Peterson’s Stress Concentration Factors

TABLE OF CONTENTS
Index to the Stress Concentration Factors xv

Preface for the Fourth Edition xxxi

Preface for the Third Edition xxxiii

Preface for the Second Edition xxxv

1 Fundamentals of Stress Analysis 1

1.1 Stress Analysis in Product Design 2

1.2 Solid Objects Under Loads 4

1.3 Types of Materials 6

1.4 Materials Properties and Testing 7

1.4.1 Tensile and Compression Tests 8

1.4.2 Hardness Tests 8

1.4.3 Shear Tests 13

1.4.4 Fatigue Tests 14

1.4.5 Impact Tests 16

1.5 Static and Fatigue Failures 17

1.6 Uncertainties, Safety Factors, and Probabilities 19

1.7 Stress Analysis of Mechanical Structures 21

1.7.1 Procedure of Stress Analysis 21

1.7.2 Geometric Discontinuities of Solids 21

1.7.3 Load Types 23

1.7.4 Stress and Representation 24

1.7.4.1 Simple Stress 26

1.7.4.2 General Stresses 26

1.7.4.3 Principal Stresses and Directions 27

1.8 Failure Criteria of Materials 30

1.8.1 Maximum Shear Stress (MSS) Theory 30

1.8.2 Distortion Energy (DE) Theory 32

1.8.3 Maximum Normal Stress (MNS) Theory 34

1.8.4 Ductile and Brittle Coulomb-Mohr (CM) Theory 36

1.8.5 Modified-Mohr (MM) Theory 37

1.8.6 Guides for Selection of Failure Criteria 37

1.9 Stress Concentration 39

1.9.1 Selection of Nominal Stresses as Reference 42

1.9.2 Accuracy of Stress Concentration Factors 45

1.9.3 Decay of Stress away from the Peak Stress 46

1.10 Stress Concentration as a Two-Dimensional Problem 46

1.11 Stress Concentration as a Three-Dimensional Problem 47

1.12 Plane and Axisymmetric Problems 49

1.13 Local and Nonlocal Stress Concentration 52

1.14 Multiple Stress Concentration 57

1.15 Principle of Superposition for Combined Loads 61

1.16 Notch Sensitivity 64

1.17 Design Relations for Static Stress 69

1.17.1 Ductile Materials 69

1.17.2 Brittle Materials 71

1.18 Design Relations for Alternating Stress 72

1.18.1 Ductile Materials 72

1.18.2 Brittle Materials 73

1.19 Design Relations for Combined Alternating and Static Stresses 74

1.19.1 Ductile Materials 74

1.19.2 Brittle Materials 77

1.20 Limited Number of Cycles of Alternating Stress 78

1.21 Stress Concentration Factors and Stress Intensity Factors 79

1.22 Selection of Safety Factors 83

References 85

2 Notches and Grooves 89

2.1 Notation 89

2.2 Stress Concentration Factors 90

2.3 Notches in Tension 92

2.3.1 Opposite Deep Hyperbolic Notches in an Infinite Thin Element; Shallow Elliptical, Semicircular, U-Shaped, or Keyhole-Shaped Notches in Semi-Infinite Thin Elements; Equivalent Elliptical Notch 92

2.3.2 Opposite Single Semicircular Notches in a Finite-Width Thin Element 94

2.3.3 Opposite Single U-Shaped Notches in a Finite-Width Thin Element 94

2.3.4 Finite-Width Correction Factors for Opposite Narrow Single Elliptical Notches in a Finite-Width Thin Element 95

2.3.5 Opposite Single V-Shaped Notches in a Finite-Width Thin Element 95

2.3.6 Single Notch on One Side of a Thin Element 96

2.3.7 Notches with Flat Bottoms 96

2.3.8 Multiple Notches in a Thin Element 96

2.3.9 Analytical Solutions for Stress Concentration Factors for Notched Bars 98

2.4 Depressions in Tension 98

2.4.1 Hemispherical Depression (Pit) in the Surface of a Semi-Infinite Body 98

2.4.2 Hyperboloid Depression (Pit) in the Surface of a Finite-Thickness Element 98

2.4.3 Opposite Shallow Spherical Depressions (Dimples) in a Thin Element 99

2.5 Grooves in Tension 100

2.5.1 Deep Hyperbolic Groove in an Infinite Member (Circular Net Section) 100

2.5.2 U-Shaped Circumferential Groove in a Bar of Circular Cross Section 100

2.5.3 Flat-Bottom Grooves 100

2.5.4 Closed-Form Solutions for Grooves in Bars of Circular Cross Section 100

2.6 Bending of Thin Beams with Notches 101

2.6.1 Opposite Deep Hyperbolic Notches in an Infinite Thin Element 101

2.6.2 Opposite Semicircular Notches in a Flat Beam 101

2.6.3 Opposite U-Shaped Notches in a Flat Beam 101

2.6.4 V-Shaped Notches in a Flat Beam Element 102

2.6.5 Notch on One Side of a Thin Beam 102

2.6.6 Single or Multiple Notches with Semicircular or Semielliptical Notch Bottoms 102

2.6.7 Notches with Flat Bottoms 103

2.6.8 Closed-Form Solutions for Stress Concentration Factors for Notched Beams 103

2.7 Bending of Plates with Notches 103

2.7.1 Various Edge Notches in an Infinite Plate in Transverse Bending 103

2.7.2 Notches in a Finite-Width Plate in Transverse Bending 104

2.8 Bending of Solids with Grooves 104

2.8.1 Deep Hyperbolic Groove in an Infinite Member 104

2.8.2 U-Shaped Circumferential Groove in a Bar of Circular Cross Section 104

2.8.3 Flat-Bottom Grooves in Bars of Circular Cross Section 105

2.8.4 Closed-Form Solutions for Grooves in Bars of Circular Cross Section 105

2.9 Direct Shear and Torsion 106

2.9.1 Deep Hyperbolic Notches in an Infinite Thin Element in Direct Shear 106

2.9.2 Deep Hyperbolic Groove in an Infinite Member 106

2.9.3 U-Shaped Circumferential Groove in a Bar of Circular Cross Section Subject to Torsion 106

2.9.4 V-Shaped Circumferential Groove in a Bar of Circular Cross Section Under Torsion 108

2.9.5 Shaft in Torsion with Grooves with Flat Bottoms 108

2.9.6 Closed-Form Formulas for Grooves in Bars of Circular Cross Section Under Torsion 109

2.10 Test Specimen Design for Maximum Kt for a Given r/D or r/H 109

References 109

Charts 113

3 Shoulder Fillets 167

3.1 Notation 167

3.2 Stress Concentration Factors 169

3.3 Tension (Axial Loading) 170

3.3.1 Opposite Shoulder Fillets in a Flat Bar 170

3.3.2 Effect of Length of Element 170

3.3.3 Effect of Shoulder Geometry in a Flat Member 170

3.3.4 Effect of a Trapezoidal Protuberance on the Edge of a Flat Bar 171

3.3.5 Fillet of Noncircular Contour in a Flat Stepped Bar 172

3.3.6 Stepped Bar of Circular Cross Section with a Circumferential Shoulder Fillet 175

3.3.7 Tubes 176

3.3.8 Stepped Pressure Vessel Wall with Shoulder Fillets 176

3.4 Bending 177

3.4.1 Opposite Shoulder Fillets in a Flat Bar 177

3.4.2 Effect of Shoulder Geometry in a Flat Thin Member 177

3.4.3 Elliptical Shoulder Fillet in a Flat Member 177

3.4.4 Stepped Bar of Circular Cross Section with a Circumferential Shoulder Fillet 177

3.5 Torsion 178

3.5.1 Stepped Bar of Circular Cross Section with a Circumferential Shoulder Fillet 178

3.5.2 Stepped Bar of Circular Cross Section with a Circumferential Shoulder Fillet and a Central Axial Hole 178

3.5.3 Compound Fillet 179

3.6 Methods of Reducing Stress Concentration at a Shoulder 180

References 182

Charts 184

4 Holes 209

4.1 Notation 209

4.2 Stress Concentration Factors 211

4.3 Circular Holes with In-Plane Stresses 214

4.3.1 Single Circular Hole in an Infinite Thin Element in Uniaxial Tension 214

4.3.2 Single Circular Hole in a Semi-Infinite Element in Uniaxial Tension 217

4.3.3 Single Circular Hole in a Finite-Width Element in Uniaxial Tension 218

4.3.4 Effect of Length of Element 218

4.3.5 Single Circular Hole in an Infinite Thin Element under Biaxial In-Plane Stresses 219

4.3.6 Single Circular Hole in a Cylindrical Shell with Tension or Internal Pressure 220

4.3.7 Circular or Elliptical Hole in a Spherical Shell with Internal Pressure 223

4.3.8 Reinforced Hole Near the Edge of a Semi-Infinite Element in Uniaxial Tension 223

4.3.9 Symmetrically Reinforced Hole in a Finite-Width Element in Uniaxial Tension 226

4.3.10 Nonsymmetrically Reinforced Hole in a Finite-Width Element in Uniaxial Tension 227

4.3.11 Symmetrically Reinforced Circular Hole in a Biaxially Stressed Wide, Thin Element 227

4.3.12 Circular Hole with Internal Pressure 235

4.3.13 Two Circular Holes of Equal Diameter in a Thin Element in Uniaxial Tension or Biaxial In-Plane Stresses 236

4.3.14 Two Circular Holes of Unequal Diameter in a Thin Element in Uniaxial Tension or Biaxial In-Plane Stresses 241

4.3.15 Single Row of Equally Distributed Circular Holes in an Element in Tension 243

4.3.16 Double Row of Circular Holes in a Thin Element in Uniaxial Tension 243

4.3.17 Symmetrical Pattern of Circular Holes in a Thin Element in Uniaxial Tension or Biaxial In-Plane Stresses 244

4.3.18 Radially Stressed Circular Element with a Ring of Circular Holes, with or without a Central Circular Hole 245

4.3.19 Thin Element with Circular Holes with Internal Pressure 246

4.4 Elliptical Holes in Tension 247

4.4.1 Single Elliptical Hole in Infinite- and Finite-Width Thin Elements in Uniaxial Tension 250

4.4.2 Width Correction Factor for a Cracklike Central Slit in a Tension Panel 252

4.4.3 Single Elliptical Hole in an Infinite, Thin Element Biaxially Stressed 253

4.4.4 Infinite Row of Elliptical Holes in Infinite- and Finite-Width Thin Elements in Uniaxial Tension 263

4.4.5 Elliptical Hole with Internal Pressure 263

4.4.6 Elliptical Holes with Bead Reinforcement in an Infinite Thin Element under Uniaxial and Biaxial Stresses 263

4.5 Various Configurations with In-Plane Stresses 263

4.5.1 Thin Element with an Ovaloid; Two Holes Connected by a Slit under Tension; Equivalent Ellipse 263

4.5.2 Circular Hole with Opposite Semicircular Lobes in a Thin Element in Tension 265

4.5.3 Infinite Thin Element with a Rectangular Hole with Rounded Corners Subject to Uniaxial or Biaxial Stress 266

4.5.4 Finite-Width Tension Thin Element with Round-Cornered Square Hole 267

4.5.5 Square Holes with Rounded Corners and Bead Reinforcement in an Infinite Panel under Uniaxial and Biaxial Stresses 267

4.5.6 Round-Cornered Equilateral Triangular Hole in an Infinite Thin Element Under Various States of Tension 267

4.5.7 Uniaxially Stressed Tube or Bar of Circular Cross Section with a Transverse Circular Hole 267

4.5.8 Round Pin Joint in Tension 268

4.5.9 Inclined Round Hole in an Infinite Panel Subjected to Various States of Tension 269

4.5.10 Pressure Vessel Nozzle (Reinforced Cylindrical Opening) 270

4.5.11 Spherical or Ellipsoidal Cavities 271

4.5.12 Spherical or Ellipsoidal Inclusions 272

4.6 Holes in Thick Elements 274

4.6.1 Countersunk Holes 276

4.6.2 Cylindrical Tunnel 277

4.6.3 Intersecting Cylindrical Holes 278

4.6.4 Rotating Disk with a Hole 279

4.6.5 Ring or Hollow Roller 281

4.6.6 Pressurized Cylinder 281

4.6.7 Pressurized Hollow Thick Cylinder with a Circular Hole in the Cylinder Wall 282

4.6.8 Pressurized Hollow Thick Square Block with a Circular Hole in the Wall 283

4.6.9 Other Configurations 283

4.7 Orthotropic Thin Members 284

4.7.1 Orthotropic Panel with an Elliptical Hole 284

4.7.2 Orthotropic Panel with a Circular Hole 286

4.7.3 Orthotropic Panel with a Crack 286

4.7.4 Isotropic Panel with an Elliptical Hole 286

4.7.5 Isotropic Panel with a Circular Hole 286

4.7.6 More Accurate Theory for a/b < 4 287

4.8 Bending 288

4.8.1 Bending of a Beam with a Central Hole 288

4.8.2 Bending of a Beam with a Circular Hole Displaced from the Center Line 289

4.8.3 Curved Beams with Circular Holes 289

4.8.4 Bending of a Beam with an Elliptical Hole; Slot with Semicircular Ends (Ovaloid); or Round-Cornered Square Hole 290

4.8.5 Bending of an Infinite- and a Finite-Width Plate with a Single Circular Hole 290

4.8.6 Bending of an Infinite Plate with a Row of Circular Holes 291

4.8.7 Bending of an Infinite Plate with a Single Elliptical Hole 291

4.8.8 Bending of an Infinite Plate with a Row of Elliptical Holes 291

4.8.9 Tube or Bar of Circular Cross Section with a Transverse Hole 291

4.9 Shear and Torsion 292

4.9.1 Shear Stressing of an Infinite Thin Element with Circular or Elliptical Hole, Unreinforced and Reinforced 292

4.9.2 Shear Stressing of an Infinite Thin Element with a Round-Cornered Rectangular Hole, Unreinforced and Reinforced 293

4.9.3 Two Circular Holes of Unequal Diameter in a Thin Element in Pure Shear 293

4.9.4 Shear Stressing of an Infinite Thin Element with Two Circular Holes or a Row of Circular Holes 294

4.9.5 Shear Stressing of an Infinite Thin Element with an Infinite Pattern of Circular Holes 294

4.9.6 Twisted Infinite Plate with a Circular Hole 294

4.9.7 Torsion of a Cylindrical Shell with a Circular Hole 294

4.9.8 Torsion of a Tube or Bar of Circular Cross Section with a Transverse Circular Hole 294

References 296

Charts 307

5 Miscellaneous Design Elements 439

5.1 Notation 439

5.2 Shaft with Keyseat 441

5.2.1 Bending 442

5.2.2 Torsion 442

5.2.3 Torque Transmitted Through a Key 443

5.2.4 Combined Bending and Torsion 443

5.2.5 Effect of Proximity of Keyseat to Shaft Shoulder Fillet 443

5.2.6 Fatigue Failures 444

5.3 Splined Shaft in Torsion 445

5.4 Gear Teeth 445

5.5 Press- or Shrink-Fitted Members 447

5.6 Bolt and Nut 450

5.7 Bolt Head, Turbine-Blade, or Compressor-Blade Fastening (T-Head) 452

5.8 Lug Joint 454

5.8.1 Lugs with h∕d < 0.5 455

5.8.2 Lugs with h∕d > 0.5 456

5.9 Curved Bar 457

5.10 Helical Spring 458

5.10.1 Round or Square Wire Compression or Tension Spring 458

5.10.2 Rectangular Wire Compression or Tension Spring 460

5.10.3 Helical Torsion Spring 461

5.11 Crankshaft 461

5.12 Crane Hook 462

5.13 U-Shaped Member 462

5.14 Angle and Box Sections 463

5.15 Cylindrical Pressure Vessel with Torispherical Ends 463

5.16 Welds 464

5.17 Parts with Inhomogeneous Materials or Composites 471

5.18 Parts with Defects 471

5.19 Parts with Threads 474

5.20 Frame Stiffeners 475

5.21 Discontinuities with Additional Considerations 476

5.22 Pharmaceutical Tablets with Holes 477

5.23 Parts with Residual Stresses 478

5.24 Surface Roughness 479

5.25 New Approaches for Parametric Studies 480

References 481

Charts 489

6 Finite Element Analysis (FEA) for Stress Analysis 517

6.1 Structural Analysis Problems 518

6.2 Types of Engineering Analysis Methods 519

6.3 Structural Analysis Theory 520

6.3.1 Trusses and Frame Structures 523

6.3.1.1 Trusses 523

6.3.1.2 Boundary Conditions (BCs) and Loads 526

6.3.1.3 Frame Structure 527

6.3.2 Plane Stress and Strain Problems 530

6.3.2.1 Plane Stresses 530

6.3.2.2 Plane Strain Problems 535

6.3.3 Modal Analysis 535

6.3.3.1 Two-Dimensional Truss Member in LCS 537

6.3.3.2 Two-Dimensional Beam Member in LCS 538

6.3.3.3 Modeling of Two-Dimensional Frame Element 540

6.3.4 Fatigue Analysis 542

6.3.4.1 Strain-Life Method 543

6.3.4.2 Linear Elastic Fracture Mechanics Method 544

6.3.4.3 Stress-Life Method 545

6.3.4.4 Selection of Fatigue Analysis Methods 546

6.4 Finite Element Anlaysis (FEA) for Structural Analysis 547

6.4.1 CAD/CAE Interface 551

6.4.2 Materials Library 552

6.4.3 Meshing Tool 554

6.4.4 Analysis Types 558

6.4.5 Tools for Boundary Conditions 559

6.4.6 Solvers to FEA Models 559

6.4.7 Postprocessing 562

6.5 Planning V&V in FEA Modeling 562

6.5.1 Sources of Errors 563

6.5.1.1 Error Quantification 563

6.5.1.2 System Inputs 564

6.5.1.3 Errors of Idealization 565

6.5.1.4 Errors of Mathematic Models 566

6.5.1.5 Errors of Model or Analysis Type 567

6.5.2 Verification 567

6.5.2.1 Code Verification 568

6.5.2.2 Calculation Verification 571

6.5.2.3 Meshing Verification 572

6.5.2.4 Convergence Study 575

6.5.2.5 Benchmarking 576

6.6 Finite Element Analysis for Verification of Structural Analysis 577

6.7 FEA for Stress Analysis of Assembly Models 580

6.8 Parametric Study for Stress Analysis 582

6.9 FEA on Study of Stress Concentration Factors 586

References 586

Index 589