WALTER D. PILKEY, PHD, was the Frederick Morse Professor in the Department of Mechanical, Aerospace, and Nuclear Engineering at the University of Virginia and a leading authority in the areas of stress and strain in mechanical and civil engineering. He is the author of Formulas for Stress, Strain, and Structural Matrices, Second Edition, and Analysis and Design of Elastic Beams, all from Wiley.

DEBORAH F. PILKEY, PHD, is an Engineer and Scientist in the Loads & Environments Department at Orbital ATK in Utah. She has been involved with structures technology, loads, dynamics, and production stress analysis of the Space Shuttle's main engines and their solid rocket motors.

ZHUMING BI, PHD, is a professor of Mechanical Engineering. He is affiliated with the School of Electromechanical Engineering and Automation, Shanghai University and the Department of Civil and Mechanical Engineering at Purdue University. He has over 25 years of experience in Machine Design, Modelling & Simulation, and CAD/CAM.

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...