CHAPTER 1 BASIC ASSUMPTIONS AND MATHEMATICAL PRELIMINARIES 1
1.1 Introduction 1
1.2 Basic Assumptions 2
1.3 Coordinate Systems and Transformations 4
1.4 Vector and Matrix Notations and Their Operations 5
1.5 Divergence Theorem 8
Problems/Tutorial Questions 9
CHAPTER 2 STRESSES 10
2.1 Stress and the Stress Tensor 10
2.2 Equilibrium Equations 14
2.3 Traction Boundary Conditions 15
2.4 Stresses on an Oblique Plane 17
2.5 Principal Stresses 18
2.6 Stationary and Octahedral Shear Stresses 22
2.7 Equilibrium Equations in Curvilinear Coordinates 25
Problems/Tutorial Questions 27
CHAPTER 3 STRAINS 30
3.1 Strains 30
3.2 Finite Deformations 33
3.3 Strains in a Given Direction and Principal Strains 39
3.4 Stationary Shear Strains 43
3.5 Compatibility 45
3.6 Kinematic and Compatibility Equations in Curvilinear Coordinates 47
3.7 Concluding Remarks 50
Problems/Tutorial Questions 51
CHAPTER 4 FORMULATION OF ELASTICITY PROBLEMS 54
4.1 Strain Energy Density Function 54
4.2 Generalised Hooke's Law 58
4.3 Initial Stresses and Initial Strains 72
4.4 Governing Equations and Boundary Conditions 73
4.5 General Solution Techniques 75
4.6 St.Venant's Principle 76
Problems/Tutorial Questions 77
CHAPTER 5 TWO-DIMENSIONAL ELASTICITY 80
5.1 Plane Strain Problems 80
5.2 Plane Stress Problems 83
5.3 Similarities and Differences Between Plane Strain/Plane Stress Problems 86
5.4 Airy Stress Function and Polynomial Solutions 86
5.5 Polar Coordinates 93
5.6 Axisymmetric Stress Distributions 96
5.7 Rotating Discs 99
5.8 Stresses Around a Circular Hole in a Plate Subjected to Equal Biaxial Tension-Compression(Pure Shear in the 45°Direction) 103
5.9 Stress Concentration Around a Circular Hole in a Plate Subjected to Uniaxial Tension 105
5.10 Concluding Remarks 107
Problems/Tutorial Questions 107
CHAPTER 6 TORSION OF BARS 114
6.1 Torsion of Bars in Strength of Materials 114
6.2 Warping 114
6.3 Prandtl's Stress Function 119
6.4 Torque 120
6.5 Bars of Circular and Elliptical Cross-Sections 121
6.6 Thin-Walled Structures in Torsion 124
6.7 Analogies 127
Problems/Tutorial Questions 128
CHAPTER 7 BENDING OF BARS 130
7.1 Bending Theory in Strength of Materials 130
7.2 Elasticity Formulation of Bending of Bars 131
7.3 Stress Resultants and Shear Centre 135
7.4 Bending of a Bar of a Circular Cross-Section 136
7.5 Bending of a Bar of an Elliptical Cross-Section 137
7.6 Analogies 138
Problems/Tutorial Questions 139
CHAPTER 8 THE STATE SPACE METHOD OF 3D ELASTICITY 140
8.1 Concept of State and State Variables 140
8.2 Solution for a Linear Time-Invariant System 142
8.3 Calculation of e[A]t 144
8.4 Solution of Linear Time-Variant System 149
8.5 State Variable Equation of Elasticity 153
8.6 Application of State Space Method 157
8.7 Conclusions 166
Problems/Tutorial Questions 166
CHAPTER 9 BENDING OF PLATES 168
9.1 Love-Kirchhoff Hypotheses 168
9.2 The Displacement Fields 169
9.3 Strains and Generalised Strains 170
9.4 Bending Moments 171
9.5 The Governing Equation 172
9.6 Generalised Forces 173
9.7 Boundary Conditions 175
9.8 Rectangular Plates 179
9.9 Circular Plates 183
Problems/Tutorial Questions 188
CHAPTER 10 ENERGY PRINCIPLES 191
10.1 Introduction 191
10.2 Work,Strain Energy and Strain Complementary Energy 191
10.3 Principle of Virtual Work 197
10.4 Application of the Principle of Virtual Work 200
10.5 The Reciprocal Law of Betti 201
10.6 Principle of Minimum Potential Energy 203
10.7 Principle of Virtual Complementary Work 205
10.8 Principle of Minimum Complementary Energy 207
10.9 Castigliano's Theorems 208
10.10 Application of the Principles of Minimum Strain Energy 211
10.11 Rayleigh-Ritz Method 213
Problems/Tutorial Questions 216
CHAPTER 11 SPECIAL TOPICS FOR ELASTICITY 219
11.1 Thermal Elasticity 219
11.2 Propagation of Elastic Waves 226
11.3 Strength Theory,Crack and Fracture 230
Problems/Tutorial Questions 236
REFERENCES 238