COMPUTATIONAL ELASTICITY

Theory of Elasticity and

Finite and Boundary Element Methods

Mohammed Ameen

Alpha Science International Ltd. Harrow, U.K.

Contents Preface Notation

PART A: THEORETICAL ELASTICITY

Chapter 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11

Chapter 2 2.1 2.2 2.3 2.4 2.5

Chapter 3 3.1 3.2

3.3

3.4 3.5 3.6

Introduction Analysis and Design of Structural Systems Introduction to Elasticity Levels of Observation Problems of Elastostatics Types of Loads The Displacement, the Strain and the Stress Fields The Constitutive Relations Cartesian Tensors and Equations of Elasticity Two-Dünensional Problems of Elasticity Energy Theorems and Variational Principles Computational Elasticity

The Displacement Field and the Strain Field Introduction Elementary Concept of Strain Strain at a Point Principal Strains and Principal Axes Compatibility Conditions Summary Problems

The Stress Field Introduction State of Stress at a Point 3.2.1 Notation and sign Convention for stresses Stress Components on an Arbitrary Plane 3.3.1 Stress transformation Differential Equations of Equilibrium Principal Stresses and Principal Planes State of Stress Referred to the Principal Coordinate System 3.6.1 Stress Ellipsoid 3.6.2 Stress Quadric 3.6.3 Octahedral stresses 3.6.4 Maximum shear stress

vii xi

1 1 3 4 4 5 6 7 7 7 8 8

11 11 12 15 21 27 30 31

33 33 33 34 36 41 43 49

54 55 57 58 59

XIV COMPUTATIONAL ELASTICITY

3.6.5 Mohr's circle 60 3.7 Hydrostatic and Deviatoric Components 62 3.8 Traction boundary conditions 63

Summary 64 Problems 65

Chapter 4 The Constitutive Relations 67 4.1 Introduction 67 4.2 Generalised Hooke's Law 68 4.3 Isotropie Elasticity 73 4.4 Orthotropic Elasticity 74

Summary 76 Problems 77

Chapter 5 Cartesian Tensors and Equations of Elasticity 78 5.1 Introduction 78 5.2 Transformation Laws of Cartesian Tensors 79

5.2.1 Zeroth Order tensors—scalars 79 5.2.2 First order tensors—vectors 80 5.2.3 Second order tensor—dyadic 88 5.2.4 n"1 order tensor 90

5.3 Special Tensors and Tensor Operations 90 5.3.1 The Kronecker's symbol 90 5.3.2 The permutation symbol 90 5.3.3 The e-<Sidentity 91 5.3.4 Symmetry and skew-symmetry 91 5.3.5 Contraction 93 5.3.6 Derivatives and the comma notation 93 5.3.7 Gauss' theorem 93 5.3.8 The base vectors and some special vector

Operations 94 5.3.9 Eigenvalue problem of a Symmetrie second

order tensor 96 5.4 Equations of Elasticity 99

5.4.1 Equations of equilibrium 99 5.4.2 Stress-strain relations 102 5.4.3 Strain-displacement and compatibility relations 105 5.4.4 Boundary conditions 108

5.5 Boundary Value Problems of Elasticity 108 5.5.1 Lame-Navier equation 109 5.5.2 Beltrami-Michell equations 110

- 5.6 Coaxialityof the Principal Directions 112 Summary 112 Problems 113

CONTENTS xv

Chapter 6 Two-Dimensional Problems of Elasticity 115 6.1 Introduction 115 6.2 Plane Stress and Plane Strain Problems 116

6.2.1 Plane stress problems 116 6.2.2 Plane strain problems 118

6.3 Solution of Plane Problems in Rectangular Coordinates —The Stress Function Approach 119 6.3.1 Airy's stress function 122 6.3.2 Solution by polynomials 125 6.3.3 Saint Venant's principle 126

6.4 Two-Dimensional Problems in Polar Coordinates 134 6.4.1 Equations of equilibrium in polar coordinates 134 6.4.2 Stress function approach 137 6.4.3 Stress-strain relations 138 6.4.4 Strain-displacement relations 139 6.4.5 Problems of symmetrical stress distributions 143 6.4.6 Lame's problem 145 6.4.7 Pure bending of curved bars 148 6.4.8 Bending of a curved bar by a concentrated force 151 6.4.9 Rotating circular disk 155 6.4.10 Stress concentration around circular holes 156 6.4.11 Concentrated force at a point of a straight

boundary of a semi-infinite continuum 161 Summary 165 Problems 166

Chapter 7 Torsion of Prismatic Bars 168 7.1 Introduction 168 7.2 Saint Venant's Semi-Inverse Method 168 7.3 Prandtl's Membrane Analogy 179

7.3.1 Narrow rectangular cross-section 183 7.3.2 Torsion of Thin Rolled Profile Sections 186 7.3.3 Torsion of Rectangular Bar 187 7.3.4 Torsion of Hollow Shafts 192 7.3.5 Approximate Analysis of Torsion of Thin Tubes 196 7.3.6 Hollow Tubes with Multiple Holes 197 Summary 201 Problems 202

Chapter 8 Energy Theorems and Variational Principles of Elasticity 203

8.1 Introduction 203 8.2 Strain Energy and Complementary Energy 204 8.3 Clapeyron's Theorem 207

XVI COMPUTATIONAL ELASTICITY

8.4 Virtual Work and Potential Energy Principles 210 8.5 Principle of Complementary Potential Energy 216 8.6 Betti's Reciprocal Theorem 219 8.7 Principle of Linear Superposition 220 8.8 Uniqueness of Elasticity Solution 222

Summary 223 Some of the Classical Books on the Theory of Elasticity 224

P A R T B : COMPUTATIONAL ELASTICITY

Chapter 9 Introduction to Computational Elasticity 229 9.1 "Exact" Methods and "Approximate" Methods 229 9.2 The Finite Element and the Boundary Element

Methods—Advantages and Limitations 232 9.3 Weighted Residual Methods 234

9.3.1 Some basic terminologies and definitions 235 Summary 248

Chapter 10 Finite Element Method in a Nutshell 249 10.1 Introduction 249 10.2 Governing Equations of Elasticity 250 10.3 Basic Steps Involved in Finite Element Analysis of

Elastostatic Problems 253 10.3.1 Details of the ConstantStrainTriangle Element 261 10.3.2 Assemblyof Equations 265

10.4 Some of the Programming Preliminaries 267 10.5 A Simple Computer Program in C++ Using

Triangulär Elements 276 10.5.1 Plotting the Mesh 289 10.5.2 Another Example 292

10.6 Additional Aspects 293 10.6.1 Prescribed nonzero degrees offreedom 293 10.6.2 Sparsity of Stiffhess Matrix—

Proper node numbering 294 10.6.3 Stress Computation 297 10.6.4 Support Reactions 298 Summary 299 Problems 300

Chapter 11 Isoparametric Formulation 301 11.1 Introduction 301 11.2 Sub, Super and Isoparametric Formulations 303 11.3 The Isoparametric Formulation 304 11.4 Four-Noded Quadrilateral Element for Plane Problems 309

CONTENTS xvu

11.5 OOP Using Vector and Matrix Classes 310 11.5.1 Object Oriented Programming 311 11.5.2 A Vector Class 311 11.5.3 A Matrix Class 320

11.6 Computer Code with Isoparametric Quadrilateral Elements 328

11.7 Isoparametric Lagrangian Elements for Plane Problems 342 11.8 Serendipity Elements 343 11.9 Transition Elements 345

Summary 346 Problems 346

Chapter 12 Advanced Topics in Finite Element Analysis 347 12.1 Introduction 347 12.2 General Rule of Transformation 347 12.3 Static Condensation 349 12.4 Analysis ofLarge Structures—Substructuring 351 12.5 Skew Supports 352 12.6 Setting Identical Displacement Boundary Conditions at

Two or More Distinct Nodes 354 12.7 Analysis of Symmetrie Structures 355 12.8 Some Aspects Regarding Finite Element Mesh 357

12.8.1 Automatic mesh generation programs 357 12.8.2 Element connection and grading 362 Summary 363 Some of the Populär Books on Finite Element Method 364

Chapter 13 Boundary Element Analysis of Elastostatic Problems 366 13.1 Introduction 366 13.2 The Reciprocal Theorem and the Somigliana Identity 367 13.3 Boundary Integral Equation 372 13.4 Numerical Solution of Boundary Integral Equations 374 13.5 Boundary Elements and Interpolation of

Displacements and Tractions 377 13.6 Stresses on the Boundary 381 13.7 Body Forces 383

13.2.1 Constant gravity force 384 13.2.2 Centrifugal force 385

13.8 Piecewise Homogeneous Bodies 386 13.9 Modelling Traction Discontinuities 388

Summary 389 Problems 390

XVU1 COMPUTATIONAL ELASTICITY

Chapter 14 Boundary Elements, Interpolation Functions and Singular Integrals 391

14.1 Introduction 391 14.2 Two-Dimensional Problems 392

14.2.1 Constant dement 393 14.2.2 Linear isoparametric dement 396 14.2.3 Higher order isoparametric elements 399

14.3 Three-Dimensional Problems 401 14.3.1 Constant triangulär elements 401 14.3.2 Linear and higher order triangulär elements 403 14.3.3 Quadrilateral elements 406 14.3.4 Higher order elements 407 14.3.5 Three-dimensional volume elements 408

14.4 Discontinuous Boundary Elements 410 Summary 411

Chapter 15 Computer Codes For Two-Dimensional Boundary Element Analysis 412

15.1 Introduction 412 15.2 Computer Code with Two-Noded Linear Boundary

Elements 413 15.2.1 The main program 414 15.2.2 Thefünctions 426 15.2.3 Gauss elimination Operator for unsymmetric

matrices 430 15.3 Computer Code with Three-Noded Isoparametric

Quadratic Boundary Elements 431 15.3.1 The main program 432 15.3.2 The function qfuncs 442 15.3.3 The functiony'acs 447 15.3.4 The function boundaryStresses 448 15.3.5 Functions for plotting the boundary

element mesh 448 15.4 Sample Problems 449 15.5 An Improved Boundary Element Formulation with

Relative Displacements 462 Summary 464 Problems 465 Some of the Books on Boundary Element Method 465

Chapter 16 Coupling Finite Element and Boundary Element Methods 467

16.1 Introduction 467 16.2 Coupling Finite Element and Boundary Element

Solutions 468

CONTENTS XIX

16.2.1 Symmetrising Ä5 using a direct algorithm 470 16.2.2 Symmetrisation using an iterative algorithm 470 16.2.3 An alternative way of coupling 471

16.3 An Example Application-Analysis of Reinforced Concrete Structural Elements 472 Summary 476

Appendix A Interpolation Polynomials 477 A.l Introduction 477 A.2 Lagrangian Interpolation in One Dimension 477 A.3 Two-Dimensional Interpolation 481

Appendix B Numerical Integration 486 B.l Standard Gauss Quadrature 486 B.2 Logarithmic Gauss Quadrature 488

Appendix C Integral Equations 490 C. 1 Definition and Classification of Integral Equations 490 C.2 Cauchy Principal Value of an Integral 491

Appendix D Fundamental Solutions 493 D. 1 Laplace Equation 493 D.2 Fundamental Solution of Elastostatic Problems 497

Subject Index 501