ENGINEERINGSOLID

MECHANICSFundamentals and Applications

Abdel-Rahman RagabSalah Eldin Bayoumi

CRC PressBoca Raton London New York Washington, D.C.

Contents

Chapter 1 Analysis of Stress1.1 Rigid and Deformable Bodies ...11.2 Body Forces and Surface Tractions 11.3 Concept of Stress and Strain 21.4 The State of Stress at a Point 21.5 Cartesian Stress Components 61.6 Some Special States of Stress 8

1.6.1 Plane Stress 81.6.2 Plane Strain 91.6.3 Axial Symmetry 91.6.4 Free Torsion 9

1.7 Stress Equations of Equilibrium 91.7.1 Cartesian Coordinates 101.7.2 Cylindrical Polar Coordinates 141.7.3 Spherical Polar Coordinates 171.7.4 Curvilinear Coordinates 18

1.8 Stress Transformation Law 211.9 Plane Stress Transformation — Mohr's Circle of Stress 271.10 Principal Stresses 291.11 Maximum Shear Stresses 341.12 Octahedral Shear Stress — Pure Shear 381.13 Mean (Hydrostatic) Stress and Deviatoric Stresses 391.14 A Note on the Stress Equations 41Problems 41References 48

Chapter 2 Analysis of Strain2.1 Infinitesimal Strains 49

2.1.1 Normal Strain 492.1.2 Shear Strain 502.1.3 Volumetric Strain 52

2.2 Infinitesimal Strain-Displacement Relations 542.2.1 Cartesian Coordinates 542.2.2 Cylindrical Polar Coordinates 592.2.3 Spherical Polar Coordinates 61

2.3 Strain Compatibility Conditions 622.3.1 Cartesian Coordinates 622.3.2 Cylindrical Polar Coordinates 642.3.3 Spherical Polar Coordinates 65

2.4 Strain Tensor 672.5 Some Special States of Strain 71

2.5.1 Plane Strain 722.5.2 Plane Stress 722.5.3 Axial Symmetry 72

2.5.4 Free Torsion 722.6 Principal Strains — Maximum and Octahedral Shear Strains 742.7 Mean Strain Dilatation and Strain Deviations 762.8 Mohr's Circle of Strain 782.9 Strain Gauge Rosettes 802.10 Notes on Finite Strains 822.11 Strain Rate-Velocity Relations 87Problems 90References 99

Chapter 3 Elastic Stress-Strain Relations3.1 Introduction 1013.2 Basic Assumptions: Elasticity, Homogeneity, and Isotropy 101

3.2.1 Elasticity 1013.2.2 Homogeneity 1023.2.3 Isotropy 103

3.3 Hooke's Law for Homogeneous Isotropic Materials 1033.3.1 Simple Loading 1033.3.2 Triaxial Loading 104

3.4 Relations Among the Elastic Constants 1083.5 Inverse Form of Hooke's Law 1103.6 Dilatation and Distortion 1123.7 Thermoelastic Stress-Strain Relations 1143.8 Strain Energy for an Elastic Isotropic Solid 1163.9 Strain Energy for a Solid Obeying Hooke's Law 1223.10 Some Elastic Energy Theorems 128

3.10.1 Principle of Work 1283.10.2 Principle of Virtual Work 1293.10.3 Principle of Stationary Potential Energy 1313.10.4 Castigliano's Theorems 132

3.11 Generalized Hooke's Law 1353.11.1 Anisotropic Elasticity 1353.11.2 Application to Fiber-Reinforced Composites 140

3.12 Note on Composite Elastic Constants 1463.13 Stress-Strain Relations for Large Elastic Deformation 146Problems 150References 154

Chapter 4 Solution of the Elastic Problem4.1 The Elastic Problem 1554.2 Boundary Conditions 1564.3 Saint-Venant's Principle 1594.4 Uniqueness and Semi-Inverse Method of Elastic Solution 1604.5 Example of Solution in Terms of Stress: Pressurized Thick-Walled Sphere 1614.6 The Elastic Plane Problem •. 164

4.6.1 Plane Strain Formulation 1654.6.2 Plane Stress Formulation 1674.6.3 Deduction of Plane Stress Equations from Plane Strain Equations 169

4.7 Stress Function Formulation for Plane Elastic Problems 1704.8 Governing Equations in Terms of a Stress Function in Cartesian Coordinates 171

4.8.1 Plane Strain 1714.8.2 Plane Stress 1744.8.3 Thermoelastic Plane Problem 174

4.8.3.1 Thermoelastic Plane Strain 1744.8.3.2 Thermoelastic Plane Stress 175

4.8.4 Finding a Stress Function in Cartesian Coordinates 1824.9 Governing Equations in Terms of a Stress Function in Polar Coordinates 183

4.9.1 Plane Strain 1834.9.2 • Plane Stress 1854.9.3 Axisymmetric Plane Problems 187

4.9.3.1 Axisymmetric Problems without Body Forces 1874.9.3.2 Axisymmetric Problems with Centrifugal Body Forces 1894.9.3.3 Axisymmetric Problems with Radial Temperature Gradient 191

4.9.4 A Note on Finding a Stress Function in Polar Coordinates 1954.10 A Glossary of Stress Functions for Some Plane Problems 195

4.10.1 Cartesian Coordinates 1954.10.2 Polar Coordinates 197

Problems 200References 203

Chapter 5 Elastic Plane Problems in Cartesian Coordinates5.1 Introduction 2055.2 Problems Solved in Terms of Algebraic Polynomials 205

5.2.1 Retaining Wall Subjected to Hydrostatic Pressure 2095.2.2 Simply Supported Beam under Uniformly Distributed Load 2135.2.3 Cantilever Beam Subjected to an End Load 220

5.2.3.1 Stresses ; : 2205.2.3.2 Displacements 222

5.3 Problems Solved in Terms of Trigonometric Stress Functions 2295.3.1 Simply Supported Beam under Laterally Distributed Sinusoidal Load on

Both Sides ; 2305.3.2 Simply Supported Beam under Two Equal Lateral Loads at the Middle

of the Span 2325.3.3 Bar subjected to Two Equal and Opposite Axial Loads 233

5.4 A Note on Some Other Forms of Stress Functions 234Problems 237References 241

Chapter 6 Elastic Plane Problems in Polar Coordinates6.1 Introduction 2436.2 Axisymmetric Problems 243

6.2.1 Thick-Walled Cylinder Subjected to Uniform Internal and/orExternal Pressure 2436.2.1.1 Cylinder Subjected to Internal Pressure Only 2466.2.1.2 Cylinder Subjected to External Pressure Only 247

6.2.2 Thick-Walled Cylinder Subjected to Steady-State RadialThermal Gradient 2506.2.2.1 Plane Strain 2506.2.2.2 Plane Stress 2536.2.2.3 Other End Conditions 254

6.2.3 Cylinder Compounding by Shrink Fit 2556.2.4 Rotating Disk of Uniform Thickness 262

6.2.4.1 Annular Rotating Disk of Constant Thickness 2626.2.4.2 Solid Rotating Disk of Constant Thickness 264

6.2.5 Rotating Solid Disk of Uniform Strength (De Laval Disk) 2666.2.6 Rotating Drums and Rotors 2696.2.7 Rotating Disks and Rotors Subjected to Radial Thermal Gradients 270

6.3 Axially Nonsymmetric Problems 2736.3.1 Bending of a Circularly Curved Beam 274

6.3.1.1 Beam Subjected to an End Shearing Force 2746.3.1.2 Beam Subjected to Pure Bending 2806.3.1.3 Beam Subjected to an End Moment and a Normal Force 2826.3.1.4 Beam Subjected to an Inclined End Force 282

6.3.2 Thermal Stresses in Curved Beams 2836.3.3 Wedge Subjected to a Concentrated Load at its Vertex 286

6.3.3.1 Force Acting Along a Wedge Axis 2866.3.3.2 Force Perpendicular to the Wedge Axis 2896.3.3.3 Force Inclined to the Wedge Axis 2906.3.3.4 Bending Moment Acting at the Vertex 292

6.3.4 Concentrated Line Load Acting on the Edge of a Straight Boundary 2956.3.4.1 Force Acting Normal to the Boundary 2956.3.4.2 Force Acting Along the Boundary 2966.3.4.3 Force Acting Inclined to the Boundary 297

6.3.5 Uniformly Distributed Line Load Acting on the Edge of aStraight Boundary 297

6.3.6 Circular Solid Disk Subjected to Two Equal and OppositeDiametral Loads 299

6.3.7 Concentrated Load Acting on a Rectangular Beam 3016.4 Stresses Concentration Around a Small Circular Hole 303Problems 311References 318

Chapter 7 Elastic Rods Subjected to General Loading7.1 Introduction 3197.2 Stress Resultants 319

7.2.1 Note on Sign Convention for Stress Resultants 3217.3 Bending of Rods 322

7.3.1 Bending Stresses 3227.3.2 Elastic Curve in Bending 3267.3.3 Bending of Curved Beams..... 330

7.3.3.1 Determination of the Location of the Neutral Axis 3347.3.3.2 Approximate Determination of the Neutral Axis 3367.3.3.3 Maximum Stresses 3377.3.3.4 Bending of a Curved Beam by Lateral Forces Acting in the

Plane of Its Axis 3377.3.3.5 Strain Energy in Curved Beams 3387.3.3.6 Comparison with Exact and Other Solutions 339

7.3.4 Thermoelastic Bending of Straight Bars 3417.4 Shear Stresses in Rods 345

7.4.1 Rectangular Solid Section 347

7.4.2 Circular Solid Section 3497.4.3 Thin-Walled Open Sections 352

7.4.3.1 Shear Center 3557.4.4 Thin-Walled Closed Sections 357

7.5 Torsion of Bars 3597.5.1 Saint-Venant's Free Torsion 3607.5.2 Solid Circular Section 3637.5.3 Solid Elliptical Section 3657.5.4 Solid Rectangular Section 3667.5.5 Thin-Walled Open Sections 3687.5.6 Thin-Walled Closed Sections 3707.5.7 Effect of Internal Stiffening Webs 3727.5.8 Effect of End Constraint 373

7.5.8.1 Solid Sections 3747.5.8.2 Thin-Walled Sections 375

7.6 Displacements in Rods — Energy Approach 3767.6.1 Application of Castigliano's Theorem 3767.6.2 Mohr's Unit Load Method 3847.6.3 A Note on the Deflection of Curved Beams 3877.6.4 Application to Springs 391

7.6.4.1 Helical Compression Spring 3917.6.4.2 Spiral Helical Compression Spring 3937.6.4.3 Flat Compression Spring 3947.6.4.4 Flat Torsion Spring 395

7.7 Buckling of Rods 3987.7.1 Buckling of Columns 398

7.7.1.1 Equilibrium Approach 3987.7.1.2 Minimum Potential Energy Solution: Rayleigh-Ritz Method 403

7.7.2 Beam-Columns 4107.7.3 Lateral Buckling of Beams 412

7.8 Beams on Elastic Foundation 4157.8.1 Infinitely Long Beams 416

7.8.1.1 Concentrated Force 4167.8.1.2 Concentrated Moment 4187.8.1.3 Uniform Load 420

7.8.2 Semi-Infinite Beams 4227.8.3 Short Beams 425

Problems 425References 439

Chapter 8 Some Problems of Elastic Plates and Shells8.1 Introduction 4418.2 State of Stress in Plates and Shells 4418.3 Plate Equations in Cartesian Coordinates 442

8.3.1 Deformation Pattern ? 4428.3.2 Stress Resultants 4448.3.3 Equations of Equilibrium 4458.3.4 Method of Solution: Pure Bending of a Plate 4488.3.5 Effect of Thermal Gradient Throughout Plate Thickness 449

8.3.5.1 A Plate with Free Edges 4498.3.5.2 A Plate with Clamped Edges 450

8.3.5.3 A Plate with Simply Supported Edges 4518.4 Bending of Rectangular Plates — Energy Approach 452

8.4.1 Uniformly Loaded Rectangular Plate Simply Supported Along ItsFour Edges ..r. 452

8.4.2 Uniformly Loaded Rectangular Plate Clamped Along Its Four Edges 4588.4.3 An Approximate Strip Method for Rectangular Plates 465

8.5 Axisymmetric Bending of Flat, Circular Plates 4668.5.1 Solid Circular Plates 468

8.5.1.1 Simply Supported Plate Subjected to Uniform Pressure 4688.5.1.2 Ail-Around Clamped Plate Subjected to Uniform Pressure 4698.5.1.3 All-Around Clamped Plate Subjected to a Concentrated

Force at the Center 4728.5.1.4 Simply Supported Plate Subjected to a Concentrated

Force at the Center 4748.5.2 Annular Circular Plates 475

8.5.2.1 Simply Supported Annular Plate Subjected to Edge Moments 4758.5.2.2 Simply Supported Annular Circular Plate Subjected to a Shearing

Force at the Inner Edge 4778.5.3 Other Loadings and Edge Conditions 4788.5.4 Thermal Stresses in Circular Plates 479

8.5.4.1 Temperature Gradient Across the Thickness of a Disk withFree Edges 479

8.5.4.2 Temperature Gradient Across the Thickness of a Disk withAil-Around Clamped Edges 480

8.5.4.3 Axisymmetric Radial Temperature Gradient 4808.5.5 Comments on the Deflection of Circular Plates 482

8.5.5.1 Deflection Due to Shear 4828.5.5.2 Large Deflection 483

8.6 Membrane Stresses in Axisymmetric Shells 4868.6.1 Axisymmetric Shells Subjected to Uniform Pressure 4868.6.2 Applications to Pressurized Containers 490

8.6.2.1 Spherical Shell 4908.6.2.2 Circular Cylindrical Shell 4918.6.2.3 Conical Shell 4918.6.2.4 Toroidal Shell 492

8.6.3 Displacement in Axisymmetric Shells 4938.6.4 Axisymmetric Shells Subjected to Gravity Loading 499

8.6.4.1 Hemispherical Liquid Container Freely Supported atIts Top Edge 499

8.6.4.2 Conical Liquid Container Freely Supported at Its Top Edge 5008.6.4.3 Spherical Container on a Skirt Support 504

8.7 Bending of Thin-Walled Cylinders Subjected to Axisymmetric Loading 5078.7.1 Problem Formulation 5078.7.2 Long, Thin-Walled Pressurized Pipe with a Rigid Flange at its End 5138.7.3 Short, Thin-Walled Pressurized Pipe with Two Rigid Flanges at Both Ends 5178.7.4 Long, Thin-Walled Pipe Subjected to Uniform Radial Compression Along a

Circular Section at its Middle Length 5188.7.5 Long, Thin-Walled Pipe Subjected to a Uniform Circumferential Load Along a

Finite Length 5218.7.6 Cylindrical Pressure Vessels With End Closures 523

8.7.6.1 Case of a Flat End 524

8.7.6.2 Case of a Curved End 5278.7.6.3 Case of a Hemispherical End 529

8.7.7 Cylindrical Storage Tanks 5348.7.8 Effect of Thermal-Gradient 537

8.8 Elastic Buckling of Plates and Shells 5408.8.1 Buckling of Uniformly Compressed Rectangular Plate 540

8.8.1.1 Ail-Around Clamped Rectangular Plate 5428.8.1.2 Rectangular Plates with Other Boundary Conditions 543

8.8.2 Axisymmetric Buckling of Circular Plates 5448.8.3 Buckling of Thin-Walled Cylinders Under External Uniform Pressure 546

8.8.3.1 Effect of Out-of-Roundness, Cylinder Length, and End Constraints 550Problems.... .- 550References 559

Chapter 9 Applications to Fracture Mechanics9.1 Introduction 5619.2 Griffith Energy Criterion 5629.3 Stress Concentration Around Elliptical Holes 5659.4 The Elastic Stress Field at the Crack Tip.. 5669.5 The Stress Intensity Factor and Fracture Toughness 5709.6 Stress Intensity Factors for Various Configurations 574

9.6.1 Plates under Tensile Loading 5759.6.2 Cracks Emanating from Circular Holes in Infinite Plates 5779.6.3 Plates under Bending 5799.6.4 Circular Rods and Tubes 5799.6.5 Pressurized Thick-Walled Cylinders 5819.6.6 Rotating Solid Disks and Drums 582

9.7 Superposition under Combined Loading 5879.8 Mixed-Mode Loading 5899.9 Plastic Zone Geometry at Crack Tip 5909.10 Notes on Fracture Toughness Testing ;..'. 5959.11 Fracture Due to Crack Growth 598

9.11.1 Fatigue Crack Propagation 5989.11.1.1 Region (i) of Nonpropagating Cracks 5999.11.1.2 Region (ii) of Steady Crack Propagation 5999.11.1.3 Region (iii) of Unstable Crack Growth Rate 601

9.11.2 Safe-Life Prediction 6039.11.3 Comments on Safe-Life Predictions 609

9.11.3.1 Margin of Safety 6099.11.3.2 Variable Amplitude Loading 6099.11.3.3 Mixed-Mode Crack Growth... 6119.11.3.4 Correlation with S-N Curves 6119.11.3.5 Growth of Physically Short Cracks 6119.11.3.6 Crack Closure 612

9.12 Stress Corrosion Cracking : 6139.13 Elastic-Plastic Fracture Mechanics 616

9.13.1 /Integral 6169.13.2 Experimental Determination of J 6199.13.3 A Scheme for Fracture Estimation Using JIc 6239.13.4 Crack Opening Displacement 627

9.13.5 Experimental Determination of COD 6299.13.6 Application of CTOD to Structural Design 630

Problems 632References ."". 639

Chapter 10 Plastic Deformation10.1 Introduction 64110.2 Basic Assumptions 64210.3 Definition of Large Plastic Strains 64410.4 Strain Hardening in Simple Tension 64610.5 Empirical Relations for Stress-Strain Curves 64710.6 Idealized Stress-Strain Curves 65210.7 Yield Criteria 653

10.7.1 von Mises Yield Criterion 65410.7.2 Comments on the von Mises Criterion 65610.7.3 Tresca Yield Criterion 65810.7.4 Geometrical Representation of von Mises and Tresca Criteria 65910.7.5 Experimental Verification of Yield Criteria 662

10.8 Plastic Stress-Strain Relations — Flow Rule 66410.9 Principle of Normality and Plastic Potential 66710.10 Plastic Work, Effective Stress, and Effective Strain Increment 66910.11 Experimental Determination of the Flow Curve 67410.12 Isotropic Hardening 67810.13 Uniqueness and Path Dependence 67910.14 Complete Elastic-Plastic Stress-Strain Relations 68310.15 Plastic Deformation of Anisotropic Materials 686

10.15.1 A Yield Criterion for Anisotropic Materials 68710.15.2 A Flow Rule for Anisotropic Materials 68810.15.3 Measurement of Anisotropic Parameters 68810.15.4 Normal Anisotropy 68910.15.5 Effective Stress and Effective Plastic Strain Increment 69110.15.6 A Special Case: Rotational Symmetry (Planar Isotropy) 69210.15.7 A Modified Nonquadratic Criterion for Planar Isotropy 697

10.16 Kinematic Hardening 70010.16.1 Uniaxial Behavior under Cyclic Loading 70110.16.2 Triaxial Behavior — Yield Function and Flow Rule 709

10.17 Plastic Deformation of Porous Solids 71510.17.1 Yield Function 71610.17.2 Flow Rule 71910.17.3 Void Growth Characteristics 72010.17.4 Application to Metal Powder Compacts 722

Problems 723References '. 731

Chapter 11 Plastic Instability, Superplasticity and Creep11.1 Introduction 73311.2 Unstable Plastic Deformation 733

11.2.1 Necking of a Tensile Bar 73411.2.2 Local Necking of a Wide Strip 73811.2.3 Limit Tensile Strain for a Bar with an Imperfection 740

11.2.4 Stresses in the Neck of a Tensile Bar 74111.2.4.1 Round Bar 74211.2.4.2 Wide Strip 745

11.2.5 Biaxial Stretching — Flat and Bulged Circular Sheets 74611.2.5.1 Flat Sheet 74611.2.5.2 Bulging of a Circular Sheet 749

11.2.6 Pressurized Axisymmetric Thin-Walled Containers 75411.2.6.1 Thin-Walled Sphere 75411.2.6.2 Thin-Walled Cylinder 756

11.3 Strain-Rate Dependent Plastic Behavior — Application to Superplasticity 76011.3.1 Neck-Free Elongations 76311.3.2 Limit Tensile Strains for a Bar of Strain-Rate-Dependent Material 76411.3.3 Forming Time for a Bulged Circular Sheet of Rate-Dependent Material 765

11.4 Creep Deformation 76711.4.1 Creep Testing and Data 76711.4.2 Empirical Creep Equation of State 771

11.4.2.1 Uniaxial Behavior 77111.4.2.2 Multiaxial Behavior 773

11.4.3 Steady Creep of Beams under Bending 77311.4.4 Steady Creep of Thin-Walled Pressurized Cylinders 77711.4.5 Steady Creep of Thick-Walled Pressurized Cylinders 78011.4.6 Steady Creep in Rotating Disks 78511.4.7 Steady Creep of Circular Shafts Under Torsion 78511.4.8 Creep Buckling of Columns 78811.4.9 The Reference Stress Method 79111.4.10 Stress Relaxation 79611.4.11 Creep under Variable Loading: Time Hardening vs. Strain Hardening 79811.4.12 Creep Rupture and Damage Concept 802

11.4.12.1 Ductile Creep Rupture under Uniaxial Stress 80311.4.12.2, Creep Damage Concept 80511.4.12.3 Brittle Creep Rupture under Uniaxial Stress 807

Problems 809References 816

Chapter 12 Some Elastic-Plastic Problems12.1 Introduction 81912.2 Plane Strain Bending of Plates 820

12.2.1 Elastic State 82012.2.2 Initial Yielding 82212.2.3 Partial and Full Yielding — Shape Factor 82212.2.4 Unloading: Residual Stresses and Springback 825

12.3 Plane Stress Bending of Beams 82812.3.1 Initial Yielding, Full Yielding, and Springback 82812.3.2 Combined Bending and Tension 831

12.3.2.1 Elastic State : 83112.3.2.2 Elastic-Plastic State 83112.3.2.3 Unloading and Residual Stresses 832

12.3.3 Plastic Collapse of Beams — Plastic Hinges 833

12.3.4 Deflection and Shear Stresses 83712.3.5 Effect of Strain Hardening 839

12.4 Biaxial Bending of Flat Plates 84312.4.1 Rectangular Plates 84312.4.2 Circular Plates 847

12.5 Bending of Circularly Curved Beams 84912.6 Buckling of Bars Under Axial Compression 853

12.6.1 Tangent Modulus Formula 85412.6.2 Double-Modulus Formula 854

12.7 Bars Subjected to Torsion 85612.7.1 Circular Solid and Hollow Sections 857

12.7.1.1 Solid Circular Section 85712.7.1.2 Hollow Circular Section 858

12.7.2 Thin-Walled Tubular Sections 86112.7.2.1 Uniform Wall Thickness 86112.7.2.2 Nonuniform Wall Thickness 861

12.7.3 Combined Torsion and Tension 86112.7.3.1 Solid Circular Section 86112.7.3.2 Hollow Circular Sections 86212.7.3.3 Thin-Walled Cylinder of Uniform Thickness 86312.7.3.4 Remarks 865

12.8 Pressurized Thick-Walled Cylinders 86512.8.1 Initial and Partial Yielding 866

12.8.1.1 Stresses in the Elastic Region rp<r< r0 87012.8.1.2 Stresses in the Plastic Region r, < r < rp 87012.8.1.3 Radial Displacements in Partially Yielded Cylinders 873

12.8.2 Full Yielding and Plastic Expansion Process 87412.8.2.1 Full Yielding 87412.8.2.2 Plastic Expansion Process 875

12.8.3 Residual Stresses — The Autofrettage Process 88012.8.4 Effect of Strain Hardening and Temperature Gradient: 884

12.8.4.1 Strain Hardening 88412.8.4.2 Radial Temperature Gradient 885

12.9 Annular Rotating Disks of Uniform Thickness 88612.9.1 Initial Yielding 886

12.9.1.1 Tresca Yield Criterion 88612.9.1.2 von Mises Yield Criterion 887

12.9.2 Partial and full Yielding 88712.9.2.1 Stresses in the Plastic Region r{ < r < rp 88712.9.2.2 Stress in the Elastic Region rp < r<r0 888

12.9.3 Residual Stresses at Stoppage 89012.9.4 Shrink-Fitted Disks 890

12.10 Solid Rotating Disks of Uniform Thickness 89212.10.1 Initial, Partial, and Full Yielding 892

12.10.1.1 Initial Yielding 89212.10.1.2 Partial Yielding 89312.10.1.3 Full Yielding 893

12.10.2 Residual Stresses at Stoppage 895

12.11 Shakedown Limit: Application to Pressurized Cylinders 896Problems 899References 904

Index 905