《FINITE ELEMENT ANALYSIS THEORY AND PRACTICE》

1 Background and application1

2 Introduction to the method5

2.1 General theory5

2.2 A simple one-dimensional element: the pin-jointed bar6

2.2.1 Stress analysis of a stepped bar9

2.3 A simple one-dimensional element: the thermal rod12

2.3.1 Heat conduction through a wall13

2.4 Conclusions15

Problems15

3 Discretization of the problem19

3.1 Introduction19

3.2 Geometrical approximations19

3.3 Simplification through symmetry20

3.4 Basic element shapes and behaviour23

3.5 Choice of element type25

3.6 Size and number of elements29

3.7 Element shape and distortion31

3.8 Location of nodes33

3.9 Node and element numbering34

3.10 Conclusions37

Problems38

4 Interpolation functions and simplex elements39

4.1 Introduction39

4.2 Simplex, complex and multiplex elements41

4.3 Linear interpolation polynomials for simplex elements43

4.4 Natural coordinates49

4.5 Vector quantities54

4.6 An axisymmetric element56

4.7 Conclusions57

Problems57

5 Formulation of the element characteristic matrices and vectors for elasticity problems61

5.1 Introduction61

5.2 The variational formulation61

5.3 One-dimensional elasticity67

5.4 Two-dimensional elasticity76

5.5 Three-dimensional elasticity85

5.6 Axisymmetric elasticity87

5.7 Conclusions93

Problems94

6 Formulation of the element characteristic matrices and vectors for field problems101

6.1 Introduction101

6.2 Formulation procedures103

6.2.1 The variational formulation103

6.2.2 The weighted residual method106

6.2.3 Summary109

6.3 Thermal problems110

6.3.1 One-dimensional heat transfer110

6.3.2 Two-dimensional heat transfer123

6.3.3 Three-dimensional heat transfer128

6.3.4 Axisymmetric heat transfer130

6.4 Torsion problems138

6.5 Fluid flow problems146

6.6 Conclusions146

Problems148

7 Assembly and solution of the finite element equations153

7.1 Introduction153

7.2 Coordinate transformations153

7.3 Assembly of the element equations156

7.4 Incorporation of the boundary conditions162

7.5 Solution of the equations164

Problems166

8 Higher-order element formulations169

8.1 Introduction169

8.2 Natural coordinate systems and numerical integration171

8.3 Higher-order one-dimensional elements174

8.3.1 Quadratic and cubic elements174

8.3.2 Evaluation of the element equations176

8.3.3 An alternative formulation183

8.4 Higher-order two- and three-dimensional elements185

8.4.1 Isoparametric triangular elements185

8.4.2 Isoparametric quadrilateral elements195

8.4.3 Isoparametric solid elements201

8.4.4 Stress and heat flow calculations203

8.5 Structural beam, plate and shell elements203

8.6 Convergence requirements of interpolation functions212

8.7 Conclusions214

Problems215

9 Modelling procedures and results processing221

9.1 Introduction221

9.2 Model validity and accuracy221

9.3 Mesh design and refinement230

9.4 Element distortion239

9.5 Results processing243

9.6 Model checking245

10 Further applications of the finite element method249

10.1 Introduction249

10.2 Non-linear static elasticity problems249

10.2.1 Material non-linearity251

10.2.2 Geometric non-linearity252

10.2.3 Combined material and geometric non-linearity255

10.2.4 Non-linear contact conditions256

10.3 Buckling problems256

10.4 Dynamic problems257

10.4.1 Modal analysis258

10.4.2 Transient response analysis259

10.4.3 Harmonic response analysis262

10.4.4 Shock spectrum analysis265

10.5 Transient thermal problems267

11 Symmetry, submodelling and validation269

11.1 Introduction269

11.2 Symmetric models with non-symmetric loading269

11.3 Submodelling and substructuring275

11.4 Element and program validation278

12 Commercial finite element programs285

12.1 Introduction285

12.2 General facilities285

12.3 Pre-processors287

12.3.1 Direct user input287

12.3.2 Automatic mesh generation293

12.4 Post-processors295

Bibliography297

Glossary299

Index311