite Element Method with Applications in Engineering including 0 (zero) be entered. Once the last entry is typed the Co-ordinates button, will appear on the left side of the screen. the Co-ordinates button will open the screen shown in Figure F.3

Figure F.1 2. CO-ORDINATES Number of rows will be equal to the Number of Nodes and number of columns will be equal Number of Dimensions specified in the Control Information. For programs P1, P3, P4, P5, P6 and P7 there will be two columns and for P2 and P8 there will be three columns. In programs P6 and P7 the co-ordinates are r and z. If there is large number of nodes, then a vertical scroll bar will appear on the table to facilitate input of data.

505

Finite Element Method with Applications in Engineering

Figure F.2

506

Finite Element Method with Applications in Engineering Figure F.3

Figure F.4 Note that it is possible to import coordinate data from a Microsoft Excel file. (Note that this feature requires Microsoft Excel to be installed.) To import the coordinate data the Import Co-ordinates from Excel button. It will open the window shown in Figure F.4. Browse for the location and import the appropriate file. Typed coordinate data can also be exported to excel for modification or reuse by the Save Co-ordinates to Excel button. (Note that this feature also requires Microsoft Excel to be installed. Saving the data to Microsoft Excel can be a time consuming process; do not interrupt the process due to the risk of data loss from “crashing” the program.) It will open the window shown in Figure F.4. Browse for the desired location and save the file with an appropriate file name. As soon as the last entry is entered into the table the Element Data button will appear on the screen. the Element Data button will open the screen shown in Figure F.5. ®

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3. ELEMENT DATA In addition to Material Set and No. of nodes per element columns, there will be number of columns corresponding to Max Node per Elements specified in the Control Information. For P1, P2 and P3 the Max Nodes per Elements are 2 and the pertinent column will be highlighted. Similarly, for P4 there are 3 and for P8 there are 8. For programs P5, P6, and P7 the Max Node per Elements can be from 3 to 9 and the pertinent column will not be highlighted. If there is large number of elements and/or nodes in an element, then a vertical and/or horizontal scroll bars will appear on the table to facilitate input of data. Note that it is possible to import (export) the element data from (to) a Microsoft Excel file. (For details see the Co-ordinates section.). ®

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There are two options to view the Graphical Display by the Graphical Display button. For programs P1, P3, P4, and P5 by not selecting the Scale to panel size option

507

Finite Element Method with Applications in Engineering

with scale the 700 × 700 pixel screen in each direction. It will take and multiply the x and y co-ordinates by Nmin. While selecting the selecting the Scale to Panel option will scale the coordinates in x–direction by and in y–direction by . For programs P6 and P7 r and z replace the role played by x and y. In programs P2 and P8 and multiply x, y and z co-ordinates by Nmin when Scale to panel option is not selected. While selecting the selecting the Scale to Panel option will scale the co-ordinates in x–direction by

, in y–direction by

and z–direction by

.

Figure F.5 Displays from both options are shown in Figure F.6 and F.7, respectively. The screen contains two buttons, a Print button and Save button. To print the output, the Print button. (Note that the output will be sent to the default Microsoft Windows printer.) the Save button will open a window similar to the one shown in Figure F.4. Browse for the desired location and save the JPG file with an appropriate file name. ®

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Finite Element Method with Applications in Engineering

Figure F.6

Figure F.7 509

Finite Element Method with Applications in Engineering

Figure F.8 As soon as the last entry is entered in the table, the Boundary Conditions and Nodal Displacement Data button will appear on the screen as seen. the Boundary Conditions and Nodal Displacement Data button will open the screen shown in Figure F.8. 4. BOUNDARY CONDITIONS AND NODAL DISPLACEMENT DATA The screen contains two tables, one for boundary conditions and the other for nodal displacement data. The number of rows in the boundary conditions table is equal to the number of supported (constrained) nodes and the number of columns is one plus the degrees of freedom (DOF) specified in the Control Information. The number of rows in the nodal displacement data table is equal to the number of nodes where the displacement is specified and the number of columns is one plus the degrees of freedom specified in the Control Information. In either table, if there is large number of nodes then a vertical scroll bar will appear on the table to facilitate input of data. For programs P1, P4, P5, and P6 there are two DOF while P2, P3, P7 and P8 have three DOF. For programs P1, P2, P4, P5 and P8 except P3 the DOF refers to the displacements u, v, and w (if applicable) in Global Co-ordinates. For P3 degrees of freedom are u, v, and the rotation θ. For programs P6 and P7 the degrees of freedom refer to the displacements ur,uz, and uθ.

510

Finite Element Method with Applications in Engineering As soon as both the tables are filled Material Characteristic Data button will appear on the screen. the Material Characteristic Data button will open the screen shown in Figure F.9.

Figure F.9 5. MATERIAL CHARACTERISTIC DATA The table contains number of rows corresponding to the Number of Material Sets specified in the Control Information. The columns are for, from left to right, assigning the material set number, element type, (which are 1 to 8 for programs P1 to P8, respectively), and the material data shown on the screen for each program. Each Material Set will have different material data. If there is large number of material sets and/or material data, then a vertical and/or horizontal scroll bars will appear on the table to facilitate input of data. Note that it is possible to import (export) the material characteristic data from (to) a Microsoft Excel file. (For details, see the Co-ordinates section.) ®

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As soon as the table is filled in, the Nodal Load Data and Distributed Load button will appear on the screen. the Nodal Load Data and Distributed Load button will open the screen shown in Figure F.10. 6. NODAL LOAD DATA AND DISTRIBUTED LOAD This screen contains two tables, one for Nodal Load Data and the other Distributed Load Data. The number of rows in the Nodal Load Data table is equal to the Number of Loaded Nodes and the number of columns is one plus the number of degrees of freedom specified in the Control Information. The Distributed Load Data table if active (it is active only for 511

Finite Element Method with Applications in Engineering programs P3, P6 and P7) will have number of rows equal to the Number of Elements with Distributed Load, and number of columns for P3 will be two and for P6 and P7 are three. As soon as the last entry is filled in the table, the Execute button is enabled. Also enabled at this time is the Save Input button. To store all the input data for later use or modifications this button and the screen shown in Figure F.4 will be displayed; browse for the desired location and save the .din file with an appropriate file name. The program can be executed by the Execute button. When the execution is finished, the Output and Result buttons will be enabled. the Output button to view the output in a window that looks like Figure F.11.

Figure F.10 To view the complete output, scroll by using the scroll bar(s). To print the output, the Print button. (Note that the output will be sent to the default Microsoft Windows printer.) To save the output, the Save Output button and the screen shown in Figure F.4 will be displayed; browse for the desired location and save the file with an appropriate file name. ®

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the Result button to view the results in a window that looks like Figure F.12. To view the complete result, scroll by using the scroll bar(s). To print the results, the Print button. (Note that the results will be sent to the default Microsoft Windows printer.) To save the results, the Save Results button and the screen shown in Figure F.4 will be displayed; browse for the desired location and save the file with an appropriate file name. ®

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Finite Element Method with Applications in Engineering

Figure F.11

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Finite Element Method with Applications in Engineering Figure F.12

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Appendix G

COMPUTER PROGRAMS FOR ONEDIMENSIONAL AND TWODIMENSIONAL PROBLEMS G.1 COMPUTER PROGRAM FOR THE FLOW NETWORK ANALYSIS For the pipe network problem considered in Chapter 6, Section 6.11, a computer program in one-dimensional has been developed using the Matlab package. The detailed program listing is given below. The program is tested with the problem mentioned in Section 6.11.1. The data file is embedded in the program listing. %---------------------------Listing 1 Matlab Program------------------------------------% Program to analyse water distribution network % written by B.V. Rao and modified by T.I. Eldho, C.E.Dept, I.I.T.,Bombay % H - pressure heads in meters 1 × n % pipe_length[1:m] % Q Discharge in the pipe [1:m] % A(n,n) global matrix % n number of nodes % m number of pipes % C consumption array [1:n] % f friction factor % T transmittivity array [1:m] - (12.1*D^5/(f*L))^0.5 % HF headloss in pipes [1:m] % R right hand side array [1:n] % Conn_nodes(m,2) shows how the pipes are connected to nodes % Valp(npv,2) values to be prescribed before solving global matrix % npv numbers of nodes where the values are prescribed % p(m) temporary array % Dia(m) diameter of pipes Q=[];HF=[];p=[];R=[];R1=[]; H=[2, 1.91, 1.92, 1.93, 1.94, 1.95]; pipe_length=[1,1,1,1,1,1,1.5,1.5,1.5]; Dia=[0.01,0.01,0.01,0.01,0.01,0.01,0.012,0.012,0.012]; 515

Finite Element Method with Applications in Engineering C=[125,-25,-25,-25,-25,-25]*1.0e-6/60; Valp=[1,2]; Conn_nodes=[1 2;2 4;1 3;4 6;3 5;5 6;1 4;3 4;3 6]; m=9;n=6;npv=1;f=0.02; itmax=25; A=[1:n;1:n]; for i=1:m p(i)= sqrt(12.1*Dia(i)^5/pipe_length(i)/f); end it=0;s1=1.0; % zeros(size(A)); while s1 > 1.0e-6 & it