The Finite Element Method

Principles and Applications

P. E. Lewis and J. P. Ward Loughborough University of Technology

T T ADDISON-WESLEY PUBLISHING COMPANY

Wokingham, England • Reading, Massachusetts • Menlo Park, California New York • Don Mills, Ontario • Amsterdam • Bonn • Sydney • Singapore Tokyo • Madrid • San Juan • Milan • Paris • Mexico City • Seoul • Taipei

Contents Preliminaries 1.1 Introduct ion 1 1.2 Boundary value problems 3 1.3 Approximate Solution of boundary value problems 7 1.4 Direct approach to finite elements in one dimension 9

Galerkin's Weighted Residual Method 2.1 Introduct ion 15 2.2 Galerkin's me thod for one-dimensional boundary 15

value problems 2.3 The modified Galerkin technique 21 2.4 Matr ix formulation of the modified Galerkin me thod 26

Exercises 32 Supplement 2S.1 The Galerkin me thod — a justification 36 Supplement 2S.2 Modified Galerkin equations for general 39

second-order boundary value problems

Shape Functions for One-dimensional Elements 3.1 Division of region into elements 43 3.2 Local and global node numbers 44 3.3 The linear element 45 3.4 Proper t ies of linear shape functions 47 3.5 Local coordinate Systems 47 3.6 Quadra t ic elements 50 3.7 Some integrals involving quadrat ic shape functions 53 3.8 Some applications of shape functions 54 3.9 Numerical Integration over elements 59

Exercises 63

Finite Element Solution of One-dimensional Boundary Value Problems 4.1 Introduct ion 67 4.2 Roof function approach for Poisson's equat ion 69 4.3 A matr ix approach for Poisson's equat ion 73 4.4 Quadra t ic Solution for Poisson's equat ion 80 4.5 A general one-dimensional equilibrium problem 81 4.6 Summary of the finite element approach 88 4.7 Application of finite elements to a Vibration problem 89 4.8 Fini te elements in one-dimensional heat transfer 95

Exercises 106 Supplement 4S.1 Computer program 110 Supplement 4S.2 Basic equations of heat transfer 119

vii

Supplement 4S.3 Governing equations for vibrat ions of continuous materials

123

Finite Elements and Linear Elasticity 5.1 Introduct ion to linear elasticity 128 5.2 Introduct ion to stress 129 5.3 The strain mat r ix 134 5.4 The consti tutive equations 138 5.5 P lane s train 140 5.6 Strain energy 144 5.7 Potent ia l energy 147 5.8 The direct approach to finite elements 148 5.9 Pin-jointed elements 152 5.10 Two- and three-dimensional pin-jointed s t ructures 157 5.11 Fini te element formulation of plane s t ra in 166

Exercises 172 Supplement 5S.1 Computer programs 176

Finite Element Approximation of Line and Double Integrals 6.1 Introduct ion 183 6.2 Ordinary integrals 184 6.3 Line integrals using quadrat ic elements 186 6.4 Double integrals using tr iangulär and quadri lateral elements 193 6.5 Double integrals using curved elements 204 6.6 The approximation of surfaces 208

Exercises 213

Finite Element Solution of Two-dimensional Boundary Value Problems 7.1 Introduct ion 218 7.2 The Galerkin formulation in two dimensions 219 7.3 Roof function approach 224 7.4 Matr ix formulation for two-dimensional finite elements 231 7.5 Stages in the finite element me thod 245 7.6 Three-noded t r iangulär elements 246 7.7 Matr ix formulation of finite elements in plane strain 257

Exercises 263

Variational Formulation of Boundary Value Problems 8.1 Introduct ion 272 8.2 Functionals and variational calculus 272

8.3 Approximate Solution to variational problems 278 8.4 Construct ion of functionals I 280 8.5 The Ritz method 284 8.6 The Ritz method and finite elements 290 8.7 Matr ix formulation of the Ritz procedure 294 8.8 Equivalence of Ritz and Galerkin procedures 297 8.9 Two-dimensional problems 298 8.10 Construct ion of functionals II 302 8.11 The Ritz method in two dimensions 309

Exercises 318 Supplement 8S.1 Matr ix differentiation 321

Pre- and Post-processing, Assembly and Solution 9.1 Mesh generation 326 9.2 Assembly and Solution 334 9.3 Solution curves 341 9.4 Contour curves 343

10 Fourth-order Boundary Value Problems 10.1 One-dimensional fourth-order problems 351 10.2 Galerkin procedures 355 10.3 Variational formulation 359 10.4 Fini te element procedures 362 10.5 Alternat ive t rea tment of fourth-order problems 366 10.6 Fourth-order boundary value problems 369

in two dimensions Exercises 371

Appendix Shape Functions A L I Definite integrals and elements 375 AI .2 Domain representat ion 376 AI .3 Representat ion of the unknown function 377 AI .4 Shape function derivation 382 AI .5 Two-dimensional elements 387 AI .6 Hermit ian elements 398

Outline Solutions to Odd-numbered Exercises 403

References and Further Reading 417

Index 419