CHAPTER 1 INTRODUCTORY CONCEPTS AND MATHEMATICS 1
Part Ⅰ Introduction 1
1-1 Trends and Scopes 1
1-2 Theory of Elasticity 4
1-3 Numerical Stress Analysis 4
1-4 General Solution of the Elasticity Problem 5
1-5 Experimental Stress Analysis 6
1-6 Boundary-Value Problems of Elasticity 6
Part Ⅱ Preliminary Concepts 8
1-7 Brief Summary of Vector Algebra 8
1-8 Scalar Point Functions 12
1-9 Vector Fields 14
1-10 Differentiation of Vectors 15
1-11 Differentiation of a Scalar Field 17
1-12 Differentiation of a Vector Field 18
1-13 Curl of a Vector Field 19
1-14 Eulerian Continuity Equation for Fluids 19
1-15 Divergence Theorem 22
1-16 Divergence Theorem in Two Dimensions 24
1-17 Line and Surface Integrals (Application of Scalar Product) 25
1-18 Stokes's Theorem 27
1-19 Exact Differential 27
1-20 Orthogonal Curvilinear Coordinates in Three-Dimensional Space 28
1-21 Expression for Differential Length in Orthogonal Curvilinear Coordinates 29
1-22 Gradient and Laplacian in Orthogonal Curvilinear Coordinates 30
Part Ⅲ Elements of Tensor Algebra 33
1-23 Index Notation: Summation Convention 33
1-24 Transformation of Tensors under Rotation of Rectangular Cartesian Coordinate System 37
1-25 Symmetric and Antisymmetric Parts of a Tensor 44
1-26 Symbols of δij and ∈ijk (the Kronecker Delta and the Alternating Tensor) 45
1-27 Homogeneous Quadratic Forms 47
1-28 Elementary Matrix Algebra 50
1-29 Some Topics in the Calculus of Variations 55
References 59
Bibliography 61
CHAPTER 2 THEORY OF DEFORMATION 62
2-1 Deformable, Continuous Media 62
2-2 Rigid-Body Displacements 63
2-3 Deformation of a Continuous Region.Material Variables. Spatial Variables 65
2-4 Restrictions on Continuous Deformation of a Deformable Medium 69
Problem Set 2-4 71
2-5 Gradient of the Displacement Vector. Tensor Quantity 72
2-6 Extension of an Infinitesimal Line Element 74
Problem Set 2-6 81
2-7 Physical Significance of ∈ii Strain Definitions 82
2-8 Final Direction of Line Element. Definition of Shearing Strain. Physical Significance of eij(i≠j) 85
Problem Set 2-8 90
2-9 Tensor Character of ∈aβ. Strain Tensor 91
2-10 Reciprocal Ellipsoid. Principal Strains. Strain Invariants 92
2-11 Determination of Principal Strains. Principal Axes 96
Problem Set 2-11 102
2-12 Determination of Strain Invariants. Volumetric Strain 104
2-13 Rotation of a Volume Element. Relation to Displacement Gradients 109
Problem Set 2-13 113
2-14 Homogeneous Deformation 115
2-15 Theory of Small Strains and Small Angles of Rotation 118
Problem Set 2-15 128
2-16 Compatibility Conditions of the Classical Theory of Small Displacements 130
Problem Set 2-16 135
2-17 Additional Conditions Imposed by Continuity 136
2-18 Kinematics of Deformable Media 138
Problem Set 2-18 144
Appendix 2A Strain--Displacement Relations in Orthogonal Curvilinear Coordinates 144
2A-1 Geometrical Preliminaries 144
2A-2 Strain-Displacement Relations 146
Appendix 2B Derivation of Strain--Displacement Relations for Special Coordinates by Cartesian Methods 150
Appendix 2C Strain-Displacement Relations in General Coordinates 153
2C-1 Euclidean Metric Tensor 153
2C-2 Strain Tensors 156
References 157
Bibliography 158
CHAPTER 3 THEORY OF STRESS 159
3-1 Definition of Stress 159
3-2 Stress Notation 162
3-3 Summation of Moments. Stress at a Point. Stress on an Oblique Plane 164
Problem Set 3-3 169
3-4 Tensor Character of Stress. Transformation of Stress Components under Rotation of Coordinate Axes 173
Problem Set 3-4 176
3-5 Principal Stresses. Stress Invariants. Extreme Values 177
Problem Set 3-5 181
3-6 Mean and Deviator Stress Tensors. Octahedral Stress 182
Problem Set 3-6 187
3-7 Approximations of Plane Stress. Mohr's Circles in Two and Three Dimensions 191
Problem Set 3-7 198
3-8 Differential Equations of Motion of a Deformable Body Relative to Spatial Coordinates 199
Problem Set 3-8 203
Appendix 3A Differential Equations of Equilibrium in Curvilinear Spatial Coordinates 204
3A-1 Differential Equations of Equilibrium in Orthogonal Curvilinear Spatial Coordinates 204
3A-2 Specialization of Equations of Equilibrium 206
3A-3 Differential Equations of Equilibrium in General Spatial Coordinates 208
Appendix 3B Equations of Equilibrium Including Couple Stress and Body Couple 210
Appendix 3C Reduction of Differential Equations of Motion for Small-Displacement Theory 212
3C-1 Material Derivative. Material Derivative of a Volume Integral 212
3C-2 Differential Equations of Equilibrium Relative to Material Coordinates 216
References 222
Bibliography 223
CHAPTER 4 THREE-DIMENSIONAL EQUATIONS OF ELASTICITY 224
4-1 Elastic and Nonelastic Response of a Solid 224
4-2 Intrinsic Energy Density Function (Adiabatic Process) 228
4-3 Relation of Stress Components to Strain Energy Density Function 230
4-4 Generalized Hooke's Law 233
Problem Set 4-4 243
4-5 Isotropic Media. Homogeneous Media 243
4-6 Strain Energy Density for Elastic Isotropic Medium 244
Problem Set 4-6 250
4-7 Special States of Stress 254
Problem Set 4-7 257
4-8 Equations of Thermoelasticity 257
4-9 Differential Equation of Heat Conduction 259
4-10 Elementary Approach to Thermal-Stress Problem in One and Two Variables 261
4-11 Stress Strain-Temperature Relations 265
Problem Set 4-11 272
4-12 Thermoelastic Equations in Terms of Displacement 274
4-13 Spherically Symmetrical Stress Distribution (The Sphere) 277
Problem Set 4-13 279
4-14 Thermoelastic Compatibility Equations in Terms of Components of Stress and Temperature. Beltrami-Michell Relations 279
Problem Set 4-14 284
4-15 Boundary Conditions 286
Problem Set 4-15 290
4-16 Uniqueness Theorem for Equilibrium Problem of Elasticity 290
4-17 Equations of Elasticity in Terms of Displacement Components 295
Problem Set 4-17 297
4-18 Elementary Three-Dimensional Problems of Elasticity. Semi-Inverse Method 298
Problem Set 4-18 304
4-19 Torsion of Shaft with Constant Circular Cross Section 308
Problem Set 4-19 312
4-20 Energy Principles in Elasticity 314
4-21 Principle of Virtual Work 315
Problem Set 4-21 320
4-22 Principle of Virtual Stress (Castigliano's Theorem) 321
4-23 Mixed Virtual Stress-Virtual Strain Principles (Reissner's Theorem) 323
Appendix 4A Application of the Principle of Virtual Work to a Deformable Medium (Navier-Stokes Equations) 324
Appendix 4B Nonlinear Constitutive Relationships 327
4B-1 Variable Stress-Strain Coefficients 328
4B-2 Higher-Order Relations 328
4B-3 Hypoelastic Formulations 328
4B-4 Summary 329
References 329
Bibliography 332
CHAPTER 5 PLANE THEORY OF ELASTICITY IN RECTANGULAR CARTESIAN COORDINATES 333
5-1 Plane Strain 334
Problem Set 5-1 338
5-2 Generalized Plane Stress 340
Problem Set 5-2 344
5-3 Compatability Equation in Terms of Stress Components 346
Problem Set 5-3 350
5-4 Airy Stress Function 351
Problem Set 5-4 361
5-5 Airy Stress Function in Terms of Harmonic Functions 368
5-6 Displacement Components for Plane Elasticity 369
Problem Set 5-6 373
5-7 Polynomial Solutions of TwoDimensional Problems in Rectangular Cartesian Coordinates 377
Problem Set 5-7 380
5-8 Plane Elasticity in Terms of Displacement Components 384
Problem Set 5-8 385
5-9 Plane Elasticity Relative to Oblique Coordinate Axes 386
Appendix 5A Plane Elasticity with Couple Stresses 390
5A-1 Introduction 390
5A-2 Equations of Equilibrium 391
5A-3 Deformation in Couple-Stress Theory 391
5A-4 Equations of Compatibility 393
5A-5 Stress Functions for Plane Problems with Couple Stresses 396
Appendix 5B Plane Theory of Elasticity in Terms of Complex Variables 398
5B-1 Airy Stress Function in Terms of Analytic Functionsψ(z) and χ(z) 398
5B-2 Displacement Components in Terms of Analytic Functions, ψ(z) and χ(z) 399
5B-3 Stress Components in Terms of ψ(z) and χ(z) 400
5B-4 Expressions for Resultant Force and Resultant Moment 403
5B-5 Mathematical Form of Functions ψ(z)and χ(z) 404
5B-6 Plane Elasticity Boundary-Value Problems in Complex Form 408
5B-7 Note on Conformal Transformation 411
5B-8 Plane Elasticity Formulas in Terms of Curvilinear Coordinates 416
5B-9 Complex Variable Solution for Plane Region Bounded by Circle in the z Plane 419
Problem Set 5B 423
References 424
Bibliography 425
CHAPTER 6 PLANE ELASTICITY IN POLAR COORDINATES 427
6-1 Equilibrium Equations in Polar Coordinates 427
6-2 Stress Components in Terms of Airy Stress Function F = F(γ, θ) 428
6-3 Strain Displacement Relations in Polar Coordinates 430
Problem Set 6-3 432
6-4 Stress-Strain-Temperature Relations 433
Problem Set 6-4 435
6-5 Compatibility Equation for Plane Elasticity in Terms of Polar Coordinates 435
Problem Set 6-5 436
6-6 Axially Symmetric Problems 438
Problem Set 6-6 449
6-7 Plane-Elasticity Equations in Terms of Displacement Components 451
6-8 Plane Theory of Thermoelasticity 455
Problem Set 6-8 458
6-9 Disk of Variable Thickness and Nonhomogeneous Anisotropic Material 460
Problem Set 6-9 465
6-10 Stress Concentration Problem of Circular Hole in Plate 465
Problem Set 6-10 472
6-11 Examples 473
Problem Set 6-11 478
Appendix 6A Stress-Couple Theory of Stress Concentration Resulting from Circular Hole in Plate 487
Appendix 6B Stress Distribution of a Diametrically Compressed Plane Disk 492
References 494
CHAPTER 7 PRISMATIC BAR SUBJECTED TO END LOAD 496
7-1 General Problem of Three-Dimensional Elastic Bars Subjected to Transverse End Loads 496
7-2 Torsion of Prismatic Bars. Saint-Venant's Solution. Warping Function 499
Problem Set 7-2 505
7-3 Prandtl Torsion Function 505
Problem Set 7-3 509
7-4 A Method of Solution of the Torsion Problem: Elliptic Cross Section 510
Problem Set 7-4 514
7-5 Remarks on Solutions of the Laplace Equation, ▽2F = 0 515
Problem Set 7-5 517
7-6 Torsion of Bars with Tubular Cavities 520
Problem Set 7-6 523
7-7 Transfer of Axis of Twist 523
7-8 Shearing-Stress Component in Any Direction 524
Problem Set 7-8 529
7-9 Solution of Torsion Problem by the Prandtl Membrane Analogy 529
Problem Set 7-9 538
7-10 Solution by Method of Series. Rectangular Section 538
Problem Set 7-10 543
7-11 Bending of a Bar Subjected to Transverse End Force 544
Problem Set 7-11 556
7-12 Displacement of a Cantilever Beam Subjected to Transverse End Force 556
Problem Set 7-12 559
7-13 Center of Shear 560
Problem Set 7-13 561
7-14 Bending of a Bar with Elliptic Cross Section 563
7-15 Bending of a Bar with Rectangular Cross Section 565
Problems Set 7-15 570
Review Problems 571
Appendix 7A Analysis of Tapered Beams 572
References 576
CHAPTER 8 GENERAL SOLUTIONS OF ELASTICITY 578
8-1 Introduction 578
Problem Set 8-1 579
8-2 Equilibrium Equations 579
Problem Set 8-2 581
8-3 The Helmholtz Transformation 581
Problem Set 8-3 583
8-4 The Galerkin (Papkovich) Vector 583
Problem Set 8-4 585
8-5 Stress in Terms of the Galerkin Vector F 585
Problem Set 8-5 586
8-6 The Galerkin Vector: A Solution of the Equilibrium Equations of Elasticity 586
Problem Set 8-6 588
8-7 The Galerkin Vector kZ and Love's Strain Function for Solids of Revolution 588
Problem Set 8-7 591
8-8 Kelvin's Problem: Single Force Applied in the Interior of an Infinitely Extended Solid 591
Problem Set 8-8 593
8-9 The Twinned Gradient and Its Application to Determine the Effects of a Change of Poisson's Ratio 593
8-10 Solutions of the Boussinesq and Cerruti Problems by the Twinned Gradient Method 597
Problem Set 8-10 600
8-11 Additional Remarks on ThreeDimensional Stress Functions 600
References 601
Bibliography 601
INDEX 603