COMPUTATIONAL ELASTICITY

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1 COMPUTATIONAL ELASTICITY Theory of Elasticity and Finite and Boundary Element Methods Mohammed Ameen Alpha Science International Ltd. Harrow, U.K.

Contents Preface Notation vii xi 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.

2 Contents Preface Notation vii xi PART A: THEORETICAL ELASTICITY Chapter Chapter Chapter 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 Notation and sign Convention for stresses Stress Components on an Arbitrary Plane Stress transformation Differential Equations of Equilibrium Principal Stresses and Principal Planes State of Stress Referred to the Principal Coordinate System Stress Ellipsoid Stress Quadric Octahedral stresses Maximum shear stress

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.

3 XIV COMPUTATIONAL ELASTICITY Mohr's circle Hydrostatic and Deviatoric Components Traction boundary conditions 63 Summary 64 Problems 65 Chapter 4 The Constitutive Relations Introduction Generalised Hooke's Law Isotropie Elasticity Orthotropic Elasticity 74 Summary 76 Problems 77 Chapter 5 Cartesian Tensors and Equations of Elasticity Introduction Transformation Laws of Cartesian Tensors Zeroth Order tensors scalars First order tensors vectors Second order tensor dyadic n" 1 order tensor Special Tensors and Tensor Operations The Kronecker's symbol The permutation symbol The e-<sidentity Symmetry and skew-symmetry Contraction Derivatives and the comma notation Gauss' theorem The base vectors and some special vector Operations Eigenvalue problem of a Symmetrie second order tensor Equations of Elasticity Equations of equilibrium Stress-strain relations Strain-displacement and compatibility relations Boundary conditions Boundary Value Problems of Elasticity Lame-Navier equation Beltrami-Michell equations 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.

4 CONTENTS xv Chapter 6 Two-Dimensional Problems of Elasticity Introduction Plane Stress and Plane Strain Problems Plane stress problems Plane strain problems Solution of Plane Problems in Rectangular Coordinates The Stress Function Approach Airy's stress function Solution by polynomials Saint Venant's principle Two-Dimensional Problems in Polar Coordinates Equations of equilibrium in polar coordinates Stress function approach Stress-strain relations Strain-displacement relations Problems of symmetrical stress distributions Lame's problem Pure bending of curved bars Bending of a curved bar by a concentrated force Rotating circular disk Stress concentration around circular holes 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 Introduction Saint Venant's Semi-Inverse Method Prandtl's Membrane Analogy Narrow rectangular cross-section Torsion of Thin Rolled Profile Sections Torsion of Rectangular Bar Torsion of Hollow Shafts Approximate Analysis of Torsion of Thin Tubes Hollow Tubes with Multiple Holes 197 Summary 201 Problems 202 Chapter 8 Energy Theorems and Variational Principles of Elasticity Introduction Strain Energy and Complementary Energy 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.

5 XVI COMPUTATIONAL ELASTICITY 8.4 Virtual Work and Potential Energy Principles Principle of Complementary Potential Energy Betti's Reciprocal Theorem Principle of Linear Superposition Uniqueness of Elasticity Solution 222 Summary 223 Some of the Classical Books on the Theory of Elasticity 224 PARTB: COMPUTATIONAL ELASTICITY Chapter 9 Introduction to Computational Elasticity "Exact" Methods and "Approximate" Methods The Finite Element and the Boundary Element Methods Advantages and Limitations Weighted Residual Methods Some basic terminologies and definitions 235 Summary 248 Chapter 10 Finite Element Method in a Nutshell Introduction Governing Equations of Elasticity Basic Steps Involved in Finite Element Analysis of Elastostatic Problems Details of the ConstantStrainTriangle Element Assemblyof Equations Some of the Programming Preliminaries A Simple Computer Program in C++ Using Triangulär Elements Plotting the Mesh Another Example Additional Aspects Prescribed nonzero degrees offreedom Sparsity of Stiffhess Matrix Proper node numbering Stress Computation Support Reactions 298 Summary 299 Problems 300 Chapter 11 Isoparametric Formulation Introduction Sub, Super and Isoparametric Formulations The Isoparametric Formulation 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 CONTENTS xvu 11.5 OOP Using Vector and Matrix Classes Object Oriented Programming A Vector Class A Matrix Class Computer Code with Isoparametric Quadrilateral Elements Isoparametric Lagrangian Elements for Plane Problems Serendipity Elements Transition Elements 345 Summary 346 Problems 346 Chapter 12 Advanced Topics in Finite Element Analysis Introduction General Rule of Transformation Static Condensation Analysis oflarge Structures Substructuring Skew Supports Setting Identical Displacement Boundary Conditions at Two or More Distinct Nodes Analysis of Symmetrie Structures Some Aspects Regarding Finite Element Mesh Automatic mesh generation programs 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 Introduction The Reciprocal Theorem and the Somigliana Identity Boundary Integral Equation Numerical Solution of Boundary Integral Equations Boundary Elements and Interpolation of Displacements and Tractions Stresses on the Boundary Body Forces Constant gravity force Centrifugal force Piecewise Homogeneous Bodies 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.

7 XVU1 COMPUTATIONAL ELASTICITY Chapter 14 Boundary Elements, Interpolation Functions and Singular Integrals Introduction Two-Dimensional Problems Constant dement Linear isoparametric dement Higher order isoparametric elements Three-Dimensional Problems Constant triangulär elements Linear and higher order triangulär elements Quadrilateral elements Higher order elements Three-dimensional volume elements Discontinuous Boundary Elements 410 Summary 411 Chapter 15 Computer Codes For Two-Dimensional Boundary Element Analysis Introduction Computer Code with Two-Noded Linear Boundary Elements The main program Thefünctions Gauss elimination Operator for unsymmetric matrices Computer Code with Three-Noded Isoparametric Quadratic Boundary Elements The main program The function qfuncs The functiony'acs The function boundarystresses Functions for plotting the boundary element mesh Sample Problems 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 Introduction Coupling Finite Element and Boundary Element Solutions 468

CONTENTS XIX 16.2.1 Symmetrising Ä 5 using a direct algorithm 470 16.2.2 Symmetrisation using an iterative algorithm 470 16.2.3 An alternative way of coupling 471 16.

8 CONTENTS XIX Symmetrising Ä 5 using a direct algorithm Symmetrisation using an iterative algorithm An alternative way of coupling 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

Contents as of 12/8/2017. Preface. 1. Overview...1

Contents as of 12/8/2017. Preface. 1. Overview...1 Contents as of 12/8/2017 Preface 1. Overview...1 1.1 Introduction...1 1.2 Finite element data...1 1.3 Matrix notation...3 1.4 Matrix partitions...8 1.5 Special finite element matrix notations...9 1.6 Finite