Advanced Mechanics of Materials and Applied Elasticity

Preface xvii
Acknowledgments xx
About the Authors xxi
List of Symbols xxii

Chapter 1: Analysis of Stress 1
1.1 Introduction 1
1.2 Scope of the Book 3
1.3 Analysis and Design 4
1.4 Conditions of Equilibrium 8
1.5 Definition and Components of Stress 9
1.6 Internal Force Resultant and Stress Relations 13
1.7 Stresses on Inclined Sections 17
1.8 Variation of Stress within a Body 20
1.9 Plane-Stress Transformation 23
1.10 Principal Stresses and Maximum In-Plane Shear Stress 26
1.11 Mohr’s Circle for Two-Dimensional Stress 28
1.12 Three-Dimensional Stress Transformation 35
1.13 Principal Stresses in Three Dimensions 38
1.14 Normal and Shear Stresses on an Oblique Plane 42
1.15 Mohr’s Circles in Three Dimensions 45
1.16 Boundary Conditions in Terms of Surface Forces 49
1.17 Indicial Notation 50
References 51
Problems 51

Chapter 2: Strain and Material Properties 68
2.1 Introduction 68
2.2 Deformation 69
2.3 Strain Defined 70
2.4 Equations of Compatibility 75
2.5 State of Strain at a Point 76
2.6 Engineering Materials 83
2.6.1 General Properties of Some Common Materials 84
2.7 Stress-Strain Diagrams 86
2.8 Elastic versus Plastic Behavior 91
2.9 Hooke’s Law and Poisson’s Ratio 92
2.10 Generalized Hooke’s Law 96
2.11 Orthotropic Materials 101
2.12 Measurement of Strain: Strain Gage 103
2.13 Strain Energy 107
2.14 Strain Energy in Common Structural Members 111
2.15 Components of Strain Energy 113
2.16 Saint-Venant’s Principle 115
References 117
Problems 118

Chapter 3: Problems in Elasticity 133
3.1 Introduction 133
3.2 Fundamental Principles of Analysis 134
Part A: Formulation and Methods of Solution 135
3.3 Plane Strain Problems 135
3.4 Plane Stress Problems 138
3.5 Comparison of Two-Dimensional Isotropic Problems 140
3.6 Airy’s Stress Function 141
3.7 Solution of Elasticity Problems 143
3.8 Thermal Stresses 149
3.9 Basic Relations in Polar Coordinates 152
Part B: Stress Concentrations 157
3.10 Stresses Due to Concentrated Loads 157
3.11 Stress Distribution Near a Concentrated Load Acting on a Beam 161
3.12 Stress Concentration Factors 163
Part C: Contact Mechanics 169
3.13 Contact Stresses and Deflections 169
3.14 Spherical and Cylindrical Contacts 171
3.15 Contact Stress Distribution 174
3.16 General Contact 178
References 181
Problems 182

Chapter 4: Failure Criteria 192
4.1 Introduction 192
Part A: Static Loading 193
4.2 Failure by Yielding 193
4.3 Failure by Fracture 195
4.4 Yield and Fracture Criteria 197
4.5 Maximum Shearing Stress Theory 198
4.6 Maximum Distortion Energy Theory 199
4.7 Octahedral Shearing Stress Theory 200
4.8 Comparison of the Yielding Theories 204
4.9 Maximum Principal Stress Theory 205
4.10 Mohr’s Theory 206
4.11 Coulomb—Mohr Theory 207
4.12 Introduction to Fracture Mechanics 210
4.13 Fracture Toughness 213
Part B: Repeated and Dynamic Loadings 216
4.14 Fatigue: Progressive Fracture 216
4.15 Failure Criteria for Metal Fatigue 217
4.16 Fatigue Life 223
4.17 Impact Loads 225
4.18 Longitudinal and Bending Impact 227
4.19 Ductile—Brittle Transition 230
References 232
Problems 233

Chapter 5: Bending of Beams 242
5.1 Introduction 242
Part A: Exact Solutions 243
5.2 Pure Bending of Beams of Symmetrical Cross Section 243
5.3 Pure Bending of Beams of Asymmetrical Cross Section 246
5.4 Bending of a Cantilever of Narrow Section 251
5.5 Bending of a Simply Supported Narrow Beam 254
Part B: Approximate Solutions 256
5.6 Elementary Theory of Bending 256
5.7 Normal and Shear Stresses 260
5.8 Effect of Transverse Normal Stress 268
5.9 Composite Beams 270
5.10 Shear Center 276
5.11 Statically Indeterminate Systems 281
5.12 Energy Method for Deflections 284
Part C: Curved Beams 286
5.13 Elasticity Theory 286
5.14 Curved Beam Formula 289
5.15 Comparison of the Results of Various Theories 293
5.16 Combined Tangential and Normal Stresses 296
References 300
Problems 300

Chapter 6: Torsion of Prismatic Bars 315
6.1 Introduction 315
6.2 Elementary Theory of Torsion of Circular Bars 316
6.3 Stresses on Inclined Planes 321
6.4 General Solution of the Torsion Problem 324
6.5 Prandtl’s Stress Function 326
6.6 Prandtl’s Membrane Analogy 333
6.7 Torsion of Narrow Rectangular Cross Section 338
6.8 Torsion of Multiply Connected Thin-Walled Sections 340
6.9 Fluid Flow Analogy and Stress Concentration 344
6.10 Torsion of Restrained Thin-Walled Members of Open Cross Section 346
6.11 Torsion Bar Springs 350
6.12 Curved Circular Bars 351
Problems 355

Chapter 7: Numerical Methods 364
7.1 Introduction 364
Part A: Finite Difference Analysis 365
7.2 Finite Differences 365
7.3 Finite Difference Equations 368
7.4 Curved Boundaries 370
7.5 Boundary Conditions 373
Part B: Finite Element Analysis 377
7.6 Fundamentals 377
7.7 The Bar Element 379
7.8 Arbitrarily Oriented Bar Element 380
7.9 Axial Force Equation 384
7.10 Force-Displacement Relations for a Truss 386
7.11 Beam Element 393
7.12 Properties of Two-Dimensional Elements 399
7.13 General Formulation of the Finite Element Method 402
7.14 Triangular Finite Element 407
7.15 Case Studies in Plane Stress 414
7.16 Computational Tools 423
References 423
Problems 424

Chapter 8: Thick-Walled Cylinders and Rotating Disks 434
8.1 Introduction 434
8.2 Thick-Walled Cylinders Under Pressure 435
8.3 Maximum Tangential Stress 441
8.4 Application of Failure Theories 442
8.5 Compound Cylinders: Press or Shrink Fits 443
8.6 Rotating Disks of Constant Thickness 446
8.7 Disk Flywheels 449
8.8 Rotating Disks of Variable Thickness 453
8.9 Rotating Disks of Uniform Stress 456
8.10 Thermal Stresses in Thin Disks 458
8.11 Thermal Stress in Long Circular Cylinders 460
8.12 Finite Element Solution 464
References 466
Problems 466

Chapter 9: Beams on Elastic Foundations 473
9.1 Introduction 473
9.2 General Theory 473
9.3 Infinite Beams 475
9.4 Semi-Infinite Beams 480
9.5 Finite Beams 483
9.6 Classification of Beams 484
9.7 Beams Supported by Equally Spaced Elastic Elements 485
9.8 Simplified Solutions for Relatively Stiff Beams 486
9.9 Solution by Finite Differences 488
9.10 Applications 490
Problems 493

Chapter 10: Applications of Energy Methods 496
10.1 Introduction 496
Part A: Energy Principles 497
10.2 Work Done in Deformation 497
10.3 Reciprocity Theorem 498
10.4 Castigliano’s Theorem 499
10.5 Unit- or Dummy-Load Method 506
10.6 Crotti—Engesser Theorem 508
10.7 Statically Indeterminate Systems 510
Part B: Variational Methods 514
10.8 Principle of Virtual Work 514
10.9 Principle of Minimum Potential Energy 515
10.10 Deflections by Trigonometric Series 517
10.11 Rayleigh—Ritz Method 522
References 524
Problems 525

Chapter 11: Stability of Columns 534
11.1 Introduction 534
11.2 Critical Load 534
11.3 Buckling of Pin-Ended Columns 536
11.4 Deflection Response of Columns 539
11.5 Columns with Different End Conditions 540
11.6 Critical Stress: Classification of Columns 543
11.7 Design Formulas for Columns 548
11.8 Imperfections in Columns 550
11.9 Local Buckling of Columns 552
11.10 Eccentrically Loaded Columns: Secant Formula 552
11.11 Energy Methods Applied to Buckling 554
11.12 Solution by Finite Differences 562
11.13 Finite Difference Solution for Unevenly Spaced Nodes 567
References 568
Problems 569

Chapter 12: Plastic Behavior of Materials 578
12.1 Introduction 578
12.2 Plastic Deformation 579
12.3 Idealized Stress—Strain Diagrams 580
12.4 Instability in Simple Tension 582
12.5 Plastic Axial Deformation and Residual Stress 585
12.6 Plastic Deflection of Beams 588
12.7 Analysis of Perfectly Plastic Beams 590
12.8 Collapse Load of Structures: Limit Design 600
12.9 Elastic—Plastic Torsion of Circular Shafts 605
12.10 Plastic Torsion: Membrane Analogy 610
12.11 Elastic—Plastic Stresses in Rotating Disks 612
12.12 Plastic Stress—Strain Relations 614
12.13 Plastic Stress—Strain Increment Relations 620
12.14 Stresses in Perfectly Plastic Thick-Walled Cylinders 623
Problems 628

Chapter 13: Stresses in Plates and Shells 635
13.1 Introduction 635
Part A: Bending of Thin Plates 635
13.2 Basic Assumptions 635
13.3 Strain—Curvature Relations 636
13.4 Stress, Curvature, and Moment Relations 638
13.5 Governing Equations of Plate Deflection 640
13.6 Boundary Conditions 642
13.7 Simply Supported Rectangular Plates 644
13.8 Axisymmetrically Loaded Circular Plates 648
13.9 Deflections of Rectangular Plates by the Strain-Energy Method 650
13.10 Sandwich Plates 652
13.11 Finite Element Solution 654
Part B: Membrane Stresses in Thin Shells 657
13.12 Theories and Behavior of Shells 657
13.13 Simple Membrane Action 658
13.14 Symmetrically Loaded Shells of Revolution 660
13.15 Some Typical Cases of Shells of Revolution 662
13.16 Thermal Stresses in Compound Cylinders 668
13.17 Cylindrical Shells of General Shape 670

Appendix A: Problem Formulation and Solution 679
A.1 Basic Method 679

Appendix B: Solution of the Stress Cubic Equation 682
B.1 Principal Stresses 682

Appendix C: Moments of Composite Areas 687
C.1 Centroid 687
C.2 Moments of Inertia 690

Appendix D: Tables and Charts 699
D.1 Charts of Stress Concentration Factors 705

Appendix E Introduction to MATLAB 710

Answers to Selected Problems 713
Index 722