Pb 6) 4.31 do = 25 mm 4.31 What bending moment is required to produce a maximum normal stress of 400 MPa: (a) In a straight round rod of 40-mm diameter? (b) In a straight square rod, 40 mm on a side (with bending about the X axis as shown for a rectangular section in Appendix B-2)?
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
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- A vertical pole of solid, circular cross section is twisted by horizontal forces P = 5kN acting at the ends of a rigid horizontal arm AB (see figure part a). The distance from the outside of the pole to the line of action of each force is c = 125 mm (sec figure part b) and the pole height L = 350 mm. (a) If the allowable shear stress in the pole is 30 MPa, what is the minimum required diameter dminof the pole? (b) What is the torsional stiffness of the pole (kN · m/rad)? Assume that G = 28 GPa. (c) If two translation al springs, each with stiffness k =2550 kN/m, are added at 2c/5 from A and B (see figure part c), repeat part (a) to find dmin. Hint: Consider the pole and pair of springs as "springs in parallel."Solve the preceding problem if the length L = 56 in., the inner radius r1— 1.25 in., the angle of twist is 0.5°, and the allowable shear strain is 0.0004 rad.A tubular bar with outside diameterd2= 4.0 in, is twisted by torques T = 70,0 kip-in. (see figure). Under the action of these torques, the maximum tensile stress in the bar is found to be 6400 psi. Determine the inside diameter rtf of the bar. If the bar has length L = 48.0 in. and is made of aluminum with shear modulus G = 4,0 × 106 psi, what is the angle of twist d (in degrees) between the ends of the bar? (c) Determine the maximum shear strain y (in radians)?
- When drilling a hole in a table leg, a furniture maker uses a hand-operated drill (see figure) with a bit of diameter d = 4.0 mm. If the resisting torque supplied by the table leg is equal to 0.3 N · m, what is the maximum shear stress in the drill bit? If the allowable shear stress in the drill bit is 32 MPa, what is the maximum resisting torque before the drill binds up? If the shear modulus of elasticity of the steel is G = 75 GPa, what is the rate of twist of the drill bit (degrees per meter)?-8 An aluminum bar of solid circular cross section is twisted by torques T acting at the ends (see figure). The dimensions and shear modulus of elasticity arc L = 1.4 m, d = 32 mm, and G = 28 GPa. Determine the torsional stiffness of the bar. If the angle of twist of the bar is 5º, what is the maximum shear stress? What is the maximum shear strain (in radians)? If a hole of diameter d/2 is drilled longitudinally through the bar, what is the ratio of the torsional stiffnesses of the hollow and solid bars? What is the ratio of their maximum shear stresses if both arc acted on by the same torque? If the hole diameter remains at d/2, what new outside diameter d2will result in equal stiffnesses of the hollow and solid bars?An aluminum tube has inside diameter dx= 50 mm, shear modulus of elasticity G = 27 GPa, v = 0.33, and torque T = 4.0 kN · m. The allowable shear stress in the aluminum is 50 MPa, and the allowable normal strain is 900 X 10-6. Determine the required outside diameter d2 Re-compute the required outside diameter d2, if allowable normal stress is 62 MPa and allowable shear strain is 1.7 X 10-3.
- A circular aluminum tube subjected to pure torsion by torques T(sec figure) has an outer radius r2equal to 1.5 times the inner radius r1. (a) If the maximum shear strain in the tube is measured as 400 × 10-6 rad, what is the shear strain y1at the inner surface? (b) If the maximum a1lo-abk rate of twist is 0.125 °/m and the maximum shear strain is to be kept at 400 × 10-6 rad by adjusting the torque T, that is the minimum required outer radius ( r2)Min?A long, thin-walled tapered tube AB with a circular cross section (see figure) is subjected to a torque T. The tube has length L and constant wall thickness t. The diameter to the median lines of the cross sections at the ends A and B are dAand dB, respectively. Derive the following formula for the angle of twist of the tube: Hint: If the angle of taper is small, you may obtain approximate results by applying the formulas for a thin-walled prismatic tube to a differential element of the tapered tube and then integrating along the axis of the tube.A copper tube with circular cross section has length L = 1.25 m, thickness t = 2 mm, and shear modulus of elasticity G = 45 GPa. The bar is designed to carry a 300 N·m torque acting at the ends. If the allowable shear stress is 25 MPa and the allowable angle of twist between the ends is 2.5°, what is the minimum required outer diameter d?
- A circular steel tube of length L = 1.0 m is loaded in torsion by torques T (see figure). (a) If the inner radius of the tube is r1= 45 mm and the measured angle of twist between the ends is 0.5°, what is the shear strain y1(in radians) at the inner surface? (b) If the maximum allowable shear strain is 0.0004 rad and the angle of twist is to be kept at 0.45° by adjusting the torque T, what is the maxi mum permissible outer radius (r2)max?A vertical pole of solid, circular cross section is twisted by horizontal forces P = 1100 lb acting at the ends of a rigid horizontal arm AB (see figure part a). The distance from the outside of the pole to the line of action of each force is c = 5.0 in. (see figure part b) and the pole height is L = 14in. (a) If the allowable shear stress in the pole is 4500 psi, what is the minimum required diameter dminof the pole? Find the torsional stiffness of the pole (kip-in./rad). Assume that G = 10,800 ksi. If two translational springs, each with stiffness k = 33 kips/in., are added at 2(75 from A and B (see figure part c), repeat part (a) to find dmin. Hint: Consider the pole and pair of springs as "springs in parallel."A solid copper bar of circular cross section has length L = 1.25 m and shear modulus of elasticity G = 45 GPa. The bar is designed to carry a 250 N · m torque acting at the ends. If the allowable shear stress is 30 M Pa and the allowable angle of twist between the ends is 2.5°, what is the minimum required diameter?
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