Mechanics of Materials

PREFACE vii

LIST OF SYMBOLS xix

Chapter 1 INTRODUCTION 1

1.1 Mechanics of Materials 2

1.2 Scope of the Book 3

1.3 Methods of Analysis 4

1.4 Engineering Design 5

1.5 Review of Static Equilibrium 6

1.6 Internal Force Resultants 10

1.7 Problem Formulation and Solution 13

1.8 Application to Simple Structures 15

Chapter Summary 26

References 27

Chapter 2 CONCEPT OF STRESS 28

2.1 Introduction 29

2.2 Internal Axial Forces 29

2.3 Normal Stress 31

2.4 Bearing Stress in Connections 37

2.5 Shearing Stress 38

2.6 Stresses in Simple Structures 43

2.7 Allowable Stress and Factor of Safety 52

2.8 Design of Bars for Axial Loading 56

2.9 Case Studies 60

2.10 Stress under General Loading 68

Chapter Summary 77

References 78

Chapter 3 STRAIN AND MATERIAL PROPERTIES 79

3.1 Introduction 80

3.2 Deformation 80

3.3 Strain 81

3.4 Components of Strain 84

3.5 Materials 89

3.6 Stress–Strain Diagrams 90

3.7 True Stress and True Strain 97

3.8 Elastic versus Plastic Behavior 98

3.9 Hooke’s Law 99

3.10 Poisson’s Ratio 102

3.11 Generalized Hooke’s Law 108

3.12 Strain Energy 113

3.13 Impact Strength 115

3.14 Fatigue 116

3.15 Permanent Deformation 119

3.16 General Properties of Materials 121

3.17 Selecting Materials 122

Chapter Summary 127

References 129

Chapter 4 AXIALLY LOADED MEMBERS 130

4.1 Introduction 131

4.2 Deformation of Axially Loaded Members 131

4.3 Statically Indeterminate Structures 143

4.4 Method of Superposition 147

4.5 Thermal Deformation and Stress 148

4.6 Stresses on Inclined Planes 156

4.7 Saint-Venant’s Principle 159

4.8 Stress Concentrations 160

4.9 Ductility and Design 164

4.10 Plastic Deformation and Residual Stress 165

Chapter Summary 172

References 173

Chapter 5 TORSION 174

5.1 Introduction 175

5.2 Deformation of a Circular Shaft 175

5.3 The Torsion Formula 178

5.4 Axial and Transverse Shear Stresses 180

5.5 Stresses on Inclined Planes 183

5.6 Angle of Twist 188

5.7 Statically Indeterminate Shafts 196

5.8 Design of Circular Shafts 202

5.9 Stress Concentrations 206

5.10 Inelastic Torsion of Circular Shafts 211

5.11 Torsion of Noncircular Solid Bars 215

5.12 Thin-Walled Hollow Members 218

Chapter Summary 228

References 230

Chapter 6 SHEAR AND MOMENT IN BEAMS 231

6.1 Introduction 232

6.2 Classification of Beams 232

6.3 Calculation of Beam Reactions 233

6.4 Shear Force and Bending Moment 238

6.5 Load, Shear, and Moment Relationships 243

6.6 Shear and Moment Diagrams 245

6.7 Discontinuity Functions 259

Chapter Summary 268

References 270

Chapter 7 STRESSES IN BEAMS 271

7.1 Introduction 272

PART A Pure Bending 273

7.2 Beam Deformation in Pure Bending 273

7.3 Assumptions of Beam Theory 275

7.4 Normal Strain in Beams 276

7.5 Normal Stress in Beams 280

7.6 Stress Concentrations in Bending 285

PART B Shear and Bending 292

7.7 Shear Stresses in Beams 292

7.8 Shear Stress Distribution in Rectangular Beams 297

7.9 Shear Stresses in Beams of Circular Cross Section 299

7.10 Shear Stress Distribution in Flanged Beams 300

7.11 Comparison of Shear and Bending Stresses 304

7.12 Design of Prismatic Beams 308

7.13 Design of Beams of Constant Strength 311

PART C Special Topics 321

7.14 Composite Beams 321

7.15 Reinforced Concrete Beams 325

7.16 Unsymmetric Bending 327

7.17 Shear Center 332

7.18 Inelastic Bending 334

7.19 Curved Beams 341

Chapter Summary 356

References 359

Chapter 8 TRANSFORMATION OF STRESS AND STRAIN 360

8.1 Introduction 361

8.2 Plane Stress 361

8.3 Principal Stresses 367

8.4 Maximum Shear Stress 368

8.5 Mohr’s Circle for Plane Stress 370

8.6 Absolute Maximum Shear Stress 383

8.7 Principal Stresses for a General State of Stress 384

8.8 Thin-Walled Pressure Vessels 385

8.9 Thick-Walled Pressure Vessels 393

8.10 Plane Strain 402

8.11 Mohr’s Circle for Plane Strain 405

8.12 Measurement of Strain;Strain Rosette 409

8.13 Relation Involving E, n, and G 412

Chapter Summary 416

References 418

Chapter 9 COMBINED LOADINGS AND FAILURE CRITERIA 419

9.1 Introduction 420

PART A Combined Stresses 421

9.2 Axial and Torsional Loads 421

9.3 Direct Shear and Torsional Loads: Helical Springs 427

9.4 Axial, Transverse, and Torsional Loads 431

9.5 Transverse Shear and Bending MomentLoads: Principal Stresses in Beams 437

9.6 Eccentric Axial Loads 440

PARTB Failure Theories inDesign 450

9.7 Material Failure 450

9.8 Yield Criteria for Ductile Materials 451

9.9 Fracture Criteria for Brittle Materials 454

9.10 Design of Transmission Shafts 460

Chapter Summary 468

References 470

Chapter 10 DEFLECTIONS OF BEAMS 471

10.1 Introduction 472

10.2 The Elastic Curve 472

10.3 Boundary Conditions 473

10.4 Method of Integration 476

10.5 Use of Discontinuity Functions 487

10.6 Method of Superposition 493

10.7 Statically Indeterminate Beams 496

10.8 Statically Indeterminate Beams—Method of Integration 497

10.9 Statically Indeterminate Beams—Method of Superposition 507

10.10 Moment–Area Method 514

10.11 Statically Indeterminate Beams—Moment–Area Method 525

10.12 Continuous Beams 527

Chapter Summary 535

References 536

Chapter 11 BUCKLING OF COLUMNS 537

11.1 Introduction 538

11.2 Stability of Structures 538

11.3 Pin-Ended Columns 540

11.4 Columns with Other End Conditions 542

11.5 Critical Stress: Classification of Columns 547

11.6 Eccentric Loaded Columns and the Secant Formula 558

11.7 Design of Columns for Centric Loading 564

11.8 Design of Columns for Eccentric Loading 569

Chapter Summary 574

References 575

Chapter 12 ENERGY METHODS AND IMPACT 576

12.1 Introduction 577

12.2 Strain Energy under Axial Loading 577

12.3 Strain Energy in Circular Shafts 580

12.4 Strain Energy in Beams 581

12.5 Strain Energy for a General State of Stress 584

12.6 Conservation of Energy 589

12.7 Displacement under a Single Load by the Work–Energy Method 590

12.8 Displacements by Castigliano’s Theorem 593

12.9 Unit-Load Method 599

12.10 Statically Indeterminate Structures 602

12.11 Impact Loading 609

12.12 Longitudinal and Bending Impact 610

Chapter Summary 621

References 623

Chapter 13 FINITE ELEMENT ANALYSIS 624

13.1 Introduction 625

13.2 The Bar Element 626

13.3 Two-Dimensional Bar Element 627

13.4 Axial Force in the Bar Element 630

13.5 Formulation of the Finite Element Method 631

13.6 Beam Elements 644

Chapter Summary 652

References 653

Appendix A PROPERTIES OF AREAS 655

A.1 Centroid of an Area 655

A.2 Moments of Inertia and Radius of Gyration 658

A.3 Parallel-Axis Theorem 660

A.4 Principal Moments of Inertia 662

Appendix B TABLES 667

B.1 Principal SI Units Used in Mechanics 668

B.2 SI Prefixes 668

B.3 Conversion Factors between U.S. Customary and SI Units 669

B.4 Properties of Selected Engineering Materials 670

B.5 Materials and Selected Members of Each Class 672

B.6 Properties of Areas 674

B.7 Properties of Selected Steel Pipe and Tubing 675

B.8 Properties of Steel W Shapes, Wide-Flange Sections 676

B.9 Properties of Steel S Shapes, American Standard I-Beams 678

B.10 Properties of Steel C Shapes, American Standard Channels 680

B.11 Properties of Steel L Shapes, Angles with Equal Legs 682

B.12 Properties of Steel L Shapes, Angles with Unequal Legs 684

B.13 Properties of Structural Lumber 686

B.14 Deflections and Slopes of Beams 687

B.15 Reactions and Deflections of Statically Indeterminate Beams 689

Appendix C MATRIX ALGEBRA 690

C.1 Definition of a Matrix 690

C.2 Determinant of a Matrix 691

C.3 Matrix Operations 693

C.4 Simultaneous Linear Equations 695

Appendix D FUNDAMENTALS OF ENGINEERING EXAMINATION 697

ANSWERS TO SELECTED EVEN-NUMBERED PROBLEMS 698

INDEX 711