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Strength of Materials
Second Edition
A.k. Srivastava
Manager (Design)
Aircraft Upgrade Research and Design Centre
Hindustan Aeronautics Limited
Nasik
P.C. GOPE
Professor
College of Technology
G.B. Pant University of Agriculture and Technology
Pantnager
و المحتوى كما يلي :
Contents
Foreword xiii
Preface xv
Preface to the First Edition xvii
1. SIMPLE AND COMPOUND STRESS 1–48
1.1 Introduction 1
1.2 Stress 1
1.3 Uniformly Distributed Stress 3
1.3.1 Tensile and Compressive Stresses 3
1.3.2 Stress Due to Bending Moment 4
1.3.3 Stress Due to Twisting Moment 5
1.4 Complex Stresses 6
1.4.1 Plane Stress 6
1.4.2 Stresses on an Inclined Plane 7
Exercises 46
2. ANALYSIS OF STRESS AND STRAIN 49–94
2.1 Introduction 49
2.2 Force Distribution 49
2.3 The State of Stress at a Point 50
2.4 Stress Notations 50
2.5 Stress Tensor at a Point 50
2.6 Stress Gradient 51
2.7 Differential Equations of Equilibrium 51
2.8 Equilibrium Equations for Plane Stress State 53
2.9 Generalized Hooke’s Law 53
2.10 Direction Cosines 55
2.11 Normal and Shear Stresses 56
Contents
2.12 Principal Directions 57
2.13 Stress Components on an Arbitrary Plane 57
2.14 Principal Stress 59
2.15 Stress Invariants 60
2.16 Principal Directions 61
2.17 Octahedral Stress 68
2.18 Mean and Deviator Stresses 70
2.19 Strain Analysis 74
2.20 Strain-Displacement Relation 75
2.21 Three-dimensional Strains 79
2.22 Normal and Shearing Strains 80
2.23 Principal Strains 80
2.24 Principal Strain Directions 81
2.25 Concept of Compatibility 81
2.26 St-Venant’s Equations of Compatibility 83
2.27 Solution of Stress Differential Equation 86
2.28 Types of Airy’s Stress Function 88
2.29 Application of Airy’s Stress Function 90
2.30 Mohr’s Circle for the Three-dimensional State of Stress 92
Exercises 94
3. THEORY OF FAILURE 95–139
3.1 Introduction 95
3.2 Failure Theory for Ductile Material 96
3.2.1 Maximum Shear Stress Theory 96
3.2.2 Maximum Distortion Energy Theory 102
3.2.3 Strain Energy Density or Total Strain Energy Criterion 105
3.3 Theory of Failure or Yield Criterion for Brittle Materials 106
3.3.1 Maximum Principal Stress Criterion 106
3.3.2 Maximum Principal Strain Criterion 107
3.4 Mohr’s Theory 108
3.5 Experimental Verification of Theory of Failure 111
3.5.1 Comparison of Failure Criteria 113
3.6 Theory of Failure for Cyclic Loads 123
3.6.1 Stress Parameters 123
3.6.2 Strength Parameter 124
Exercises 139
4. ENERGY METHODS 140–157
4.1 Introduction 140
4.2 Strain Energy 140
4.2.1 Strain Energy with Simple Loading 143
4.2.2 Strain Energy due to Moment M 143Contents
4.2.3 Strain Energy due to Torsional Loading 143
4.2.4 Strain Energy due to Transverse Shear 144
4.3 Castigliano’s First Theorem 147
Exercises 156
5. DEFLECTION OF BEAMS 158–195
5.1 Introduction 158
5.2 Relation between Slope, Deflection and Radius of Curvature 158
5.3 Method for Slope and Deflection 159
5.3.1 Double Integration Method for Slope and Deflection 160
5.3.2 Macaulay’s Method 172
5.3.3 Moment Area Method 179
5.3.4 Mohr’s Theorems 181
5.4 Indeterminate Structure 185
5.5 Continuous Beam 186
5.6 Clapeyron’s Theorem of Three Moments 186
Exercises 194
6. CURVED BEAM 196–224
6.1 Introduction 196
6.2 Stresses in Curved Beam (Winkler–Bach Theory) 196
6.3 Position of Neutral Axis 200
6.4 Values of h2 200
6.4.1 Rectangular Cross-section 201
6.4.2 Circular Cross-section 201
6.4.3 I-section 202
6.4.4 T-section 202
6.4.5 Trapezoidal Cross-section 203
6.5 Stresses in a Ring 214
6.6 Stresses in a Chain Link 220
Exercises 223
7. THIN CYLINDER AND SPHERE 225–240
7.1 Introduction 225
7.2 Classification of Pressure Vessels 225
7.3 Stresses in a Thin Cylindrical Shell due to an Internal Pressure 225
7.4 Circumferential or Hoop Stress 226
7.5 Longitudinal Stress 227
7.6 Effect of Internal Pressure on the Dimensions of a Thin Cylindrical Shell 230
7.7 The Spherical Shells Subjected to an Internal Pressure 236
7.8 Change in Dimensions of Thin Spherical Shell due to an Internal Pressure 237
Exercises 240 Contents
8. THICK AND COMPOUND CYLINDER 241–262
8.1 Introduction 241
8.2 Lame’s Theory 241
8.3 Application of Theories of Failure 250
8.4 Compound Cylindrical Shell 252
8.5 Thick Spherical Shells 256
Exercises 262
9. UNSYMMETRICAL BENDING AND SHEAR CENTRE 263–281
9.1 Introduction 263
9.2 Definitions 263
9.3 Stresses due to Unsymmetrical Bending 264
9.4 Deflection of Beam due to Unsymmetrical Bending 266
9.5 Shear Centre 275
9.5.1 Shear Centre for Channel Section 275
9.5.2 Shear Centre of Unequal I-section 278
Exercises 280
10. COLUMNS AND STRUTS 282–307
10.1 Introduction 282
10.2 Definitions 282
10.3 Classification of Column 282
10.4 Assumptions Made in the Euler’s Column Theory 283
10.5 Expressions for Crippling Load of Different Cases 283
10.5.1 Both the Ends are Hinged or Pinned 283
10.5.2 One End is Fixed and Other is Free 284
10.5.3 Both Ends are Fixed 286
10.5.4 One End is Fixed, Other is Hinged 287
10.6 Effective Length of a Column 288
10.7 Slenderness Ratio 289
10.8 Crippling Stress in Terms of Effective Length and Radius of Gyration 289
10.9 Limitation of Euler’s Formula 290
10.10 Rankine’s Formula 296
10.11 Eccentric Loading 302
10.12 Johnson’s Formula for Columns 304
10.12.1 Johnson’s Straight Line Formula for Columns 304
10.12.2 Johnson’s Parabolic Formula for Columns 305
Exercises 305
11. SPRING 308–333
11.1 Introduction 308
11.2 Definitions 308
11.3 Types of Springs 309Contents
11.4 Helical Spring 309
11.4.1 Closely-coiled Helical Springs 309
11.4.2 Open-coiled Helical Springs 312
11.5 Strain Energy in the Spring 314
11.6 Springs under Impact Load 315
11.7 Springs in Series 315
11.8 Springs in Parallel 315
11.9 Leaf Springs or Carriage Springs 326
11.9.1 Semi-elliptical Spring 326
11.9.2 Quarter-elliptical Leaf Spring 330
Exercises 333
12. ROTATING DISCS AND CYLINDERS 334–349
12.1 Introduction 334
12.2 Rotating Disc 334
12.2.1 Strain Considerations 335
12.3 Hollow Disc (Disc with a Central Hole) 337
12.4 Solid Disc 338
12.5 Disc of Uniform Strength 338
12.6 Rotating Cylinder 340
12.7 Solid Cylinder 342
12.8 Hollow Cylinder 343
Exercises 348
13. FINITE ELEMENT METHOD AND ITS APPLICATION USING ANSYS
SOFTWARE 350–391
13.1 Introduction 350
13.2 The Steps 350
13.3 Principle of Minimum Potential Energy 351
13.3.1 Potential Energy 352
13.4 Computer Aided Stress Analysis Technique 353
13.5 Elements Type and Shapes 354
13.6 One-dimensional Problems 358
13.6.1 Natural Coordinate (Intrinsic Coordinate) 358
13.6.2 Isoparametric Element 359
13.6.3 Element Strain Displacement Matrix 359
13.6.4 Element Stiffness Matrix 360
13.6.5 Forces 362
13.7 Application of Finite Element Analysis Using the ANSYS Software 367
13.7.1 Application of Finite Element Analysis Using 1D Element 367
13.7.2 Application of Finite Element Analysis Using 2D Element 379
13.7.3 Application of Finite Element Analysis Using 3D Element 386
INDEX 393–39
7
reduction factor, 127
Finite element method, 350
Generalized Hooke’s law, 53
Gerber equation, 128
Goodman diagram, 128–129
Helical spring, 308
Hooke’s law, 53
Hoop or circumferential stress, 226, 236
Indeterminate structure, 185
Lame’s
equations, 243
theory, 241
Load factor, 126
Long column, 283
Longitudinal stress, 225–226
Macaulay’s method, 159, 172
Maximum
distortion energy theory, 102
normal stress, 10 Index
principal strain criterion, 107
principal stress criterion, 106
shear stress, 10
shear stress theory, 96
Mean
and deviator stresses, 70
stress, 123
Medium column, 283
Mohr’s
circle, 15
theory, 108
Mohr–Coulomb criterion, 111
Moment area method, 159, 179
Neutral axis, 196, 200
Normal stress, 2
Octahedral
plane, 68
stress, 68
Open-coiled spring, 308
PI plane, 99
Plane stress, 6
Principal stress, 14
Radial stress, 226
Radius of gyration, 289
Rankine’s formula, 296
Safe load, 282
Second moment area, 263
Shear stress, 5, 8
Short column, 282
Size factor, 125
Slenderness ratio, 228–283
Soderberg criterion, 132, 136
Solution of stress differential equation, 86
Spring index, 312
St. Venant’s equations, 83
State of pure shear, 71
State of stress, 71
Statically indeterminate
problems, 185
structures, 185
Stiffness, 158, 315
Strain energy density, 105
Strain-displacement relation, 74–75
Stress, 1–2
gradient, 51
invariants, 60
on an inclined plane, 7
tensor, 60
Strut, 282
Surface
finish factor, 125
forces, 125
Tensile and compressive stresses, 3
Theory of failure, 132
Thick shell, 225
Thin shell, 225
Torsional rigidity, 309
Tresca criterion, 96
Two-dimensional state of stress, 41
Unsymmetrical bending, 263
Variable stress, 124
von Mises theory, 102
Wahl’s correction factor, 312
Winkler-Bach theory, 196
Yield criterion, 96
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