Thomas Grant Content
Chapter 1 Analysis of Stress
1.1 Introduction 1
1.1.1 Mechanics of Materials and Theory of Elasticity 1
1.1.2 Historical Development 2
1.2 Scope of the Book 3
1.3 Analysis and Design 4
1.3.1 Role of Analysis in Design 6
1.3.2 Selection of Factor of Safety 6
1.3.3 Case Studies 7
1.4 Conditions of Equilibrium 8
1.5 Definition and Components of Stress 9
1.5.1 Sign Convention 11
1.5.2 Equality of Shearing Stresses 12
1.5.3 Some Special Cases of Stress 12
1.6 Internal Force Resultant and Stress Relations 13
1.6.1 Basic Formulas for Stress 15
1.6.2 Combined Stresses 17
1.7 Stresses on Inclined Sections 17
1.7.1 Axially Loaded Members 18
1.8 Variation of Stress within a Body 20
1.8.1 Equations of Equilibrium 20
1.9 Plane-Stress Transformation 23
1.9.1 Stress Tensor 25
1.9.2 Polar Representations of State of Plane Stress 25
1.9.3 Cartesian Representation of State of Plane Stress 25
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
1.13.1 Invariants for Three-Dimensional Stress 40
1.14 Normal and Shear Stresses on an Oblique Plane 42
1.14.1 Octahedral Stresses 44
1.15 Mohr’s Circles in Three Dimensions 45
1.15.1 Absolute Maximum Shear Stress 46
1.15.2 Equations of Three Mohr’s Circles for Stress 47
1.16 Boundary Conditions in Terms of Surface Forces 49
1.17 Indicial Notation 50
Chapter 2 Strain and Material Properties
2.1 Introduction 68
2.2 Deformation 69
2.2.1 Superposition 69
2.3 Strain Defined 70
2.3.1 Plane Strain 70
2.3.2 Three-Dimensional Strain 72
2.3.3 Eulerian and Lagrangian Coordinates 73
2.3.4 Large Strains 74
2.4 Equations of Compatibility 75
2.5 State of Strain at a Point 76
2.5.1 Transformation of Two-Dimensional Strain 76
2.5.2 Transformation of Three-Dimensional Strain 78
2.5.3 Invariants in Three-Dimensional Strain 79
2.5.4 Mohr’s Circle for Plane Strain 80
2.6 Engineering Materials 83
2.6.1 General Properties of Some Common Materials 84
2.7 Stress-Strain Diagrams 86
2.7.1 Ductile Materials in Tension 86
2.7.2 Geometry Change of Specimen 87
2.7.3 True Stress and True Strain 88
2.7.4 Brittle Materials in Tension 89
2.7.5 Materials in Compression 89
2.7.6 Materials in Shear 90
2.7.7 Short-Time Effects of Temperature on Stress-Strain Properties 90
2.8 Elastic versus Plastic Behavior 91
2.9 Hooke’s Law and Poisson’s Ratio 92
2.9.1 Volume Change 93
2.9.2 Deflection of Axially Loaded Members 93
2.10 Generalized Hooke’s Law 96
2.11 Orthotropic Materials 101
2.11.1 Generalized Hook’s Law for Orthotropic Material 102
2.12 Measurement of Strain: Strain Gage 103
2.12.1 Strain Rosette of Three Gages 104
2.12.2 Rectangular and Delta Strain Rosettes 106
2.13 Strain Energy 107
2.13.1 Strain Energy Density for Normal and Shear Stresses 108
2.13.2 Strain Energy Density for Three-Dimensional Stresses 110
2.14 Strain Energy in Common Structural Members 111
2.14.1 Strain Energy for Axially Loaded Bars 111
2.14.2 Strain Energy of Circular Bars in Torsion 112
2.14.3 Strain Energy for Beams in Bending 113
2.15 Components of Strain Energy 113
2.16 Saint-Venant’s Principle 115
2.16.1 Confirmation of Saint-Venant’s Rule 116
Chapter 3 Problems in Elasticity
3.1 Introduction 133
3.2 Fundamental Principles of Analysis 134
3.2.1 Three-Dimensional Problems 134
3.2.2 Two-Dimensional Problems 134
Part A: Formulation and Methods of Solution 135
3.3 Plane Strain Problems 135
3.4 Plane Stress Problems 138
3.4.1 Stress–Strain Relations for Orthotropic Materials 139
3.5 Comparison of Two-Dimensional Isotropic Problems 140
3.6 Airy’s Stress Function 141
3.6.1 Generalized Plane Strain Problems 142
3.6.2 Antiplane Shear Deformations 142
3.7 Solution of Elasticity Problems 143
3.7.1 Polynomial Solutions 144
3.8 Thermal Stresses 149
3.8.1 Equations of Thermoelasticity 149
3.9 Basic Relations in Polar Coordinates 152
3.9.1 Equations of Equilibrium 153
3.9.2 Stress Function 153
3.9.3 Strain-Displacement Relations 154
3.9.4 Hooke’s Law 155
3.9.5 Transformation Equations 155
3.9.6 Compatibility Equation 156
Part B: Stress Concentrations 157
3.10 Stresses Due to Concentrated Loads 157
3.10.1 Compression of a Wedge (Fig. 3.10a) 157
3.10.2 Bending of a Wedge (Fig. 3.10b) 159
3.10.3 Concentrated Load on a Straight Boundary (Fig. 3.11a) 160
3.11 Stress Distribution Near a Concentrated Load Acting on a Beam 161
3.11.1 Accuracy of Results 163
3.12 Stress Concentration Factors 163
3.12.1 Circular Hole in a Large Plate in Simple Tension 165
3.12.2 Circular Hole in a Large Plate in Biaxial Tension 167
3.12.3 Elliptic Hole in a Large Plate in Tension 167
3.12.4 Graphs for Stress Concentration Factors 168
Part C: Contact Mechanics 169
3.13 Contact Stresses and Deflections 169
3.13.1 Hertz Theory 170
3.13.2 Johnson–Kendall–Roberts Theory 170
3.14 Spherical and Cylindrical Contacts 171
3.14.1 Two Spheres in Contact 171
3.14.2 Two Parallel Cylinders in Contact 173
3.15 Contact Stress Distribution 174
3.15.1 Two Spheres in Contact (Figure 3.18a) 174
3.15.2 Two Parallel Cylinders in Contact (Figure 3.20a) 174
3.16 General Contact 178
Chapter 4 Failure Criteria
4.1 Introduction 192
4.1.1 Failure 192
Part A: Static Loading 193
4.2 Failure by Yielding 193
4.2.1 Creep: Time-Dependent Deformation 194
4.3 Failure by Fracture 195
4.3.1 Types of Fracture in Tension 196
4.4 Yield and Fracture Criteria 197
4.5 Maximum Shearing Stress Theory 198
4.6 Maximum Distortion Energy Theory 199
4.6.1 Yield Surfaces for Triaxial Stress 200
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
4.12.1 Stress-Intensity Factors 211
4.13 Fracture Toughness 213
Part B: Repeated and Dynamic Loadings 216
4.14 Fatigue: Progressive Fracture 216
4.14.1 Fatigue Tests 216
4.14.2 Estimating the Endurance Limit and Fatigue Strength 217
4.15 Failure Criteria for Metal Fatigue 217
4.15.1 Uniaxial State of Stress 218
4.15.2 Comparison of Fatigue Failure Criteria 219
4.15.3 Design for Uniaxial Stress 219
4.15.4 Combined State of Stress 221
4.16 Fatigue Life 223
4.17 Impact Loads 225
4.17.1 Strain Rate 226
4.17.2 Basic Assumptions of Impact Analysis 227
4.18 Longitudinal and Bending Impact 227
4.18.1 Freely Falling Weight 227
4.18.2 Horizontally Moving Weight 228
4.19 Ductile–Brittle Transition 230
Chapter 5 Bending of Beams
5.1 Introduction 242
Part A: Exact Solutions 243
5.2 Pure Bending of Beams of Symmetrical Cross Section 243
5.2.1 Kinematic Relationships 244
5.2.2 Timoshenko Beam Theory 246
5.3 Pure Bending of Beams of Asymmetrical Cross Section 246
5.3.1 Stress Distribution 248
5.3.2 Transformation of Inertia Moments 248
5.4 Bending of a Cantilever of Narrow Section 251
5.4.1 Comparison of the Results with the Elementary Theory Results 253
5.5 Bending of a Simply Supported Narrow Beam 254
5.5.1 Use of Stress Functions 255
5.5.2 Comparison of the Results with the Elementary Theory Results 256
Part B: Approximate Solutions 256
5.6 Elementary Theory of Bending 256
5.6.1 Assumptions of Elementary Theory 257
5.6.2 Method of Integration 258
5.7 Normal and Shear Stresses 260
5.7.1 Rectangular Cross Section 262
5.7.2 Various Cross Sections 262
5.7.3 Beam of Constant Strength 267
5.8 Effect of Transverse Normal Stress 268
5.9 Composite Beams 270
5.9.1 Transformed Section Method 270
5.9.2 Equation of Neutral Axis 271
5.9.3 Stresses in the Transformed Beam 272
5.9.4 Composite Beams of Multi Materials 272
5.10 Shear Center 276
5.10.1 Thin-Walled Open Cross Sections 277
5.10.2 Arbitrary Solid Cross Sections 281
5.11 Statically Indeterminate Systems 281
5.11.1 The Method of Superposition 282
5.12 Energy Method for Deflections 284
5.12.1 Form Factor for Shear 285
Part C: Curved Beams 286
5.13 Elasticity Theory 286
5.13.1 Equations of Equilibrium and Compatibility 286
5.13.2 Boundary Conditions 287
5.13.3 Stress Distribution 288
5.13.4 Deflections 289
5.14 Curved Beam Formula 289
5.14.1 Basic Assumptions 289
5.14.2 Location of the Neutral Axis 290
5.14.3 Tangential Stress 291
5.14.4 Winkler’s Formula 293
5.15 Comparison of the Results of Various Theories 293
5.15.1 Correction of σ θ for Beams with Thin-Walled Cross Sections 294
5.16 Combined Tangential and Normal Stresses 296
Chapter 6 Torsion of Prismatic Bars
6.1 Introduction 315
6.2 Elementary Theory of Torsion of Circular Bars 316
6.2.1 Shearing Stress 317
6.2.2 Angle of Twist 317
6.2.3 Axial and Transverse Shear Stresses 320
6.3 Stresses on Inclined Planes 321
6.3.1 Stress Transformation 321
6.3.2 Transmission of Power by Shafts 323
6.4 General Solution of the Torsion Problem 324
6.4.1 Geometry of Deformation 324
6.4.2 Equations of Equilibrium 325
6.4.3 Equations of Compatibility 325
6.5 Prandtl’s Stress Function 326
6.5.1 Boundary Conditions 326
6.5.2 Force and Moments over the Ends 327
6.5.3 Circular Cross Section 331
6.6 Prandtl’s Membrane Analogy 333
6.6.1 Equation of Equilibrium 333
6.6.2 Shearing Stress and Angle of Twist 335
6.7 Torsion of Narrow Rectangular Cross Section 338
6.7.1 Thin-Walled Open Cross Sections 339
6.8 Torsion of Multiply Connected Thin-Walled Sections 340
6.8.1 Shearing Stress 340
6.8.2 Angle of Twist 341
6.9 Fluid Flow Analogy and Stress Concentration 344
6.10 Torsion of Restrained Thin-Walled Members of Open Cross Section 346
6.10.1 Torsional and Lateral Shears 347
6.10.2 Boundary Conditions 348
6.10.3 Long Beams Under Torsion 348
6.10.4 Angle of Twist 348
6.11 Torsion Bar Springs 350
6.12 Curved Circular Bars 351
6.12.1 Helical Springs 352
Chapter 7 Numerical Methods
7.1 Introduction 364
Part A: Finite Difference Analysis 365
7.2 Finite Differences 365
7.2.1 Central Differences 366
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
7.7.1 Equilibrium Method 379
7.7.2 Energy Method 379
7.8 Arbitrarily Oriented Bar Element 380
7.8.1 Coordinate Transformation 380
7.8.2 Force Transformation 381
7.8.3 Displacement Transformation 383
7.8.4 Governing Equations 383
7.9 Axial Force Equation 384
7.10 Force-Displacement Relations for a Truss 386
7.10.1 The Assembly Process 386
7.11 Beam Element 393
7.12 Properties of Two-Dimensional Elements 399
7.12.1 Displacement Matrix 399
7.12.2 Strain, Stress, and Elasticity Matrices 401
7.13 General Formulation of the Finite Element Method 402
7.13.1 Outline of General Finite Element Analysis 403
7.14 Triangular Finite Element 407
7.14.1 Element Nodal Forces 410
7.15 Case Studies in Plane Stress 414
7.16 Computational Tools 423
Chapter 8 Thick-Walled Cylinders and Rotating Disks
8.1 Introduction 434
8.1.1 Basic Relations 434
8.2 Thick-Walled Cylinders Under Pressure 435
8.2.1 Special Cases 438
8.2.2 Closed-Ended Cylinder 440
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
8.6.1 Annular Disk 447
8.6.2 Solid Disk 448
8.7 Disk Flywheels 449
8.7.1 Design Factors 450
8.7.2 Stresses and Displacement 450
8.8 Rotating Disks of Variable Thickness 453
8.9 Rotating Disks of Uniform Stress 456
8.10 Thermal Stresses in Thin Disks 458
8.10.1 Annular Disk 459
8.10.2 Solid Disk 459
8.11 Thermal Stress in Long Circular Cylinders 460
8.11.1 Solid Cylinder 460
8.11.2 Cylinder with a Central Circular Hole 461
8.11.3 Special Case 464
8.12 Finite Element Solution 464
8.12.1 Axisymmetric Element 464
Chapter 9 Beams on Elastic Foundations
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
9.10.1 Grid Configurations of Beams 490
Chapter 10 Applications of Energy Methods
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
10.4.1 Application to Bars and Beams 500
10.4.2 Application to Trusses 500
10.4.3 Use of a Fictitious Load 501
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
10.8.1 Variation in Strain Energy 514
10.8.2 Virtual Work Done by Forces 515
10.9 Principle of Minimum Potential Energy 515
10.10 Deflections by Trigonometric Series 517
10.10.1 Strain Energy 518
10.10.2 Virtual Work 518
10.11 Rayleigh–Ritz Method 522
Chapter 11 Stability of Columns
11.1 Introduction 534
11.2 Critical Load 534
11.2.1 Equilibrium Method 535
11.2.2 Energy Method 536
11.3 Buckling of Pin-Ended Columns 536
11.3.1 Modes of Buckling 538
11.4 Deflection Response of Columns 539
11.4.1 Effects of Large Deflections 539
11.4.2 Effects of Imperfections 540
11.4.3 Effects of Inelastic Behavior 540
11.5 Columns with Different End Conditions 540
11.6 Critical Stress: Classification of Columns 543
11.6.1 Long Columns 543
11.6.2 Short Columns 544
11.6.3 Intermediate Columns: Inelastic Buckling 544
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
11.10.1 Simplified Formula for Short Columns 554
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
Chapter 12 Plastic Behavior of Materials
12.1 Introduction 578
12.2 Plastic Deformation 579
12.2.1 Slip Action: Dislocation 579
12.3 Idealized Stress–Strain Diagrams 580
12.3.1 True Stress–True Strain Relationships 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
12.7.1 Shape Factor 593
12.7.2 Plastic Hinge 593
12.8 Collapse Load of Structures: Limit Design 600
12.8.1 Collapse Mechanism 600
12.8.2 Ultimate Load by the Energy Method 601
12.9 Elastic–Plastic Torsion of Circular Shafts 605
12.9.1 Yield Torque 606
12.9.2 Elastic–Plastic Torque 606
12.9.3 Ultimate Torque 607
12.9.4 Residual Rotation and Stress 608
12.10 Plastic Torsion: Membrane Analogy 610
12.10.1 Membrane–Roof Analogy 610
12.10.2 Sand Hill Analogy 611
12.11 Elastic–Plastic Stresses in Rotating Disks 612
12.11.1 Initial Yielding 612
12.11.2 Partial Yielding 612
12.11.3 Complete Yielding 614
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
12.14.1 Complete Yielding 624
12.14.2 Partial Yielding 626
Chapter 13 Stresses in Plates and Shells
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
13.10.1 Design of Sandwich Beams and Plates 653
13.11 Finite Element Solution 654
13.11.1 Strain, Stress, and Elasticity Matrices 655
13.11.2 Displacement Function 655
13.11.3 Stiffness Matrix 657
13.11.4 External Nodal Forces 657
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
13.14.1 Equations of Equilibrium 661
13.14.2 Conditions of Compatibility 662
13.15 Some Typical Cases of Shells of Revolution 662
13.15.1 Spherical Shell 662
13.15.2 Conical Shell 663
13.15.3 Circular Cylindrical Shell 664
13.16 Thermal Stresses in Compound Cylinders 668
13.17 Cylindrical Shells of General Shape 670
References 673
Problems 673
Appendix A Problem Formulation and Solution 679
A.1 Basic Method 679
A.1.1 Numerical Accuracy 680
A.1.2 Daily Planning 680
Appendix B Solution of the Stress Cubic Equation 682
B.1 Principal Stresses 682
B.1.1 Direction Cosines 683
Appendix C Moments of Composite Areas 687
C.1 Centroid 687
C.2 Moments of Inertia 690
C.2.1 Parallel Axis Theorem 690
C.2.2 Principal Moments of Inertia 692
Appendix D Tables and Charts 699
D.1 Charts of Stress Concentration Factors 705
Appendix E Introduction to MATLAB 710
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