Contents
Chapter 1 Introduction to Mechanics of Materials 1
1.1 Introduction 1
1.1.1 Strength, Stiffness, and Stability 1
1.1.2 Research Objects of Mechanics of Materials 2
1.2 Basic Assumptions of Mechanics of Materials 2
1.3 External Force, Internal Force and Stress 3
1.3.1 External Force 3
1.3.2 Internal Force and Method of Section 4
1.3.3 Stress 6
1.4 Displacement, Deformation, and Strain 7
1.4.1 Displacement and Deformation 7
1.4.2 Strain 7
1.5 Basic Patterns of Deformation of Prismatic Bar 8
Problems 9
Chapter 2 Axial Tension and Compression 11
2.1 Introduction 11
2.2 Axial Forces and Axial Force Diagrams 11
2.2.1 Normal Force 11
2.2.2 Axial Force Diagram 12
2.3 Stress in an Axially Loaded Bar 14
2.3.1 Stresses on Cross Section 14
2.3.2 Stresses on Oblique Section 15
2.3.3 Saint-Venant Principle 17
2.4 Mechanical Behaviour of Materials Under Tension and Compression 18
2.4.1 Introduction of Tensile and Compressive Tests 19
2.4.2 Mechanical Behaviour of Materials Under Tension 20
2.4.3 Mechanical Behaviour of Materials Under Compression 23
2.5 Strength Condition of Axially Loaded Bar 25
2.6 Deformation of Axially Loaded Bar 29
2.7 Statically Indeterminate Problems of Axial Tension and Compression 34
2.7.1 Analysis of Statically Indeterminate Problem in Tension and Compression 35
2.7.2 Assembly Stress 39
2.7.3 Thermal Stress 40
2.8 Stress Concentration 43
Problems 44
Chapter 3 Shearing in Connections 50
3.1 Introduction 50
3.2 Strength Check of Shearing Deformation in Connections 50
3.3 Strength Check of Bearing Deformation 52
Problems 56
Chapter 4 Torsion 59
4.1 Introduction 59
4.2 Internal Torque and Internal Torque Diagram 59
4.2.1 Twisting Couple on a Transmission Shaft 59
4.2.2 Internal Torque and Torque Diagrams 60
4.3 Pure Shear, Shearing Stress, Hooke’s Law in Shear Situations 62
4.3.1 Shear Stress on Cross Sections of a Thin-Walled Tube Induced by Torsion 62
4.3.2 Shearing Stress 63
4.3.3 Hooke’s Law in Shearing 63
4.4 Shear Stress in a Circular Shaft 64
4.4.1 Calculation of Shear Stress 64
4.4.2 Stress on Oblique Plane of a Circular Shaft Subjected to Torsion 69
4.4.3 Strength Design 70
4.5 Torsional Deformation of a Circular Shaft and Stiffness Design 72
4.5.1 Torsional Deformation of a Circular Shaft 72
4.5.2 Stiffness Design 72
4.6 Statically Indeterminate Shaft in Torsion 76
4.7 The Strength of Helical Springs 78
4.8 Torsion of Noncircular Shafts 80
4.8.1 Free Torsion and Restrained Torsion 80
4.8.2 Rectangular Shaft in Torsion 81
Problems 82
Chapter 5 Bending 86
5.1 Introduction 86
5.1.1 Bending 86
5.1.2 Boundary Supports of Beams 86
5.1.3 Basic Types of Beams 87
5.2 Shear Force and Bending Moment 88
5.3 Shear Force Diagram and Bending Moment Diagram 92
5.4 Relationships Between Shear Forces, Bending Moments and External Loads 97
5.4.1 Differential Relations Among Shear Force, Bending Moments and External Loads 97
5.4.2 Draw the Shear Force and the Bending Moment Diagram According to Differential Relations 98
5.4.3 Draw Bending Moment Diagram by the Principle of Superposition 104
5.5 Internal Forces in Plane Frames and Curved Beams 105
5.5.1 Internal Forces of Plane Frame 105
5.5.2 Internal Force of Curved Beams 106
Problems 107
Chapter 6 Bending Stress 113
6.1 Introduction 113
6.2 Bending Normal Stress 113
6.2.1 Geometric Aspects 114
6.2.2 Physical Law 115
6.2.3 Statics 115
6.2.4 Bending Normal Stress 116
6.2.5 Normal Stress in Transverse Bending 118
6.3 Shear Stress in Bending 118
6.3.1 Shear Stress of the Beam with Rectangular Cross Section 118
6.3.2 Shear Stress of a Beam with I-Shape Cross Section 121
6.3.3 Shear Stress of a Beam with Circular and Circular Tubular Cross Section 122
6.3.4 Magnitude of Normal Stress and Shear Stress in Transverse Bending 123
6.3.5 Effects of Shear Stress on Normal Stress in Transverse Bending 124
6.4 Strength Condition of the Beam 125
6.4.1 Strength Condition of Normal Stress 125
6.4.2 Strength Condition of Shear Stress 126
6.5 Plane Bending and Bending Centre of Asymmetric Section Beams 131
6.6 Measures to Increase the Bending Strength of Beams 133
6.6.1 Selection of an Optimal Cross Section 133
6.6.2 Tapered Beams and Beams of Uniform Strength 134
6.6.3 Rearrange the External Loading of the Beam 135
Problems 138
Chapter 7 Bending Deformation 143
7.1 Introduction 143
7.2 Approximated Differential Equation of Deflection Curve 144
7.3 Calculation of Beam Deformation by the Integral Method 145
7.4 Calculate the Displacement of Beams by the Principle of Superposition 152
7.5 Stiffness Condition and Design of Beams 158
7.5.1 Stiffness Condition of Beam 158
7.5.2 Approaches to Improve Bending Stiffness 159
7.6 Simple Statically Indeterminate Beams 160
Problems 167
Chapter 8 State of Stress and Strength Theory 174
8.1 Introduction 174
8.1.1 Concept of State of Stress 174
8.1.2 General Description of State of Stress 174
8.1.3 Strength Theory 176
8.2 Plane State of Stress Analysis 176
8.2.1 Stress on Any Oblique Plane 177
8.2.2 Principal Plane 178
8.2.3 Principal Stress 178
8.2.4 Extreme Shear Stress 179
8.3 Stress Circles in the State of Plane Stress 181
8.4 Maximum Stress in the Three-Dimensional State of Stress 182
8.5 Generalized Hooke’s Law 184
8.5.1 Mathematical Formulation of the Generalized Hooke’s Law 184
8.5.2 Volumetric Strain 186
8.6 Strain Energy and Distortion Energy 187
8.6.1 Strain Energy and Strain Energy Density 187
8.6.2 Dilatational Strain Energy Density and Distortion Energy Density 188
8.7 Four Commonly Used Classical Strength Theories 189
8.7.1 Strength Theory on Brittle Fracture 189
8.7.2 Strength Theory of Plastic Yield 191
Problems 196
Chapter 9 Combined Deformation 202
9.1 Introduction 202
9.2 Unsymmetric Bending 202
9.3 Combination of Tension (or Compression) and Bending 208
9.4 Eccentric Load and Core of Section 211
9.4.1 Strength Calculation of an Eccentric Axially Loaded Bar 211
9.4.2 Core of Section 214
9.5 Combination of Bending and Torsion 215
Problems 219
Chapter 10 Stability of Compressive Columns 224
10.1 Introduction 224
10.2 Critical Force of Slender Compressive Columns and Euler’s Formula 226
10.2.1 Critical Force of a Column Hinged at Both Ends 226
10.2.2 Critical Force of the Compressive Columns with Other Types of Boundary Conditions 228
10.2.3 General Form of Euler’s Formula 230
10.2.4 Application Scope of the Euler’s Formula 230
10.3 Critical Force and Critical Stress of the Column with Medium or Small
Slenderness Ratio 232
10.4 Stability Check of Compressive Columns 235
10.5 Methods to Improve the Stability of Compressive Columns 237
10.5.1 Select a Proper Section Shape 237
10.5.2 Change the Constraints of Compressive Columns 238
10.5.3 Reasonably Select Materials 238
Problems 238
Chapter 11 Energy Method 241
11.1 Introduction 241
11.2 External Work Done and Strain Energy 241
11.2.1 The External Work Done 241
11.2.2 Strain Energy of Member 243
11.2.3 Strain Energy Density 244
11.3 Castigliano’s Theorem 246
11.3.1 Complementary Work and Complementary Strain Energy 246
11.3.2 Introduction of Castigliano’s Theorem 248
11.3.3 Calculation of the Displacement by Castigliano’s Theorem 249
11.4 Principle of Virtual Work on a Deformable Body 253
11.5 Unit Load Method 254
11.5.1 Introduction of Unit Load Method 255
11.5.2 Mohr Integral 255
11.5.3 Graph Multiplication Method 260
11.6 Reciprocal Theorem 265
11.6.1 Maxwell-Betti Reciprocal Work Theorem 265
11.6.2 Theorem of Reciprocal Displacements 267
Problems 269
Chapter 12 Statically Indeterminate Structures 275
12.1 Introduction 275
12.2 Force Method and General Equation of the Force Method 276
12.2.1 Basic Unknown Variables, the Equivalent Statically Determinate System 276
12.2.2 General Formulation of the Force Method 277
12.2.3 Solution to Externally Statically Indeterminate Problem 280
12.2.4 Solution to Internally Statically Indeterminate Problem 284
12.3 Symmetric Statically Indeterminate Structure 287
Problems 295
Chapter 13 Dynamic Loads 299
13.1 Introduction 299
13.2 Stress and Strain of Members Under Motion with a Constant Acceleration 299
13.2.1 Stress of Members in a Uniformly Accelerated Motion 299
13.2.2 Stresses and Deformations of Members in a Uniform Circular Motion 302
13.3 Stress and Deformation Under Impact 305
13.4 Measures to Improve the Capability of Resisting Impact 314
Problems 314
Chapter 14 Fatigue 318
14.1 Introduction 318
14.1.1 Alternating Stress 318
14.1.2 Fatigue 320
14.2 Endurance Limit 321
14.2.1 The Endurance Limit of Materials 321
14.2.2 The Endurance Limit of a Member 322
14.3 The Fatigue Strength Check of Structures Under Symmetric Cycling Loads 326
14.4 The Fatigue Strength Check of Structures Under Asymmetric Cycling Loads 327
14.5 Measures to Improve the Fatigue Strength 328
Problems 329
Appendix A1 Geometric Properties of a Cross Section 331
A1.1 First Moment 331
A1.1.1 First Moment and Centroid 331
A1.1.2 First Moment and Centroid of a Composite Area 333
A1.2 Moment of Inertia and Product Moment of Inertia 334
A1.3 Parallel-Axis Theorem 337
A1.4 Rotation of Axes and Principal Axes 339
A1.4.1 The Rotation of Axes 339
A1.4.2 Principal Axes and Principal Moments of Inertia 340
A1.5 Principal Centroidal Moments of Inertia of Composite Areas 342
Problems 343
Appendix A2 Cross-Sectional Properties of Shapes of Steel 346
Appendix A3 Mechanical Properties of Materials 357
Appendix A4 Geometric Properties of Plane 358
Bibliography 360
Answers to Problems 361