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CHAPTER ONE BASIC CONCEPTS OF FORCE,STRESS,STRAIN,AND DISPLACEMENT 1

1．0 Introduction 1

1．1 Force Diagrams 2

1．2 Force Distributions 2

1．3 Stress 10

1．4 Strain,Stress—Strain Relations 20

1．4．1 Normal Strains 20

1．4．2 Shear Strains 23

1．4．3 Thermal Strains 24

1．5 Displacements, Strain-Displacement Relations 24

1．5．1 Rectangular Coordinates 24

1．5．2 Cylindrical Coordinates 29

1．6 Summary of Important Relationships 33

1．7 Problems 37

CHAPTER TWO STRESS AND STRAIN.TRANSFORMATIONS,EQUILIBRIUM,AND COMPATIBILITY 46

2．0 Introduction 46

2．1．1 General Three-Dimensional Stress Transformations 47

2．1 Stress Transformations 47

2．1．2 Plans Stress Transformations 56

2．1．3 Mohr s Circle for Plane Stress 57

2．1．4 Three-Dimensional Stress Transformation Simplified 62

2．1．5 Principal Stresses 66

2．1．6 Mohr s Circles in Three Dimensions 72

2．1．7 Maximum Shear Stress 75

2．2．1 Strain Transformations,General 78

2．2 Strain Transformations 78

2．2．2 Principal Strains 81

2．3 Generalized Stress-Strain Relations 82

2．4 The Equilibrium Equations 86

2．5 Compatibility 89

2．6 Summary of Important Equations 97

2．7 Problems 101

CHAPTER THREE A REVIEW OF THE FUNDAMENTAL FORMULATIONS OF STRESS,STRAIN,AND DEFLECTION 113

3．0 Introduction 113

3．2 Axial Loading 114

3．2．1 Axial Stresses 114

3．1 Assumptions and Limitations 114

3．2．2 Axial Strains and Deflections 117

3．3 Torsion of Circular Shafts 129

3．3．1 Torsional Stresses 129

3．3．2 Torsional Strains and Deflections 130

3．4 Beams in Bending 132

3．4．1 Shear Force and Bending Moment Equations and Diagrams 132

3．4．2 Bending Stresses 138

3．4．3 Transverse Shear Stresses 143

3．4．4 Bending Strains and Deflections 152

3．5 Bending of Symmetric Beams in Two Planes 158

3．6 Thin-Walled Pressure Vessels 162

3．6．1 Stresses 162

3．6．2 Strains and Deflections in a Circular Cylinder 166

3．7 Superposition 166

3．8 Statically Indeterminate Problems 175

3．9 Stress and Strain Transformations 185

3．9．1 Plane Stress 185

3．9．2 Principal Stress 186

3．9．3 Maximum In-Plane Shear Stress 189

3．9．4 Strain Transformations 191

3．10 Buckling Instability of Columns in Compression 192

3．11 Problems 199

3．12References 219

CHAPTER FOUR CONCEPTS FROM THE THEORY OF ELASTICITY 220

4．0 Introduction 220

4．1．1 Definition 221

4．1 Plane Elastic Problems 221

4．1．2 Governing Equations 222

4．1．3 Conversion between Plane Stress and Plane Strain Problems 224

4．2 The Airy Stress Function 225

4．2．1 Rectangular Coordinates 225

4．2．2 Polar Coordinates 230

4．2．3 Curved Beam in Bending 233

4．2．4 Circular Hole in a Plate Loaded in Tension 235

4．2．5 Concentrated Force on a Flat Boundary (Flamant Solution) 238

4．2．6 Disk with Opposing Concentrated Forces 240

4．3．1 General Formulation 244

4．3 Prandtl s Stress Function for Torsion 244

4．3．2 Torsion on a Rectangular Cross Section 251

4．4 Discussion 255

4．5 Problems 255

4．6 References 260

CHAPTER FIVE TOPICS FROM ADVANCED MECHANICS OF MATERIALS 261

5．0 Introduction 261

5．1 Shear Flow 261

5．2 Torsion of Closed Thin-Walled Tubes 262

5．2．1 Single Cell Sections in Torsion 263

5．2．2 Multiple Cell Sections in Torsion 267

5．3 Bending of Unsymmetrical Beams 272

5．3．1 Stresses 272

5．3．2 Deflections 279

5．4 Further Discussion of Transverse Shear Stresses 280

5．4．1 Shear Flow in Open Thin-Walled Beams 280

5．4．2 Shear Center for Open Thin-Walled Beams with One Axis of Symmetry 283

5．4．3 Shear Center for Open Unsymmetric Thin-Walled Beams 291

5．4．4 Shear in Closed Thin-Walled Sections 296

5．5 Composite Beams in Bending 302

5．6 Curved Beams 309

5．6．1 Tangential Stresses 309

5．6．2 Approximate and Numerical Calculations of e 317

5．6．3 Radial Stresses 322

5．7 Bending of Thin Flat Plates 324

5．7．1 Governing Equations in Rectangular Coordinates 324

5．7．2 Tabulated Solutions of Uniformly Loaded Rectangular Plates 329

5．7．3 Governing Equations for Axisymmetric Circular Plates in Bending 330

5．7．4 Tabulated Solutions of Circular Plates 336

5．7．5 Superposition 341

5．8 Thick-Walled Cylinders and Rotating Disks 348

5．9 Contact Stresses 357

5．9．1 Distributed Contact Loading 357

5．9．2 Contact Between Curved Surfaces 361

5．10 Stress Concentrations 364

5．11 Problems 371

5．12 References 403

6．0 Introduction 404

CHAPTER SIX ENERGY TECHNIQUES IN STRESS ANALYSIS 404

6．1 Work 409

6．2 Strain Energy 410

6．2．1 Uniaxial Case 410

6．2．2 Additional Normal Stresses 411

6．2．3 Shear Stress 411

6．2．4 General State of Stress 412

6．2．5 Plane Stress 413

6．3．2 Torsional Loading of a Solid Circular Bar 414

6．3．1 Axial Loading 414

6．3 Total Strain Energy in Bars with Simple Loading Conditions 414

6．3．3 Transverse Loading 415

6．4 Castigliano s First Theorem 419

6．5 The Complementary-Energy Theorem 424

6．6 Castigliano s Second Theorem 428

6．6．1 Deflections of Statically Determinate Problems 428

6．6．2 Deflections Due to Temperature Changes 445

6．7 Deflections of Thick-Walled Curved Beams 446

6．8 Castigliano s Second Theorem Applied to Statically Indeterminate Problems 450

6．9 The Virtual Load Method 456

6．9．1 Axial Loading 458

6．9．2 Torsional Loading 459

6．9．3 Bending 460

6．10 The Virtual Load Method Applied to Statically Indeterminate Problems 462

6．11 Rayleigh s Method Applied to Beams in Bending 464

6．12 The Rayleigh-Ritz Technique Applied to Beams in Bending 469

6．13 Straight Beams Undergoing the Combined Effects of Axial and Transverse Loading 471

6．13．1 Unconstrained Beams 471

6．13．2 Constrained Beams 479

6．14 Problems 483

CHAPTER SEVEN STRENGTH，FAILURE MODES，AND DESIGN CONSIDERATIONS 498

7．0 Introduction 498

7．1 Strength 498

7．2 The Design Factor 500

7．3 Strength Theories 505

7．3．1 Basis of Theories 505

7．3．2 Tresca （Maximum-Shear-Stress）Theory for Ductile Materials 509

7．3．3 von Mises （Maximum-Energy-of-Distortion）Theory for Ductile Materials 509

7．3．4 Comparison between the Tresca and von Mises Theories（Plane Stress） 512

7．3．5 Coulomb-Mohr Theory for Brittle Materials 513

7．3．6 Design Equations 515

7．4 Fracture Mechanics 518

7．4．1 Introduction 518

7．4．2 Crack Modes and the Stress Intensity Factor 520

7．4．3 The Plastic Zone Correction 526

7．5 Fatigue Analysis 532

7．5．1 Fatigue Strength and Endurance Limit 532

7．5．2 Cyclic Stress with a Static Component 534

7．5．3 Fatigue Strength Reduction Factors 538

7．5．4 Equivalent Stresses（Plane Stress） 541

7．5．5 Estimating Life for Nonreversing or Nonrepetitive Stress Cycles 543

7．6 Structural Stability 546

7．6．1 Column Buckling 547

7．6．2 Buckling of Plates 556

7．7 Inelastic Behavior 561

7．7．1 EPP Materials 561

7．7．2 Plastic Behavior of Straight Beams in Bending 564

7．7．3 Depth of the Plastic Zone（Rectangular Beam） 568

7．7．4 Residual Stresses（Rectangular Beam） 569

7．7．5 Residual Stresses and Fatigue（Rectangular Beam） 571

7．8 Engineering Approximations Used in Statically Indeterminate Problems 573

7．8．1 Considering Deflections of Flexible Elements Only 573

7．8．2 Limit Analysis 579

7．9 Problems 584

7．10 References 596

8．0 Introduction 597

CHAPTER EIGHT EXPERIMENTAL STRESS ANALYSIS 597

8．1 Dimensional Analysis 598

8．2 Analysis Techniques 601

8．2．1 Symmetry 601

8．2．2 When a Surface Perpendicular to a Free Surface Exists Without Shear Stress 602

8．3 Strain Gages,General 604

8．4 Strain Gage Configurations 607

8．5 Strain Gage Instrumentation 610

8．5．1 The Wheatstone Bridge 610

8．5．2 Commercial Strain Gage Indicator Systems 613

8．6 Characteristics of Strain Gage Measurements 616

8．6．1 Linearity of the Grid Material 617

8．6．2 Transverse Sensitivity of the Gage 617

8．6．3 Temperature Effects 620

8．6．4 Lead-Wire Connection 623

8．6．5 Strain Gradient 624

8．6．6 Zero Shift and Hysteresis Effects 624

8．6．7 Dynamic Response 624

8．6．8 Gage-Current Heating Effects 625

8．7．1 Electromagnetic Wave Representation of Light 626

8．6．9 Noise from Electric and/or Magnetic Fields 626

8．7 The Theory of Photoelasticity 626

8．7．2 Polarization 628

8．7．3 Refraction 630

8．7．4 Birefringence 631

8．7．5 Stress and Birefringence 634

8．7．6 Isoclinic Fringe Analysis 637

8．7．7 Isochromatic Fringe Analysis 642

8．8．1 Photoelastic Material Calibrations 645

8．8 Techniques Used in Photoelastic Applications 645

8．8．2 Fractional Fringe Orders 647

8．8．3 Separation of the Principal Stresses,σ1 andσ2 650

8．8．4 Reflection Photoelasticity 654

8．8．5 Stress Freezing in Three-Dimensional Photoelasticity 660

8．9 Problems 662

8．10 References 672

CHAPTER NINE INTRODUCTION TO THE FINITE ELEMENT METHOD 673

9．0 Introduction 673

9．2 The Truss Element 677

9．1 Node and Element Subscript Notation 677

9．2．1 The One-Dimensional Truss Element-Direct Stiffness Method 678

9．2．2 The One-Dimensional Truss Element-The Rayleigh-Ritz Method 681

9．2．3 The Assembly Process 683

9．2．4 Distributed Loads 692

9．2．5 Thermal Stress 694

9．2．6 The Two-Dimensional Truss Element 698

9．2．7 Skew Supports 708

9．2．8 The Three-Dimensional Truss Element 711

9．3 Beam and Frame Elements 718

9．3．1 The Planar Beam Element 718

9．3．2 Distributed Loading 727

9．3．3 Pin Releases(Hinges)in Beam Elements 736

9．3．4 Beams in Two-Plane Bending 741

9．3．5 The Frame Element 742

9．3．6 Three-Dimensional Transformation of the Frame Element 743

9．3．7 Load-Stiffening and Buckling of Beams 747

9．4 Two-Dimensional Elastic Elements 757

9．4．1 The Two-Dimensional Constant Strain Triangle(CST)Element 758

9．4．2 The Two-Dimensional Isoparametric Quadrilateral Element 767

9．5 Higher-Order and Three-Dimensional Elastic Elements 777

9．6 Problems 778

9．7 References 788

CHAPTER TEN FINITE ELEMENT MODELING TECHNIQUES 789

10．0 Introduction 789

10．1 Planning and Creating the Finite Element Model(Preprocessing) 791

10．2．1 Introductory Remarks 792

10．2 Element Selection and Mesh Strategy 792

10．2．2 Element Selection 796

10．2．3 Element Input Information 802

10．2．4 Mesh Generation 803

10．2．5 Two-Dimensional Meshing Strategies 808

10．2．6 Submodeling 811

10．2．7 Symmetry 812

10．3 Load Application 818

10．3．1 Nodal Loads 819

10．3．3 Forcing Specified Nonzero Boundary Conditions with Boundary Elements 820

10．3．2 Element Loads 820

10．3．4 Multiple Load Cases 822

10．3．5 Load Scale Factors 823

10．4 Constraints 823

10．5 Preprocessing Checks 829

10．6 Processing the Model 830

10．7 Postprocessing 830

10．7．1 Graphic Output 831

10．8 Closure 834

10．7．2 Text Output 834

10．9 Problems 835

10．10 References 842

APPENDIX A SI AND USCU CONVERSIONS 843

APPENDIX B PROPERTIES OF CROSS SECTIONS 845

B．1 Tables 845

B．2 Combinations of Sections 847

APPENDIX C BEAMS IN BENDING 849

D．0 Introduction 858

APPENDIX D SINGULARITY FUNCTIONS 858

D．1 Integral Relations for Beams in Bending 859

D．2 Singularity Functions 860

APPENDIX E PRINCIPAL SECOND-AREA MOMENTS 868

E．1 Second-Area Moments 868

E．2 Principal Second-Area Moments 870

APPENDIX F STRESS CONCENTRATION FACTORS 873

APPENDIX G STRAIN GAGE ROSETTE EQUATIONS 879

G．1 Three-Element Rectangular Rosette 879

G．2 Three-Element Delta Rosette 881

H．0 Introduction 883

APPENDIX H CORRECTIONS FOR THE TRANSVERSE SENSITIVITY OF STRAIN GAGES 883

H．1 Corrections for the Two-Gage Rectangular Rosette 884

H．2 Corrections for the Three-Gage Rectangular Rosette 885

H．3 Corrections for the Three-Gage Delta(120°)Rosette 887

APPENDIX I MATRIX ALGEBRA AND CARTESIAN TENSORS 889

I．0 Introduction 889

I．1 Matrix Algebra 890

I．1．1 Addition 890

I．1．3 Matrix Multiplication 891

I ．1．2 Scalar Multiplication 891

I．1．4 Transposition 892

I．1．5 Determinant of a Matrix 892

I．1．6 Cofactor Matrix 893

I．1．7 Matrix Inversion 894

I．1．8 Eigenvalues and Eigenvectors 895

I．2 Cartesian Tensors 896

APPENDIX J ANSWERS TO MOST ODD-NUMBERED PROBLEMS 900

INDIX 913