a. Determine the torque T and the radius of the elastic-plastic boundary. b. Determine the residual stresses at the outer radius and at the elastic-plastic boundary. c. Determine the permanent set us in the member.
Q: a) Determine the "torsional" shear stress (resulting from torque) at point A. (absolute value)
A: ζ =TJRζ = torsional shearT = torsionJ = Polar MOIR = maxiumum radius
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Q: the stresses at point A.
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- Two sections of steel drill pipe, joined by bolted flange plates at Ä are being tested to assess the adequacy of both the pipes. In the test, the pipe structure is fixed at A, a concentrated torque of 500 kN - m is applied at x = 0.5 m, and uniformly distributed torque intensity t1= 250 kN m/m is applied on pipe BC. Both pipes have the same inner diameter = 200 mm. Pipe AB has thickness tAB=15 mm, while pipe BC has thickness TBC= 12 mm. Find the maximum shear stress and maximum twist of the pipe and their locations along the pipe. Assume G = 75 GPa.A tapered bar AB with a solid circular cross section is twisted by torques T = 36,000 lb-in. (sec figure). The diameter of the bar varies linearly from dAat the left-hand end to dBat the right-hand end. The bar has length L = 4,0 ft and is made of an aluminum alloy having shear modulus of elasticity G = 3.9 × 106 psi. The allowable shear stress in the bar is 15,000 psi and the allowable angle of twist is 3.0°. If the diameter at end B is 1.5 times the diameter at end A, what is the minimum required diameter dAat end A?.17 A mountain-bike rider going uphill applies torque T = Fd(F = l5lb, d = 4 in.) to the end of the handlebars ABCD by pulling on the handlebar extenders DE. Consider the right half of the handlebar assembly only (assume the bars are fixed at the fork at A). Segments AB and CD are prismatic with lengths L, = 2 in.andL3 = 8.5 in, and with outer diameters and thicknesses d01 = 1.25 in. 101 = 0.125 in. and d03 = O.87in.,i03 = 0.ll5in, respectively as shown. Segment BC’ of length L, = 1.2 in. however. is tapered, and outer diameter and thickness vary linearly between dimensions at B and C. Consider torsion effects only. Assume G = 4000 ksi is constant. Derive an integral expression for the angle of twist of half of the handlebar tube when it is subjected to torque T = Fd acting at the end. Evaluate ‘b1-, for the given numerical1ues.
- 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?A solid circulai' aluminum bar AB is fixed at both ends and loaded by a uniformly distributed torque 150N·n/m. The bar has diameter d = 30 mm. Calculate the reactive torques at the supports and the angle of twist at midspan. Assume that G = 28 GPa.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 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)?A thin-walled rectangular tube has uniform thickness t and dimensions a x b to the median line of the cross section (see figure). How does the shear stress in the tube vary with the ratio = a/b if the total length Lmof the median line of the cross section and the torque T remain constant? From your results, show that the shear stress is smallest when the tube is square (ß = 1).The non prismatic, cantilever circular bar shown has an internal cylindrical hole from 0 to y, so the net polar moment of inertia of the cross section for segment 1 is (7/8 )Ip. Torque Tis applied at _y and torque 772 is applied at .v = L. Assume that G is constant. Find the reaction moment Ry. Find internal torsional moments Tiin segments 1 and 2. Find x required to obtain twist at joint 3 of tf3 = TLtGIp. What is the rotation at joint 2, Draw the torsional moment (TMD:7(.,0 _v L) and displacement (TDD: M_y),0 x L) diagrams.
- 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?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 statically indeterminate stepped shaft ACE is fixed at ends A and B and loaded by a torque TQat point C (see figure). The two segments of the bar are made of the same material, have lengths L4and LB, and have polar moments of inertia IAand Ipb. Determine the angle of rotation 4>of the cross section at Cby using strain energy. Hint: Use Eq, (3-55b) to determine the strain energy Urn terms of the angle d?. Then equate the strain energy to the work done by the torque to. Compare your result with Eq. (3-52) of Example 3-9.