Strength and Stiffness of Engineering Systems
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1 Numerical SOLUTIONS to Odd-Numbered roblems* of Chapters 1 10 of: Strength and Stiffness of Engineering Systems by: Frederick A. Leckie Dominic J. Dal Bello Solutions Version: Solutions by: D. J. Dal Bello domdalbello@yahoo.com *Note: Chapters 3 and 8 include several multi-part problems both odd- and even-numbered for which each part repeats the question of art (a), but with different numerical values. For such problems (including even-numbered problems), the numerical solutions are given for arts (a), (c), (e), etc. The problems are: 3.20, 8.1, 8.3, 8.4, and 8.20.
2 .2 Copyright 2009 by Dominic J. Dal Bello Santa Barbara, California All rights reserved. No part of this booklet (numerical solutions) may be reproduced in any form, printed, electronic or otherwise, without the written permission of the author (Dominic J. Dal Bello). These numerical solutions may not be distributed commercially or otherwise sold as a product, or as part of a larger work. However, students, instructors and others using the text, Strength and Stiffness of Engineering Systems (Leckie and Dal Bello, Springer, 2009), for academic and learning purposes may print a copy of these numerical solutions for use in learning the material.
3 .3 Chapter Ultimate, roof, Working kips 1.5 W min = 4710 lb, 3-ton (6000-lb) crane ; 64% lb; 12.8 kips, 11.2 kips, 3200 lb; 18.4 kips; 38.4 kips 1.11 two 8 kn loads, R Ax, R Ay, R By Chapter CD =2.5kips; BC =1.5kips; EF = 3.0 kips; BE =2.50kips 2.3 T 2 :T 1 =2: lb; 2400 lb-ft 2.7 M max = 41.4 kn m for a =4m or 6m lb; 180 lb/ft, 90 lb/ft; 3240 lb-ft 2.11 AB = 3.23 kips; T = kipin. CCW; N = kips up Chapter %, 31.3 ksi; in 2 ; ksi 3.5 lot; S p ~ 225 Ma; E = 200 Ga; S y = 255 Ma; S u = 450 Ma Ma; 1.88; mm; mm in.; in kn m/m 3 ; N m ,000 cycles; 40%; 40.8 kn Ma; 131 Ma ksi kn; 94.3 kn, 3.62 kn m, kn m 3.20 Strain ( 10 3 ) art x y z xy (a) (c) (e) (g) Strain ( 10 6 ) art x y z (a) (b) (c) Chapter Ma; ; 686 m Ma, 40.7 Ma; 1.16 mm, mm = 582 m ksi, 3.00 ksi; in., in.; in. down, in. down psi; in kips left; 8.57 kips left; in. right 4.11 FL 2AE ; plot; FL 3AE ; 34.1 m x Ma, x in meters m right, 582 m up Ma, 56.6 Ma; 4.2; mm, mm; mm right, 0 mm Ma, 80 Ma; 3.0; mm, 2.29 mm; 4.58 mm right 4.21 AB =56.6Ma, BC = 56.6 Ma, BD = 0 Ma; 4.2; AB = mm; BC = mm; BD = 0 mm; mm right 4.23 s = 4.80 ksi; c = ksi; in.; 8330 ksi in 2 ; in Ma ksi, 8.57 ksi; 317 F Ma (matrix), 727 Ma Ma; 5.00 Ma; mm minimum; 18.3 mm lb; ; Ma; 62.8 kn/m kips ave ; 2.67a (n =2.67 radii) N m, 22.9 N m; 8.75 mm down; 2.29 MN/m N m, 22.6 N m; 3.06 mm right; 6.53 MN/m FL AE down; EA 3L ; m = 9K E L ; minimize E or maximize E
4 U AB = U BC = 3EAv 2 16L, U BD = EAv 2 2 3L ; in.; 1195 kips/in N m; mm down lb-in.; in. down 4.57 AD = kn, BD = kn, CD = 28.0 kn; kn right, kn down; kn Chapter in 4 ; 3.98 ksi; 5.0; m 4 ; m 4 ; 1.2% mm; mm, 45.8 mm; 62.0 mm, 6.20 mm; 6.0%, 63.2% savings ksi; ksi, ksi; kip-in.; 2.74 in., 3.0 in.; 63% savings N m; ksi; 1.12 (approx.) in kn m; 73.0 Ma; 3.34 ; 128 mm N m; 32.3 mm; 41.9 mm; 41.9 mm Chapter (a) 0 < x <4 m: V(x) = 6.0 kn, M(x) =6x kn m; 4< x <10 m: V(x) =+4.0 kn; M(x) = 4x+40 kn m; (b) V max =6kN; M max =24kN m at x =4m in.; 2.18% 6.5 S12 31, W14 26; W kip-in.; 6.00 ksi; in Ma ksi at x =21ft psi Ma; w o M = L x ; 29.4 ksi 6L mm Ma; 2.47 mm difference 6.23 v = x ; beam hits structure 6EI 3Lx L=a+b; answers will vary based on length variables (a, b, L) used v 1 v 2 If a < b: 6.31 R A = 3wL 8, R B = 5wL 8, M B = wl 2 8 CW ; w 2x ; 48EI 4 3Lx 3 + L 3 x 6.33 R A = R B = wl 2 up, 6.35 v 1 v 2 bx = L 6EIL 2 b 2 x 2 ; 0 x a = ax 6EIL 3 3aLx 2 + 3a 2 L bl 2 + b 3 x a 3 L ; b L 2 b 2 3/ 2 max = at x = 9 3EIL v 1 v 2 M A = wl 2 12 CCW, M B = wl 2 12 CW ; kn; 59.2 Ma ka; 533 ka; 416 ka a x a+ b = L = x 6EI 3 3Lx 2 ; 0 x L L = x 2L ; L x 3L 2 6EI L b 2 3 = x 6EI 3 3aa + s x ; 0 x a = ax 6EI 2 3a 2a+ s x + a 3 ; a x a+ s v = vl 2 = wl 4 192EI max = wl 4 384EI = 12 Ma at m = 0.3 Ma at m max = 2.00 mm down
5 Ma; 2.39 Ma in in 3 ; 21.3 in 3 ; 6.28 in AD/ mm; S ; 68.6% or 69.4 lb/ft savings lb, deflection 6.55 Silicon Carbide (SiC), but it is brittle; Wood is next best 6.57 m wl = -- 1/ 2 L ; 96 E st = 64.7 ; al = 22.2 Chapter Ma; 2.60 Ma 7.3 at A: = psi; B: = 28.6 psi 7.5 at A: y = 5728 psi; B: y = 5731 psi; C: y = 1.6 Ma, xy = 32. psi Note: a = 1.6 psi (at A, B, C); b = psi (+ at A, atb) 7.7 at A: Ma; B: 1.33 Ma; C: 9.33 Ma; D: 1.33 Ma Ma; 1.90 Ma 7.11 R/ d/(3L 2 ) 7.15 b =82.8Ma (+atb); T =6.90Ma (+ata, B; at C); V =0.28Ma ( ata, C) 7.17 a = /A =67.9psi ( ata, B, C); b,x = FeD/(2I) = 1207 psi (+ at B); b,y = hd/(2i) = 2897 psi (+ at A, atc); T = FhD/(2J) =966psi(+atA, B; atc); V =4F/(3A) = 60.4 psi ( at A, C) 7.19 v wx L3 ; 24EI 1 2L 1 x 2 x 3 wl 2 x = AE max = in. down at x =121.25in. wx 7.21 v = x, 48EI 3 + 3Lx 2 L 3 w v = x ; 48EI 3 + 9Lx 2 L 3 3wL 5wL A y = R = , B ; 8 y = wl M 2 B = CW 8 Chapter (a) Ma, Ma, 32.1 Ma (c) Ma, 55.7 Ma, 3.3 Ma (e) Ma, 76.5 Ma, Ma (g) Ma, 57.7 Ma, 19.6 Ma (i) Ma, Ma, 89.7 Ma 8.3 (a) 17.1 ksi@ 19.3, 2.1 ksi@70.7 (c) 16.4 ksi@12.8, 11.4 ksi@102.8 (e) 19.0 ksi@ 33.7, 6.0 ksi@56.3 (g) 11.7 ksi@ 15.5, 11.7 ksi@ (a) 9.6 ksi@115.7, 9.6 ksi@25.7, 7.5 ksi (c) 13.9 ksi@ 32.3, 13.9 ksi@57.8, 2.5 ksi (e) 6.5 ksi@101.3, 6.5 ksi@11.3, 12.5 ksi (g) 11.7 ksi@119.5, 11.7 ksi@29.5, 0ksi Ma, 26.0 Ma; 30.0 Ma 8.7 = 2.60 Ma, = Ma; I, II = Ma, 90.4 Ma; max = Ma Ma; 0Ma; 0Ma, 42.8Ma t. B; 1.14 t. B (a) 7.7 ksi, 7.70 ksi, ksi (b) 15.4 ksi, 15.4 ksi, 4.74 ksi (c) 15.0 ksi, 15.8 ksi, 0 ksi (d) 16.8 ksi, 17.8 ksi, 0 ksi (normal stresses along y-axis, at points not on x- or z- axes) (e) 3.85 ksi, 8.61 ksi, 7.9 ksi, 8.88 ksi, corresponding to (a)-(d) 8.13 (a) ksi, ksi, 1.61 ksi (c) 3.55 ksi, 1.45 ksi, ksi
6 .6 (e) Ma, 59.0 Ma, 17.8 Ma (g) 70.9 Ma, 30.9 Ma, 94.9 Ma 8.14 Note: s defines plane where max >0. (a) 15 ksi@0, 10 ksi@90 ; max = 2.5 ksi, ave = 12.5 ksi, s = 45 (c) 16.4 ksi@12.8, 11.4 ksi@102.8 ; max =13.9ksi, ave = 2.5 ksi, s = 32.2 (e) Ma@31.7, 55.3 Ma@121.7 ; max =44.7Ma, ave =100Ma, s = 13.3 (g) Ma@ 10.9, 87.7 Ma@79.1 ; max = Ma, ave =20Ma, s = (a) , , (c) , , (a) ; , 7.08 Ma Ga, (a) , , x = A, y = --, 3 A + 2 B + 2 C 2 xy = B C (a) 5.0 ksi; 7.5 ksi; I-III (out-of-plane) (c) 2.5 ksi; 3.5 ksi; I-III (out-of-plane) Chapter Tresca or von Mises; Maximum Normal Stress 9.3 I = 3.1 ksi, does not fail Try: T =120lb and M =160lb-ft ( I = 14.4 ksi) U D = E I II + III 2 I II + II III + III I 1 U D = E I + II III I II + II III + III I I II 2 + II III 2 + III I 2 o = Chapter in. = 11.7 ft kn, AE and CD in. = 8.84 ft 10.7 (a) 708 kips (y-axis buckling); 760 kips (z-axis buckling); buckling about weak-axis; (b) 1390 kips (y-axis buckling); 760 kips (z-axis buckling); buckling about strong-axis K T ; plot Lsin + ecos m m = ; 2 cr -- L E R 2 1st: Ceramic, 2nd: Composite 12 /
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