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### Analysis of Stress and Deformation

Analysis of Stress and Deformation
George W. Housner
Thad Vreeland, Jr.
Division of Engineering and Applied Science, California Institute
447 Pages

Open: Analysis of Stress and Deformation
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Introduction:

The mechanics of deformable bodies deals with the stresses and strains produced in bodies by external actions. In its practical aspects, the subject deals with such questions as how large a force can a body withstand without collapsing; how far out of shape will the body be deformed by the action of prescribed forces; what is the most efficient shape of the body for withstanding the forces? The answers to these and related questions are required in all phases of a technically advanced society. We come into contact with numerous examples of the application of stress analysis every day. Bridges and buildings are examples, as are machines, airplanes, missiles, etc. In short, any solid body whose weight, strength, or deformation is an item for consideration must be studied from the point of view of stress analysis. In its theoretical aspects, the subject is concerned with investigating the differential equations, and their solutions, that describe the states of stress and strain in bodies of various shapes and materials under the actions of various external agencies.

TOC

Chapter 1. Basic Principles of Stress and Strain 1
1-1 Introduction 1
1-2 Definition of Stress 3
1-3 Properties of Stresses 11
1-4 Properties of Planar Stress Systems 15
1-5 Displacements and Strains in a Continuum 24
1-6 Relations Between Stresses and Strains 36
1-7 Strain Energy 48

Chapter 2. Equations of the Theory of Elasticity 74
2-1 Introduction 74
2-2 Equations of Elasticity: Plane Stress and Plane Strain 78
2-3 A Uniqueness Theorem 83
2-4 Equations of Equilibrium in Terms of Displacements 84
2-5 The Equations of Hydrodynamics and Elasticity 85
2-6 St. Venant's Principle 86

Chapter 3. Applications to Beams 90
3-1 Introduction 90
3-2 Extension of a Bar 9 1
3-3 Pure Bending of Prismatic Bars 92
3-4 Cantilever Beam Carrying a Concentrated Load 98
3-5 The Technical Theory of Bending 106
3-6 Composite Beams 116
3-7 Deflection of Transversely Loaded Beams 119
3-8 Statically Indeterminate Systems 133
3-9 Beam on an Elastic Foundation 141
3-10 Footing on an Elastic Foundation 145
3-11 Thin-walled Tubes 146
3-12 Moving Load on a Beam on an Elastic Foundation 149

Chapter 4. Elastic Instability 156
4-1 An Example of Elastic Instability 156
4-2 Buckling of a Simply-Supported Column 159
4-3 Column With Initial Curvature 166
4-4 Column With Eccentric Loading 169
4-5 Considerations in the Design of Columns 171
4-6 Combined Axial and Lateral Loading of Slender Members 172
4-7 Rayleigh-Ritz Method 175
4-8 Other Types of Buckling Problems 178

Chapter 5. Applications to Axially Symmetrical Problems, Curved Beams and Stress Concentrations 190
5-1 Axially Symmetrical Problems 190
5-2 Thick- Walled Cylinders 193
5-3 Rotating Disc of Uniform Thickness 198
5-4 Bending of Curved Beams 200
5-5 The Technical Theory of Bending for Curved Bars 204
5-6 Stress Concentrations 217
5-7 Contact Stresses 221

Chapter 6. Applications to Torsion Problems 228
6-1 Torsion of Prismatical Bars 228
6-2 Solution for a Circular Bar 233
6-3 Curved Circular Bars 235
6-4 Bars of Noncircular Cross Section 237
6-5 Membrane Analogy 239
6-6 Torsion of Tubular Sections 244
6-7 Restraint of Warping 246

Chapter 7. Applications to Plates and Shells 253
7-1 The Bending of Plates 253
7-2 Bending Moments and Twisting Moments 260
7-3 Transverse Shear Forces 262 7-4 Equations of Equilibrium 262
7-5 Boundary Conditions for a Plate 264
7-6 Circular Plates 267
7-7 The Load Carrying Action of a Shell 274
7-8 Cylindrical Shell 281

Chapter 8. Applications to Viscous and Plastic Behavior of Materials 289
8-1 Deviations from Linear Elastic Behavior 289
8-2 Simplified Stress-Strain Relations 292
8-3 The Yield Surface in Stress Space 302
8-4 Yield Hinge in a Beam 306
8-5 Plastic Collapse of Beams and Frames 308
8-6 Failures Due to Plastic Straining 310
8-7 Plastic Torsion 312
8-8 Inelastic Buckling of Columns 314
8-9 Plastic Extension, Drawing, and Rolling 316

Chapter 9. Elastic Wave Propagation 323
9-1 The Wave Equation 323
9-2 General Equations of Motion 335
9-3 Displacement Potential Functions 338
9-4 Plane Waves in an Infinite Continuum 339
9-5 Waves in Nonhomogeneous Media 341
9-6 Surface Waves 344
9-7 Longitudinal Waves in Rods 347
9-8 Vibration of Beams 348

Appendix I. Stresses and Strains in Tensor Notation 381
1-1 Tensor Notation 381
1-2 Transformation of Coordinates 386
1-3 Principal Stresses 388
1-4 Transformation of Strains 391
1-5 Derivation of the Compatibility Equations 393
1-6 Definition of a Tensor 396
1-7 Nonisotropic Stress-Strain Relations 399

Appendix II. The Measurement of Strain 404
II- 1 Strain Measurement 404
II-2 Bonded Strain Gage 404
II-3 Brittle Coating Method 409

Appendix III. Photoelastic Strain Measurement 413

Appendix IV. Variational Methods 423 IV- 1
Variation of a Function 423
IV-2 Derivation of the Differential Equation and Boundary Conditions 427

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