Preface. Symbols and abbreviations. 1.1 Approach A brief historical note Notation and layout Organization 9

Transcription

1 Contents Preface Symbols and abbreviations Software agreement v xv xix 1. Introduction Approach 1 The engineering approach 1 The mathematical approach A brief historical note Notation and layout Organization 9 2. Summary of matrix structural analysis Conventions, assupmtions and simple beam theory 15 Reference coordinates and kinematic degrees of freedom 15 Basic assumptions 17 EULER-BERNOULLI beam theory The fundamental requirements in matrix notation 21 Computational model 21 Force-displacement 23 Static equilibrium 25 Kinematic compatibility 27 Discussion Principle of virtual displacements (PVD) The system stiffness relation 33 vii

2 2.5 Distributed loading - element load vector Transformations Summary of linear elasticity theory Three-dimensional stress analysis 53 Stress and equilibrium 53 Strains and kinematic compatibility 61 Stress-strain relationship 65 Initial strain Axisymmetric stress and strain Stress and strain in two dimensions 73 Plane stress 73 Plane strain Stress and strain in beams Plate bending 81 Plate kinematics 81 Stress-strain 85 Equilibrium 87 Strain energy Mathematical basics of FEM The approximate nature of FEM - the basic assumption Example problem Strong and weak form Principle of minimum potential energy (PMPE) 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 Weighted residual methods - GALERKIN General formulation of FEM using PVD Element analysis I Natural coordinates and interpolation 137 viii

3 5.1 Types and classifications of elements 137 Characteristics of an individual element 137 Classification of elements Natural coordinates 139 One-dimensional element 141 Plane (2D) rectangle 143 The cuboid 143 Plane triangle - area coordinates 145 Tetrahedron Polynomials Nodal points and degrees of freedom The shape functions Indirect interpolation - generalized displacements Direct interpolation 169 C 0 elements 169 C 1 elements 179 Hierarchic C 0 elements (2D) 181 Summary of chapter Element analysis II Mapping and numerical integration Mapping - isoparametric formulation node C 0 quadrilateral 193 Higher order C 0 quadrilaterals - curved edges 203 Triangular, isoparametric C 0 elements Numerical integration 207 One-dimensional element 209 Two- and three-dimensional elements Integration schemes 219 Full integration 219 Reduced integration 221 Selective reduced integration Integration and convergence Element instabilities - mechanisms 225 ix

4 Summary of chapter Element analysis III Element loads and stresses Static equivalent nodal point loads 235 Consistent element load vector 235 Load lumping 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 Accuracy and convergence Energy bounds in R-R consistent FEM solutions Error and rate of convergence Error estimates Convergence criteria Element tests 281 Eigenvalue test 283 The patch test Exact solution at the nodal points 291 Summary of chapter System analysis Mesh generation Storage formats and node renumbering 299 Staorage formats 299 Renumbering schemes Assembly of K and R Boundary conditions - an overview 305 Definitions 305 Implementation 309 Formal treatment 309 x

5 9.5 Boundary conditions - elimination of dofs Boundary conditions - LAGRANGE multipliers Boundary conditions - penalty functions Boundary conditions - rigid elements Solution of Kr = R Static condensation and substructure analysis Numerical issues Programming issues Programming paradigms and languages Data structures and storage formats Stiffness matrix for an isoparametric element Two typical FEM programs 365 Program CrossX 365 Program FEMplate Plane stress and plane strain Triangular elements 371 The linear triangle 371 The quadratic triangle 373 The cubic triangle 373 Formulation for computer implementation Quadrilateral elements node element - the basic version node element - incompatibleincompatible version 391 Stabilization - hourglass stiffness 399 Higher order quadrilateral elements 401 Discussion Boundary conditions and singularities Axisymmetric stress analysis Axisymmetric loading 409 Isoparametric elements 415 Load vectors 415 xi

6 Boundary conditions 417 Convergence Non-symmetric loading Three-dimensional stress analysis The basics 425 Theory of elasticity 425 Element shapes and natural coordinates Common solid elements 429 Hexahedral elements 429 Tetrahedral elements Bending of beams and plates The two-dimensional beam problem 435 EULER-BERNOULLI beam theory 437 TIMOSHENKO beam theory 437 MINDLIN beam theory 443 A discrete KIRCHHOFF element 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 Triangular (thin) plate bending elements 461 The MORLEY triangle - T6 463 Cubic triangle - T Quartic triangle - T The quintic triangle - T21 and T MINDLIN plate bending elements Discrete KIRCHHOFF elements A hybrid 9-node triangular element A note on boundary conditions Comparison of some plate elements 491 xii

7 15. Arches and shells Curved beams and arches The shell problem 507 Shells of revolution 507 Flat shell elements 511 Thick shell elements 517 Solid elements in shell analysis ST. VENANT torsion ST. VENANT torsion - theoretical basis Finite element torsion analysis 525 Shear stress distribution and torsional stiffness 529 Position of shear centre Finite element shear analysis of prismatic beam 533 Theoretical approach Numerical examples 537 Rectangular, massive sections 537 Massive circular sections Practical use of FEM The Sleipner accident Advice and guidelines 547 Know your program and your own limitations 547 From structure to FEM model 551 Interpretation and assessment of results Personal comments The past The present The future References 575 xiii

8 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 MPQ D.3 Subroutine MPQ D.4 Subroutine MPQ D.5 Subroutine MPQ D.6 Subroutine SHPQ D.7 Subroutine GAUSQ2 647 Index 649 xiv