ch5-4 - 338 CHAPTER 5 Stresses in Beams (Basic Topics)...

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Shear Stresses in Circular Beams Problem 5.9-1 A wood pole of solid circular cross section ( d 5 diameter) is subjected to a horizontal force P 5 450 lb (see figure). The length of the pole is L 5 6 ft, and the allowable stresses in the wood are 1900 psi in bending and 120 psi in shear. Determine the minimum required diameter of the pole based upon (a) the allowable bending stress, and (b) the allowable shear stress. Solution 5.9-1 Wood pole of circular cross section 338 CHAPTER 5 Stresses in Beams (Basic Topics) L P d d P 5 450 lb L 5 6 ft 5 72 in. s allow 5 1900 psi t allow 5 120 psi Find diameter d (a) B ASED UPON BENDING STRESS M max 5 PL 5 (450 lb)(72 in.) 5 32,400 lb-in. d min 5 5.58 in. (b) B ASED UPON SHEAR STRESS V max 5 450 lb d min 5 2.52 in. (Bending stress governs.) t 5 4 V 3 A 5 16 V 3 p d 2 Ê d 2 5 16 V max 3 p t allow 5 6.366 in. 2 s 5 M S 5 32 M p d 3 Ê d 3 5 32 M max p s allow 5 173.7 in. 3 L P d d Problem 5.9-2 A simple log bridge in a remote area consists of two parallel logs with planks across them (see figure). The logs are Douglas fir with average diameter 300 mm. A truck moves slowly across the bridge, which spans 2.5 m. Assume that the weight of the truck is equally distributed between the two logs. Because the wheelbase of the truck is greater than 2.5 m, only one set of wheels is on the bridge at a time. Thus, the wheel load on one log is equivalent to a concentrated load W acting at any position along the span. In addition, the weight of one log and the planks it supports is equivalent to a uniform load of 850 N/m acting on the log. Determine the maximum permissible wheel load W based upon (a) an allowable bending stress of 7.0 MPa, and (b) an allowable shear stress of 0.75 MPa. W 2.5 m x 850 N/m 300 m
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Solution 5.9-2 Log bridge SECTION 5.9 Shear Stresses in Circular Beams 339 Diameter d 5 300 mm s allow 5 7.0 MPa t allow 5 0.75 MPa Find allowable load W (a) B ASED UPON BENDING STRESS Maximum moment occurs when wheel is at midspan ( x 5 L /2). 5 0.625 W 1 664.1 (N ? m) ( W 5 newtons) M max 5 S s allow 5 (2.651 3 10 2 3 m 3 )(7.0 MPa) 5 18,560 N ? m [ 0.625 W 1 664.1 5 18,560 W 5 28,600 N 5 28.6 kN S 5 p d 3 32 5 2.651 3 10 2 3 m 3 M max 5 WL 4 1 qL 2 8 5 W 4 (2.5 m) 1 1 8 (850 N / m)(2.5 m) 2 (b) B ASED UPON SHEAR STRESS Maximum shear force occurs when wheel is adjacent to support ( x 5 0). 5 W 1 1062.5 N ( W 5 newtons) 5 39,760 N [ W 1 1062.5 N 5 39,760 N W 5 38,700 N 5 38.7 kN V max 5 3 A t allow 4 5 3 4 (0.070686 m 2 )(0.75 MPa) t max 5 4 V max 3 A A 5 p d 2 4 5 0.070686 m 2 V max 5 W 1 qL 2 5 W 1 1 2 (850 N / W L = 2.5 m x q = 850 N/m Problem 5.9-3 A sign for an automobile service station is supported by two aluminum poles of hollow circular cross section, as shown in the figure. The poles are being designed to resist a wind pressure of 75 lb/ft 2 against the full area of the sign. The dimensions of the poles and sign are h 1 5 20 ft, h 2 5 5 ft, and b 5 10 ft. To prevent buckling of the walls of the poles, the thickness t is specified as one-tenth the outside diameter d .
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ch5-4 - 338 CHAPTER 5 Stresses in Beams (Basic Topics)...

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