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1 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the shaft. The bearings at and exert only vertical reactions on the shaft. 250 mm 800 mm 24 kn 6 2. Draw the shear and moment diagrams for the simply supported beam. M 2 kn m 4 kn 2 m 2 m 2 m 329

2 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The engine crane is used to support the engine, which has a weight of 1200 lb. Draw the shear and moment diagrams of the boom C when it is in the horizontal position shown. 3 ft 5 ft C a+ M = 0; 4 5 F (3) (8) = 0; F = 4000 lb 4 ft + c F y = 0; - y (4000) = 0; y = 2000 lb ; + F x = 0; x (4000) = 0; x = 2400 lb *6 4. Draw the shear and moment diagrams for the cantilever beam. 2 kn/m 2 m 6 kn m The free-body diagram of the beam s right segment sectioned through an arbitrary point shown in Fig. a will be used to write the shear and moment equations of the beam. + c F y = 0; V - 2(2 - x) = 0 V = {4-2x} kn (1) a + M = 0; -M - 2(2 - x)c 1 (2) 2 (2 - x) d - 6 = 0 M = {-x2 + 4x - 10}kN # m The shear and moment diagrams shown in Figs. b and c are plotted using Eqs. (1) and (2), respectively. The value of the shear and moment at x = 0 is evaluated using Eqs. (1) and (2). V x = 0 = 4-2(0) = 4 kn M x= 0 = C (0) - 10D = -10kN # m 330

3 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the beam. 10 kn 8 kn 15 kn m 2 m 3 m 6 6. Draw the shear and moment diagrams for the overhang beam. 8 kn/m 4 m 2 m C 6 7. Draw the shear and moment diagrams for the compound beam which is pin connected at. 6 kip 8 kip C 4 ft 6 ft 4 ft 4 ft 331

4 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 332 *6 8. Draw the shear and moment diagrams for the simply supported beam. 150 lb/ft 300 lb ft 12 ft The free-body diagram of the beam s left segment sectioned through an arbitrary point shown in Fig. b will be used to write the shear and moment equations. The intensity of the triangular distributed load at the point of sectioning is w = 150a x 12 b = 12.5x Referring to Fig. b, + c F y = 0; (1) 2 (12.5x)(x) - V = 0 V = { x2 }lb a + M = 0; M + 1 (2) 2 (12.5x)(x)a x 3 b - 275x = 0 M = {275x x3 }lb # ft The shear and moment diagrams shown in Figs. c and d are plotted using Eqs. (1) and (2), respectively. The location where the shear is equal to zero can be obtained by setting V = 0 in Eq. (1). 0 = x 2 x = ft The value of the moment at x = ft (V = 0) is evaluated using Eq. (2). M x= ft = 275(6.633) (6.633) 3 = 1216 lb # ft 332

5 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the beam. Hint: The 20-kip load must be replaced by equivalent loadings at point C on the axis of the beam. 15 kip 1 ft 20 kip C 4 ft 4 ft 4 ft Members C and D of the counter chair are rigidly connected at and the smooth collar at D is allowed to move freely along the vertical slot. Draw the shear and moment diagrams for member C. Equations of Equilibrium: Referring to the free-body diagram of the frame shown in Fig. a, P 150 lb + c F y = 0; y = 0 y = 150 lb a + M = 0; N D (1.5) - 150(3) = 0 N D = 300 lb 1.5 ft 1.5 ft D 1.5 ft C Shear and Moment Diagram: The couple moment acting on due to N D is M = 300(1.5) = 450 lb # ft. The loading acting on member C is shown in Fig. b and the shear and moment diagrams are shown in Figs. c and d. 333

6 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The overhanging beam has been fabricated with a projected arm D on it. Draw the shear and moment diagrams for the beam C if it supports a load of 800 lb. Hint: The loading in the supporting strut DE must be replaced by equivalent loads at point on the axis of the beam. 800 lb D E 5 ft Support Reactions: 2 ft C a + M C = 0; 800(10) F DE(4) F DE(2) = 0 6 ft 4 ft F DE = 2000 lb + c F y = 0; (2000) - C y = 0 C y = 400 lb : + F x = 0; -C x (2000) = 0 C x = 1600 lb Shear and Moment Diagram: *6 12. reinforced concrete pier is used to support the stringers for a bridge deck. Draw the shear and moment diagrams for the pier when it is subjected to the stringer loads shown. ssume the columns at and exert only vertical reactions on the pier. 60 kn 35 kn 35 kn 35 kn 60 kn 1 m 1 m 1.5 m 1.5 m 1 m 1 m 334

7 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the compound beam. t is supported by a smooth plate at which slides within the groove and so it cannot support a vertical force, although it can support a moment and axial load. P P Support Reactions: From the FD of segment D C D a+ M C = 0; y (a) - P(a) = 0 y = P + c F y = 0; C y - P - P = 0 C y = 2P a a a a : + F x = 0; x = 0 From the FD of segment a+ M = 0; P(2a) - P(a) - M = 0 M = Pa + c F y = 0; P - P = 0 (equilibrium is statisfied!) The industrial robot is held in the stationary position shown. Draw the shear and moment diagrams of the arm C if it is pin connected at and connected to a hydraulic cylinder (two-force member) D. ssume the arm and grip have a uniform weight of 1.5 lb in. and support the load of 40 lb at C. 4 in. 10 in. 50 in. C 120 D 335

8 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 336 *6 16. Draw the shear and moment diagrams for the shaft and determine the shear and moment throughout the shaft as a function of x. The bearings at and exert only vertical reactions on the shaft. 800 lb 500 lb x 3 ft 2 ft 0.5 ft 0.5 ft For 0 6 x 6 3 ft + c F y = V = 0 V = 220 lb a + M N = 0. M - 220x = 0 M = (220x) lb ft For 3 ft 6 x 6 5 ft + c F y = 0; V = 0 V = -580 lb a + M N = 0; M + 800(x - 3) - 220x = 0 M = {-580x } lb ft For 5 ft 6 x 6 ft + c F y = 0; V = 0 V = 500 lb a + M N = 0; -M - 500(5.5 - x) = 0 M = (500x ) lb ft 336

9 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the cantilevered beam. 300 lb 200 lb/ft 6 ft The free-body diagram of the beam s left segment sectioned through an arbitrary point shown in Fig. b will be used to write the shear and moment equations. The intensity of the triangular distributed load at the point of sectioning is w = 200a x 6 b = 33.33x Referring to Fig. b, + c F y = 0; (33.33x)(x) - V = 0 V = { x2 } lb (1) a + M = 0; M + 1 (2) 2 (33.33x)(x)a x 3 b + 300x = 0 M = {-300x x3 } lb # ft The shear and moment diagrams shown in Figs. c and d are plotted using Eqs. (1) and (2), respectively. 337

10 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the beam, and determine the shear and moment throughout the beam as functions of x. 2 kip/ft 10 kip 8 kip 40 kip ft Support Reactions: s shown on FD. Shear and Moment Function: For 0 x 6 6 ft: + c F y = 0; x - V = 0 x 6 ft 4 ft V = {30.0-2x} kip a + M N = 0; M xa x 2 b x = 0 M = {-x x - 216} kip # ft For 6 ft 6 x 10 ft: + c F y = 0; V - 8 = 0 V = 8.00 kip a + M N = 0; -M - 8(10 - x) - 40 = 0 M = {8.00x - 120} kip # ft Draw the shear and moment diagrams for the beam. 2 kip/ft 30 kip ft 5 ft 5 ft 5 ft 338

11 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 339 *6 20. Draw the shear and moment diagrams for the simply supported beam. 10 kn 10 kn/m 3 m 3 m Since the area under the curved shear diagram can not be computed directly, the value of the moment at x = 3 m will be computed using the method of sections. y referring to the free-body diagram shown in Fig. b, a + M = 0; M x= 3 m (10)(3)(1) - 20(3) = 0 M x= 3m = 45 kn # m 339

12 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The beam is subjected to the uniform distributed load shown. Draw the shear and moment diagrams for the beam. 2 kn/m Equations of Equilibrium: Referring to the free-body diagram of the beam shown in Fig. a, a + M = 0; F C a 3 b(2) - 2(3)(1.5) = m F C = 7.5 kn + c F y = 0; y + 7.5a 3 5 b - 2(3) = 0 C 2 m 1 m y = 1.5 kn Shear and Moment Diagram: The vertical component of F C is F C y = 7.5a 3 5 b = 4.5 kn. The shear and moment diagrams are shown in Figs. c and d. 340

13 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the overhang beam. 4 kn/m 3 m 3 m Since the loading is discontinuous at support, the shear and moment equations must be written for regions 0 x 6 3 m and 3 m 6 x 6 m of the beam. The free-body diagram of the beam s segment sectioned through an arbitrary point within these two regions is shown in Figs. b and c. Region 0 x 6 3 m, Fig. b + c F y = 0; a 4 3 xb(x) - V = 0 V = e x2-4 f kn (1) a + M = 0; M + 1 (2) 2 a 4 3 xb(x)a x 3 b + 4x = 0 M = e x3-4x f kn # m Region 3 m 6 x 6 m, Fig. c + c F y = 0; V - 4(6 - x) = 0 V = {24-4x} kn (3) a + M = 0; -M - 4(6 - x)c 1 (4) 2 (6 - x) d = 0 M = {-2(6 - x)2 }kn # m The shear diagram shown in Fig. d is plotted using Eqs. (1) and (3). The value of shear just to the left and just to the right of the support is evaluated using Eqs. (1) and (3), respectively. V x= 3 m - = (32 ) - 4 = -10 kn V x=3 m + = 24-4(3) = 12 kn The moment diagram shown in Fig. e is plotted using Eqs. (2) and (4). The value of the moment at support is evaluated using either Eq. (2) or Eq. (4). or M x=3 m = (33 ) - 4(3) = -18 kn # m M x= 3 m = -2(6-3) 2 = -18 kn # m 341

14 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the beam. t is supported by a smooth plate at which slides within the groove and so it cannot support a vertical force, although it can support a moment and axial load. w L *6 24. Determine the placement distance a of the roller support so that the largest absolute value of the moment is a minimum. Draw the shear and moment diagrams for this condition. w + c F y = 0; wl - wl2 2a - wx = 0 a L x = L - L2 2a a + M = 0; M max (+) + wxa x wl2 b - awl - 2 2a bx = 0 Substitute x = L - L2 ; 2a M max (+) = awl - wl2 L2 bal - 2a 2a b - w 2 al - L2 2a b 2 = w 2 al - L2 2a b 2 (L - a) M = 0; M max (-) - w(l - a) 2 = 0 M max (-) = w(l - a)2 2 To get absolute minimum moment, M max (+) = M max (-) w L2 (L - 2 2a )2 = w (L - a)2 2 L - L2 2a = L - a a = L

15 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The beam is subjected to the uniformly distributed moment m ( moment>length). Draw the shear and moment diagrams for the beam. Support Reactions: s shown on FD. Shear and Moment Function: + c F y = 0; V = 0 a+ M N = 0; M + mx - ml = 0 M = m(l - x) Shear and Moment Diagram: L m Draw the shear and moment diagrams for the beam. w 0 + c F y = 0; w 0L a w 0x L b(x) = 0 x = L a + M N = 0; M a w 0x L b(x)a x 3 b - w 0L ax - L 4 3 b = 0 L 3 2L 3 Substitute x = L, M = w 0 L 2 343

16 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 344 *6 28. Draw the shear and moment diagrams for the beam. w 0 L 3 L 3 L 3 Support Reactions: s shown on FD. Shear and Moment Diagram: Shear and moment at x = L>3 can be determined using the method of sections. + c F y = 0; w 0 L 3 - w 0 L 6 - V = 0 V = w 0 L 6 a + M N = 0; M + w 0 L 6 a L 9 b - w 0 L 3 M = 5w 0 L 2 54 a L 3 b = 0 344

17 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the beam. 5 kn/m 5 kn/m 4.5 m 4.5 m From FD(a) + c F y = 0; x 2 = 0 x = m a From FD(b) + M N = 0; M + (0.5556) a b (4.108) = 0 3 M = kn # m a + M N = 0; M (1.5) (4.5) = 0 M = kn # m 345

18 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the compound beam. 150 lb/ft 150 lb/ft C 6 ft 3 ft Support Reactions: From the FD of segment a+ M = 0; 450(4) - y (6) = 0 y = lb + c F y = 0; y = 0 y = lb : + F x = 0; x = 0 From the FD of segment C a + M C = 0; 225(1) (3) - M C = 0 M C = lb # ft + c F y = 0; C y = 0 C y = lb : + F x = 0; C x = 0 Shear and Moment Diagram: The maximum positive moment occurs when V = 0. + c F y = 0; x 2 = 0 x = ft a + M N = 0; 150(3.464) a b - M 3 max = 0 M max = lb # ft 346

19 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the beam and determine the shear and moment in the beam as functions of x. w 0 Support Reactions: s shown on FD. Shear and Moment Functions: For 0 x 6 L>2 x L 2 L 2 + c F y = 0; 3w 0 L 4 -w 0 x - V = 0 V = w 0 (3L - 4x) 4 a + M N = 0; 7w 0 L2 24-3w 0 L x + w 4 0 xa x 2 b + M = 0 M = w x2 + 18Lx - 7L 2 ) For L>2 6 x L + c F y = 0; V c 2w 0 L (L - x) d(l - x) = 0 V = w 0 (L - x)2 L a + M N = 0; -M c 2w 0 L L - x (L - x) d(l - x)a b = 0 3 w 0 M = - (L - x)3 3L 347

20 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 348 *6 32. The smooth pin is supported by two leaves and and subjected to a compressive load of 0.4 kn m caused by bar C. Determine the intensity of the distributed load w 0 of the leaves on the pin and draw the shear and moment diagram for the pin. C w 0 w kn/m + c F y = 0; 2(w 0 )(20)a 1 2 b - 60(0.4) = 0 20 mm 60 mm 20 mm w 0 = 1.2 kn>m The ski supports the 180-lb weight of the man. f the snow loading on its bottom surface is trapezoidal as shown, determine the intensity w, and then draw the shear and moment diagrams for the ski. 3 ft 180 lb w w 1.5 ft 3 ft 1.5 ft Ski: + c F y = 0; 1 2 w(1.5) + 3w + 1 w(1.5) = 0 2 w = 40.0 lb>ft Segment: + c F y = 0; 30 - V = 0; V = 30.0 lb a+ M = 0; M - 30(0.5) = 0; M = 15.0 lb # ft 348

21 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the compound beam. 3 kn/m 5 kn C D 3 m 3 m 1.5 m 1.5 m Draw the shear and moment diagrams for the beam and determine the shear and moment as functions of x. 200 N/m 400 N/m Support Reactions: s shown on FD. Shear and Moment Functions: For 0 x 6 3 m: x 3 m 3 m + c F y = 0; V = 0 V = 200 N a + M N = 0; M x = 0 M = (200 x) N # m For 3 m 6 x 6 m: + c F y = 0; (x - 3) c 200 (x - 3) d(x - 3) - V = V = e - 3 x f N Set V = 0, x = m a + M N = 0; M c x - 3 (x - 3) d(x - 3)a b (x - 3)a x - 3 b - 200x = M = e - 9 x x f N # m Substitute x = 3.87 m, M = 691 N # m 349

22 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 350 *6 36. Draw the shear and moment diagrams for the overhang beam. 18 kn 6 kn 2 m 2 m 2 m M 10 kn m Draw the shear and moment diagrams for the beam. 50 kn/m 50 kn/m 4.5 m 4.5 m 350

23 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The dead-weight loading along the centerline of the airplane wing is shown. f the wing is fixed to the fuselage at, determine the reactions at, and then draw the shear and moment diagram for the wing lb 250 lb/ft 400 lb/ft Support Reactions: + c F y = 0; y = 0 8 ft 2 ft lb 3 ft y = kip a + M = 0; 1.00(7.667) + 3(5) - 15(3) (2.5) (1.667) + M = 0 M = kip # ft = 18.6 kip # ft : + F x = 0; x = 0 Shear and Moment Diagram: The compound beam consists of two segments that are pinned together at. Draw the shear and moment diagrams if it supports the distributed loading shown. w + c F y = 0; 2wL 27-1 w 2 L x2 = 0 C x = 4 27 L = L a + M = 0; M + 1 w 2 L (0.385L)2 a 1 2wL b(0.385l) (0.385L) = 0 2/3 L 1/3 L M = wl 2 351

24 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 352 *6 40. Draw the shear and moment diagrams for the simply supported beam. 10 kn 10 kn 15 kn m 2 m 2 m 2 m Draw the shear and moment diagrams for the compound beam. The three segments are connected by pins at and E. 3 kn 0.8 kn/m 3 kn E F 2 m 1 m C 1 m 2 m D 1 m 1 m 2 m 352

25 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the compound beam. 5 kn/m Support Reactions: C D From the FD of segment 2 m 1 m 1 m a+ M = 0; y (2) (1) = 0 y = 5.00 kn + c F y = 0; y = 0 y = 5.00 kn From the FD of segment D a + M C = 0; 5.00(1) (0) - D y (1) = 0 D y = 5.00 kn + c F y = 0; C y = 0 C y = 20.0 kn : + F x = 0; x = 0 From the FD of segment : + F x = 0; x = 0 Shear and Moment Diagram: Draw the shear and moment diagrams for the beam. The two segments are joined together at. 8 kip 3 kip/ft C 3 ft 5 ft 8 ft 353

26 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 354 *6 44. Draw the shear and moment diagrams for the beam. w x = 8 F R = 1 x 2 dx = kip 8 L x 3 dx = 6.0 ft w x kip/ft x 8 ft Draw the shear and moment diagrams for the beam. w L F R = d = wdx = w 0 L L 0 L 2 L0 L x 2 dx = w 0 L 3 w w 0 L 2 x 2 w 0 x = L xd d L = w 0 L L 2 x 3 dx L0 w 0 L 3 = 3L 4 L x + c F y = 0; w 0L 12 - w 0x 3 3L 2 = 0 x = a 1 4 b 1>3L = L a + M = 0; w 0L 12 (x) - w 0x 3 3L 2 a 1 4 xb - M = 0 M = w 0Lx 12 - w 0x 4 12L 2 Substitute x = 0.630L M = w 0 L 2 354

27 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Draw the shear and moment diagrams for the beam. w w 0 p w w0 sin x L L F R = d = w 0 sina p L L L xbdx = 2w 0 L p 0 x L 2 L member having the dimensions shown is used to resist an internal bending moment of M = 90 kn # m. Determine the maximum stress in the member if the moment is applied (a) about the z axis (as shown) (b) about the y axis. Sketch the stress distribution for each case. 200 mm y 150 mm The moment of inertia of the cross-section about z and y axes are M z = 1 12 (0.2)(0.153 ) = 56.25(10-6 ) m 4 z x y = 1 12 (0.15)(0.23 ) = 0.1(10-3 ) m 4 For the bending about z axis, c = m. s max = Mc = 90(103 ) (0.075) = 120(10 6 )Pa = 120 MPa z (10-6 ) For the bending about y axis, C = 0.1 m. s max = Mc = 90(103 ) (0.1) y 0.1 (10-3 = 90 (10 6 )Pa = 90 MPa ) The bending stress distribution for bending about z and y axes are shown in Fig. a and b respectively. 355

28 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 356 *6 48. Determine the moment M that will produce a maximum stress of 10 ksi on the cross section. 0.5 in. 0.5 in. 3 in. 3 in. C 0.5 in. 10 in. M D 0.5 in. Section Properties: y = y = 0.25(4)(0.5) + 2[2(3)(0.5)] + 5.5(10)(0.5) 4(0.5) + 2[(3)(0.5)] + 10(0.5) = 3.40 in. N = 1 12 (4) (0.5)( ) 2 = in 4 Maximum ending Stress: pplying the flexure formula s max = Mc + 2c 1 12 (0.5)(33 ) + 0.5(3)(3.40-2) 2 d (0.5) (10)( ) 2 M ( ) 10 = M = kip # in = 10.8 kip # ft 356

29 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Determine the maximum tensile and compressive bending stress in the beam if it is subjected to a moment of M = 4 kip # ft. 0.5 in. 0.5 in. 3 in. 3 in. C 0.5 in. 10 in. M D 0.5 in. Section Properties: y = y = 0.25(4)(0.5) + 2[2(3)(0.5)] + 5.5(10)(0.5) 4(0.5) + 2[(3)(0.5)] + 10(0.5) = 3.40 in. N = 1 12 (4) (0.5)( ) 2 = in 4 + 2c 1 12 (0.5)(33 ) + 0.5(3)(3.40-2) 2 d (0.5) (10)( ) 2 Maximum ending Stress: pplying the flexure formula s max = Mc (s t ) max = 4(103 )(12)( ) = psi = 3.72 ksi (s c ) max = 4(103 )(12)(3.40) = psi = 1.78 ksi 357

30 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The channel strut is used as a guide rail for a trolley. f the maximum moment in the strut is M = 30 N # m, determine the bending stress at points,, and C. 2.5(50)(5) + 7.5(34)(5) + 2[20(5)(20)] + 2[(32.5)(12)(5)] y = 50(5) + 34(5) + 2[5(20)] + 2[(12)(5)] 50 mm C 5 mm 5 mm 30 mm = mm = c 1 12 (50)(53 ) + 50(5)( ) 2 d + c 1 12 (34)(53 ) + 34(5)( ) 2 d + 2c 1 12 (5)(203 ) + 5(20)( ) 2 d + 2c 1 12 (12)(53 ) + 12(5)( ) 2 d = (10-6 ) m 4 5 mm 5 mm 5 mm 7 mm 10 mm7 mm s = 30( )(10-3 ) (10-6 ) s = 30( )(10-3 ) (10-6 ) = 6.81 MPa = 1.01 MPa The channel strut is used as a guide rail for a trolley. f the allowable bending stress for the material is s allow = 175 MPa, determine the maximum bending moment the strut will resist. s C = 30(13.24)(10-3 ) (10-6 ) y = y2 = c 1 12 (50)(53 ) + 50(5)( ) 2 d + c 1 12 (34)(53 ) + 34(5)( ) 2 d = (10-6 ) m 4 = 4.14 MPa = 2.5(50)(5) + 7.5(34)(5) + 2[20(5)(20)] + 2[(32.5)(12)(5)] 50(5) + 34(5) + 2[5(20)] + 2[(12)(5)] s = Mc ; 175(106 ) = M( )(10-3 ) (10-6 ) = mm + 2c 1 12 (5)(203 ) + 5(20)( ) 2 d + 2c 1 12 (12)(53 ) + 12(5)( ) 2 d 5 mm 50 mm C 5 mm 5 mm 5 mm 7 mm 10 mm7 mm 5 mm 30 mm M = 771 N # m 358

31 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 359 *6 52. The beam is subjected to a moment M. Determine the percentage of this moment that is resisted by the stresses acting on both the top and bottom boards, and, of the beam. 25 mm Section Property: D M = 1 12 (0.2) (0.15)0.153 = m 4 ending Stress: pplying the flexure formula 150 mm 25 mm 25 mm 150 mm 25 mm s = My s E = M(0.1) (10-6 ) = M s D = M(0.075) (10-6 ) = M Resultant Force and Moment: For board or F = M(0.025)(0.2) + 1 ( M M)(0.025)(0.2) 2 = M M = F( ) = 4.80M( ) = M s c a M b = (100%) = 84.6 % M Determine the moment M that should be applied to the beam in order to create a compressive stress at point D of s D = 30 MPa. lso sketch the stress distribution acting over the cross section and compute the maximum stress developed in the beam. 25 mm D M Section Property: = 1 12 (0.2) (0.15)0.153 = m 4 ending Stress: pplying the flexure formula 150 mm 25 mm 25 mm 150 mm 25 mm s = My = M(0.075) (10-6 ) M = N # m = 36.5 kn # m s max = Mc = 36458(0.1) = 40.0 MPa (10-6 ) 359

32 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The beam is made from three boards nailed together as shown. f the moment acting on the cross section is M = 600 N # m, determine the maximum bending stress in the beam. Sketch a three-dimensional view of the stress distribution acting over the cross section. (0.0125)(0.24)(0.025) + 2 (0.1)(0.15)(0.2) y = = m 0.24 (0.025) + 2 (0.15)(0.02) 200 mm 25 mm 150 mm 20 mm M 600 N m = 1 12 (0.24)( ) + (0.24)(0.025)( ) 20 mm + 2 a 1 12 b(0.02)(0.153 ) + 2(0.15)(0.02)( ) = (10-6 ) m 4 s max = s = Mc = 600 ( ) (10-6 ) = 2.06 MPa s C = My = 600 ( ) (10-6 ) = MPa The beam is made from three boards nailed together as shown. f the moment acting on the cross section is M = 600 N # m, determine the resultant force the bending stress produces on the top board. 25 mm 150 mm y = (0.0125)(0.24)(0.025) + 2 (0.15)(0.1)(0.02) 0.24 (0.025) + 2 (0.15)(0.02) = m 200 mm 20 mm M 600 N m = 1 12 (0.24)( ) + (0.24)(0.025)( ) 20 mm + 2 a 1 12 b(0.02)(0.153 ) + 2(0.15)(0.02)( ) = (10-6 ) m 4 s 1 = My = 600( ) = MPa (10-6 ) s b = My = 600( ) (10-6 ) = MPa F = 1 2 (0.025)( )(106 )(0.240) = 4.56 kn 360

33 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 361 *6 56. The aluminum strut has a cross-sectional area in the form of a cross. f it is subjected to the moment M = 8 kn # m, determine the bending stress acting at points and, and show the results acting on volume elements located at these points. 20 mm 100 mm 100 mm 20 mm 50 mm 50 mm M 8 kn m Section Property: = 1 12 (0.02) (0.1)0.023 = m 4 ending Stress: pplying the flexure formula s = My s = 8(103 )(0.11) = 49.4 MPa (C) (10-6 ) s = 8(103 )(0.01) = 4.49 MPa (T) (10-6 ) The aluminum strut has a cross-sectional area in the form of a cross. f it is subjected to the moment M = 8 kn # m, determine the maximum bending stress in the beam, and sketch a three-dimensional view of the stress distribution acting over the entire cross-sectional area. 20 mm 100 mm 100 mm 20 mm 50 mm 50 mm M 8 kn m Section Property: = 1 12 (0.02) (0.1)0.023 = m 4 ending Stress: pplying the flexure formula s and s = My max = Mc, s max = 8(103 )(0.11) = 49.4 MPa (10-6 ) s y = 0.01m = 8(103 )(0.01) = 4.49 MPa (10-6 ) 361

34 06 Solutions 46060_Part1 5/27/10 3:51 PM Page f the beam is subjected to an internal moment of M = 100 kip # ft, determine the maximum tensile and compressive bending stress in the beam. 3 in. 3 in. M 6 in. 2 in. 1.5 in. Section Properties: The neutral axis passes through centroid C of the cross section as shown in Fig. a. The location of C is Thus, the moment of inertia of the cross section about the neutral axis is = + d 2 y = y 4(8)(6) - 2cp1.5 2 d = 8(6) - p1.5 2 = 1 12 (6)a83 b + 6(8) pa1.54 b + pa1.5 2 b R = in 4 = in. Maximum ending Stress: The maximum compressive and tensile bending stress occurs at the top and bottom edges of the cross section. s max T = Mc = 100(12)(4.3454) = 23.8 ksi (T) s max C = My = 100(12)( ) = 20.0 ksi (C) 362

35 06 Solutions 46060_Part1 5/27/10 3:51 PM Page f the beam is made of material having an allowable tensile and compressive stress of (s allow ) t = 24 ksi and (s allow ) c = 22 ksi, respectively, determine the maximum allowable internal moment M that can be applied to the beam. 3 in. 3 in. M 6 in. 2 in. 1.5 in. Section Properties: The neutral axis passes through centroid C of the cross section as shown in Fig. a. The location of C is Thus, the moment of inertia of the cross section about the neutral axis is = + d 2 = 1 12 (6)83 + 6(8) p p R = in 4 llowable ending Stress: The maximum compressive and tensile bending stress occurs at the top and bottom edges of the cross section. For the top edge, (s allow ) c = My For the bottom edge, s max t = Mc y = y 4(8)(6) - 2cp1.5 2 d = 8(6) - p1.5 2 ; 22 = M( ) M(4.3454) ; 24 = = in. M = kip # ina 1 ft 12 in. b = kip # ft M = kip # ina 1 ft 12 in. b = 101 kip # ft (controls) 363

36 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 364 *6 60. The beam is constructed from four boards as shown. f it is subjected to a moment of M z = 16 kip # ft, determine the stress at points and. Sketch a three-dimensional view of the stress distribution. C y 1 in. 10 in. 1 in. 2[5(10)(1)] (16)(1) + 16(10)(1) y = 2(10)(1) + 16(1) + 10(1) = in. = 2c 1 12 (1)(103 ) + 1(10)( ) 2 d (16)(13 ) + 16(1)( ) (1)(103 ) + 1(10)( ) 2 = in 4 z 10 in. M z 16 kip ft 14 in. 1 in. 1 in. x s = Mc = 16(12)( ) = 2.05 ksi s = My = 16(12)(9.3043) = 1.63 ksi The beam is constructed from four boards as shown. f it is subjected to a moment of M z = 16 kip # ft, determine the resultant force the stress produces on the top board C. C y 1 in. 10 in. 1 in. 10 in. 2[5(10)(1)] (16)(1) + 16(10)(1) y = = in. 2(10)(1) + 16(1) + 10(1) = 2c 1 12 (1)(103 ) + (10)( ) 2 d (16)(13 ) + 16(1)( ) (1)(103 ) + 1(10)( ) 2 = in 4 z M z 16 kip ft 14 in. 1 in. 1 in. x s = Mc = 16(12)( ) = ksi s D = My = 16(12)( ) = ksi (F R ) C = 1 ( )(10)(1) = 11.8 kip 2 364

37 06 Solutions 46060_Part1 5/27/10 3:51 PM Page box beam is constructed from four pieces of wood, glued together as shown. f the moment acting on the cross section is 10 kn # m, determine the stress at points and and show the results acting on volume elements located at these points. 20 mm 160 mm 20 mm 25 mm 250 mm 25 mm M 10 kn m The moment of inertia of the cross-section about the neutral axis is = (0.2)(0.33 ) (0.16)(0.253 ) = (10-3 ) m 4 For point, y = C = 0.15 m. s = My = 10(103 ) (0.15) (10-3 ) = 6.207(106 )Pa = 6.21 MPa (C) For point, y = m. s = My = 10(103 )(0.125) (10-3 ) = 5.172(10 6 )Pa = 5.17 MPa (T) The state of stress at point and are represented by the volume element shown in Figs. a and b respectively. 365

38 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Determine the dimension a of a beam having a square cross section in terms of the radius r of a beam with a circular cross section if both beams are subjected to the same internal moment which results in the same maximum bending stress. r a a Section Properties: The moments of inertia of the square and circular cross sections about the neutral axis are S = 1 12 aa3 = a4 12 C = 1 4 pr4 Maximum ending Stress: For the square cross section, c = a>2. s max S = Mc S = M(a>2) a 4 >12 = 6M a 3 For the circular cross section, c = r. s max c = Mc = Mr - 4M c 1 pr 3 4 pr4 t is required that s max S = s max C 6M a 3 = 4M pr 3 a = 1.677r *6 64. The steel rod having a diameter of 1 in. is subjected to an internal moment of M = 300 lb # ft. Determine the stress created at points and. lso, sketch a three-dimensional view of the stress distribution acting over the cross section. = p 4 r4 = p 4 (0.54 ) = in 4 s = Mc = 300(12)(0.5) = 36.7 ksi s = My 300(12)(0.5 sin 45 ) = = 25.9 ksi in. M 300 lb ft 366

39 06 Solutions 46060_Part1 5/27/10 3:51 PM Page f the moment acting on the cross section of the beam is M = 4 kip # ft, determine the maximum bending stress in the beam. Sketch a three-dimensional view of the stress distribution acting over the cross section. 1.5 in. 12 in. The moment of inertia of the cross-section about the neutral axis is = 1 12 (12)(153 ) (10.5)(123 ) = 1863 in 4 long the top edge of the flange y = c = 7.5 in. Thus 12 in. M 1.5 in. 1.5 in. s max = Mc = 4(103 )(12)(7.5) = 193 psi 1863 long the bottom edge to the flange, y = 6 in. Thus s = My = 4(103 )(12)(6) 1863 = 155 psi f M = 4 kip # ft, determine the resultant force the bending stress produces on the top board of the beam. The moment of inertia of the cross-section about the neutral axis is 1.5 in. 12 in. = 1 12 (12)(153 ) (10.5)(123 ) = 1863 in 4 long the top edge of the flange y = c = 7.5 in. Thus s max = Mc = 4(103 )(12)(7.5) = psi in. M 1.5 in. 1.5 in. long the bottom edge of the flange, y = 6 in. Thus s = My = 4(103 )(12)(6) 1863 = psi The resultant force acting on board is equal to the volume of the trapezoidal stress block shown in Fig. a. F R = 1 ( )(1.5)(12) 2 = lb = 3.13 kip 367

40 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The rod is supported by smooth journal bearings at and that only exert vertical reactions on the shaft. f d = 90 mm, determine the absolute maximum bending stress in the beam, and sketch the stress distribution acting over the cross section. 12 kn/m d 3 m 1.5 m bsolute Maximum ending Stress: The maximum moment is M max = kn # m as indicated on the moment diagram. pplying the flexure formula s max = M max c = 11.34(103 )(0.045) p 4 ( ) = 158 MPa 368

41 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 369 *6 68. The rod is supported by smooth journal bearings at and that only exert vertical reactions on the shaft. Determine its smallest diameter d if the allowable bending stress is s allow = 180 MPa. 12 kn/m d 3 m 1.5 m llowable ending Stress: The maximum moment is M max = kn # m as indicated on the moment diagram. pplying the flexure formula s max = s allow = M max c = 11.34(103 ) d 2 p 4 d 2 4 d = m = 86.3 mm Two designs for a beam are to be considered. Determine which one will support a moment of M = 150 kn # m with the least amount of bending stress. What is that stress? 200 mm 15 mm 200 mm 30 mm Section Property: For section (a) 300 mm 30 mm 300 mm 15 mm = 1 12 (0.2) (0.17)(0.3)3 = (10-3 ) m 4 15 mm 30 mm For section (b) (a) (b) = 1 12 (0.2) (0.185)0.33 = (10-3 ) m 4 Maximum ending Stress: pplying the flexure formula s max = Mc For section (a) s max = 150(103 )(0.165) (10-3 ) = MPa For section (b) s max = 150(103 )(0.18) (10-3 ) = MPa = 74.7 MPa 369

42 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The simply supported truss is subjected to the central distributed load. Neglect the effect of the diagonal lacing and determine the absolute maximum bending stress in the truss. The top member is a pipe having an outer diameter of 1 in. 100 lb/ft 5.75 in. 3 and thickness of 16 in., and the bottom member is a solid rod having a diameter of 1 2 in. 6 ft 6 ft 6 ft y = y = 0 + (6.50)(0.4786) = in = c 1 4 p(0.5)4-1 4 p(0.3125)4 d ( ) p(0.25) (4.6091) 2 = in 4 M max = 300(9-1.5)(12) = lb # in. s max = Mc = ( ) = 22.1 ksi The axle of the freight car is subjected to wheel loadings of 20 kip. f it is supported by two journal bearings at C and D, determine the maximum bending stress developed at the center of the axle, where the diameter is 5.5 in. C D 60 in. 10 in. 10 in. 20 kip 20 kip s max = Mc = 200(2.75) = 12.2 ksi 1 p(2.75)

43 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 371 *6 72. The steel beam has the cross-sectional area shown. Determine the largest intensity of distributed load w 0 that it can support so that the maximum bending stress in the beam does not exceed s max = 22 ksi. w 0 Support Reactions: s shown on FD. nternal Moment: The maximum moment occurs at mid span. The maximum moment is determined using the method of sections. Section Property: = 1 12 (8) (7.7)103 = in 4 bsolute Maximum ending Stress: The maximum moment is M max = 48.0w 0 as indicated on the FD. pplying the flexure formula s max = M max c 22 = 48.0w 0 (12)(5.30) ft 12 ft 0.3 in. 8 in in. 10 in in. w 0 = 1.10 kip>ft The steel beam has the cross-sectional area shown. f w 0 = 0.5 kip>ft, determine the maximum bending stress in the beam. w 0 Support Reactions: s shown on FD. nternal Moment: The maximum moment occurs at mid span. The maximum moment is determined using the method of sections. Section Property: = 1 12 (8) (7.7)103 = in 4 bsolute Maximum ending Stress: The maximum moment is as indicated on the FD. pplying the flexure formula s max = M max c = 24.0(12)(5.30) M max = 24.0 kip # ft 12 ft 12 ft 0.3 in. 8 in in. 10 in in. = 10.0 ksi 371

44 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The boat has a weight of 2300 lb and a center of gravity at G. f it rests on the trailer at the smooth contact and can be considered pinned at, determine the absolute maximum bending stress developed in the main strut of the trailer. Consider the strut to be a box-beam having the dimensions shown and pinned at C. 3 ft D G 1 ft 1 ft 5 ft 4 ft C 1.75 in. 3 in in. 1.5 in. oat: : + F x = 0; x = 0 a + M = 0; -N (9) (5) = 0 + c F y = 0; y = 0 ssembly: N = lb y = lb a + M C = 0; -N D (10) (9) = 0 N D = 2070 lb + c F y = 0; C y = 0 C y = 230 lb = 1 12 (1.75)(3) (1.5)(1.75)3 = in 4 s max = Mc = (12)(1.5) = 21.1 ksi

45 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The shaft is supported by a smooth thrust bearing at and smooth journal bearing at D. f the shaft has the cross section shown, determine the absolute maximum bending stress in the shaft. C D 40 mm 25 mm 0.75 m 1.5 m 0.75 m Shear and Moment Diagrams: s shown in Fig. a. 3 kn 3 kn Maximum Moment: Due to symmetry, the maximum moment occurs in region C of the shaft. Referring to the free-body diagram of the segment shown in Fig. b. Section Properties: The moment of inertia of the cross section about the neutral axis is = p = m 4 bsolute Maximum ending Stress: s allow = M maxc = (0.04) = 52.8 MPa *6 76. Determine the moment M that must be applied to the beam in order to create a maximum stress of 80 MPa.lso sketch the stress distribution acting over the cross section. 20 mm 300 mm The moment of inertia of the cross-section about the neutral axis is Thus, = 1 12 (0.3)(0.33 ) (0.21)(0.263 ) = (10-3 ) m 4 s max = Mc ; 80(106 ) = M(0.15) (10-3 ) 260 mm 20 mm 30 mm M 30 mm 30 mm M = (10 3 ) N # m = 196 kn # m The bending stress distribution over the cross-section is shown in Fig. a. 373

46 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The steel beam has the cross-sectional area shown. Determine the largest intensity of distributed load w that it can support so that the bending stress does not exceed s max = 22 ksi. w w = 1 12 (8)(10.6) (7.7)(103 ) = in 4 s max = Mc 22 = 32w(12)(5.3) ft 8 ft 8 ft 0.3 in. 8 in in. 10 in in. w = 1.65 kip>ft The steel beam has the cross-sectional area shown. f w = 5 kip>ft, determine the absolute maximum bending stress in the beam. w w 8 ft 8 ft 8 ft 0.3 in. 8 in in. 10 in in. From Prob. 6-78: M = 32w = 32(5)(12) = 1920 kip # in. = in 4 s max = Mc = 1920(5.3) = 66.8 ksi

47 06 Solutions 46060_Part1 5/27/10 3:51 PM Page f the beam C in Prob. 6 9 has a square cross section, 6 in. by 6 in., determine the absolute maximum bending stress in the beam. 15 kip 1 ft 20 kip C 4 ft 4 ft 4 ft M max = 46.7 kip # ft s max = Mc = 46.7(103 )(12)(3) 1 12 (6)(63 ) = 15.6 ksi *6 80. f the crane boom C in Prob. 6 3 has a rectangular cross section with a base of 2.5 in., determine its 1 required height h to the nearest 4 in. if the allowable bending stress is s allow = 24 ksi. 4 ft 3 ft 5 ft C a+ M = 0; 4 5 F (3) (8) = 0; F = 4000 lb + c F y = 0; - y (4000) = 0; y = 2000 lb ; + F x = 0; x (4000) = 0; x = 2400 lb s max = Mc = 6000(12)h (2.5)(h3 ) = 24(10) 3 h = 2.68 in. Use h = 2.75 in. 375

48 06 Solutions 46060_Part1 5/27/10 3:51 PM Page f the reaction of the ballast on the railway tie can be assumed uniformly distributed over its length as shown, determine the maximum bending stress developed in the tie. The tie has the rectangular cross section with thickness t = 6 in. 15 kip 15 kip 1.5 ft 5 ft 1.5 ft 12 in. t w Support Reactions: Referring to the free - body diagram of the tie shown in Fig. a, we have + c F y = 0; w(8) - 2(15) = 0 w = 3.75 kip>ft Maximum Moment: The shear and moment diagrams are shown in Figs. b and c.s indicated on the moment diagram, the maximum moment is. bsolute Maximum ending Stress: M max = 7.5 kip # ft s max = M maxc = 7.5(12)(3) = 1.25 ksi 1 12 (12)(63 ) 376

49 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The reaction of the ballast on the railway tie can be assumed uniformly distributed over its length as shown. f the wood has an allowable bending stress of s allow = 1.5 ksi, determine the required minimum thickness t of the 1 rectangular cross sectional area of the tie to the nearest in kip 15 kip 1.5 ft 5 ft 1.5 ft 12 in. t w Support Reactions: Referring to the free-body diagram of the tie shown in Fig. a,we have + c F y = 0; w(8) - 2(15) = 0 w = 3.75 kip>ft Maximum Moment: The shear and moment diagrams are shown in Figs. b and c.s indicated on the moment diagram, the maximum moment is. bsolute Maximum ending Stress: M max = 7.5 kip # ft s max = Mc 7.5(12)a t ; 1.5 = 2 b 1 12 (12)t3 t = 5.48 in. Use t = in. 377

50 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Determine the absolute maximum bending stress in the tubular shaft if d i = 160 mm and d o = 200 mm. 15 kn/m 60 kn m d i d o 3 m 1 m Section Property: = p = m 4 bsolute Maximum ending Stress: The maximum moment is M max = 60.0 kn # m as indicated on the moment diagram. pplying the flexure formula s max = M maxc = 60.0(103 )(0.1) (10-6 ) = 129 MPa 378

51 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 379 *6 84. The tubular shaft is to have a cross section such that its inner diameter and outer diameter are related by d i = 0.8d o. Determine these required dimensions if the allowable bending stress is s allow = 155 MPa. 15 kn/m 60 kn m d i d o 3 m 1 m Section Property: = p 4 a d o 2 b 4 - a d l 2 b 4 R = p 4 d o 4 llowable ending Stress: The maximum moment is M max = 60.0 kn # m as indicated on the moment diagram. pplying the flexure formula s max = s allow = M max c = 60.0(103 ) d o pd o a 0.8d o b R = pd o Thus, d o = m = 188 mm d l = 0.8d o = 151 mm 379

52 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The wood beam has a rectangular cross section in the proportion shown. Determine its required dimension b if the allowable bending stress is s allow = 10 MPa. 500 N/m 2 m 2 m b 1.5b llowable ending Stress: The maximum moment is M max = N # m as indicated on the moment diagram. pplying the flexure formula s max = s allow = M max c = 562.5(0.75b) 1 12 (b)(1.5b)3 b = m = 53.1 mm 380

53 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Determine the absolute maximum bending stress in the 2-in.-diameter shaft which is subjected to the concentrated forces. The journal bearings at and only support vertical forces. 800 lb 600 lb 15 in. 15 in. 30 in. The FD of the shaft is shown in Fig. a. The shear and moment diagrams are shown in Fig. b and c, respectively. s indicated on the moment diagram, M max = lb # in. The moment of inertia of the cross-section about the neutral axis is = p 4 (14 ) = 0.25 p in 4 Here, c = 1 in. Thus s max = M max c = 15000(1) 0.25 p = 19.10(10 3 ) psi = 19.1 ksi 381

54 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The journal bearings at and only support vertical forces. The allowable bending stress is s allow = 22 ksi. 800 lb 600 lb 15 in. 15 in. 30 in. The FD of the shaft is shown in Fig. a The shear and moment diagrams are shown in Fig. b and c respectively. s indicated on the moment diagram, M max = 15,000 lb # in The moment of inertia of the cross-section about the neutral axis is = p 4 a d 2 b 4 = p 64 d4 Here, c = d>2. Thus s allow = M max c ; 22(10 3 ) = 15000(d > 2 ) pd 4 >64 d = in = 2 in. 382

55 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 383 *6 88. f the beam has a square cross section of 9 in. on each side, determine the absolute maximum bending stress in the beam. 800 lb/ft 1200 lb 8 ft 8 ft bsolute Maximum ending Stress: The maximum moment is M max = 44.8 kip # ft as indicated on moment diagram. pplying the flexure formula s max = M max c = 44.8(12)(4.5) 1 12 (9)(9)3 = 4.42 ksi f the compound beam in Prob has a square cross section, determine its dimension a if the allowable bending stress is s allow = 150 MPa. llowable ending Stress: The maximum moments is M max = 7.50 kn # m as indicated on moment diagram. pplying the flexure formula s max = s allow = M max c = 7.50(103 ) a a4 a = m = 66.9 mm 383

56 06 Solutions 46060_Part1 5/27/10 3:51 PM Page f the beam in Prob has a rectangular cross section with a width b and a height h, determine the absolute maximum bending stress in the beam. bsolute Maximum ending Stress: The maximum moments is M max = 23w 0 L as indicated on the moment diagram. pplying the flexure formula s max = M max c = 23w 0 L h 2 = 23w 0 L 1 12 bh3 36bh

57 06 Solutions 46060_Part1 5/27/10 3:51 PM Page Determine the absolute maximum bending stress in the 80-mm-diameter shaft which is subjected to the concentrated forces. The journal bearings at and only support vertical forces. 0.5 m 0.4 m 0.6 m The FD of the shaft is shown in Fig. a The shear and moment diagrams are shown in Fig. b and c, respectively. s M max = 6 kn # m indicated on the moment diagram,. The moment of inertia of the cross-section about the neutral axis is 12 kn 20 kn = p 4 (0.044 ) = 0.64(10-6 )p m 4 Here, c = 0.04 m. Thus s max = M max c = 6(103 )(0.04) 0.64(10-6 )p = (10 6 ) Pa = 119 MPa 385

58 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 386 *6 92. Determine the smallest allowable diameter of the shaft which is subjected to the concentrated forces. The journal bearings at and only support vertical forces. The allowable bending stress is s allow = 150 MPa. 0.5 m 0.4 m 0.6 m 12 kn 20 kn The FD of the shaft is shown in Fig. a. The shear and moment diagrams are shown in Fig. b and c, respectively. s M max = 6 kn # m indicated on the moment diagram,. The moment of inertia of the cross-section about the neutral axis is = p 4 a d 2 b 4 = pd4 64 Here, c = d>2. Thus s allow = M max c ; 150(10 6 ) = 6(103 )( d > 2 ) pd 4 >64 d = m = mm = 75 mm 386

59 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The man has a mass of 78 kg and stands motionless at the end of the diving board. f the board has the cross section shown, determine the maximum normal strain developed in the board. The modulus of elasticity for the material is E = 125 GPa. ssume is a pin and is a roller. 1.5 m 2.5 m C 350 mm 30 mm 20 mm 10 mm 10 mm 10 mm nternal Moment: The maximum moment occurs at support. The maximum moment is determined using the method of sections. Section Property: y = y = 0.01(0.35)(0.02) (0.03)(0.03) 0.35(0.02) (0.03) = m = 1 12 (0.35) (0.02)( ) (0.03) (0.03)( ) 2 = m 4 bsolute Maximum ending Stress: The maximum moment is as indicated on the FD. pplying the flexure formula s max = M max c M max = N # m = ( ) (10-6 ) = MPa bsolute Maximum Normal Strain: pplying Hooke s law, we have e max = s max E = 88.92(106 ) 125(10 9 = mm>mm ) 387

60 06 Solutions 46060_Part1 5/27/10 3:51 PM Page The two solid steel rods are bolted together along their length and support the loading shown. ssume the support at is a pin and is a roller. Determine the required diameter d of each of the rods if the allowable bending stress is sallow = 130 MPa. 20 kn/m 80 kn 2m Section Property: = 2 2m p d 4 p d 2 5p 4 a b + d2 a b R = d llowable ending Stress: The maximum moment is Mmax = 100 kn # m as indicated on moment diagram. pplying the flexure formula smax = sallow = = Mmax c 100(103)(d) 5p 32 d4 d = m = 116 mm Solve Prob if the rods are rotated 90 so that both rods rest on the supports at (pin) and (roller). 20 kn/m Section Property: = 2 p d 4 p 4 a b R = d smax = sallow = 2m Mmax c 100(103)(d) p 32 2m llowable ending Stress: The maximum moment is Mmax = 100 kn # m as indicated on the moment diagram. pplying the flexure formula = 80 kn d4 d = m = 199 mm 388

61 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 389 *6 96. The chair is supported by an arm that is hinged so it rotates about the vertical axis at. f the load on the chair is 180 lb and the arm is a hollow tube section having the dimensions shown, determine the maximum bending stress at section a a. 180 lb 1 in. a 3 in. 2.5 in. 8 in. a 0.5 in. c + M = 0; M - 180(8) = 0 M = 1440 lb # in. x = 1 12 (1)(33 ) (0.5)(2.53 ) = in 4 s max = Mc = 1440 (1.5) = 1.35 ksi portion of the femur can be modeled as a tube having an inner diameter of in. and an outer diameter of 1.25 in. Determine the maximum elastic static force P that can be applied to its center. ssume the bone to be roller supported at its ends. The s P diagram for the bone mass is shown and is the same in tension as in compression. s (ksi) P 4 in. 4 in P (in./in.) = 1 4 p C D = in 4 M max = P (4) = 2P 2 Require s max = 1.25 ksi s max = Mc 1.25 = 2P(1.25>2) P = kip = 119 lb 389

62 06 Solutions 46060_Part1 5/27/10 3:51 PM Page f the beam in Prob has a rectangular cross section with a width of 8 in. and a height of 16 in., determine the absolute maximum bending stress in the beam. bsolute Maximum ending Stress: The maximum moment is as indicated on moment diagram. pplying the flexure formula M max = 216 kip # ft 16 in. s max = M max c = 216(12)(8) 1 12 (8)(163 ) = 7.59 ksi 8 in f the beam has a square cross section of 6 in. on each side, determine the absolute maximum bending stress in the beam. 400 lb/ft 6 ft 6 ft The maximum moment occurs at the fixed support. Referring to the FD shown in Fig. a, a + M = 0; M max - 400(6)(3) (400)(6)(8) = 0 M max = lb # ft The moment of inertia of the about the neutral axis is = 1.Thus, 12 (6)(63 ) = 108 in 4 s max = Mc = 16800(12)(3) 108 = 5600 psi = 5.60 ksi 390

63 06 Solutions 46060_Part1 5/27/10 3:51 PM Page 391 * The steel beam has the cross-sectional area shown. Determine the largest intensity of the distributed load w 0 that it can support so that the maximum bending stress in the beam does not exceed s allow = 22 ksi. w 0 9 ft 9 ft Support Reactions. The FD of the beam is shown in Fig. a. The shear and moment diagrams are shown in Fig. a and b, respectively. s indicated on the moment diagram, M max = 27w o. The moment of inertia of the cross-section about the neutral axis is = 1 12 (9)(12.53 ) (8.75)(123 ) = in 4 Here, = 6.25 in. Thus, s allow = M max c ; 22(10 3 ) = (27w o)(12)(6.25) w o = lb>ft = 2.23 kip>ft 0.25 in. 9 in in. 12 in in. 391

64 06 Solutions 46060_Part2 5/26/10 1:17 PM Page The steel beam has the cross-sectional area shown. f w 0 = 2 kip>ft, determine the maximum bending stress in the beam. w 0 9 ft 9 ft 0.25 in. 9 in in. 12 in in. The FD of the beam is shown in Fig. a The shear and moment diagrams are shown in Fig. b and c, respectively. s indicated on the moment diagram, M max = 54 kip # ft. The moment of inertia of the cross-section about the bending axis is = 1 12 (9) (8.75)123 = in 4 Here, c = 6.25 in. Thus s max = M max c = 54 (12)(6.25) = ksi = 19.8 ksi 392

65 06 Solutions 46060_Part2 5/26/10 1:17 PM Page The bolster or main supporting girder of a truck body is subjected to the uniform distributed load. Determine the bending stress at points and. 1.5 kip/ft 8 ft 12 ft F in. 6 in. F 2 12 in. 0.5 in in. Support Reactions: s shown on FD. nternal Moment: Using the method of sections. + M N = 0; M (4) (8) = 0 Section Property: M = 72.0 kip # ft = 1 12 (6) (5.5)123 = in 4 ending Stress: pplying the flexure formula s = My s = 72.0(12)(6.75) = 13.3 ksi s = 72.0(12)(6) = 11.8 ksi 393

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