Principles of Solid Mechanics

Principles of Solid Mechanics
اسم المؤلف
Rowland Richards
التاريخ
27 يناير 2018
المشاهدات
57
التقييم
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Principles of Solid Mechanics
Rowland Richards, Jr.
Contents
1. Introduction 1
1.1 Types of Linearity . 1
1.1.1 Linear Shapes—The “Elastic Line” . 1
1.1.2 Linear Displacement (Plane Sections) . 2
1.1.3 Linear Stress Strain Behavior (Hooke’s Law) 3
1.1.4 Geometric Linearity . 4
1.1.5 Linear Tangent Transformation 4
1.2 Displacements—Vectors and Tensors . 5
1.3 Finite Linear Transformation 6
1.4 Symmetric and Asymmetric Components . 9
1.4.1 Asymmetric Transformation 9
1.4.2 Symmetric Transformation . 10
1.5 Principal or Eigenvalue Representation . 13
1.6 Field Theory 17
1.7 Problems and Questions . 19
2. Strain and Stress . 23
2.1 Deformation (Relative Displacement) . 23
2.2 The Strain Tensor 24
2.3 The Stress Tensor 28
2.4 Components at an Arbitrary Orientation . 30
(Tensor Transformation)
2.4.1 Invariants and Principal Orientation 33
2.5 Isotropic and Deviatoric Components 37
2.6 Principal Space and the Octahedral Representation . 39
2.7 Two-Dimensional Stress or Strain 42
2.8 Mohr’s Circle for a Plane Tensor . 46
2.9 Mohr’s Circle in Three Dimensions 50
2.10 Equilibrium of a Differential Element . 53
2.11 Other Orthogonal Coordinate Systems 55
2.11.1 Cylindrical Coordinates (r, , z) . 57
2.11.2 Spherical Coordinates (r, , ) 58
2.11.3 Plane Polar Coordinates (r, ) . 58
2.12 Summary . 59
2.13 Problems and Questions . 61
3. Stress–Strain Relationships (Rheology) 65
3.1 Linear Elastic Behavior 65
3.2 Linear Viscous Behavior 723.3 Simple Viscoelastic Behavior 74
3.4 Fitting Laboratory Data with Viscoelastic Models 80
3.5 Elastic–Viscoelastic Analogy 83
3.6 Elasticity and Plasticity . 86
3.7 Yield of Ductile Materials 87
3.8 Yield (Slip) of Brittle Materials . 90
3.9 Problems and Questions . 93
4. Strategies for Elastic Analysis and Design . 99
4.1 Rational Mechanics 99
4.2 Boundary Conditions 101
4.3 Tactics for Analysis 102
4.3.1 Direct Determination of Displacements . 102
4.3.2 Direct Determination of Stresses 103
4.4 St. Venant’s Principle . 105
4.5 Two- Dimensional Stress Formulation 106
4.6 Types of Partial Differential Field Equations . 108
4.7 Properties of Elliptic Equations 109
4.8 The Conjugate Relationship Between Mean 112
Stress and Rotation
4.9 The Deviatoric Field and Photoelasticity 120
4.10 Solutions by Potentials 123
4.11 Problems and Questions . 124
5. Linear Free Fields . 127
5.1 Isotropic Stress 127
5.2 Uniform Stress 128
5.3 Geostatic Fields . 130
5.4 Uniform Acceleration of the Half-space . 133
5.5 Pure Bending of Prismatic Bars 135
5.6 Pure Bending of Plates 140
5.7 Problems and Questions . 142
6. Two-Dimensional Solutions for Straight 145
and Circular Beams
6.1 The Classic Stress-Function Approach 145
6.2 Airy’s Stress Function in Cartesian Coordinates . 146
6.3 Polynomial Solutions and Straight Beams . 148
6.4 Polar Coordinates and Airy’s Stress Function . 157
6.5 Simplified Analysis of Curved Beams 162
6.6 Pure bending of a Beam of Circular Arc . 165
6.7 Circular Beams with End Loads 171
6.8 Concluding Remarks . 174
6.9 Problems and Questions . 1757. Ring, Holes, and Inverse Problems 181
7.1 Lamés Solution for Rings under Pressure 181
7.2 Small Circular Holes in Plates, Tunnels, and Inclusions 187
7.2.1 Isotropic Field . 187
7.2.2 Deviatoric Field 194
7.2.3 General Biaxial Field 197
7.3 Harmonic Holes and the Inverse Problem . 198
7.3.1 Design Condition . 198
7.4 Harmonic Holes for Free Fields . 203
7.4.1 Harmonic Holes for Biaxial Fields . 203
7.4.2 Harmonic Holes for Gradient Fields . 209
7.5 Neutral Holes 213
7.6 Solution Tactics for Neutral Holes—Examples 220
7.6.1 Isotropic Field . 222
7.6.2 Deviatoric Field 223
7.6.3 General Biaxial Field 225
7.6.4 Gradient Fields with an Isotropic Component 226
7.6.5 Summary . 229
7.7 Rotating Disks and Rings 233
7.7.1 Disk of Constant Thickness 233
7.7.2 Variable Thickness and the Inverse Problem . 236
7.8 Problems and Questions . 238
8. Wedges and the Half-Space . 243
8.1 Concentrated Loadings at the Apex 243
8.2 Uniform Loading Cases 251
8.3 Uniform Loading over a Finite Width 256
8.4 Nonuniform Loadings on the Half-Space 257
8.5 Line Loads within the Half-Space . 259
8.6 Diametric Loading of a Circular Disk . 261
8.7 Wedges with Constant Body Forces 263
8.8 Corner Effects—Eigenfunction Strategy . 270
8.9 Problems and Questions . 272
9. Torsion 291
9.1 Elementary (Linear) Solution . 291
9.2 St. Venant’s Formulation (Noncircular Cross-Sections) . 292
9.2.1 Solutions by St. Venant 295
9.3 Prandtl’s Stress Function . 297
9.4 Membrane Analogy . 301
9.5 Thin-Walled Tubes of Arbitrary Shape . 307
9.6 Hydrodynamic Analogy and Stress Concentration 311
9.7 Problems and Questions . 31510. Concepts of Plasticity 321
10.1 Plastic Material Behavior 321
10.2 Plastic Structural Behavior 323
10.3 Plasticity Field Equations 324
10.4 Example—Thick Ring 326
10.5 Limit Load by a “Work” Calculation 329
10.6 Theorems of Limit Analysis 332
10.7 The Lower-Bound Theorem . 332
10.8 The Upper-Bound Theorem . 335
10.9 Example—the Bearing Capacity (Indentation) Problem 337
10.9.1 Circular Mechanisms . 337
10.9.2 Sliding Block Mechanisms 339
10.10 Problems and Questions . 341
11. One-Dimensional Plasticity for Design 347
11.1 Plastic Bending . 347
11.2 Plastic “Hinges” . 352
11.3 Limit Load (Collapse) of Beams . 354
11.4 Limit Analysis of Frames and Arches . 357
11.5 Limit Analysis of Plates . 361
11.6 Plastic Torsion . 369
11.6.1 Sand-Hill and Roof Analogies 370
11.6.2 Sections with Holes and Keyways . 372
11.7 Combined Torsion with Tension and/or Bending 375
11.8 Problems and Questions . 378
12. Slip-Line Analysis . 389
12.1 Mohr-Coulomb Criterion (Revisited) 389
12.2 Lateral “Pressures” and the Retaining Wall Problem . 394
12.3 Graphic Analysis and Minimization . 399
12.4 Slip-Line Theory . 402
12.5 Purely Cohesive Materials ( 0) 405
12.6 Weightless Material ( 0) 407
12.7 Retaining Wall Solution for 0 (EPS Material) . 408
12.8 Comparison to the Coulomb Solution ( 0) 412
12.9 Other Special Cases: Slopes and Footings ( 0) 414
12.10 Solutions for Weightless Mohr-Coulomb Materials 417
12.11 The General Case . 422
12.12 An Approximate “Coulomb Mechanism” 425
12.13 Problems and Questions . 430
Index
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