STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS 4.14 CHAPTER FOUR Stress concentration factor theoretical/empirical or otherwise Particular Extreme value Formula FIGURE 4-22 Reproduced with permission Stress-concentration factor K for notched flat bar in tension (R E Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol 23, Nos to 7, 1951.) (iii) Bar with shallow V-groove in tension for r >1 h1 ( Kv ẳ ỵ Kv 1ị 180 ỵ 2:4p r=h1 ) ð4-25Þ (iv) Elliptical groove at the edge of plate in tension K ẳ ỵ 2h1 b r h K ẳ ỵ r (v) Bar with symmetrical U, semicircular shallow grooves in bending (Fig 4-23) K ẳ ỵ h1 0:85 D r5 4:27 À d ð4-26aÞ ð4-26bÞ ð4-27aÞ or 30:85 D À1 d7 d K ẳ ỵ D r5 4:27 À d Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð4-27bÞ STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS 4.15 Stress concentration factor theoretical/empirical or otherwise Particular Extreme value FIGURE 4-23 Reproduced with permission Stress-concentration factor K for notched flat bar in bending (R E Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol 23, Nos to 7, 1951.) For stress-concentration factors for small grooves in a shaft subjected to torsion (o) Bar containing shoulders (i) Bar with shoulders in tension (Fig 4-24) TABLE 4-4 Stress-concentration factors for relatively grooves in a shaft subject to torsion, K small Formula FIGURE 4-24 Reproduced with permission Stress-concentration factor K for filleted flat bar in tension (R E Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol 23, Nos 2–7, 1951.) Refer to Table 4-4 h1 0:85 K ẳ ỵ D r5 2:8 À d or 30:85 D À1 d7 d K ẳ ỵ D r5 2:8 À d h1 r Included angle of V, deg 0.5 60 90 120 1.85 1.84 1.81 1.66 2.01 2.00 1.95 1.75 2.66 2.54 2.40 1.95 3.23 3.06 2.40 2.00 4.54 3.99 3.12 2.13 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð4-28aÞ ð4-28bÞ STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS 4.16 CHAPTER FOUR Stress concentration factor theoretical/empirical or otherwise Particular (ii) Bar with shoulders in bending (Fig 4-25) Extreme value K ẳ ỵ or Formula h1 0:85 D r5 5:37 À 4:8 d ð4-29aÞ 30:85 D À1 d7 d K ẳ ỵ D r5 5:37 À 4:8 d ð4-29bÞ (p) Press-fitted or shrink-fitted members (Table 4-5): (i) Plain member K ¼ 1:95 (ii) Grooved member K ¼ 1:34 (iii) Plain member Kf ¼ 2:00 (iv) Grooved member Kf ¼ 1:70 (q) Bolts and nuts (Tables 4-6 and 4-7) Bolt and nut of standard proportions K ¼ 3:85 Bolt and nut having lip K ¼ 3:00 TABLE 4-5 Stress-concentration factors in shrink-fitted members Particular K Kf Plain Grooved 1.95 1.34 2.00 1.70 FIGURE 4-25 Reproduced with permission Stress-concentration factor K for stepped bar in bending (R E Peterson, ‘‘Design Factors for Stress Concentration,’’ Machine Design, Vol 23, Nos to 7, 1951.) Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS 4.17 TABLE 4-6 Stress-concentration factors for screw threads Analysis Types of thread Seely and Smith Square Sharp V Whitworth US standard Medium Carbon steel National coarse thread Heat-treated Nickel steel Black (8) 2.0 3.0 2.0 Suggested value 3.35 to 2.5 2.84 3.85 TABLE 4-7 Stress-concentration factors Kf for screw threads Annealed Type of thread Peterson (1) Hardened Rolled Cut Rolled Cut TABLE 4-8 Stress-concentration factors for welds K Location 2.2 Whitworth rounded roots 1.4 2.8 1.8 3.0 2.6 3.8 3.3 2.7 Reinforced butt 1.2 Tee of transverse fillet weld Sellers, American National, square thread End or parallel fillet weld 1.5 T-butt weld with sharp corners 2.0 TABLE 4-9 Index of sensitivity for repeated stress Average index of sensitivity q Material Annealed or soft Armco iron, 0.02% C Carbon steel 0.10% C 0.20% C (also cast steel) 0.30% C 0.50% C 0.85% C Spring steel, 0.56% C, 2.3 Si, rolled SAE 3140, 0.73 C; 0.6 Cr; 1.3 Ni Cr–Ni steel 0.8 Cr; 3.5 Ni Stainless steel, 0.3 C; 8.3 Cr, 19.7 Ni Cast iron Copper, electrolitic Duraluminum Heat-treated and drawn at 921 K (6488C) Heat-treated and drawn at 755 K (4828C) 0.35 0.40 0.45 0.38 0.45 0.25 0.45 0.50 0.57 0.15–0.20 0.05–0.10 0.10 0.18 0.26 0.25 0.70 0.16 0–0.05 0.07 0.05–0.13 Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS 4.18 CHAPTER FOUR Stress concentration factor theoretical/empirical or otherwise Particular Extreme value Formula (r) Crane hook: For crane hook under tensile load K ¼ 1:56 (s) Rotating disk: For rotating disk with a hole for For thin disk (ring) Ri !0 Ro K ¼ K ¼ (t) Eye bar: For eye bar subjected to tensile load K ¼ 2:8 Stress concentration factors for welds Refer to Table 4-8 (u) Notch sensitivity factors (Table 4-9): For index of sensitivity for repeated stresses (ii) Fatigue stress concentration factor for normal stress (iii) Notch sensitivity factor for shear stress (iv) Fatigue stress-concentration factor for shear stress q ¼ Kf À K À ð4-30aÞ q ¼ (i) Notch sensitivity factor for normal stress Kf À K À ð4-30bÞ Refer to Table 4-9 Kf ẳ ỵ q K 1ị 4-31aị Kf ẳ ỵ q K 1ị 4-31bị q ẳ Kf K Kf ẳ ỵ q K 1ị ð4-32Þ ð4-33Þ STRESS CONCENTRATION IN FLANGED PIPE SUBJECTED TO AXIAL EXTERNAL FORCE The stress in the pipe due to external load F (Fig 4-25A) F 4-33aị A where f ẳ depends on the distance x from the flange of the pipe, MPa (psi) ẳ f ỵ fm ẳ maximum stress at x ¼ 0, MPa (psi) A ¼ area of the cross section of pipe, m2 (in2 ) F ¼ external load, kN (lbf) FIGURE 4-25A Pipe and flange under the axial force F Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS Particular Formula sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3ð1 À ị ẳ 10 R2 h2 The value of constant 4.19 4-33bị where 2R ẳ 2Ri ỵ h ẳ mean diameter of pipe, m (in) 2Ro ¼ outer diameter of pipe, m (in) 2Ri ¼ inner diameter of pipe, m (in) h ¼ thickness of pipe, m (in) ¼ Poisson’s ratio of material For plot of the stress ratio f versus x fm Refer to Fig 4-25B FIGURE 4-25B Stress concentration region in flanged pipe under axial external force F Courtesy: Douglas C Greenwood, Engineering Data for Product Design, McGraw-Hill Publishing Company, New York, 1961 REDUCTION OR MITIGATION OF STRESS CONCENTRATIONS In designing a machine part, one has to take into consideration the stress concentration occurring in such parts and eliminate or reduce stress concentration Fig 4-25C shows various methods used to reduce stress concentration Stream line flowing analogy in a channel can be applied to force flow lines of a flat plate without any type of flow subject to uniform uniaxial tensile stress as shown in Fig 4-25C i(a) The stream line flow of water or any fluid is Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS 4.30 CHAPTER FOUR Particular The localized stress components and crack tip displacement fields for Mode III in terms of polar coordinates The critical applied tensile stress necessary for crack extension according to Griffith theory for brittle metals The modified Griffith’s equation for a small amount of plastic deformation according to Orowan which can be applied to ductile materials at low temperature, high strain rate and localized geometric constraint The elastic energy release rate for Mode I Formula KIII r ẳ p sin 2r KIII 4-36pị ¼ pffiffiffiffiffiffiffi cos 2r rffiffiffiffiffiffi 2K r sin uz ẳ III 4-36qị G 2 r EU 4-36rị c / a where c ẳ critical applied stress E ¼ Young’s modulus U ¼ surface energy per unit area a ẳ crack length r EU ỵ pị ẳ 4-36sị a where p ẳ plastic deformation energy per unit area for metallic solid, p ) U 2 4-36tị GI ẳ KI2 for plane strain E ¼ KI2 =E The elastic energy release rate for Mode II The elastic energy release rate for Mode III The stress-intensity factor for a centrally located straight crack in an infinite plate subjected to uniform shear stress ð4-36nÞ GII ¼ for plane stress ð1 À Þ KII E ỵ ị KIII E p p KI À iKII ¼ Ài a GIII ¼ The stress-intensity magnification factor for a centrally located straight crack of length 2a in a flat plate whose length 2h and width 2b are very large compared with the crack length subjected to uniform uniaxial tensile stress K MF ¼ pffiffiffi Ipffiffiffi a For stress-intensity magnification factors of plates with straight crack located at various positions in the plate and cylinders subjected to various types of rate of loadings and for various values of a=b, a=d, a=h, a=ðro À ri ị, and other ratios nẳ 4-36vị 4-36wị 4-37aị Refer to Figs from 4-28, 4-29 to 4-34 The factor of safety ð4-36uÞ KIc K Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website ð4-37bÞ ð4-38Þ STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS 4.31 FIGURE 4-28 Stress intensity magnification factor pffiffiffi pffiffiffi KI = a for various ratios a=b of a flat plate with a centrally located straight crack under the action of uniform uniaxial tensile stress FIGURE 4-29 Stress intensity magnification factor pffiffiffi pffiffiffi KI = a for an off-center straight crack in a flat plate subjected to uniform unidirectional tensile stress ; solid curves are for the crack tip at A; dashed curves for tip at B FIGURE 4-30 Stress intensity magnification factor pffiffiffi pffiffiffi KI = a for an edge straight crack in a flat plate subjected to uniform uniaxial tensile stress for solid curves there are no constraints to bending; the dashed curve was obtained with bending constraints added FIGURE 4-31 Stress intensity magnification factor pffiffiffi pffiffiffi KI = a for a rectangular cross-sectional beam subjected to bending Mb Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS 4.32 CHAPTER FOUR FIGURE 4-32 Stress intensity magnification factor pffiffiffi pffiffiffi KI = a for a flat plate with a centrally located circular hole with two straight cracks under uniform uniaxial tensile stress FIGURE 4-33 Stress intensity magnification factor pffiffiffi pffiffiffi KI = a for axially tensile loaded cylinder with a radial crack of a depth extending completely around the circumference of the cylinder pffiffiffi pffiffiffi FIGURE 4-34 Stress intensity magnification factor KI = a for a cylinder subjected to internal pressure pi having a radial crack in the longitudinal direction of depth a Use equation of tangential stress of thick cylinder subjected to internal pressure to calculate the stress at r ¼ ro Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com) Copyright © 2004 The McGraw-Hill Companies All rights reserved Any use is subject to the Terms of Use as given at the website STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS STRESS CONCENTRATION AND STRESS INTENSITY IN MACHINE MEMBERS Particular 4.33 Formula KIc sy =2 Critical crack length ac ¼ For values of critical stress-intensity factor (KIc ) for some engineering materials Refer to Table 4-10 TABLE 4-10 Plane-strain fracture toughness or stress intensity factor KIc for some engineering materials KIc Material Previous designation Aluminum 2014-T651 2024-T851 7075-T651 7178 UNS designation A92024-T851 A97075-T651 Titanium Ti-6Al-4V Ti-6Al-4VÃ R56401 R56401Ã Steel 4340 4340Ã H-11 H-11 52100 G43400 G43400Ã – – G52986 Ã pffiffiffiffi MPa m 24.2 26 24 13 115 55 99 60 38.5 27.8 14 Yield strength, xy pffiffiffiffi kpsi in Critical crack length, ac MPa kpsi mm in 22 24 22 30 455 455 495 490 66 66 72 71 3.6 4.3 3.0 5.8 0.14 0.17 0.12 0.23 105 50 910 1035 132 150 20.5 3.6 0.81 0.14 90 55 35 27 13 860 1515 1790 2070 2070 125 220 260 300 300 16.8