Ce4257 1 Introduction [PDF]

Zitiervorschau

CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

The National University of Singapore Department of Civil Engineering

CE 4257 - LINEAR FINITE ELEMENT ANALYSIS Venue: Room E3-06-08 Course Lecturers: a. Professor Quek Ser Tong Room E1A-02-15 Email: [email protected], Tel: 6516 2263. b. Professor Koh Chan Ghee Room E1A-05-06 Email: [email protected], Tel: 6516 2163. Teaching Assistant: a. Ms Anastasia Santoso (first part of CE4257) Room E1A-02-04, Email: [email protected], Tel: 6516 4838. Consultation from 5 p.m. to 6 p.m. (before lecture starts) Assessment: -

Marked Tutorial Assignments: Project 1: Project 2 (or Quiz): Final exam:

Modular Credit Workload Prerequisites

10% + 10% 10% 10% 60% on 24 Nov 10

4 3-0-0-0-7 CE3155 or equivalent

Matlab cluster (last year) E1-04-09, E1-04-10, E2-03-06, E2-03-07, E2-03-08, E2-03-09. _____________________________________________________________________________ D:\...\CE4257-1-INTRODUCTION 1 Last printed 8/7/2010 11:13 PM

CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

The objective of this module is to equip students with fundamentals of finite element principles so as to enable them to understand the behavior of various finite elements and to be able to select appropriate elements to solve physical and engineering problems. Upon completion of this course, student will be able to:  analyse linear, axisymmetric and/or field problems using appropriate finite elements  engage in further studies on advanced finite element procedures

BROAD OUTLINE 1. 2. 3. 4 5 6 7 8

Introduction One-Dimensional Elements Equations in Elastic Mechanics Two-Dimensional Elements Weighted Residual Methods Three-Dimensional Stress Analysis Field Problems Special Topics (subject to change)

References 1. Grandin, H., “Fundamentals of the Finite Element Method”, Macmillan Publishing Company, 1986. 2. Zienkiewicz, O.C. and Morgan, K., “Finite Elements and Approximation”, John Wiley and Sons, 1983. 3. Cook, R.D., “Finite Element Modelling for Stress Analysis”, John Wiley and Sons, 1995. 4. Weaver, W. And Johnston, P.R., “Finite Elements for Structural Analysis”, Prentice-Hall, 1984. 5. Beer, G. And Watson, J.O., “Introduction to Finite and Boundary Element Methods for Engineers”, John Wiley and Sons, 1992. 6. D.L. Logan, “A First Course in the Finite Element Method”, Third Edition, Thomson Learning, 2001, TA347.F5L 64 7. J.N. Reddy, “An Introduction to the Finite Element Method”, Second Edition, McGraw-Hill International Editions, Singapore.

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1 INTRODUCTION This module equips students with the fundamentals of finite element principles to enable them to understand the behavior of various finite elements and to be able to select appropriate elements to solve physical and engineering problems. It covers weak formulation element shape function, isoparametric concepts, 1-D, 2-D, 3-D and axisymmetric elements, field problems, modelling, and practical considerations of the finite element method. In this introductory lecture, the motivation for and basic concept behind the finite element method is presented using simple examples. Despite this, key steps in the formulation and resulting equations introduced are general and will be repeatedly encountered throughout this course. Analytical solutions are amenable only for certain simplified situations. For the general case, numerical methods are often employed. Some reasons for using numerical method over analytical approach include: - the problem (e.g. structure) is complicated - there are more than one kind of material within the structure - the materials are non-homogenous and/or nonlinear - the geometry of the structure is complex - the boundary conditions are not simple - the loading is not regular In engineering, there are 3 approaches to solve problems: - displacement-based (displacements are unknowns) - force-based (stresses are the primary unknowns) - mixed formulation (some displacements & stresses are unknowns) We shall illustrate the first approach with a simple problem formulated in 3 different ways. This is then extended to a 2D problem after which a general discussion is given on the finite element method. The objective of CE4257 and the course outline are presented. For part 1 of this course, the proposed schedule of lectures, homework and project assignments are also given.

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.1 Example Problem – Structure with Spring or Rod or Truss Elements k2 k1 P

k3

E1,A1,L1

E2,A2,L2 P E3,A3,L2

How to find the displacement of the central rigid element and the forces in the springs or rod elements?

Reference: D.L. Logan, “ A First Course in the Finite Element Method”, Third Edition, Thomson Learning, 2001, TA347.F5L 64 Chapter 2 (Introduction to Stiffness Method) Chapter 3 (Development of Truss Equations)

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.1.1 Spring (or Truss or Axial) Element f1x

1

k

d1x

2

f2x

d2x

f1x  k (d1x  d 2 x ) f2 x  k (d 2 x  d1x )

 f1x   k     f  2 x   k

 k  d1x    k  d 2 x 

For truss element, k  EA / L where E = Young’s modulus; A = cross-sectional area; L = length of element This is the force-displacement relationship.

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.1.2 Solution by Direct Equilibrium Approach Use all of the following (most important set of steps in mechanics): (1) Nodal equilibrium conditions (2) Compatibility or continuity conditions (3) Material property (or Force-displacement relationship) (4) Boundary conditions (5) Solve for unknown displacements (6) Use the displacement solution to find the forces in the elements using (3) k2 k1 P

k3

Free Body Diagram

f3x f2x(el 2)

2

F3x

k2

3 f3x

d2x(el 2) F1x

d3x 2

f1x k1

2 f (el 1) 2x

1 f1x

f2x(el 2)

f2x(el 1) P

d2x(el 1)

f2x (el 3)

d1x

k3

2 f2x (el 3)

3

d2x(el 3)

f4x d4x

f4x

F4x

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

(1) Nodal equilibrium conditions At node 1: F1x  f1x At node 2: P  f2x (el 1)  f2x (el 2)  f2x (el 3) At node 3: F3 x  f3 x At node 4: F4 x  f4 x (2) Compatibility or continuity conditions Displacements at node 2, d2x, for all 3 springs are the same. (3) Material property (or Force-displacement relationship) Using the relationship in Section 1.1.1 and substitute into the nodal equilibrium equations, we have

F1x  k1d1x  k1d 2x P  k1d1x  k1d 2x  k 2d 2x  k 2d 3 x  k3d 2x  k3d 4 x F3 x  k 2d 2x  k 2d 3 x F4 x  k3d 2x  k3d 4 x  k1  F1x   k1     P   k1 k1  k 2  k 3 F    0  k2  3 x    k3  F4 x   0 (4) Boundary conditions d1x  0, d 3 x  0,

0  k2 k2 0

0  d1x     k 3  d 2 x  0  d 3 x    k 3  d 4 x 

d 4x  0

(5) Solve for unknown displacements By imposing the boundary conditions, we have d 2 x 

P k1  k 2  k 3

(6) Use the displacement solution to find the forces in the elements using (3)

F1x  k1d 2x  k1P /(k1  k 2  k3 ) F3 x  k 2d 2x  k 2P /(k1  k 2  k3 ) F4 x  k3d 2x  k3P /(k1  k 2  k3 )

Why negative sign?

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.1.3 Solution by Direct Stiffness Approach Approach is based on proper superposition of individual element stiffness matrices (such as the one in Section 1.1.1) making up a structure. (1) write element matrix for each element in the structure (2) expand the matrix to accommodate all the degrees of freedom (i.e. displacements at the node) of the structure (3) assemble matrices in (2) to form a global stiffness matrix (actually result of considering force equilibrium at each node) (4) put in the boundary conditions (at each node, either the displacement or the force is unknown but not both) (5) solve for the unknowns, giving the displacements and the forces

1

2

k2

3

k1

4 k3

(1) write element matrix for each element in the structure

 f1x   k1  k1  d1x         f2 x   k1 k1  d 2 x   f2 x     f  4x 

 k3  k  3

 f2 x   k 2     f 3 x    k 2

 k 2  d 2 x    k 2  d 3 x 

 k 3  d 2 x    k 3  d 4 x 

(2) expand the matrix to accommodate all the degrees of freedom (i.e. displacements at the node) of the structure

 f1x   k1  k1     f 2 x    k1 k1  0  0 0    0  0   0

0 0 d1x    0 0 d 2 x  0 0 d 3 x    0 0 d 4 x 

0  0  0     f2 x  0 k 2  f   0  k 2  3 x    0  0

0  k2 k2 0

0 d1x    0 d 2 x  0 d 3 x    0 d 4 x 

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CE4257 LINEAR FINITE ELEMENT METHOD

0  0  0     f2 x  0 k 3  0   0 0     f 4 x  0  k 3

by Professor Quek Ser Tong

0  d1x    0  k 3  d 2 x  0 0  d 3 x    0 k 3  d 4 x  0

(3) assemble matrices in (2) to form a global stiffness matrix

 k1  F1x   k1     P   k1 k1  k 2  k 3 F    0  k2  3 x    k3  F4 x   0

0  k2 k2 0

0  d1x     k 3  d 2 x  0  d 3 x    k 3  d 4 x 

The square matrix is known as the global stiffness matrix. It is the total stiffness matrix of the whole structure with respect to the degrees of freedom d1x , d 2x , d 3 x , d 4 x

(4) put in the boundary conditions (at each node, either the displacement or the force is unknown but not both)

 k1  F1x   k1     P   k1 k1  k 2  k 3 F    0  k2  3 x    k3  F4 x   0

0  k2 k2 0

0  0     k 3  d 2 x  0  0    k 3  0 

(5) solve for the unknowns, giving the displacements and the forces see Section 1.1.2

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.1.4 Solution by Energy Approach We can use the principle of minimum potential energy (applicable only for elastic materials) or the principle of virtual work (applicable for any material behaviour). Both are included in the general category of variational methods. For problems where a variational formulation is not clearly definable, methods of weighted residuals (one of which is the Galerkin method) are often used. The principle of minimum potential energy can be stated as follows: Of all geometrically possible shapes that a body can assume, the true one, corresponding to the satisfaction of stable equilibrium of the body, is identified by a minimum value of the total potential energy. The total potential energy p can be expressed in terms of displacements and for FE formulation, nodal displacements are most convenient. Minimizing p produces the equilibrium equations. Total potential energy p =

internal strain energy U + potential energy of external forces 

Strain energy U = capacity of internal forces (or stresses) to do work through deformation (strains) in the structure For a spring element with force displacement relationship given by F = kx, the differential internal work or strain energy dU for a small change in length is given by the internal force multiplied by the change in displacement through which the force moves, i.e. dU = F.dx = kx.dx

F

The total strain energy is x U  0 kx.dx  1 kx2  1 Fx 2 2

dU k F

k

dx x

x

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

Potential energy of external forces  = capacity of forces (body, surface traction or applied nodal forces) to do work through deformation of the structure Potential energy of external force = - (external work done by force) Negative is because potential energy is lost when work is done by external force, e.g. vertical position of a mass.

F1x

1

k1

k2

2

3

F3x P

k3

4 4

F4x

 p  21 k1(d 2 x  d1x )2  21 k 2 (d 3 x  d 2 x )2  21 k 3 (d 4 x  d 2 x )2  F1x d1x  Pd 2 x  F3 x d 3 x  F4 x d 4 x To minimize total potential energy, p(d1x, d2x, d3x, d4x), set p/dix = 0.

 p / d1x  k1(d 2x  d1x )  F1x  0

 p / d 2x  k1(d 2x  d1x )  k 2 (d 3 x  d 2x )  k 3 (d 4 x  d 2x )  P  0  p / d 3 x  k 2 (d 3 x  d 2x )  F3 x  0  p / d 4 x  k 3 (d 4 x  d 2x )  F4 x  0  k1  F1x   k1     P   k1 k1  k 2  k 3 F    0  k2  3 x    k3  F4 x   0

0  k2 k2 0

0  d1x     k 3  d 2 x  0  d 3 x    k 3  d 4 x 

which is the same as obtained by the first two approaches.

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.2 Another Example – Truss Problem Above example is limited to spring (or rod or truss) elements aligned in the same direction. We shall now consider inclined elements and forces. For example, the truss in the figure below – NEED TO CHANGE THE EQUATIONS SUCH THAT DIRECTION IS TAKEN CARE OF 2

3

k1 k2 

θ

4

1

k3

F 1.2.1 Transformation Matrix y

y x’

d

dy

dx’ y’

dy’

i’

i’sin θ θ

θ

θ dx

x

x i’cos θ

 hand figure above, the (displacement)  In the right vector d can be expressed using either the x-y or the x’-y’ coordinate systems. That is,

d  d x i  d y j  d x ' i 'd y ' j ' where i, j, i’ and j’ are unit vectors in the respective directions. i’ and j’ can be expressed in terms of i and j through angle θ. That is, _____________________________________________________________________________ D:\...\CE4257-1-INTRODUCTION 12 Last printed 8/7/2010 11:13 PM

CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

i '  cos .i  sin . j j '   sin .i  cos . j Substituting into d gives

d  d x i  d y j  d x ' (cos .i  sin . j )  d y ' ( sin .i  cos . j ) Hence, d x  d x ' cos  d y ' sin

 d x  cos Or      d y   sin

d y  d x ' sin  d y ' cos

 sin  d x '  C  S  d x '     S C  d  cos  d y '    y ' 

where C = cos θ and S = sin θ. It can also be shown that

 d x'   C S  d x         dy'     S C  d y     C S The matrix   is known as the transformation (or rotation)  S C  matrix. For example, it can be used to relate the global displacement (or force which is in a certain orientation) to local displacement (or force which may be in a different orientation).

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.2.2 Global Stiffness Matrix for Inclined Element We now formulate the global stiffness matrix (meaning wrt global coordinate system) of an inclined bar (or spring or truss) element y

x’

d2x’

 f1x '   k    f2 x '   k

k   d1x '    k   d2 x ' 

f’ = ke.d’ k d1x’ θ

θ x

d1x '  d1x cos  d1y sin

 f1x '   k    f 1 y '  0  f   k  2x '    f2 y '   0  

0 k 0   d1x '    0 0 0   d1y '  0 k 0   d2 x '    0 0 0   d2 y ' 

d 2x '  d 2x cos  d 2y sin

 d1x     d1x '  C S 0 0  d1y         d 2 x '   0 0 C S  d 2 x   d 2y    C S 0 0  or d’=T*d where T     0 0 C S However, T is not a square matrix and to make future manipulation easier, we can introduce d1y’ and d2y’ (they are zero quantities wrt to figure above).

 d1x '   C S 0 0  d1x   C S 0 0       S C 0 0    d1y '   S C 0 0  d1y     d    0 0 C S  d  with T   0 0 C S   2x'    2x      d 2y '     0 0  S C    0 0  S C  d 2 y 

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

Similarly the forces can be transformed in the same manner as displacements,

 f1x '   C S 0 0  f1x        f1y '   S C 0 0  f1y   f    0 0 C S  f  2x  2x'    f   f2 y '     0 0  S C  2 y   C S 0 0  S C 0 0   or f’=T*f where T    0 0 C S    0 0  S C Hence, f’ = ke*d’ = ke*T*d

or T*f = ke*T*d

giving f = T-1*ke*T*d = kg*d where T-1 is the inverse of T . It can be shown that T-1= TT where TT is the transpose of T . Hence, kg = TT*ke*T For the spring (or rod or truss) element, it can be shown that  C 2 CS C 2 CS    2 2 S  CS  S   kg  k   C2 CS   Sym S 2  

0 1  1  0 0 For the case where θ = 0, k g  k   1  Sym which is to be expected.

0 0  0  0

Hence, for general 2D truss problems, all the members are formulated with respect to the global coordinate system before assembling to get the total stiffness matrix. Solve homework Q1 & Q2. _____________________________________________________________________________ D:\...\CE4257-1-INTRODUCTION 15 Last printed 8/7/2010 11:13 PM

CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.3 Finite Element Method In the above 2 examples, we basically do the following:  split the entire structures into elements (discretization),  formulate the equation for each element,  assemble the equations resulting in a set of simultaneous equations involving unknown nodal displacements,  solve for the nodal displacements  compute the boundary forces from the known displacements  compute the internal forces in each element using the element equations. In the 2 examples, the structure is in reality constructed by assembling physical elements. So, the approach to solve the problem is easy to accept as it reflects reality. The question arises whether such approach can be used to solve more complicated problems. In fact, this has been well-developed to solve not only discrete structures such as truss and frames, but also 2D and 3D problems such as dams, slabs, shell structures, details of joints, cracks in bearings, cooling towers, flow problems, biomechanics such as blood cells and blood flow, linear and nonlinear problems, static, dynamic and impact problems, etc. This numerical method is commonly known as the FINITE ELEMENT METHOD.

Three-dimensional view of structure on top of depot

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

Finite Element Model of Spherical Ball and Steel Bearing

(a)

(b) 2

Von Mises Stress (N/mm ) Distribution in Upper Hemispherical Cap

(a)

(b)

Comparison of FE model with half-space model. (a) deformed FE mesh (Rp * = 0.5 and e* = 0.06) of cell aspirated into micropipette. (b) half-space model. _____________________________________________________________________________ D:\...\CE4257-1-INTRODUCTION 17 Last printed 8/7/2010 11:13 PM

CE4257 LINEAR FINITE ELEMENT METHOD

Hybrid-fiber ECC target

by Professor Quek Ser Tong

Steel projectile

Support

Void

Support

Shared nodes between the hybrid-fiber ECC target and surrounding void area Meshes for hybrid-fiber ECC panel and the surrounding void area.

Void space Void space

Projectile

Hybrid-fiber ECC

Penetration depth

Projectile

Void space (a) 3D view

(b) Side view

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

Finite element softwares are now used in every engineering consulting companies, including those involving structures, geotechnics, hydraulics, pavement, manufacturing, aircraft, ships, chemical, electronics and offshore companies. Why such power? Versatility within one single computer problem. It can almost analysed problems with arbitrary shapes, loads and boundary conditions. It contains element library of different shapes, types and physical properties. Danger? Reliability of program. Intelligent use. Experience and good engineering judgment needed to define a good model. Proper inputs are required. Voluminous output must be sorted and understood. Feeling of correctness of solution is important for complex problems. Why not directly use the FE software but go through a FE course? Many types of elements are available to solve different problems and practitioners must understand how various elements behave to use them and interpret the results properly. Treatment of loads and boundary conditions in the program must be understood. Users must know the limitations, such as material models, so that the problem is appropriately modeled. Examples of FE packages: STRUDL (1965), NASTRAN (1966), PAFEC (1969), SESAM (1969), ANSYS (1970), SAP (1970), DIANA (1972), ADINA (1975), FEAP (1975), DYNA2D & DYNA3D (1978), ABAQUS (1979), COSMOS/M (1982), ALGOR (1984), LSDYNA, SAP2000, ETABS, GTSTRUDL, etc.

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.4 Objective of CE 4257 Linear Finite Element Method The objective of this module is to equip students with fundamentals of finite element principles so as to enable them to understand the behavior of various finite elements and to be able to select appropriate elements to solve physical and engineering problems with emphasis on structural and geotechnical engineering applications. Upon completion of this course, student will be able to:  analyse linear, axisymmetric and/or field problems in structural and geotechnical disciplines using appropriate finite elements  engage in further studies on advanced finite element procedures

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.5 Course Outline 1. 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.2 1.2.1 1.2.2 1.3 1.4 1.5 1.6 1.7 1.8

INTRODUCTION Example – Structure with Spring/Rod/Truss Elements Spring or Truss Element Solution by Direct Equilibrium Approach Solution by Stiffness Approach Solution by Energy Approach Example – 2-D Truss Structure Transformation Matrix Global Stiffness Matrix for Inclined Element Finite Element Method Objective of CE4257 Course Outline References Proposed Schedule Homework & Assignments

2. One-Dimensional Elements 2.1 Uniaxial Stress (or Truss or Axial) Element 2.1.1 Simple Axial Example – Analytical Solution 2.1.2 Constant Strain Element (Linear Element) 2.1.3 Principle of Virtual Work 2.1.4 Summary of Equations for Constant Strain Formulation 2.1.5 Using Finer Elements 2.1.6 Linear Strain Element (Quadratic Element) 2.1.7 Comparison of Results 2.2 Special Considerations 2.2.1 h and p Convergence 2.2.2 Normalized Coordinates 2.2.3 Linear Isoparametric Element 2.2.4 Sub-parametric Element 2.2.5 Numerical Integration 2.3 Beam (flexural) element 2.3.1 Displacement Shape Function 2.3.2 Strain-Displacement Relation 2.3.3 Element Stiffness Matrix 2.3.4 Equivalent (consistent) Nodal Load Vector 2.3.5 Inclined Element 2.3.6 Numerical Example _____________________________________________________________________________ D:\...\CE4257-1-INTRODUCTION 21 Last printed 8/7/2010 11:13 PM

CE4257 LINEAR FINITE ELEMENT METHOD

3. 3.1 3.2 3.3 3.4 3.5

by Professor Quek Ser Tong

Equations in Elastic Mechanics Equilibrium Differential Equations Strain-Displacement and Compatibility Equations Constitutive (Stress-Strain) Relationships Boundary Conditions (Stresses) Summary of equations for 2-D problems

4 Two-Dimensional Elements 4.1 Triangular 3-node element (CST) 4.2 Formulation using normalized (area) coordinates 4.3 Integration using Area Coordinates 4.4 Numerical Example 4.5 Stress Averaging 4.6 Triangular 6-node element (LST) 4.7 Complete, Compatible and Conforming Elements 4.8 Modelling Considerations 4.8.1 Symmetry 4.8.2 Discretization 4.8.3 Stress Concentration Areas 4.8.4 Substructures 4.8.5 Consistent versus Lumped Formulation 4.8.6 Element and Node Numbering 4.9 Bilinear Quadrilateral (Q-4) Element 4.9.1 Isoparametric Quadrilateral 4-node (Q-4) Element 4.9.2 Interpolation Function by Lagrange Family of Elements 4.9.3 Interpolation Function by Lines of Zero Value Method 4.9.4 Strain-Displacement Relationship 4.9.5 Element Stiffness Matrix 4.9.6 Numerical Integration in 2-D 4.9.7 Body Forces, Surfaces Forces and Element Stresses 4.9.8 Numerical Example – Computing B and K matrix 4.10 Quadrilateral 8-node (Q-8 or Serendipity) Element 4.11 Bi-Quadratic (Q-9) Element 4.12 Transition Elements 4.13 Comparison of Performances of Various Elements 4.14 Special Considerations – symmetry, physical checks, stress plots, meshing, infinite elements, patch test, Gauss quadrature, skew boundary conditions, temperature-induced strain 4.15 Axisymmetric Stress Analysis _____________________________________________________________________________ D:\...\CE4257-1-INTRODUCTION 22 Last printed 8/7/2010 11:13 PM

CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

Part 2 – actual details to be decided by Professor Koh Chan Ghee 5 5.1 5.2 5.3 5.4 5.5

Weighted Residual Methods Introduction to Various Methods - collocation, least squares, Galerkin, subdomain Galerkin FE Method Compatibility Requirements Weak Formulation Applications

6

Three-Dimensional Stress Analysis a. Four-Noded Tetrahedra b. Eight-Noded Hexahedron c. Brief Discussion of Higher-Order 3-D Elements d. Applications

7

Field Problems a. Fluid Flow using Laplace Equation b. Finite Element Formulation using Galerkin Method c. Seepage Analysis d. Brief Discussion of Other Applications : Torsion, Heat Transfer,

8

Special Topics (subject to change) a. Special Elements in Common Use : Plate Elements, Interface Elements b. Drained, Undrained and Consolidation Analysis in SoilStructure Interaction Analysis c. Small Strain vs Updated Mesh Analysis d. An example analysis involving superstructure, foundation, and soil to illustrate how structural and geotechnical engineers model the problem from different angles. e. Limitations of Linear Material Behavior Considerations, especially for sands and soft clays. f. Importance of Sensitivity Studies and other Sanity Checks

Note – items in yellow may or may not be taught depending on the lecturer who is teaching that part of the course. _____________________________________________________________________________ D:\...\CE4257-1-INTRODUCTION 23 Last printed 8/7/2010 11:13 PM

CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.6 References 1.

Grandin, H., “Fundamentals of the Finite Element Method”, Macmillan Publishing Company, 1986.

2.

Zienkiewicz, O.C. and Morgan, K., “Finite Elements and Approximation”, John Wiley and Sons, 1983.

3.

Cook, R.D., “Finite Element Modelling for Stress Analysis”, John Wiley and Sons, 1995.

4.

Weaver, W. And Johnston, P.R., “Finite Elements for Structural Analysis”, Prentice-Hall, 1984.

5.

Beer, G. And Watson, J.O., “Introduction to Finite and Boundary Element Methods for Engineers”, John Wiley and Sons, 1992.

6.

D.L. Logan, “A First Course in the Finite Element Method”, Third Edition, Thomson Learning, 2001, TA347.F5L 64.

7.

J.N. Reddy, “An Introduction to the Finite Element Method”, Second Edition, McGraw-Hill International Editions, Singapore.

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

1.7 Proposed Schedule Venue: Room E3-06-08

Day & Time: Wednesday, 6 to 9 p.m.

Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 Term break Lecture 7

Chapter 1, 2 Chapter 2 Chapter 3, 4.5 Chapter 4.6 – 4.10 SAP2000 & Discuss Q1-8 Chapter 4.11 – 4.15

11/8/07 18/8/07 25/8/07 1/9/07 8/9/07 15/9/07 22/9/07 29/9/07

1.8 Homework & Project Assignments Question 1 2 3 4 5 6 7 8 9 10 11 12 13 Project 1 Exam

Group 1 X

Group 2 X

X X X X X X X X (opt) X

X X X X (opt) X

Date to submit 18 August 18 August 25 August 25 August 1 Sept 1 Sept 8 Sept 8 Sept 15 Sept 15 Sept 15 Sept 22 Sept 22 Sept 6 October 24 Nov 10

Chapter 2.1.4 2.1.4 2.3.6 2.3.6 4.4 4.4 4.5 4.5 4.9 4.9 4.9 4.10 4.10

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

Project – topic of your choice but must make use of FE program To include discussion on at least the following: – description of problem (plane stress or strain, 2D or 3D, etc) – choice of FE program, element to use and no. of integration points – mesh size study (uniform vs non-uniform, coarse vs fine mesh) – regions of high and low stresses (mesh distribution) – correctness of solution (what checks you use) – other issues of interest Some examples of problems that have been used – cannot be used again!

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CE4257 LINEAR FINITE ELEMENT METHOD

by Professor Quek Ser Tong

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