ANSWERS - M. Vable F Mechanics of Materials: Answers to...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: M. Vable F Mechanics of Materials: Answers to Selected problems APPENDIX F ANSWERS TO SELECTED PROBLEMS CHAPTER 1 1.1 σ = 1019 psi (T) 1.3 W max = 125.6 lb 1.6 d min = 1.5 mm 1.8 σ = 2.57 MPa ( T ) 1.12 (a) σ col = 232.8 MPa (C); (b) σ b = 20 MPa ( C ) 1.15 (a) σ col = 156 MPa (C); (b) σ b = 8.33 MPa (C) 1.19 σ b = 3 MPa (C) 1.25 P max = 10.8 kips 1.28 P = τπ ( d o + d i ) t 1.31 W max = 125.6 lb 1.44 σ AA = 3.286 ksi (T); τ AA = 1.53 ksi 1.51 σ = 11.9 psi (T); V = 19 lbs 1.52 (a) σ HA = 38 MPa (C); σ HB = 16 MPa (T); σ HG = 22 MPa (C); σ HC = 16 MPa (C) (b) ( τ H ) max = 53.76 MPa 1.56 σ BD = 100 MPa (T); τ max = 259 MPa 1.62 P max = 70.6 kN 1.67 P max = 5684 lb 1.69 L = 10.4 in 1.71 (a) d CG = 30 mm; d CD = 27 mm; d CB = 23 mm (b) d C = 22 mm; sequence: CB, CG, CD 1.74 τ = 9947 P a 1.75 P = 3 a L τ 1.77 τ = 3.18 MPa 1.79 τ = 226.3 MPa 1.82 (a) τ = 8.5 psi; (b) T = 6.7 in-lb Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm y 85 75 85 75 85 100 x 1.86 27 x 18 18 18 18 1.84 100 January, 2010 y 75 75 27 12 12 25 1.93 18 18 85 25 12 12 569 M. Vable F Mechanics of Materials: Answers to Selected problems y z 175 225 1.94 20 200 200 125 125 225 100 25 1.97 x 15 25 x r 1.99 15 22 25 10 y 150 x 20 32 32 10 22 25 z r 18 1.101 25 20 25 10 18 CHAPTER 2 2.1 ε = 0.9294 cm/cm 2.4 ε = 0.321 in/in 2.7 u D – u A = 2.5 mm 2.8 ε A = 393.3 μ in/in; ε B = – 150 μ in/in 2.11 ε A = – 0.0125 in/in 2.13 ε A = – 0.0108 in/in 2.15 ε A = – 0.0108 in/in; ε F = – 0.003 in/in 2.19 δ B = 2 mm to the left 2.21 δ B = 2.5 mm to the left 2.22 ε A = – 416.7 μ mm/mm; ε F = 400 μ mm/mm 2.29 γ A = – 3000 μ rad 2.32 γ A = 5400 μ rad 2.34 γ A = 1296 μ rad 2.38 γ A = – 928 μ rad 2.48 γ A = – 1332 μ rad 2.51 (a) ε AP = 1174.7 μ mm/mm; (b) ε AP = 1174.6 μ mm/mm; (c) ε AP = 1174.6 μ mm/mm 2.54 δ AP = 0.0647 mm extension; δ BP = 0.2165 mm extension 2.57 δ AP = 0.0035 in contraction; δ BP = 0.0188 in contraction 2.61 ε BC = 4200 μ mm/mm; ε CF = – 2973 μ mm/mm; ε FE = – 2100 μ mm/mm Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 2.64 ε BC = 500 μ mm/mm; ε CG = – 833 μ mm/mm; ε GB = 0; ε CD = 667.5 μ mm/mm 2.68 ε xx = – 128 μ mm/mm; ε yy = – 666.7 μ mm/mm; γ xy = 3600 μ rad 2.71 ε xx = 1750 μ mm/mm; ε yy = – 1625 μ mm/mm; γ xy = – 1125 μ rad 2.74 ε xx ( 24 ) = 555 μ in/in 2.77 u ( 20 ) = 0.005 in 2.80 u ( 1250 ) = 1.516 mm 2.85 ε = 42.2 μ mm/mm 2.87 ε = 47% January, 2010 570 M. Vable F Mechanics of Materials: Answers to Selected problems CHAPTER 3 3.1 (a) σ u lt = 510 MPa; (b) σ frac = 480 MPa (c) E = 150 ( 7.5 ) GPa; (d) σ prop = 300 MPa (e) σ yield = 300 MPa (f) E t = 2.5 GPa; (g) E s = 6.5 GPa 3.2 (a) P = 23.56 kN; (b) P = 35.34 kN 3.3 δ = 3.25 mm 3.4 ε total = 0.065; ε elas = 0.0028; ε plas = 0.0622 3.5 P = 36.9 kN 3.12 (a) E = 300 GPa; (b) σ prop = 1022 MPa; (c) σ yield = 1060 MPa; (d) E t = 1.72 GPa; (e) E s = 11.2 GPa; (f ) ε plas = 0.1203 3.16 E = 25,000 ksi; ν = 0.2 3.18 G = 4000 ksi 3.25 P = 70.7 kN; Δ d = – 0.008mm 3.27 0.0327% 3.31 U = 125 in.-lbs 3.36 (a) 300 kN-m/m3; (b) 21,960 kN-m/m3; (c) 5,340 kN-m/m3; (d) 57,623 kN-m/m3. 3.38 (a) 1734 kN-m/m3; (b) 157 MN-m/m3; (c) 18 MN-m/m3; (d) 264 MN-m/m3 3.41 F = 22.1 kN 3.44 F = 16.7 kN 3.45 F = 0.795 lb; θ = 65.96 ° 3.50 P 1 = 0; P 2 = 2 kN --3.53 h = 4 3 in; d = 1 1 in 8 8 3.59 d min = 23 mm 3.65 N = 60 kips; M z = 30 in-kips 3.66 (a) a = 1062.1 MPa; b = 4493.3 MPa; c = – 12993.1 MPa; (b) E T = 1.621 GPa 3.68 P = 70.1 lbs 3.74 (a) σ zz = 0; ε xx = – 3661 μ ; ε yy = 2589 μ ; γ xy = 5357 μ rad; ε zz = 357 μ ; (b) ε zz = 0; σ zz = 25 MPa (C); ε xx = – 3571 μ ; ε yy = 2679 μ ; γ xy = 5357 μ rad 3.78 (a) σ zz = 0; (b) ε zz = 0; ε xx = – 0.06875; σ zz = 12.50 psi ( T ) ; ε yy = 0.0875; ε xx = – 0.0703; ε zz = – 0.00625; ε yy = 0.08594; γ xy = 0.125 γ xy = 0.125 3.81 σ zz = 0; σ yy = 40.9 ksi (C); σ xx = 36.26 ksi (C); ε zz = 771 μ in/in; τ xy = – 5.77 ksi 3.83 σ zz = 0; σ yy = 60 MPa (T); σ xx = 60 MPa (C); ε zz = 0; τ xy = 18 MPa 3.86 σ xx = 16 ksi (C); σ yy = 4 ksi (C) 3.92 a = 50.06 mm; b = 50.1725 mm 3.111 ε xx = – 936 μ ; ε yy = – 2180 μ ; γ xy = – 5333 μ Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 3.115 σ xx = 19.07 ksi (C); σ yy = 0.99 ksi (C); τ xy = 0.6 ksi 3.119 K = 2.4 3.121 σ max = 45.3 ksi 3.127 θ = 0.34 ° 3.130 σ xx = 47.4 ksi (C); σ yy = 52.02 ksi (C); ε zz = 1254 μ ; τ xy = – 5.77 ksi 3.137 (a) T = 33.33 hours; (b) T = 133.33 hours; (c) T = ∞ 3.139 n = 400,000 cycles 3.142 (a) E 1 = 15,000 ksi; (b) E 2 = 64.15 ksi; (c) n = 0.1694; E = 56.2 ksi January, 2010 571 M. Vable Mechanics of Materials: Answers to Selected problems CHAPTER 4 4.2 F 1 = 108.5 kN; F 2 = 45.2 kN; F 3 = 94.3 kN 4.4 F = 11.25 kips 4.9 u D – u A = – 0.175 in 4.13 (a) u D – u A = – 0.0234 in; (b) σ max = 3.75 ksi (C) 4.18 u B – u A = 0.126 mm 4.20 u = 0.4621 P ⁄ EK 4.21 (a) u C – u A = 0.034 in; (b) σ max = 33.95 ksi (T) 4.24 u B = – γ L ⁄ 2 E 2 4.27 δ = 0.045 in 4.31 F max = 4886 lb 5 -4.33 d p = 0.5 in; a b = 1 1 in; b s = 1 ----- in 8 16 4.41 (a) Δ u = 0.60 mm; (b) σ max = 62.2 MPa (T) 4.44 a = 224.40 ; b = – 23.60 ; c = – 0.40; u A = 0.017 in to the left 4.46 F = 46.9 kips 4.50 δ P = 0.23 mm 4.51 σ A = 8.0 ksi (C); δ B = 0.0021 in 4.61 δ P = 0.24 mm; σ A = 118 MPa (C) 4.65 (a) δ p = 0.0265 in; (b) Δ d s = 0.00074 in; Δ d al = – 0.00066 in 4.67 σ A = 22.5 ksi (C); σ B = 17.2 ksi (T) 4.70 F max = 555 kN 4.74 F max = 17.2 kN 4.77 w max = 9.4 MPa 4.83 P max = 106.7 kips 2 4.85 A BC = 1.1 in ; d = 1.3 in 4.87 F max = 148.6 kN 4.89 F max = 181.9 kN 4.90 σ A = 5.2 ksi (T); σ B = 3.5 ksi (T) 4.94 σ xx = 0; u ( L ⁄ 2 ) = α T L L ⁄ 24 4.95 σ xx = E α T L ⁄ 3 ( C ) ; u ( L ⁄ 2 ) = – α T L L ⁄ 8 4.99 σ A = 25.70 ksi (T) Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 4.100 σ θθ = 10 MPa (T); τ r = 40 MPa 4.104 t min = 0.05 in; d noz = 0.206 in 4.106 p max = 500 psi; d riv = 0.85 in CHAPTER 5 5.1 γ D = 2400 μ rad 5.2 T = 64.8 in-kips 5.7 φ 1 = 0.0400 rad; φ 2 = 0.0243 rad; φ 3 = 0.0957 rad 5.11 T = – 495.2 in-kips January, 2010 F 572 M. Vable Mechanics of Materials: Answers to Selected problems 5.13 T = 10.9 kN-m 5.23 (a) ( τ xy ) A > 0; (b) ( τ xy ) B < 0 5.26 (a) ( τ xy ) A > 0; (b) ( τ xy ) B < 0 5.29 φ D – φ A = 0.00711 rads CW 5.32 φ D = 0.0163 rads CW; γ max = – 1094 μ ; ( τ x θ ) E = – 4.4 ksi 5.35 φ A = 1676 μ rads CW; ( τ x θ ) E = 15.1 MPa 5.38 (a) φ B = 0.1819 ( T ext L ⁄ Gr ) CCW; (b) τ max = 0.275 T ext ⁄ r 4 3 5.40 φ A = ( qL ⁄ GJ ) CW 2 5.41 T = 69.2 in-kips 5.43 ( r i ) max = 24 mm 5.47 d min = 21 mm; τ AB = 52.5 MPa -5.57 R o = 2 3 in 8 5.59 Δ φ = 0.085 rad; τ max = 172 MPa 5.60 Δ φ = 0.088 rad 5.63 φ B = 0.0516 rads ccw; τ max = 25.8 ksi 5.66 φ C = 0.006 rads CCW; T = 200.5 in-kips 5.67 φ B = 0.0438 rads CW TL T 5.71 φ B = 5.659 --------- CCW; τ max = 2.83 ---4 3 Gd d 5.74 T max = 32 kN-m; φ B = 0.048 rads CCW; τ max = 130.4 MPa 5.75 d min = 89 mm; φ B = 0.0487 rads CCW; τ max = 116 MPa 5.76 d min = 108 mm; φ B = 0.025 rads CCW; τ max = 58.62 MPa 5.95 τ max = 10.8 MPa 5.101 τ max = 21.65 MPa CHAPTER 6 6.1 ψ = 2.41 ° 6.3 ε 1 = 182 μ m/m; ε 3 = – 109.1 μ m/m; ε 4 = – 654 μ m/m; ε 6 = 393 μ m/m 6.6 P = 1454 N ; M z = 123.6 N-m 6.7 P 1 = 14.58 kN; M 1 = 130.3 N-m; P 2 = 9.88 kN; M 2 = 64.0 N-m 6.12 M z = 9.13 in-kips Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 6.14 M z = – 2134 kN-m 6.25 σ T = 3.73 ksi ( T ) ; σ C = 6.93 ksi ( C ) 6.29 σ A = 1224 psi (C); σ B = 735 psi (C); σ D = 1714 ksi (T) 6.35 σ A is (C); σ B is (T) 6.38 σ A is (T); σ B is (C) 6.42 (a) σ 3.0 = 2.96 ksi (C); (b) σ max = 6.93 ksi (C) or (T) 6.45 σ A = 4.17 ksi (C); σ max = 12.5 ksi (C) or (T) 6.49 σ A = 6.68 ksi (C); σ max = 28.9 ksi (T) 6.51 ε A = – 1500 μ 6.53 ε A = – 327 μ January, 2010 F 573 M. Vable F Mechanics of Materials: Answers to Selected problems 2 6.60 (a) V y = 3 ( 72 – x ) kips; (b) M z = 1.5 ( 72 – x ) in-kips 2 3 1 1 6.62 (a) V y = [ 108 – ----- x ] kips; (b) M z = [ 5184 – 108 x + -------- x ] in-kips 48 144 2 0 ≤ x < L ; M z = ( wLx – wL ) in-kips 6.64 V y = – wL kips V y = [ w ( x – L ) – wL ] kips 6.68 V y = ( 76 – 12 x ) kN L < x ≤ 2L; M z = [ wLx – 2 2 0 ≤ x < 3 m; M z = 3 x kN-m 6.69 V y = – 6 x kN 2 2 – L ) – wL ] in-kips 5 m < x < 9 m; M z = ( 6 x – 76 x + 154 ) kN-m 9 m < x < 12 m; M z = ( 20 x – 240 ) kN-m V y = – 20 kN L < x ≤ 2L 5 m < x < 9 m; 9 m < x < 12 m 0 ≤ x < 3 m; 3 m < x < 5 m; M z = ( 8 x – 7 ) kN-m V y = – 8 kN 0 ≤ x < L; w --- ( x 2 3m<x<5m 6.74 ( V y ) max = ± 7.5 kN; ( M z ) max = 5.625 kN-m 6.79 ( V y ) max = 36 kN; ( M z ) max = – 86.67 kN-m 6.83 ( V y ) max = 9 kN; ( M z ) max = – 23.625 kN-m 6.86 ( V y ) max = ± 6 kips; ( M z ) max = – 16 in-kips 6.92 w max = 154.3 lb ⁄ in -6.99 a i = 11 7 in 8 6.100 r = 3.75 mm 6.102 P = 165.7 N 6.107 Point A: negative τxz; Point B: positive τxy; Point C: negative τxz; Point D: positive τxz 6.112 Point A: positive τxy; Point B: negative τxz; Point C: zero; Point D: positive τxy 6.117 σ max = 348.4 MPa; τ max = 6.84 MPa; ( σ xx ) A = 48 MPa (C); ( τ xy ) A = – 3.2 MPa 6.120 V AB = 614.4 lbs; V BC = 921.6 lbs 6.124 M ext = 8333.33 in-lbs 6.125 P max = 202 N; Δ s = 16 cm 36.137 R O = 2 ----- in 16 6.138 σ max = 9185 psi; τ max = 295 psi CHAPTER 7 3 2 3 4 7.2 v ( x ) = – [ wx ⁄ ( 24 EI ) ] ( x – 2 Lx + L ) ; v ( L ⁄ 2 ) = – [ 5 wL ⁄ ( 384 EI ) ] 2 2 2 4 7.4 v ( x ) = – [ wx ⁄ ( 24 EI ) ] ( x – 4 Lx + 6 L ) ; v ( L ) = – [ wL ⁄ ( 8 EI ) ] 3 7.9 v A = PL ⁄ ( 3 EI ) 2 2 Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 0≤x≤L ⎧ [ wLx ⁄ ( 9 EI ) ] ( x – 5 L ) 7.17 ( x ) = ⎨ ; 2 2 ⎩ [ wLx ⁄ ( 9 EI ) ] ( x – 5 L ) – [ wL ⁄ ( 6 EI ) ] ( x – L ) 3 L ≤ x ≤ 3 2 2 ⎧ ( wLx ⁄ 48 EI ) ( 2 x – 7 L ) 7.19 v ( x ) = ⎨ ⎩ ( w ⁄ EI ) ( 2 Lx 3 – 7 L 3 x ) ⁄ 48 – ( w ⁄ 24 EI ) ( x – L ) 4 7.25 h ( x ) = 4 v ( L ) = – ( 4 wL ⁄ 9 EI ) 0≤x≤L 4 v ( L ) = – [ 5 wL ⁄ ( 48 EI ) ] ; L ≤ x ≤ 2L 6 Px ⁄ ( b σ ) ; v max = – 8 b σ L ⁄ ( 27 PE ) 33 2 7.28 σ max = 128 PL ⁄ ( 27 π d 0 ) ; v max = – 8 PL ⁄ ( 3 E π d 0 ) 3 3 4 7.32 R A = 16.2 kips up; M A = 10.8 in-kips CCW 3 3 2 3 ⎧ ( P ⁄ 12 EI ) [ 2 ( x – 2 L ) – 5 ( x – L ) – 9 L x + 11 L ] 7.34 R A = 5 P /2; v ( x ) = ⎨ ⎩ ( P ⁄ 12 EI ) [ 2 ( x – 2 L ) 3 – 9 L 2 x + 11 L 3 ] January, 2010 0≤x≤L L ≤ x ≤ 2L 574 M. Vable F Mechanics of Materials: Answers to Selected problems 3 dv wL 61 wL 2 7.36 ----- ( L ) = ----------- ; R A = ------------- up; M A = 11 wL ⁄ 120 CW 80 EI dx 120 4 7.57 v A = – 41 wL ⁄ 24 EI 3 7.61 v A = – P L ⁄ 96 EI 4 2 7.64 v A = – w L ⁄ 136 EI ; R C = 11 wL ⁄ 17 ; M C = 5 wL ⁄ 34 3 3 3 7.67 v ( x ) = ( w ⁄ 18 EI ) [ 2 Lx – 3 L 〈 x – L〉 – 10 L x ] ; 4 4 3 4 v ( L ) = – 4 wL ⁄ 9 EI 2 2 3 4 4 7.71 v ( x ) = ( w ⁄ 24 EI ) [ x – 〈 x – L〉 – 4 L 〈 x – 2 L〉 – 12 L 〈 x – 2 L〉 – 40 L x + 71 L ] ; v ( 2 L ) = ( wL ) ⁄ ( 4 EI ) 2 3 3 7.72 v ( x ) = ( P ⁄ 12 EI ) [ 3 Lx – 3 x + 5 〈 x – L〉 ] ; R A = 5 P ⁄ 2 3 7.76 v ′ ( x A ) = – ( wL ⁄ 6 EI ) ; 4 v ( x A ) = – ( wL ⁄ 8 EI ) CHAPTER 8 8.1 σ nn is ( C ) ; τ nt is positive 8.6 σ nn is ( C ) ; τnt can’t say 8.12 σ nn = 8.66 ksi ( C ) τ nt = 5.0 ksi 8.15 Compression 8.19 σ nn = 50 MPa ( C ) ; τ nt = 40 MPa 8.24 σ nn = 45.36 ksi ( C ) ; τ nt = 1.84 ksi 8.26 P max = 84.9 lb 8.33 (a) σ nn = σ ( T ) ; τ nt = 0; (b) σ nn = 0; τ nt = – σ 8.42 σ nn = 7 MPa ( T ) ; τ nt = – 59.7 MPa; σ 1 = 75.2 MPa (T); σ 2 = 45.2 MPa ( C ) ; σ 3 = 0; θ 1 = 69.2 ° ; τ max = 60.2 MPa 8.44 σ nn = 45.4 ksi ( C ) ; τ nt = 1.84 ksi; σ 1 = 15.4 ksi ( T ) ; σ 2 = 45.4 ksi ( C ) ; σ 3 = 0; θ 1 = 40.3 ° ; τ max = 30.4 ksi 8.47 σ nn = 0.63 ksi ( C ) ; τ nt = – 7.06 ksi; σ 1 = 0.62 ksi ( T ) ; σ 2 = 40.62 ksi ( C ) ; σ 3 = 12 ksi ( C ) θ 1 = 128 ° ; τ max = 20.62 ksi 8.50 σ 1 = 67.9 MPa ( T ) ; σ 2 = 207.9 MPa ( C ) ; σ 3 = 0; θ 1 = 78 ° ; τ max = 137.9 MPa 8.54 σ xx = 7.54 ksi ( C ) ; σ yy = 9.46 ksi ( C ) ; τ xy = 1.15 ksi 8.60 σ nn = 16.5 ksi ( C ) ; τ nt = – 9.55 ksi Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 8.71 Pmax = 30.6 kN CHAPTER 9 9.3 ε nn = – 234.7 μ ; φ = 196.96 μ rad CW 9.6 ε nn = 150 μ ; ε tt = 450 μ ; γ nt = – 519.6 μ 9.8 ε nn = – 70.2 μ ; ε tt = – 529.8 μ ; γ nt = 385.67 μ 9.13 ε nn = – 295.4 μ ; ε tt = 295.4 μ ; γ nt = – 104.2 μ 9.16 Sectors 8 and 2 or Sectors 4 and 6 9.31 ε 1 = 659 μ ; ε 2 = – 459 μ ; ε 3 = 0; γ max = 1118 μ ; θ 1 = 103.3 ° ; ε nn = 643.7 μ ; ε tt = – 443.7 μ ; γ nt = – 259.8 μ 9.37 ε 1 = 1246.5 μ ; ε 2 = – 196.5 μ ; ε 3 = 0; γ max = 1443 μ ; January, 2010 575 M. Vable Mechanics of Materials: Answers to Selected problems ε xx = 1027 μ ; ε yy = 23 μ ; γ xy = – 1037 μ 9.42 ε xx = – 466 μ ; ε yy = 1266 μ ; γ xy = – 1000 μ 9.44 ε 1 = 767.9 μ ; ε 2 = – 125 μ ; ε 3 = – 214.3 μ ; γ max = 982.2 μ ; θ 1 = – 26.57 9.46 ε 1 = 681.4 μ ; ε 2 = – 604.3 μ ; ε 3 = 0; γ max = 642.9 μ ; θ 1 = 26.57 9.49 ( θ 1 ) strain = 103.3 ; o o o ( θ 1 ) stress = 98.8 ° 9.54 ε a = 33.49 μ ; ε b = 400 μ ; ε c = 166.5 μ 9.58 ε a = 687.5 μ ; ε b = – 406.3 μ ; ε c = – 656.9 μ 9.62 ε 1 = 685.9 μ ; ε 2 = – 185.9 μ ; ε 3 = – 166.7; γ max = 871.8 μ ; θ 1 = 48.3 ° 9.64 ε = 392.9 μ 9.68 ε = 716.7 μ 9.74 ε = – 112.5 μ CHAPTER 10 10.1 σ nn = 4.6 ksi (C); τ nt = – 16.4 ksi 10.4 P = 60.76 kN 10.6 ε a = 1696 μ ; ε b = – 1176 μ 10.12 ε a = 1333 μ ; ε b = – 666.66 μ 10.24 ( σ xx ) A = 0; ( σ xx ) B = – σ bend -y = 85.39 MPa ( C ) ; ( τ xz ) A = τ tor + τ bend -y =42.89 MPa ; ( τ xy ) B = – τ tor = – 25.62 MPa 10.27 ( σ xx ) A = 0; ( σ xx ) B = – σ bend -y = 222 MPa ( C ) ; ( τ xz ) A = τ bend -y = 17.27 MPa ; ( τ xy ) B = 0 10.30 ( σ max ) A = 102.7 MPa ( T ) or ( C ) ; ( σ max ) B = 137.33 MPa ( C ) ; ( τ max ) A = 51.35 MPa ; ( τ max ) B = 91.79 MPa ; 10.35 ( σ xx ) A = 23.1 ksi (C); ( τ xy ) A = – 7.2 ksi 10.36 σ nn = 8219 psi (C); τ nt = 13180 psi 10.44 P max = 4.3 kN 10.48 w = 791.2 N/m 10.53 σ BD = σ CE = 5.13 psi (C); σ BC = 10 psi (T); σ AB = 167.4 psi (C) 10.55 W max = 67 lb 10.62 R o = 2.405 in 10.65 (a) P max = 5 kN; (b) P max = 5.75 kN Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 10.66 P max = 9.5 kips 10.69 K = 1.22 CHAPTER 11 11.2 P cr = 5/4 kL 11.6 P cr = 153.3 lb 11.15 L ⁄ r = 72.7; P cr = 215.4 kip; σ cr = 3.36 ksi (C) 11.21 K = 1.106 11.25 K = 3.633 January, 2010 F 576 M. Vable Mechanics of Materials: Answers to Selected problems 11.40 L max = 42 in 11.43 w max = 12 kN/m; K BD = 2.3 11.55 σ max = 2.68 ksi (C); v max = 0.0458 in 11.59 L max = 2.09 m 11.63 P max = 39.45 kip Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 11.64 P cr = 17.0 kip January, 2010 F 577 M. Vable F Mechanics of Materials: Answers to Selected problems FORMULA SHEET σ av = N ⁄ A ΔF σ ij = lim ⎛ --------j⎞ Δ A i → 0⎝ Δ A i⎠ τ av = V ⁄ A δ ε = ----- L –L Lo f o ε = --------------- u –u xB – xA Lo γ xy = τ xy ⁄ G ε xx = [ σ xx – ν ( σ yy + σ zz ) ] ⁄ E EG = ------------------2( 1 + ν ) N ( x2 – x1 ) u 2 – u 1 = ------------------------EA NL δ = ------- 2 J My I zz V y Qz τ xs = – ------------ Mz I yy Vz Qy τ xs = – ------------ z σ xx = – --------- I zz t y σ xx = – --------- I yy t x2 d Mz ---------- = V dx dV =p dx V y = –V A Tρ τ x θ = ------ GJ dv M z = EI zz ------2 dx N σ xx = ---- EA T ( x2 – x1 ) φ 2 – φ 1 = ------------------------ dφ T= -----dx GJ 2 γ xy tan 2 θ p = -----------------------( ε xx – ε yy ) τ nt = – σ xx cos θ sin θ + σ yy sin θ cos θ + τ xy ( cos θ – sin θ ) 2 2 2 Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm January, 2010 1- 3 I = ----- ab 12 14 I = -- π r 4 σ1 – σ2 σ2 – σ3 σ3 – σ1 τ max = max ⎛ -----------------, -----------------, -----------------⎞ ⎝2 2 2⎠ 2 ( ε xx + ε yy ) ε xx – ε yy 2 γ xy 2 ε 1, 2 = ------------------------ ± ⎛ -------------------⎞ + ⎛ ------⎞ ⎝⎠ ⎝ ⎠ 2 2 γ nt = – 2 ε xx sin θ cos θ + 2 ε yy sin θ cos θ + γ xy ( cos θ – sin θ ) π EI P cr = ----------2 L 4r η C = ----3π x1 ( σ xx + σ yy ) σ xx – σ yy 2 2 σ 1, 2 = -------------------------- ± ⎛ ---------------------⎞ + τ xy ⎝ ⎠ 2 2 ε nn = ε xx cos θ + ε yy sin θ + γ xy sin θ cos θ 2 M 2 = M 1 + ∫ V dx x1 2 2 τ xy tan 2 θ p = -------------------------( σ xx – σ yy ) x2 V 2 = V 1 + ∫ p dx σ nn = σ xx cos θ + σ yy sin θ + 2 τ xy sin θ cos θ 2 dx νε zz = – ⎛ -----------⎞ ( ε xx + ε yy ) ⎝ 1 – ν⎠ E σ xx = [ ε xx + νε yy ] -----------------2 (1 – ν ) Ndu = -----dx EA du ( x ) ε xx = ------------- γ = π⁄2–α B A ε = ----------------- 14 J = -- π r 2 2 γ max 2 ε1 – ε2 ε2 – ε3 ε3 – ε1 --------- = max ⎛ --------------- , --------------- , ---------------⎞ ⎝2 2 2 2⎠ 578 ...
View Full Document

Ask a homework question - tutors are online