# Advanced Mechanics of Materials and Applied Elasticity, 6th edition

Published by Prentice Hall (August 1st 2019) - Copyright © 2020

6th edition

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Advanced Mechanics of Materials and Applied Elasticity

ISBN-13: 9780134859286

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Hardcover

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## Overview

**The Leading Practical Guide to Stress Analysis–Updated with State-of-the-Art Methods, Applications, and Problems**

This widely acclaimed exploration of real-world stress analysis reflects advanced methods and applications used in today’s mechanical, civil, marine, aeronautical engineering, and engineering mechanics/science environments. Practical and systematic,

*has been updated with many new examples, figures, problems, MATLAB solutions, tables, and charts.*

**Advanced Mechanics of Materials and Applied Elasticity, Sixth Edition,**The revised edition balances discussions of advanced solid mechanics, elasticity theory, classical analysis, and computer-oriented approaches that facilitate solutions when problems resist conventional analysis. It illustrates applications with case studies, worked examples, and problems drawn from modern applications, preparing readers for both advanced study and practice.

Readers will find updated coverage of analysis and design principles, fatigue criteria, fracture mechanics, compound cylinders, rotating disks, 3-D Mohr’s circles, energy and variational methods, buckling of various columns, common shell types, inelastic materials behavior, and more. The text addresses the use of new materials in bridges, buildings, automobiles, submarines, ships, aircraft, and spacecraft. It offers significantly expanded coverage of stress concentration factors and contact stress developments. This book aims to help the reader

- Review fundamentals of statics, solids mechanics, stress, and modes of load transmission
- Master analysis and design principles through hands-on practice to illustrate their connections
- Understand plane stress, stress transformations, deformations, and strains
- Analyze a body’s load-carrying capacity based on strength, stiffness, and stability
- Learn and apply the theory of elasticity
- Explore failure criteria and material behavior under diverse conditions, and predict component deformation or buckling
- Solve problems related to beam bending, torsion of noncircular bars, and axisymmetrically loaded components, plates, or shells
- Use the numerical finite element method to economically solve complex problems
- Characterize the plastic behavior of materials

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## Table of contents

*Preface xvii*

Acknowledgments xx

About the Authors xxi

List of Symbols xxii

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

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

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

Index 722

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