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HW05 - Mechanics of Materials Homework#5 Problem 1 A...

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Unformatted text preview: Mechanics of Materials Homework #5 Problem 1: A circular aluminum tube is subjected to pure torsion by equal and opposite torques, T, at both ends. The tube has an outer radius r2 equal to twice the inner radius r1. (a) If the maximum shear strain in the tube is measured as 350 x 10‐6 rad, what is the shear strain at the inner surface? (b) If the maximum allowable rate of twist is 0.20 degrees per foot and the maximum shear strain is to be kept at 350 x 10‐6 rad by adjusting the torque, T, what is the minimum required outer radius, r2,min? Problem 2: While removing a wheel to change a tire, a driver applies forces P = 100 N at the ends of the arms of a lug wrench, the geometry of which is shown in the figure. The wrench is made of steel with shear modulus of elasticity G = 78 GPa. Each arm of the wrench is 225 mm long and has a solid circular cross section of diameter d = 12 mm. Determine (a) the maximum shear stress in the arm that is turning the lug nut (arm A) and (b) the angle of twist (in degrees) of this same arm. Problem 3: A circular tube of outer diameter d3 = 2.75 in and inner diameter d2 = 2.35 in is welded at the right‐hand end to a fixed plate and at the left‐hand end to a rigid end plate. A solid circular bar of diameter d1 = 1.60 in is inside of, and concentric with, the tube. The bar passes through a hole in the fixed plate and is welded to the end plate. The bar is 40 in long and the tube is half as long as the bar. A torque T = 10,000 lb‐in acts at end A of the bar. Both the bar and tube are made of an aluminum alloy with shear modulus of elasticity G = 3.9 x 106 psi. Determine (a) the maximum shear stresses in both the bar and tube and (b) the angle of twist (in degrees) at the end A of the bar. Problem 4: A steel shaft ABC connecting three gears consists of a solid bar of diameter d between gears A and B and a hollow bar of outside diameter 1.25d and inside diameter d between gears B and C. Both bars have length L = 0.6 m. The gears transmit torques T1 = 240 N∙m, T2 = 540 N∙m, and T3 = 300 N∙m, acting in the directions shown in the figure. The shear modulus of elasticity for the shaft is 80 GPa. (a) What is the minimum permissible diameter d if the allowable shear stress in the shaft is 80 MPa? (b) What is the minimum permissible diameter d if the angle of twist between any two gears is limited to 4.0°? Problem 5: A bar AB of length L and solid circular cross section (diameter d) is loaded by a distributed torque of constant intensity t per unit distance. (a) What is the maximum shear stress τmax in the bar? (b) What is the angle of twist φ between the ends of the bar? Problem 6: A solid circular bar ABCD with fixed supports at ends A and D is acted upon by two equal and oppositely directed torques T0, as shown in the figure. The torques are applied at points B and C, each of which is located at a distance x from one end of the bar (the distance x may vary from zero to L/2). (a) For what distance x will the angle of twist at points B and C be a maximum? (b) What is the corresponding angle of twist φmax? Problem 7: A solid steel bar of diameter 30 mm is enclosed by a steel tube of outer diameter 45 mm and inner diameter 36 mm. Both bar and tube are held rigidly at end A and joined securely to a rigid plate at end B. The composite bar, which is 500 mm long, is twisted by a torque T = 500 N∙m acting on the end plate. (a) Find the maximum shear stresses τ1 and τ2 in the bar and tube, respectively. (b) Determine the angle of rotation φ (in degrees) of the end plate, assuming that the shear modulus of the steel is G = 80 GPa. ...
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