Peterson's Stress Concentration Factors

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 xvPreface for the Fourth Edition xxxiPreface for the Third Edition xxxiiiPreface for the Second Edition xxxv1 Fundamentals of Stress Analysis 11.1 Stress Analysis in Product Design 21.2 Solid Objects Under Loads 41.3 Types of Materials 61.4 Materials Properties and Testing 71.4.1 Tensile and Compression Tests 81.4.2 Hardness Tests 81.4.3 Shear Tests 131.4.4 Fatigue Tests 141.4.5 Impact Tests 161.5 Static and Fatigue Failures 171.6 Uncertainties, Safety Factors, and Probabilities 191.7 Stress Analysis of Mechanical Structures 211.7.1 Procedure of Stress Analysis 211.7.2 Geometric Discontinuities of Solids 211.7.3 Load Types 231.7.4 Stress and Representation 241.7.4.1 Simple Stress 261.7.4.2 General Stresses 261.7.4.3 Principal Stresses and Directions 271.8 Failure Criteria of Materials 301.8.1 Maximum Shear Stress (MSS) Theory 301.8.2 Distortion Energy (DE) Theory 321.8.3 Maximum Normal Stress (MNS) Theory 341.8.4 Ductile and Brittle Coulomb-Mohr (CM) Theory 361.8.5 Modified-Mohr (MM) Theory 371.8.6 Guides for Selection of Failure Criteria 371.9 Stress Concentration 391.9.1 Selection of Nominal Stresses as Reference 421.9.2 Accuracy of Stress Concentration Factors 451.9.3 Decay of Stress away from the Peak Stress 461.10 Stress Concentration as a Two-Dimensional Problem 461.11 Stress Concentration as a Three-Dimensional Problem 471.12 Plane and Axisymmetric Problems 491.13 Local and Nonlocal Stress Concentration 521.14 Multiple Stress Concentration 571.15 Principle of Superposition for Combined Loads 611.16 Notch Sensitivity 641.17 Design Relations for Static Stress 691.17.1 Ductile Materials 691.17.2 Brittle Materials 711.18 Design Relations for Alternating Stress 721.18.1 Ductile Materials 721.18.2 Brittle Materials 731.19 Design Relations for Combined Alternating and Static Stresses 741.19.1 Ductile Materials 741.19.2 Brittle Materials 771.20 Limited Number of Cycles of Alternating Stress 781.21 Stress Concentration Factors and Stress Intensity Factors 791.22 Selection of Safety Factors 83References 852 Notches and Grooves 892.1 Notation 892.2 Stress Concentration Factors 902.3 Notches in Tension 922.3.1 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 922.3.2 Opposite Single Semicircular Notches in a Finite-Width Thin Element 942.3.3 Opposite Single U-Shaped Notches in a Finite-Width Thin Element 942.3.4 Finite-Width Correction Factors for Opposite Narrow Single Elliptical Notches in a Finite-Width Thin Element 952.3.5 Opposite Single V-Shaped Notches in a Finite-Width Thin Element 952.3.6 Single Notch on One Side of a Thin Element 962.3.7 Notches with Flat Bottoms 962.3.8 Multiple Notches in a Thin Element 962.3.9 Analytical Solutions for Stress Concentration Factors for Notched Bars 982.4 Depressions in Tension 982.4.1 Hemispherical Depression (Pit) in the Surface of a Semi-Infinite Body 982.4.2 Hyperboloid Depression (Pit) in the Surface of a Finite-Thickness Element 982.4.3 Opposite Shallow Spherical Depressions (Dimples) in a Thin Element 992.5 Grooves in Tension 1002.5.1 Deep Hyperbolic Groove in an Infinite Member (Circular Net Section) 1002.5.2 U-Shaped Circumferential Groove in a Bar of Circular Cross Section 1002.5.3 Flat-Bottom Grooves 1002.5.4 Closed-Form Solutions for Grooves in Bars of Circular Cross Section 1002.6 Bending of Thin Beams with Notches 1012.6.1 Opposite Deep Hyperbolic Notches in an Infinite Thin Element 1012.6.2 Opposite Semicircular Notches in a Flat Beam 1012.6.3 Opposite U-Shaped Notches in a Flat Beam 1012.6.4 V-Shaped Notches in a Flat Beam Element 1022.6.5 Notch on One Side of a Thin Beam 1022.6.6 Single or Multiple Notches with Semicircular or Semielliptical Notch Bottoms 1022.6.7 Notches with Flat Bottoms 1032.6.8 Closed-Form Solutions for Stress Concentration Factors for Notched Beams 1032.7 Bending of Plates with Notches 1032.7.1 Various Edge Notches in an Infinite Plate in Transverse Bending 1032.7.2 Notches in a Finite-Width Plate in Transverse Bending 1042.8 Bending of Solids with Grooves 1042.8.1 Deep Hyperbolic Groove in an Infinite Member 1042.8.2 U-Shaped Circumferential Groove in a Bar of Circular Cross Section 1042.8.3 Flat-Bottom Grooves in Bars of Circular Cross Section 1052.8.4 Closed-Form Solutions for Grooves in Bars of Circular Cross Section 1052.9 Direct Shear and Torsion 1062.9.1 Deep Hyperbolic Notches in an Infinite Thin Element in Direct Shear 1062.9.2 Deep Hyperbolic Groove in an Infinite Member 1062.9.3 U-Shaped Circumferential Groove in a Bar of Circular Cross Section Subject to Torsion 1062.9.4 V-Shaped Circumferential Groove in a Bar of Circular Cross Section Under Torsion 1082.9.5 Shaft in Torsion with Grooves with Flat Bottoms 1082.9.6 Closed-Form Formulas for Grooves in Bars of Circular Cross Section Under Torsion 1092.10 Test Specimen Design for Maximum Kt for a Given r/D or r/H 109References 109Charts 1133 Shoulder Fillets 1673.1 Notation 1673.2 Stress Concentration Factors 1693.3 Tension (Axial Loading) 1703.3.1 Opposite Shoulder Fillets in a Flat Bar 1703.3.2 Effect of Length of Element 1703.3.3 Effect of Shoulder Geometry in a Flat Member 1703.3.4 Effect of a Trapezoidal Protuberance on the Edge of a Flat Bar 1713.3.5 Fillet of Noncircular Contour in a Flat Stepped Bar 1723.3.6 Stepped Bar of Circular Cross Section with a Circumferential Shoulder Fillet 1753.3.7 Tubes 1763.3.8 Stepped Pressure Vessel Wall with Shoulder Fillets 1763.4 Bending 1773.4.1 Opposite Shoulder Fillets in a Flat Bar 1773.4.2 Effect of Shoulder Geometry in a Flat Thin Member 1773.4.3 Elliptical Shoulder Fillet in a Flat Member 1773.4.4 Stepped Bar of Circular Cross Section with a Circumferential Shoulder Fillet 1773.5 Torsion 1783.5.1 Stepped Bar of Circular Cross Section with a Circumferential Shoulder Fillet 1783.5.2 Stepped Bar of Circular Cross Section with a Circumferential Shoulder Fillet and a Central Axial Hole 1783.5.3 Compound Fillet 1793.6 Methods of Reducing Stress Concentration at a Shoulder 180References 182Charts 1844 Holes 2094.1 Notation 2094.2 Stress Concentration Factors 2114.3 Circular Holes with In-Plane Stresses 2144.3.1 Single Circular Hole in an Infinite Thin Element in Uniaxial Tension 2144.3.2 Single Circular Hole in a Semi-Infinite Element in Uniaxial Tension 2174.3.3 Single Circular Hole in a Finite-Width Element in Uniaxial Tension 2184.3.4 Effect of Length of Element 2184.3.5 Single Circular Hole in an Infinite Thin Element under Biaxial In-Plane Stresses 2194.3.6 Single Circular Hole in a Cylindrical Shell with Tension or Internal Pressure 2204.3.7 Circular or Elliptical Hole in a Spherical Shell with Internal Pressure 2234.3.8 Reinforced Hole Near the Edge of a Semi-Infinite Element in Uniaxial Tension 2234.3.9 Symmetrically Reinforced Hole in a Finite-Width Element in Uniaxial Tension 2264.3.10 Nonsymmetrically Reinforced Hole in a Finite-Width Element in Uniaxial Tension 2274.3.11 Symmetrically Reinforced Circular Hole in a Biaxially Stressed Wide, Thin Element 2274.3.12 Circular Hole with Internal Pressure 2354.3.13 Two Circular Holes of Equal Diameter in a Thin Element in Uniaxial Tension or Biaxial In-Plane Stresses 2364.3.14 Two Circular Holes of Unequal Diameter in a Thin Element in Uniaxial Tension or Biaxial In-Plane Stresses 2414.3.15 Single Row of Equally Distributed Circular Holes in an Element in Tension 2434.3.16 Double Row of Circular Holes in a Thin Element in Uniaxial Tension 2434.3.17 Symmetrical Pattern of Circular Holes in a Thin Element in Uniaxial Tension or Biaxial In-Plane Stresses 2444.3.18 Radially Stressed Circular Element with a Ring of Circular Holes, with or without a Central Circular Hole 2454.3.19 Thin Element with Circular Holes with Internal Pressure 2464.4 Elliptical Holes in Tension 2474.4.1 Single Elliptical Hole in Infinite- and Finite-Width Thin Elements in Uniaxial Tension 2504.4.2 Width Correction Factor for a Cracklike Central Slit in a Tension Panel 2524.4.3 Single Elliptical Hole in an Infinite, Thin Element Biaxially Stressed 2534.4.4 Infinite Row of Elliptical Holes in Infinite- and Finite-Width Thin Elements in Uniaxial Tension 2634.4.5 Elliptical Hole with Internal Pressure 2634.4.6 Elliptical Holes with Bead Reinforcement in an Infinite Thin Element under Uniaxial and Biaxial Stresses 2634.5 Various Configurations with In-Plane Stresses 2634.5.1 Thin Element with an Ovaloid; Two Holes Connected by a Slit under Tension; Equivalent Ellipse 2634.5.2 Circular Hole with Opposite Semicircular Lobes in a Thin Element in Tension 2654.5.3 Infinite Thin Element with a Rectangular Hole with Rounded Corners Subject to Uniaxial or Biaxial Stress 2664.5.4 Finite-Width Tension Thin Element with Round-Cornered Square Hole 2674.5.5 Square Holes with Rounded Corners and Bead Reinforcement in an Infinite Panel under Uniaxial and Biaxial Stresses 2674.5.6 Round-Cornered Equilateral Triangular Hole in an Infinite Thin Element Under Various States of Tension 2674.5.7 Uniaxially Stressed Tube or Bar of Circular Cross Section with a Transverse Circular Hole 2674.5.8 Round Pin Joint in Tension 2684.5.9 Inclined Round Hole in an Infinite Panel Subjected to Various States of Tension 2694.5.10 Pressure Vessel Nozzle (Reinforced Cylindrical Opening) 2704.5.11 Spherical or Ellipsoidal Cavities 2714.5.12 Spherical or Ellipsoidal Inclusions 2724.6 Holes in Thick Elements 2744.6.1 Countersunk Holes 2764.6.2 Cylindrical Tunnel 2774.6.3 Intersecting Cylindrical Holes 2784.6.4 Rotating Disk with a Hole 2794.6.5 Ring or Hollow Roller 2814.6.6 Pressurized Cylinder 2814.6.7 Pressurized Hollow Thick Cylinder with a Circular Hole in the Cylinder Wall 2824.6.8 Pressurized Hollow Thick Square Block with a Circular Hole in the Wall 2834.6.9 Other Configurations 2834.7 Orthotropic Thin Members 2844.7.1 Orthotropic Panel with an Elliptical Hole 2844.7.2 Orthotropic Panel with a Circular Hole 2864.7.3 Orthotropic Panel with a Crack 2864.7.4 Isotropic Panel with an Elliptical Hole 2864.7.5 Isotropic Panel with a Circular Hole 2864.7.6 More Accurate Theory for a/b < 4 2874.8 Bending 2884.8.1 Bending of a Beam with a Central Hole 2884.8.2 Bending of a Beam with a Circular Hole Displaced from the Center Line 2894.8.3 Curved Beams with Circular Holes 2894.8.4 Bending of a Beam with an Elliptical Hole; Slot with Semicircular Ends (Ovaloid); or Round-Cornered Square Hole 2904.8.5 Bending of an Infinite- and a Finite-Width Plate with a Single Circular Hole 2904.8.6 Bending of an Infinite Plate with a Row of Circular Holes 2914.8.7 Bending of an Infinite Plate with a Single Elliptical Hole 2914.8.8 Bending of an Infinite Plate with a Row of Elliptical Holes 2914.8.9 Tube or Bar of Circular Cross Section with a Transverse Hole 2914.9 Shear and Torsion 2924.9.1 Shear Stressing of an Infinite Thin Element with Circular or Elliptical Hole, Unreinforced and Reinforced 2924.9.2 Shear Stressing of an Infinite Thin Element with a Round-Cornered Rectangular Hole, Unreinforced and Reinforced 2934.9.3 Two Circular Holes of Unequal Diameter in a Thin Element in Pure Shear 2934.9.4 Shear Stressing of an Infinite Thin Element with Two Circular Holes or a Row of Circular Holes 2944.9.5 Shear Stressing of an Infinite Thin Element with an Infinite Pattern of Circular Holes 2944.9.6 Twisted Infinite Plate with a Circular Hole 2944.9.7 Torsion of a Cylindrical Shell with a Circular Hole 2944.9.8 Torsion of a Tube or Bar of Circular Cross Section with a Transverse Circular Hole 294References 296Charts 3075 Miscellaneous Design Elements 4395.1 Notation 4395.2 Shaft with Keyseat 4415.2.1 Bending 4425.2.2 Torsion 4425.2.3 Torque Transmitted Through a Key 4435.2.4 Combined Bending and Torsion 4435.2.5 Effect of Proximity of Keyseat to Shaft Shoulder Fillet 4435.2.6 Fatigue Failures 4445.3 Splined Shaft in Torsion 4455.4 Gear Teeth 4455.5 Press- or Shrink-Fitted Members 4475.6 Bolt and Nut 4505.7 Bolt Head, Turbine-Blade, or Compressor-Blade Fastening (T-Head) 4525.8 Lug Joint 4545.8.1 Lugs with h∕d < 0.5 4555.8.2 Lugs with h∕d > 0.5 4565.9 Curved Bar 4575.10 Helical Spring 4585.10.1 Round or Square Wire Compression or Tension Spring 4585.10.2 Rectangular Wire Compression or Tension Spring 4605.10.3 Helical Torsion Spring 4615.11 Crankshaft 4615.12 Crane Hook 4625.13 U-Shaped Member 4625.14 Angle and Box Sections 4635.15 Cylindrical Pressure Vessel with Torispherical Ends 4635.16 Welds 4645.17 Parts with Inhomogeneous Materials or Composites 4715.18 Parts with Defects 4715.19 Parts with Threads 4745.20 Frame Stiffeners 4755.21 Discontinuities with Additional Considerations 4765.22 Pharmaceutical Tablets with Holes 4775.23 Parts with Residual Stresses 4785.24 Surface Roughness 4795.25 New Approaches for Parametric Studies 480References 481Charts 4896 Finite Element Analysis (FEA) for Stress Analysis 5176.1 Structural Analysis Problems 5186.2 Types of Engineering Analysis Methods 5196.3 Structural Analysis Theory 5206.3.1 Trusses and Frame Structures 5236.3.1.1 Trusses 5236.3.1.2 Boundary Conditions (BCs) and Loads 5266.3.1.3 Frame Structure 5276.3.2 Plane Stress and Strain Problems 5306.3.2.1 Plane Stresses 5306.3.2.2 Plane Strain Problems 5356.3.3 Modal Analysis 5356.3.3.1 Two-Dimensional Truss Member in LCS 5376.3.3.2 Two-Dimensional Beam Member in LCS 5386.3.3.3 Modeling of Two-Dimensional Frame Element 5406.3.4 Fatigue Analysis 5426.3.4.1 Strain-Life Method 5436.3.4.2 Linear Elastic Fracture Mechanics Method 5446.3.4.3 Stress-Life Method 5456.3.4.4 Selection of Fatigue Analysis Methods 5466.4 Finite Element Anlaysis (FEA) for Structural Analysis 5476.4.1 CAD/CAE Interface 5516.4.2 Materials Library 5526.4.3 Meshing Tool 5546.4.4 Analysis Types 5586.4.5 Tools for Boundary Conditions 5596.4.6 Solvers to FEA Models 5596.4.7 Postprocessing 5626.5 Planning V&V in FEA Modeling 5626.5.1 Sources of Errors 5636.5.1.1 Error Quantification 5636.5.1.2 System Inputs 5646.5.1.3 Errors of Idealization 5656.5.1.4 Errors of Mathematic Models 5666.5.1.5 Errors of Model or Analysis Type 5676.5.2 Verification 5676.5.2.1 Code Verification 5686.5.2.2 Calculation Verification 5716.5.2.3 Meshing Verification 5726.5.2.4 Convergence Study 5756.5.2.5 Benchmarking 5766.6 Finite Element Analysis for Verification of Structural Analysis 5776.7 FEA for Stress Analysis of Assembly Models 5806.8 Parametric Study for Stress Analysis 5826.9 FEA on Study of Stress Concentration Factors 586References 586Index 589