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1 Background and application 1

2 Introduction to the method 5

2.1 General theory 5

2.2 A simple one-dimensional element: the pin-jointed bar 6

2.2.1 Stress analysis of a stepped bar 9

2.3 A simple one-dimensional element: the thermal rod 12

2.3.1 Heat conduction through a wall 13

2.4 Conclusions 15

Problems 15

3 Discretization of the problem 19

3.1 Introduction 19

3.2 Geometrical approximations 19

3.3 Simplification through symmetry 20

3.4 Basic element shapes and behaviour 23

3.5 Choice of element type 25

3.6 Size and number of elements 29

3.7 Element shape and distortion 31

3.8 Location of nodes 33

3.9 Node and element numbering 34

3.10 Conclusions 37

Problems 38

4 Interpolation functions and simplex elements 39

4.1 Introduction 39

4.2 Simplex, complex and multiplex elements 41

4.3 Linear interpolation polynomials for simplex elements 43

4.4 Natural coordinates 49

4.5 Vector quantities 54

4.6 An axisymmetric element 56

4.7 Conclusions 57

Problems 57

5 Formulation of the element characteristic matrices and vectors for elasticity problems 61

5.1 Introduction 61

5.2 The variational formulation 61

5.3 One-dimensional elasticity 67

5.4 Two-dimensional elasticity 76

5.5 Three-dimensional elasticity 85

5.6 Axisymmetric elasticity 87

5.7 Conclusions 93

Problems 94

6 Formulation of the element characteristic matrices and vectors for field problems 101

6.1 Introduction 101

6.2 Formulation procedures 103

6.2.1 The variational formulation 103

6.2.2 The weighted residual method 106

6.2.3 Summary 109

6.3 Thermal problems 110

6.3.1 One-dimensional heat transfer 110

6.3.2 Two-dimensional heat transfer 123

6.3.3 Three-dimensional heat transfer 128

6.3.4 Axisymmetric heat transfer 130

6.4 Torsion problems 138

6.5 Fluid flow problems 146

6.6 Conclusions 146

Problems 148

7 Assembly and solution of the finite element equations 153

7.1 Introduction 153

7.2 Coordinate transformations 153

7.3 Assembly of the element equations 156

7.4 Incorporation of the boundary conditions 162

7.5 Solution of the equations 164

Problems 166

8 Higher-order element formulations 169

8.1 Introduction 169

8.2 Natural coordinate systems and numerical integration 171

8.3 Higher-order one-dimensional elements 174

8.3.1 Quadratic and cubic elements 174

8.3.2 Evaluation of the element equations 176

8.3.3 An alternative formulation 183

8.4 Higher-order two- and three-dimensional elements 185

8.4.1 Isoparametric triangular elements 185

8.4.2 Isoparametric quadrilateral elements 195

8.4.3 Isoparametric solid elements 201

8.4.4 Stress and heat flow calculations 203

8.5 Structural beam, plate and shell elements 203

8.6 Convergence requirements of interpolation functions 212

8.7 Conclusions 214

Problems 215

9 Modelling procedures and results processing 221

9.1 Introduction 221

9.2 Model validity and accuracy 221

9.3 Mesh design and refinement 230

9.4 Element distortion 239

9.5 Results processing 243

9.6 Model checking 245

10 Further applications of the finite element method 249

10.1 Introduction 249

10.2 Non-linear static elasticity problems 249

10.2.1 Material non-linearity 251

10.2.2 Geometric non-linearity 252

10.2.3 Combined material and geometric non-linearity 255

10.2.4 Non-linear contact conditions 256

10.3 Buckling problems 256

10.4 Dynamic problems 257

10.4.1 Modal analysis 258

10.4.2 Transient response analysis 259

10.4.3 Harmonic response analysis 262

10.4.4 Shock spectrum analysis 265

10.5 Transient thermal problems 267

11 Symmetry, submodelling and validation 269

11.1 Introduction 269

11.2 Symmetric models with non-symmetric loading 269

11.3 Submodelling and substructuring 275

11.4 Element and program validation 278

12 Commercial finite element programs 285

12.1 Introduction 285

12.2 General facilities 285

12.3 Pre-processors 287

12.3.1 Direct user input 287

12.3.2 Automatic mesh generation 293

12.4 Post-processors 295

Bibliography 297

Glossary 299

Index 311