Contents Preface Symbols and abbreviations Software agreement v xv xix 1. Introduction 1 1.1 Approach 1 The engineering approach 1 The mathematical approach 3 1.2 A brief historical note 5 1.3 Notation and layout 7 1.4 Organization 9 2. Summary of matrix structural analysis 15 2.1 Conventions, assupmtions and simple beam theory 15 Reference coordinates and kinematic degrees of freedom 15 Basic assumptions 17 EULER-BERNOULLI beam theory 17 2.2 The fundamental requirements in matrix notation 21 Computational model 21 Force-displacement 23 Static equilibrium 25 Kinematic compatibility 27 Discussion 29 2.3 Principle of virtual displacements (PVD) 29 2.4 The system stiffness relation 33 vii
2.5 Distributed loading - element load vector 37 2.6 Transformations 39 3. Summary of linear elasticity theory 53 3.1 Three-dimensional stress analysis 53 Stress and equilibrium 53 Strains and kinematic compatibility 61 Stress-strain relationship 65 Initial strain 69 3.2 Axisymmetric stress and strain 71 3.3 Stress and strain in two dimensions 73 Plane stress 73 Plane strain 75 3.4 Stress and strain in beams 77 3.5 Plate bending 81 Plate kinematics 81 Stress-strain 85 Equilibrium 87 Strain energy 87 4. Mathematical basics of FEM 91 4.1 The approximate nature of FEM - the basic assumption 91 4.2 Example problem 97 4.3 Strong and weak form 97 4.4 Principle of minimum potential energy (PMPE) 103 4.5 The RAYLEIGH-RITZ method 103 Equilibrium in the R-R process 109 Accuracy and convergence of the R-R process 111 Problems with classical R-R solutions 111 RAYLEIGH-RITZ as a finite element method 113 4.6 Weighted residual methods - GALERKIN 119 4.7 General formulation of FEM using PVD 123 5. Element analysis I Natural coordinates and interpolation 137 viii
5.1 Types and classifications of elements 137 Characteristics of an individual element 137 Classification of elements 139 5.2 Natural coordinates 139 One-dimensional element 141 Plane (2D) rectangle 143 The cuboid 143 Plane triangle - area coordinates 145 Tetrahedron 151 5.3 Polynomials 153 5.4 Nodal points and degrees of freedom 155 5.5 The shape functions 157 5.6 Indirect interpolation - generalized displacements 163 5.7 Direct interpolation 169 C 0 elements 169 C 1 elements 179 Hierarchic C 0 elements (2D) 181 Summary of chapter 5 185 6. Element analysis II Mapping and numerical integration 191 6.1 Mapping - isoparametric formulation 191 4-node C 0 quadrilateral 193 Higher order C 0 quadrilaterals - curved edges 203 Triangular, isoparametric C 0 elements 205 6.2 Numerical integration 207 One-dimensional element 209 Two- and three-dimensional elements 217 6.3 Integration schemes 219 Full integration 219 Reduced integration 221 Selective reduced integration 221 6.4 Integration and convergence 221 6.5 Element instabilities - mechanisms 225 ix
Summary of chapter 6 229 7. Element analysis III Element loads and stresses 235 7.1 Static equivalent nodal point loads 235 Consistent element load vector 235 Load lumping 237 7.2 Stress recovery and stress smoothing 245 Stresses from computed displacements 247 Interpolation / extrapolation 249 Nodal point averaging 251 Global smoothing 251 Stresses from nodal point forces 255 8. Accuracy and convergence 265 8.1 Energy bounds in R-R consistent FEM solutions 269 8.2 Error and rate of convergence 271 8.3 Error estimates 275 8.4 Convergence criteria 277 8.5 Element tests 281 Eigenvalue test 283 The patch test 285 8.6 Exact solution at the nodal points 291 Summary of chapter 8 293 9. System analysis 295 9.1 Mesh generation 295 9.2 Storage formats and node renumbering 299 Staorage formats 299 Renumbering schemes 301 9.3 Assembly of K and R 303 9.4 Boundary conditions - an overview 305 Definitions 305 Implementation 309 Formal treatment 309 x
9.5 Boundary conditions - elimination of dofs 311 9.6 Boundary conditions - LAGRANGE multipliers 317 9.7 Boundary conditions - penalty functions 321 9.8 Boundary conditions - rigid elements 327 9.9 Solution of Kr = R 329 9.10 Static condensation and substructure analysis 333 9.11 Numerical issues 339 10. Programming issues 345 10.1 Programming paradigms and languages 347 10.2 Data structures and storage formats 349 10.3 Stiffness matrix for an isoparametric element 359 10.4 Two typical FEM programs 365 Program CrossX 365 Program FEMplate 367 11. Plane stress and plane strain 369 11.1 Triangular elements 371 The linear triangle 371 The quadratic triangle 373 The cubic triangle 373 Formulation for computer implementation 375 11.2 Quadrilateral elements 385 4-node element - the basic version 385 4-node element - incompatibleincompatible version 391 Stabilization - hourglass stiffness 399 Higher order quadrilateral elements 401 Discussion 403 11.3 Boundary conditions and singularities 405 12. Axisymmetric stress analysis 409 12.1 Axisymmetric loading 409 Isoparametric elements 415 Load vectors 415 xi
Boundary conditions 417 Convergence 417 12.2 Non-symmetric loading 419 13. Three-dimensional stress analysis 425 13.1 The basics 425 Theory of elasticity 425 Element shapes and natural coordinates 427 13.2 Common solid elements 429 Hexahedral elements 429 Tetrahedral elements 433 14. Bending of beams and plates 435 14.1 The two-dimensional beam problem 435 EULER-BERNOULLI beam theory 437 TIMOSHENKO beam theory 437 MINDLIN beam theory 443 A discrete KIRCHHOFF element 451 14.2 Beam element in 3D space 453 Deformation stiffness for a double symmetric cross section 453 Including the rigid body modes 455 Arbitrary cross section 455 Transformation to global axes 457 Offset nodes - eccentricity 459 14.3 Triangular (thin) plate bending elements 461 The MORLEY triangle - T6 463 Cubic triangle - T10 471 Quartic triangle - T15 471 The quintic triangle - T21 and T18 473 14.4 MINDLIN plate bending elements 479 14.5 Discrete KIRCHHOFF elements 485 14.6 A hybrid 9-node triangular element 489 14.7 A note on boundary conditions 491 14.8 Comparison of some plate elements 491 xii
15. Arches and shells 499 15.1 Curved beams and arches 501 15.2 The shell problem 507 Shells of revolution 507 Flat shell elements 511 Thick shell elements 517 Solid elements in shell analysis 517 16. ST. VENANT torsion 519 16.1 ST. VENANT torsion - theoretical basis 521 16.2 Finite element torsion analysis 525 Shear stress distribution and torsional stiffness 529 Position of shear centre 531 16.3 Finite element shear analysis of prismatic beam 533 Theoretical approach 535 16.4 Numerical examples 537 Rectangular, massive sections 537 Massive circular sections 537 17. Practical use of FEM 541 17.1 The Sleipner accident 541 17.2 Advice and guidelines 547 Know your program and your own limitations 547 From structure to FEM model 551 Interpretation and assessment of results 557 18. Personal comments 563 18.1 The past 563 18.2 The present 569 18.2 The future 573 19. References 575 xiii
APPENDIX A Matrix algebra 581 A.1 Definitions 581 A.2 Addition and multiplication 587 A.3 Matrix partitioning - submatrices 593 A.4 Determinants 593 A.5 Linear dependencies - rank 599 A.6 Liear systems of equations 601 A.7 Matrix inversion 601 A.8 Quadratic forms and definiteness 605 A.9 The eigenvalue problem 607 A.10 Matrices and differentiation 609 A.11 Solution by least square approximation 613 B Coordinate transformation 617 C Numerical integration 625 C.1 Quadrilaterals 625 C.2 Triangles 627 D Source code 629 D.1 Introduction 629 D.2 Subroutine MPQ61 630 D.3 Subroutine MPQ62 634 D.4 Subroutine MPQ63 637 D.5 Subroutine MPQ64 641 D.6 Subroutine SHPQ49 644 D.7 Subroutine GAUSQ2 647 Index 649 xiv