Advanced Mechanics of Materials and Applied Elasticity

Ansel C. Ugural, Ph.D., served for two decades as professor and chairman of the mechanical engineering department at Fairleigh Dickinson University. He has also been a visiting and research professor of solid mechanics in mechanical engineering at New Jersey Institute of Technology. He is also a National Science Foundation (NSF) Fellow and is a faculty member at the University of Wisconsin–Madison, where he earned his M.S. in mechanical engineering and Ph.D. in engineering mechanics. Saul K. Fenster, Ph.D., is professor at New Jersey Institute of Technology, where he served as a president for more than two decades. In addition to experience in industry, he has held varied positions at Fairleigh Dickinson University and taught at the City University of New York. Fenster, a Fellow of the American Society of Mechanical Engineers and the American Society for Engineering Education, is co-author of a text on mechanics.

• Preface xviiAcknowledgments xxAbout the Authors xxiList 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 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 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 713Index 722