Advanced Mechanics of Materials and Applied Elasticity

Ansel C. Ugural, Ph.D., is a visiting professor at the New Jersey Institute of Technology. He has held various faculty and administrative positions at Fairleigh Dickinson University, and previously taught at the University of Wisconsin. Ugural has extensive industrial experience, is a member of several professional societies, and is author of Mechanics of Materials (Wiley, 2007), Stresses in Beams, Plates and Shells (CRC Press, 2009), and Mechanical Design: An Integrated Approach (McGraw-Hill, 2004).   Saul K. Fenster, Ph.D., was a professor at the New Jersey Institute of Technology, where he served as president for over twenty years. He is a fellow of the American Society of Mechanical Engineers and the American Society for Engineering Education.

Preface         xii Acknowledgments         xiv

About the Authors         xv

List of Symbols         xvi



 

Chapter 1: Analysis of Stress         1

1.1   Introduction    1

1.2   Scope of Treatment   3

1.3   Analysis and Design   5

1.4   Conditions of Equilibrium   7

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   19

1.9   Plane-Stress Transformation   22

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   33

1.13 Principal Stresses in Three Dimensions   36

1.14 Normal and Shear Stresses on an Oblique Plane   40

1.15 Mohr’s Circles in Three Dimensions   43

1.16 Boundary Conditions in Terms of Surface Forces   47

1.17 Indicial Notation   48

References   49

Problems   49

 

Chapter 2: Strain and Material Properties         65

2.1   Introduction   65

2.2   Deformation   66

2.3   Strain Defined   67

2.4   Equations of Compatibility   72

2.5   State of Strain at a Point   73

2.6   Engineering Materials   80

2.7   Stress—Strain Diagrams   82

2.8   Elastic versus Plastic Behavior   86

2.9   Hooke’s Law and Poisson’s Ratio   88

2.10 Generalized Hooke’s Law   91

2.11 Hooke’s Law for Orthotropic Materials   94

2.12 Measurement of Strain: Strain Rosette   97

2.13 Strain Energy   101

2.14 Strain Energy in Common Structural Members   104

2.15 Components of Strain Energy   106

2.16 Saint-Venant’s Principle   108

References 110

Problems 111

 

Chapter 3:Problems in Elasticity         124

3.1   Introduction   124

3.2   Fundamental Principles of Analysis   125

Part A–Formulation and Methods of Solution   126

3.3   Plane Strain Problems   126

3.4   Plane Stress Problems   128

3.5   Comparison of Two-Dimensional Isotropic Problems   131

3.6   Airy’s Stress Function   132

3.7   Solution of Elasticity Problems   133

3.8   Thermal Stresses   138

3.9   Basic Relations in Polar Coordinates   142

Part B–Stress Concentrations 147

3.10 Stresses Due to Concentrated Loads   147

3.11 Stress Distribution Near Concentrated Load Acting on a Beam   151

3.12 Stress Concentration Factors   153

3.13 Contact Stresses 159

3.14 Spherical and Cylindrical Contacts   160

3.15 Contact Stress Distribution   163

3.16 General Contact   167

References   170

Problems   171

 

Chapter 4: Failure Criteria         181

4.1   Introduction   181

4.2   Failure   181

4.3   Failure by Yielding   182

4.4   Failure by Fracture   184

4.5   Yield and Fracture Criteria   187

4.6   Maximum Shearing Stress Theory   188

4.7   Maximum Distortion Energy Theory   189

4.8   Octahedral Shearing Stress Theory   190

4.9   Comparison of the Yielding Theories   193

4.10 Maximum Principal Stress Theory   195

4.11 Mohr’s Theory   195

4.12 Coulomb—Mohr Theory   196

4.13 Fracture Mechanics   200

4.14 Fracture Toughness   203

4.15 Failure Criteria for Metal Fatigue   206

4.16 Impact or Dynamic Loads   212

4.17 Dynamic and Thermal Effects   215

References   217

Problems   218

 

Chapter 5: Bending of Beams          226

5.1   Introduction   226

Part A–Exact Solutions   227

5.2   Pure Bending of Beams of Symmetrical Cross Section   227

5.3   Pure Bending of Beams of Asymmetrical Cross Section   230

5.4   Bending of a Cantilever of Narrow Section   235

5.5   Bending of a Simply Supported Narrow Beam   238

Part B–Approximate Solutions   240

5.6   Elementary Theory of Bending   240

5.7   Normal and Shear Stresses   244

5.8   Effect of Transverse Normal Stress   249

5.9   Composite Beams   250

5.10 Shear Center   256

5.11 Statically Indeterminate Systems   262

5.12 Energy Method for Deflections   264

Part C–Curved Beams   266

5.13 Elasticity Theory   266

5.14 Curved Beam Formula   269

5.15 Comparison of the Results of Various Theories   273

5.16 Combined Tangential and Normal Stresses   276

References   280

Problems   280

 

Chapter 6: Torsion of Prismatic Bars          292

6.1   Introduction   292

6.2   Elementary Theory of Torsion of Circular Bars   293

6.3   Stresses on Inclined Planes   298

6.4   General Solution of the Torsion Problem   300

6.5   Prandtl’s Stress Function   302

6.6   Prandtl’s Membrane Analogy   310

6.7   Torsion of Narrow Rectangular Cross Section   315

6.8   Torsion of Multiply Connected Thin-Walled Sections   317

6.9   Fluid Flow Analogy and Stress Concentration   321

6.10 Torsion of Restrained Thin-Walled Members of Open Cross Section   323

6.11 Curved Circular Bars: Helical Springs   327

References   330

Problems   330

 

Chapter 7: Numerical Methods         337

7.1   Introduction   337

Part A–Finite Difference Method   338

7.2   Finite Differences   338

7.3   Finite Difference Equations   341

7.4   Curved Boundaries   343

7.5   Boundary Conditions   346

Part B–Finite Element Method   350

7.6   Fundamentals   350

7.7   The Bar Element   352

7.8   Arbitrarily Oriented Bar Element  354

7.9   Axial Force Equation   357

7.10 Force-Displacement Relations for a Truss   359

7.11 Beam Element   366

7.12 Properties of Two-Dimensional Elements   372

7.13 General Formulation of the Finite Element Method   374

7.14 Triangular Finite Element   379

7.15 Case Studies in Plane Stress   386

7.16 Computational Tools   394

References   395

Problems   396

 

Chapter 8: Axisymmetrically Loaded Members          407

8.1   Introduction   407

8.2   Thick-Walled Cylinders   408

8.3   Maximum Tangential Stress   414

8.4   Application of Failure Theories   415

8.5   Compound Cylinders: Press or Shrink Fits   416

8.6   Rotating Disks of Constant Thickness   419

8.7   Design of Disk Flywheels   422

8.8   Rotating Disks of Variable Thickness   426

8.9   Rotating Disks of Uniform Stress   429

8.10 Thermal Stresses in Thin Disks   431

8.11 Thermal Stresses in Long Circular Cylinders   432

8.12 Finite Element Solution   436

8.13 Axisymmetric Element   437

References   441

Problems   442

 

Chapter 9:Beams on Elastic Foundations         448

9.1   Introduction   448

9.2   General Theory   448

9.3   Infinite Beams   449

9.4   Semi-Infinite Beams   454

9.5   Finite Beams   457

9.6   Classification of Beams   458

9.7   Beams Supported by Equally Spaced Elastic Elements   458

9.8   Simplified Solutions for Relatively Stiff Beams   460

9.9   Solution by Finite Differences   461

9.10 Applications  464

References   466

Problems   466

 

Chapter 10: Applications of Energy Methods         469

10.1   Introduction   469

10.2   Work Done in Deformation   470

10.3   Reciprocity Theorem   471

10.4   Castigliano’s Theorem   472

10.5   Unit- or Dummy-Load Method   479

10.6   Crotti—Engesser Theorem   481

10.7   Statically Indeterminate Systems   483

10.8   Principle of Virtual Work   486

10.9   Principle of Minimum Potential Energy   487

10.10 Deflections by Trigonometric Series   489

10.11 Rayleigh—Ritz Method   493

References   496

Problems   496

 

Chapter 11: Stability of Columns         505

11.1   Introduction   505

11.2   Critical Load   505

11.3   Buckling of Pinned-End Columns   507

11.4   Deflection Response of Columns   509

11.5   Columns with Different End Conditions   511

11.6   Critical Stress: Classification of Columns   513

11.7   Allowable Stress   517

11.8   Imperfections in Columns   519

11.9   Eccentrically Loaded Columns: Secant Formula   520

11.10 Energy Methods Applied to Buckling   522

11.11 Solution by Finite Differences   529

11.12 Finite Difference Solution for Unevenly Spaced Nodes   534

References   536

Problems   536

 

Chapter 12: Plastic Behavior of Materials          545

12.1   Introduction   545

12.2   Plastic Deformation   546

12.3   Idealized Stress—Strain Diagrams   546

12.4   Instability in Simple Tension   549

12.5   Plastic Axial Deformation and Residual Stress   551

12.6   Plastic Defection of Beams   553

12.7   Analysis of Perfectly Plastic Beams   556

12.8   Collapse Load of Structures: Limit Design   565

12.9   Elastic—Plastic Torsion of Circular Shafts   569

12.10 Plastic Torsion: Membrane Analogy   573

12.11 Elastic—Plastic Stresses in Rotating Disks   575

12.12 Plastic Stress—Strain Relations   578

12.13 Plastic Stress—Strain Increment Relations   583

12.14 Stresses in Perfectly Plastic Thick-Walled Cylinders   586

References   590

Problems   590

 

Chapter 13:Plates and Shells          598

13.1   Introduction   598

Part A–Bending of Thin Plates   598

13.2   Basic Assumptions   598

13.3   Strain—Curvature Relations   599

13.4   Stress, Curvature, and Moment Relations   601

13.5   Governing Equations of Plate Deflection   603

13.6   Boundary Conditions   605

13.7   Simply Supported Rectangular Plates   607

13.8   Axisymmetrically Loaded Circular Plates   610

13.9   Deflections of Rectangular Plates by the Strain-Energy Method   613

13.10 Finite Element Solution   615

Part B–Membrane Stresses in Thin Shells   618

13.11 Theories and Behavior of Shells   618

13.12 Simple Membrane Action   618

13.13 Symmetrically Loaded Shells of Revolution   620

13.14 Some Common Cases of Shells of Revolution   622

13.15 Thermal Stresses in Compound Cylinders   626

13.16 Cylindrical Shells of General Shape   628

References   631

Problems   631

 

Appendix A: Problem Formulation and Solution         637

 

Appendix B: Solution of the Stress Cubic Equation         640

B.1   Principal Stresses   640

B.2   Direction Cosines   641

 

Appendix C: Moments of Composite Areas            645

C.1   Centroid   645

C.2   Moments of Inertia   648

C.3   Parallel-Axis Theorem   649

C.4   Principal Moments of Inertia   652

 

Appendix D: Tables and Charts         659

D.1   Average Properties of Common Engineering Materials   660

D.2   Conversion Factors: SI Units to U.S. Customary Units   662

D.3   SI Unit Prefixes   662

D.4   Deflections and Slopes of Beams   663

D.5   Reactions Deflections of Statically Indeterminate Beams   664

D.6   Stress Concentration Factors for Bars and Shafts with Fillets, Grooves, and Holes   665

 

Answers to Selected Problems         669

 

Index         677