DESIGN OF WELDED JOINTS 12.20 CHAPTER TWELVE TABLE 12-15 Stress concentration factor, K Stress concentration factor, K Weld type and metal Low-carbon steel 1.2 3.5 1.4 2.5 4.5 1.2 1.5 2.7 3.5 1.4 1.9 2.5 3.3 4.5 1.5 2.7 1.5 1.9 3.3 1.9 2.7 Weld metal Butt welds with full penetration End fillet welds Parallel fillet welds Base metal Toe of machined butt weld Toe of unmachined butt weld Toe of machined end fillet weld with leg ratio : 1.5 Toe of unmachined end fillet weld with leg ratio : 1.5 Parallel fillet weld Stiffening ribs and partitions welded with end fillet welds having smooth transitions at the toes Butt and T-welded corner plates Butt and T-welded corner plates, but with smooth transitions in the shape of the plates and with machined welds Lap-welded corner plates Low-alloy steel 3.3 TABLE 12-16 Allowable stresses for welds under static loads Allowable stresses Weld type and process Tension, ta Compression, ca Shear, a Automatic and hand welding with shielded arc and butt welding Hand welding with ordinary quality electrodes Resistance spot welding t a 0.9t 0.9t t t t 0.65t 0.6t 0.5t a t is the allowable stress in tension of the base metal of the weld 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 Source: MACHINE DESIGN DATABOOK CHAPTER 13 RIVETED JOINTS SYMBOLS2;3;4 A b c d Di e or l F h hc , h1 , h2 i I J K¼ m Mb p pc pd pt Pf Z a c a F F0 area of cross-section, m2 (in2 ) the cross-sectional area of rivet shank, m2 (in2 ) breadth of cover plates (also with suffixes), m (in) distance from the centroid of the rivet group to the critical rivet, m (in) diameter of rivet, m (in) internal diameter of pressure vessel, m (mm) eccentricity of loading, m (in) force on plate or rivets (also with suffixes), kN (lbf) thickness of plate or shell, m (in) thickness of cover plate (butt strap), m (in) number of rivets in a pitch fine (also with suffixes and 2, respectively, for single shear and double shear rivets) moment of inertia, area, m4 , cm4 (in4 ) moment of inertia, polar, m4 , cm4 (in4 ) coefficient (Table 13-11) margin, m (in) bending moment, N m (lbf in) pitch on the gauge line or longitudinal pitch, m (in) pitch along the caulking edge, m (in) diagonal pitch, m (in) transverse pitch, m (in) intensity of fluid pressure, MPa (psi) section modulus of the angle section, m3 , cm3 (in3 ) hoop stress in pressure vessel or normal stress in plate, MPa (psi) allowable normal stress, MPa (psi) crushing stress in rivets, MPa (psi) shear stress in rivet, MPa (psi) allowable shear stress, MPa (psi) efficiency of the riveted joint angle between a line drawn from the centroid of the rivet group to the critical rivet and the horizontal (Fig 13-5) 13.1 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 RIVETED JOINTS 13.2 CHAPTER THIRTEEN Particular Formula PRESSURE VESSELS Thickness of main plates The thickness of plate of the pressure vessel with longitudinal joint h¼ P f Di 2 ð13-1Þ For thickness of boiler plates and suggested types of joints Refer to Tables 13-1 and 13-2 The thickness of plate of the pressure vessel with circumferential joint h¼ For allowable stress and efficiency of joints Refer to Tables 13-3, 13-4, 13-5, and 13-6 P f Di 4 ð13-2Þ PITCHES Lap joints The diagonal pitch (staggered) (Fig 13-1) for p, pt , and pd The distance between rows or transverse pitch or back pitch (staggered) pd ẳ 2p ỵ d ð13-3Þ Refer to Tables 13-7 and 13-8 for rivets for general purposes and boiler rivets sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2p ỵ d p 13-4ị pt ¼ pffiffiffi pffiffiffi d ¼ 0:19 h to 0:2 h SI ð13-5aÞ where h and d in m pffiffiffi pffiffiffi d ¼ 1:2 h to 1:4 h The rivet diameter USCS ð13-5bÞ where h and d in in pffiffiffi pffiffiffi d ¼ h to 6:3 h CM ð13-5cÞ where h and d on mm FIGURE 13-1 Pitch relation TABLE 13-1 Suggested types of joint Diameter of shell, mm (in) Thickness of shell, mm (in) Type of joint 600–1800 (24–72) 900–2150 (36–84) 1500–2750 (60–108) Double-riveted Triple-riveted Quadruple-riveted 6–12 (0.25–0.5) 7.5–25 (0.31–1.0) 9.0–44 (0.375–1.75) 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 RIVETED JOINTS RIVETED JOINTS 13.3 TABLE 13-2 Minimum thickness of boiler plates Shell plates Tube sheets of firetube boilers Diameter of shell, mm (in) Minimum thickness after flanging, mm (in) 900 (36) 900–1350 (36–54) 1350–1800 (54–72) !1800 (72) 6.0 (0.25) 8.0 (0.3125) 9.5 (0.375) 12.5 (0.5) Diameter of tube sheet, mm (in) 1050 (42) 1050–1350 (42–54) 1350–1800 (54–72) 1800 (72) Minimum thickness, mm (in) 9.5 (0.375) 11.5 (0.4375) 12.5 (0.50) 14.0 (0.5625) TABLE 13-3 Efficiency of riveted joints () % Efficiency, Normal range Type of joint Lap joints Single-riveted Double-riveted Triple-riveted Butt joints (with two cover plates) Single-riveted Double-riveted Triple-riveted Quadruple-riveted Maximum 50–60 60–72 72–80 63 77 86.6 55–60 76–84 80–88 86–94 63 87 95 98 TABLE 13-4 Allowable stresses in structural riveting (b ) Rivets acting in single shear Rivets acting in double shear Load-carrying member Type of stress Rivet-driving method Rolled steel SAE 1020 Tension Shear Power 124 93 18.0 13.5 124 93 18.0 13.5 Shear Crushing Crushing Hand Power Hand 68 165 110 10.0 24.0 16.0 68 206 137 10.0 30.0 20.0 Rivets, SAE 1010 MPa kpsi MPa kpsi 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 RIVETED JOINTS 13.4 CHAPTER THIRTEEN TABLE 13-5 Allowable stress for aluminum rivets, a Allowable stressa , a Shear Bearing Rivet alloy Procedure of drawing MPa kpsi MPa kpsi 2S (pure aluminum) 17S 17S 615–T6 53S Cold, as received Cold, immediately after quenching Hot, 500–5108C Cold, as received Hot, 515–5278C 20 68 62 55 41 3.0 10.0 9.0 8.0 6.0 48 179 179 103 103 7.0 26.0 26.0 15.0 15.0 a Actual safety factor or reliability factor is 1.5 TABLE 13-6 Values of working stressa at elevated temperatures Minimum of the specified range of tensile strength of the material, MPa (kpsi) Maximum temperatures (45) 311 (50) 344 (55) 380 (60) 413 (75) 517 8F 8C MPa kpsi MPa kpsi MPa kpsi MPa kpsi MPa kpsi 0–700 750 800 850 900 950 0–371 399 427 455 482 511 61 56 45 37 29 22 9.0 8.22 6.55 5.44 4.33 3.20 68 62 53 41 33 26 10.0 9.11 7.33 6.05 4.83 3.60 76 68 54 46 37 27 11.00 10.00 8.00 6.75 5.50 4.00 82 77 61 51 38 27 12.00 11.20 9.00 7.40 5.60 4.00 103 89 70 57 41 27 15.00 13.00 10.20 8.30 6.00 4.00 a Design stresses of pressure vessels are based on a safety factor of TABLE 13-7 Pitch of butt joints Type of joint Diameter of rivets, d, mm Pitch, p Double-riveted— use for h 12:5 mm (0.5 in) Triple-riveted— use for h 25 mm (1 in) Quadruple-riveted— use for h 31:75 mm (1.25 in) Any 5.5d (approx.) 1.75–23.80 27.00 30.15–36.50 17.50–23.80 27.00 30.15 33.30–36.50 8d–8.5d 7.5d 6.5d–7d 16d–17d 15d (approx.) 14d (approx.) 13d–14d 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 RIVETED JOINTS RIVETED JOINTS 13.5 TABLE 13-8 Transverse pitch ( pt ) as per ASME Boiler Code Value of p=d Value of pt 2d 2d 2d 2d 2d 2.2d 2.3d Particular Formula Butt joint pt ¼ 2d to 2:5d qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pt ! 0:5pd ỵ 0:25d The transverse pitch 13-6aị ð13-6bÞ For rivets, rivet holes, and strap thick Refer to Tables 13-9, 13-10, and Fig 13-2 TABLE 13-9 Rivet hole diameters TABLE 13-10 Rivet hole diameters and strap thickness Diameter of rivet, mm 12 14 16 18 20 22 24 27 30 33 36 39 42 48 Rivet hole diameters, mm (min) 13 15 17 19 21 23 25 28.5 31.5 34.5 37.5 41.0 44 50 Plate thickness, h, mm 6.25 7.20 8.00 8.75 9.50 10.30 11.10 12.00 12.50 13.50 Minimum strap thickness, hc mm Hole Plate diameter, thickness, d, mm h, mm Minimum strap thickness, hc mm 14.25 6.25 22.25 8.00 11.10 12.50 15.90 25.00 28.50 31.75 83.10 12.50 19.00 22.25 25.00 17.50 20.50 9.50 11.10 15.90 19.00 24.00 Hole diameter, d, mm 27.0 30.15 FIGURE 13-2 Quadruple-riveted double-strap butt joint 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 33.30 36.50 39.70 RIVETED JOINTS 13.6 CHAPTER THIRTEEN Particular Minimum transverse pitch as per ASME Boiler Code Formula pt ẳ 1:75d if p d 13-7aị p if > d pt ẳ 1:75d ỵ 0:001 p À dÞ SI ð13-8aÞ USCS ð13-8bÞ where pt , p, and d in m pt ẳ 1:75d ỵ 0:1 p À dÞ if p >4 d where pt , d, and p in in For transverse pitches Haven and Swett formula for permissible pitches along the caulking edge of the outside cover plate Refer to Table 13-8 sffiffiffiffiffiffi hc pc d ẳ 14 Pf CM 13-9aị where pc , d, hc in cm, and Pf in kgf/cm2 sffiffiffiffiffiffi hc pc À d ¼ 21:38 USCS Pf where pc , d, hc in in, and Pf in psi sffiffiffiffiffiffi hc pc À d ¼ 77:8 Pf SI ð13-9bÞ ð13-9cÞ where pc , d, hc in m, and Pf in N/m2 Diagonal pitch, pd , is calculated from the relation 2ð pd À dÞ ! ð p À dÞ ð13-10Þ MARGIN Margin for longitudinal seams of all pressure vessels and girth seams of power boiler having unsupported heads m ẳ 1:5d to 1:75d 13-11aị Margin for girth seams of power boilers having supported heads and all unfired pressure vessels m ! 1:25d ð13-11bÞ COVER PLATES The thickness of cover plate hc ẳ 0:6h ỵ 0:0025 if h 0:038 m SI ð13-12aÞ USCS ð13-12bÞ SI ð13-12cÞ USCS 13-12dị where hc and h in m hc ẳ 0:6h ỵ 0:1 if h 1:5 in where hc and h in in hc ¼ 0:67h if h > 0:038 m where hc and h in m hc ¼ 0:67h if h > 1:5 in where hc and h in in 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 RIVETED JOINTS RIVETED JOINTS 13.7 TABLE 13-11 Rivet groups under eccentric loading value of coefficient K K¼ Kẳ } lp ỵ p2 ỵ p2 n K ẳ s 2 6l ỵ1 n þ 1Þpt n sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Alcn Alcn þ þ1 2I 2I n K ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ln 1ịpt lp ỵ 2 ỵ 2 2 p ỵ n 1ịpt p ỵ n 1ịpt n K ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lðn À 1ịpt lp ỵ 2 ỵ p þ ðn À 1Þp2 p2 þ ðn 1ịp2 t t n K ẳ s lðn À 1Þpt lp þ þ p2 þ p2 þ ðn2 À 1Þp2 p2 ỵ p2 ỵ n2 1ịp2 t t 1 3 Key: n ¼ total number of rivets in a column F ¼ permissible load, acting with lever arm, l, kN (lbf) F ¼ permissible load on one rivet, kN (lbf) K ¼ F=F , coefficient Source: K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook ( fps Units), Engineering College Cooperative Society, Bangalore, India, 1962; K Lingaiah and B R Narayana Iyengar, Machine Design Data Handbook, Vol I (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1983; and K Lingaiah, Machine Design Data Handbook, Vol II (SI and Customary Metric Units), Suma Publishers, Bangalore, India, 1986 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 RIVETED JOINTS 13.8 CHAPTER THIRTEEN Particular Formula Thickness of the cover plate according to Indian Boiler Code Thickness of single-butt cover plate h1 ẳ 1:125h 13-13ị Thickness of single-butt cover plate omitting alternate rivet in the over rows h2 ¼ 1:25h Thickness of double-butt cover plates of equal width hc ¼ h1 ¼ h2 ¼ 0:625h Thickness of double-butt cover plates of equal width omitting alternate rivet in the outer rows hc ¼ h1 ¼ h2 ¼ 0:625h pÀd p À 2d 13-14ị 13-15ị 13-16ị h1 ẳ 0:625h for narrow strap 13-17aị h2 ¼ 0:750h for wide strap Thickness of the double-butt cover plates of unequal width pÀd p À 2d ð13-17bÞ For thickness of cover plates Refer to Table 13-10 The width of upper cover plate (narrow strap) b1 ¼ 4m þ 2pt1 ð13-18Þ The width of lower cover plate (wide strap) b2 ẳ b1 ỵ 2pt2 ỵ 4m 13-19ị The tensile strength of the solid plate F ẳ ph 13-20ị The tensile strength of the perforated strip along the outer gauge line F ẳ p dịh 13-21ị STRENGTH ANALYSIS OF TYPICAL RIVETED JOINT (Fig 13-2) The general expression for the resistance to shear of all the rivets in one pitch length F ẳ 2i2 ỵ i1 ị The general expression for the resistance to crushing of the rivets Fc ẳ i2 h ỵ i1 h2 ịdc The resistance against failure of the plate through the second row and simultaneous shearing of the rivets in the first row F 1 ẳ p 2dịh ỵ d ð13-22Þ ð13-23Þ d ð13-24Þ The resistance against failure of the plate through the second row and simultaneous crushing of the rivets in the rst row Fc1 ỵ p 2dịh ỵ dhc 13-25ị The resistance against shearing of the rivets in the outer row and simultaneous crushing of the rivets in the two inner rows F c ẳ d ỵ idhc 13-26ị 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 RIVETED JOINTS RIVETED JOINTS Particular 13.9 Formula EFFICIENCY OF THE RIVETED JOINT The efficiency of plate The efficiency of rivet in general case For efficiency of joints The diameter of the rivet in general case ¼ pÀd p d i1 ỵ 2i2 ị 4ph h i ỵ i c h ẳ h2 c ỵ i2 ỵ i1 h 13-27ị ẳ 13-28ị Refer to Table 13-3 dẳ 4hi2 ỵ i1 h2 c i1 ỵ 2i2 ị 13-29ị Note: for lap joint i2 ¼ for butt joint i1 ¼ ð2i2 þ i1 Þd þd 4h The pitch in general case p¼ For pitch of joint Refer to Table 13-7 THE LENGTH OF THE SHANK OF RIVET (Fig 13-3) 13-30ị L ẳ h ỵ h1 ỵ h2 ỵ 1:5 to 1:7ịD 13-31aị L ẳ h ỵ hc ỵ 1:5 to 1:7ÞD ð13-31bÞ for butt joint with single cover plate L ẳ 2h ỵ 1:5 to 1:7ịD for lap joint where D ¼ diameter of rivet FIGURE 13-3 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 ð13-31cÞ DESIGN OF SHAFTS 14.2 CHAPTER FOURTEEN SUFFIXES a b d e h m sc t u y max f amplitude bending design elastic limit hollow mean static strength (su or sy ), solid twisting ultimate yield strength maximum minimum endurance Other factors in performance or in special aspect are included from time to time in this chapter and, being applicable in their immediate context, are not given at this stage Note: and with the initial subscript s designates strength properties of material used in the design which will be used and observed throughout this handbook In some books on machine design and in this Machine Design Data Handbook the ratios of design stresses sd =fd and sd = fd ; and design stresses yd , yd , fd , and fd have been used instead of sy =sf , sy = sf ; and yield strengths sy , sy and fatigue strengths, sf , sf in the design equations for shafts [Eqs (14-1) to (14-65)] This has to be taken into consideration in the design of shafts while using Eqs (14-1) to (14-65) Particular Formula SOLID SHAFTS (1) Stationary shafts with static loads The diameter of shaft subjected to simple torsion The diameter of shaft subjected to simple bending D¼ D¼ 16Mt yd 32Mb yd 1=3 ð14-1Þ 1=3 ð14-2Þ The diameter of shaft subjected to combined torsion and bending: (a) According to maximum normal stress theory (b) According to maximum shear stress theory Dẳ 16 fMb ỵ Mb ỵ Mt2 ị1=2 g yd Dẳ 16 Mb ỵ Mt2 ị1=2 yd ( (c) According to maximum shear energy theory D¼ 16 yd 1=3 ð14-3Þ 1=3 1=3 ) 1=2 Mb ỵ Mt 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 ð14-4Þ ð14-5Þ DESIGN OF SHAFTS 14.3 DESIGN OF SHAFTS Particular Formula The diameter of shaft subjected to axial load, bending, and torsion:1À3 " (a) According to maximum normal theory D¼ ( 16 yd FD Mb ỵ ( Mb ỵ ỵ FD ) )# 1=2 1=3 ỵ Mt2 (b) According to maximum shear stress theory ( FD Mb ỵ 16 Dẳ4 yd (c) According to maximum shear energy theory 14-6ị ( FD Mb ỵ 16 D¼4 yd ) 31=3 1=2 þM ð14-7Þ t þ Mt2 ) 31=3 1=2 ð14-8Þ (2) Rotating shafts with dynamic loads, taking dynamic effect indirectly into consideration1À3 For empirical shafting formulas The diameter of shaft subjected to simple torsion The diameter of shaft subjected to simple bending Refer to Table 14-1 1=3 16 Kt Mt ị Dẳ yd Dẳ 32 K M ị yd b b 14-9ị 1=3 ð14-10Þ The diameter of shaft subjected to combined bending and torsion (a) According to maximum normal stress theory Dẳ 16 ẵK M ỵ fKb Mb ị2 ỵ Kt Mt Þ2 g1=2 yd b b 1=3 ð14-11Þ (b) According to maximum shear stress theory (c) According to maximum shear energy theory D¼ D¼ 16 fðKb Mb ị2 ỵ Kt Mt ị2 g1=2 yd 1=3 16 fKb Mb ị2 ỵ Kt Mt ị2 g1=2 yd ð14-12Þ 1=3 ð14-13Þ The diameter of shaft subjected to axial load, bending, and torsion ( (a) According to maximum normal stress theory Dẳ 16 yd Kb Mb ỵ " ỵ Kb Mb ỵ FD FD 1=3 #1=2 = ỵ Kt Mt ị2 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 ; ð14-14Þ DESIGN OF SHAFTS 14.4 CHAPTER FOURTEEN Particular (b) According to maximum shear stress theory Formula " 16 Dẳ yd FD Kb Mb ỵ #1=3 1=2 ỵ Kt Mt ị 14-15ị (c) According to maximum shear energy theory " 16 D¼ yd FD Kb Mb ỵ ỵ ðKt Mt Þ2 #1=3 1=2 ð14-16Þ The diameter of shaft based on torsional rigidity D¼ 584Mt L G 1=4 ð14-17Þ where Kb and Kt are taken from Table 14-2 (3) Rotating shafts and fluctuating loads, taking fatigue effect directly into consideration1À3 The diameter of shaft subjected to fluctuating torsion The diameter of shaft subjected to fluctuating bending ( D¼ ( D¼ 16 32 Mtm Mta ỵ yd fd ) 1=3 Mbm Mba þ yd fd ð14-18Þ ) 1=3 ð14-19Þ The diameter of shaft subjected to combined fluctuating torsion and bending: (a) According to maximum normal stress theory (b) According to maximum shear stress theory (c) According to maximum shear energy theory 1=3 16 02 02 1=2 fMbm ỵ Mbm ỵ Mtm ị g Dẳ yd Dẳ 16 02 02 Mbm ỵ Mtm ị1=2 yd ( Dẳ 16 yd 14-20ị 1=3 1=3 ) 02 1=2 02 Mbm ỵ Mtm 14-21ị 14-22ị where sd M fd ba 14-22aị sd M fd ta 14-22bị Mbm ẳ Mbm ỵ Mtm ẳ Mtm ỵ 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 DESIGN OF SHAFTS DESIGN OF SHAFTS Particular 14.5 Formula The diameter of shaft subjected to combined fluctuating axial load, bending, and torsion (a) According to maximum normal stress theory ( D¼ 16 yd ỵ (b) According to maximum shear stress theory " 16 D¼ yd " Mbm Mbm þ F D þ m Fm D Mbm 02 ỵ Mtm F D ỵ m ỵ 1=3 #) 1=2 ð14-23Þ 02 Mtm #1=3 1=2 ð14-24Þ " (c) According to maximum shear energy theory D¼ 16 yd Mbm ỵ Fm D 02 ỵ Mtm #1=3 1=2 14-25ị Mbm Mtm where and have the same meaning as in Eqs (14-22a) and (14-22b) and Fm ẳ Fm ỵ sd Fa 14-25aị fd HOLLOW SHAFTS (1) Stationary shafts with static loads The outside diameter of shaft subjected to simple torsion Do ¼ The outside diameter of shaft subjected to simple bending Do ¼ 16Mt yd ð1 À K Þ 32Mb yd ð1 À K Þ 1=3 ð14-26Þ 1=3 ð14-27Þ The diameter of shaft subjected to combined torsion and bending (a) According to maximum normal stress theory Do ¼ 1=3 16 fMb ỵ Mb ỵ Mt2 ị1=2 g yd ð1 À K Þ ð14-28Þ (b) According to maximum shear stress theory Do ¼ ( (c) According to maximum shear energy theory Do ẳ 16 Mb ỵ Mt2 Þ1=2 yd ð1 À K Þ 1=3 1=3 ) 16 1=2 Mb þ Mt yd ð1 À K Þ 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 ð14-29Þ ð14-30Þ DESIGN OF SHAFTS 14.6 CHAPTER FOURTEEN Particular Formula The outside diameter of shaft subjected to axial load, bending, and torsion (a) According to maximum normal stress theory ( Do ẳ 16 yd K ị þ FDo ð1 þ K Þ Mb þ FDo ð1 þ K Þ Mb þ 1=3 1=2 !) ỵ Mt2 14-31ị (b) According to maximum shear stress theory ( Do ¼ 16 yd ð1 À K Þ #1=2 ) 1=3 " FDo Mb ỵ ỵ K ị þ Mt2 (c) According to maximum shear energy theory ( 16 Do ẳ yd K ị #1=2 ) 1=3 ỵ Mt 14-32ị " FDo 2 ỵ K ị Mb ỵ ð14-33Þ (2) Rotating shafts with dynamic loads, taking dynamic effect indirectly into consideration1À3 The outside diameter of shaft subjected to simple torsion Do ¼ The outside diameter of shaft subjected to simple bending Do ¼ 16 Kt M t yd ð1 À K Þ 1=3 32 Kb Mb yd ð1 À K Þ ð14-34Þ 1=3 ð14-35Þ The outside diameter of shaft subjected to combined bending and torsion (a) According to maximum normal stress theory (b) According to maximum shear stress theory Do ¼ 16 ẵKb Mb ỵ fKb Mb ị2 yd K ị 1=3 ỵ Kt Mt ị2 g1=2 14-36ị Do ẳ 1=3 16 fKb Mb ị2 þ ðKt Mt Þ2 g1=2 yd ð1 À K Þ ð14-37Þ 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 DESIGN OF SHAFTS DESIGN OF SHAFTS Particular (c) According to maximum shear energy theory 14.7 Formula " 1=2 #1=3 16 2 Do ¼ Kb Mb ị ỵ Kt Mt ị yd À K Þ ð14-38Þ The outside diameter of shaft subjected to axial load, bending and torsion (a) According to maximum normal stress theory " ( 16 FDo ỵ K ị Do ẳ Kb Mb ỵ yd ð1 À K Þ FDo ỵ Kb Mb ỵ ỵ K ị )#1=3 1=2 ỵ Kt Mt ị2 (b) According to maximum shear stress theory " Do ẳ 14-39ị ( 16 FDo ỵ K ị Kb Mb þ yd ð1 À K Þ ) #1=3 1=2 ỵ Kt Mt ị2 (c) According to maximum shear energy theory The outside diameter of shaft based on torsional rigidity 14-40ị ( " 16 FDo ỵ K ị Do ẳ Kb Mb ỵ yd K ị # )1=3 1=2 ỵ Kt Mt ị2 14-41ị Do ẳ 584Mt L ð1 À K ÞG 1=4 ð14-42Þ (3) Rotating shaft with fluctuating loads, taking fatigue effect directly into consideration The outside diameter of shaft subjected to fluctuating torsion The outside diameter of shaft subjected to fluctuating bending " 16 Do ẳ 1 K ị " 32 Do ¼ ð1 À K Þ Mtm Mta þ yd fd #1=3 Mbm Mba þ yd fd ð14-43Þ #1=3 Please note: If the axial load does not produce column action, the constant need not be used to multiply the term [FDo (1 ỵ K )/8] throughout this chapter 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 ð14-44Þ DESIGN OF SHAFTS 14.8 CHAPTER FOURTEEN Particular Formula The outside diameter of shaft subjected to combined fluctuating torsion and bending (a) According to maximum normal stress theory Do ẳ 1=3 16 02 02 fMbm ỵ Mbm ỵ Mtm ị1=2 g yd 1K ị 14-45ị (b) According to maximum shear stress theory Do ¼ 16 02 02 Mbm ỵ Mtm ị1=2 yd K Þ 1=3 ð14-46Þ " (c) According to maximum shear energy theory #1=3 16 02 1=2 02 Do ẳ Mbm ỵ Mtm yd K Þ ð14-47Þ 0 where Mbm , Mtm have the same meaning as in Eqs (14-22a) and (14-22b) The outside diameter of shaft subjected to combined fluctuating axial load, bending, and torsion (a) According to maximum normal stress theory " Do ¼ 16 yd ð1 À K ị ỵ Mbm ỵ ( Mbm ỵ Fm Do ỵ K ị Fm Do ỵ K ị 2 02 þ Mtm 1=2 )#1=3 ð14-48Þ ( (b) According to maximum shear stress theory Do ¼ " 16 yd ð1 K ị Mbm ỵ Fm Do ỵ K ị 2 #1=2 !)1=3 ỵ ( (c) According to maximum shear energy theory Do ¼ 02 Mtm ð14-49Þ 16 yd ð1 À K Þ 02 ỵ Mtm " Fm Do ỵ K2 ị Mbm ỵ #1=2 )1=3 ð14-50Þ 0 where Mbm , Mtm , and Fm have the same meaning as in Eqs (14-22a), (14-22b), and (14-25a) 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 DESIGN OF SHAFTS DESIGN OF SHAFTS Particular 14.9 Formula COMPARISON BETWEEN DIAMETERS OF SOLID AND HOLLOW SHAFTS OF SAME LENGTH For equal strength in bending, torsion, and/or combined bending and torsion, the diameter (a) When materials of both shafts are same D ¼ Do ð1 À K Þ1=3 (b) When materials of shafts are dierent D ẳ Do 14-51ị eh K Þ1=3 es ð14-52Þ For torsional rigidity (a) When torsional rigidities are equal (b) When torsional rigidities are different D ¼ Do ð1 À K Þ1=4 D ¼ Do Gh ð1 À K Þ Gs ð14-53Þ 1=4 14-54ị For equal weight D ẳ Do K ị1=2 14-55ị w 1=2 D ẳ Do ð1 À K Þ h ws ð14-56Þ (a) For same material and machining cost for both shafts D ¼ Do ð1 À K Þ1=2 ð14-57Þ (b) For no machining cost for both shafts but with different material cost w k 1=2 D ¼ Do ð1 À K Þ h h w s ks ð14-58Þ (c) When machining costs are different and material cost negligible D¼ (a) When material of both shafts is same (b) When materials of both shafts are different For equal cost (d) When machining and material costs are different D¼ ch cs 1=2 1=2