Problem 2.48

2.48. Please ignore the question in the text, and modify the figure by letting T_B be independent of T. 

(a) Determine U as a function of T and T_B.

(b)  Determine the twisting rotations, j at C, and j_B at B, in terms of T and T_B, and verify that

j = ∂U/∂T

j_B = ∂U/∂T_B

c) Let T_B = 4T in the result of part (a), so U becomes a function of T.  Can you determine or guess what would be obtained by the operation ∂U/∂T?

a) For a uniform bar of circular cross section in uniform torsion T, U = T²L/2GJ, J = pr⁴/2 = pd⁴/32.

Bar BC:  U_BC = T²a/2GJ

Bar AB: Torque = (T - T_B ), positive contr-clkwise as seen from the right.

U_BC = (T - T_B )²(1.2a)/2GJ_AB

J_AB = p(1.4d)⁴/32 = (1.4)⁴J

U = T²a/2GJ + (T - T_B )²(1.2a)/2G(1.4)⁴J

b) For a uniform bar of circular cross section in uniform torsion T, the relative twist between the ends is

Dj = TL/GJ, J = pd⁴/32

j_B = (T - T_B )(1.2a)/G(1.4)⁴J,  positive contr-clkwise as seen from the right.

j_C = j_B + j_C/B = (T - T_B )(1.2a)/G(1.4)⁴J +  Ta/GJ,  positive contr-clkwise as seen from the right.

∂U/∂T = Ta/GJ + (T - T_B )(1.2a)/G(1.4)⁴J, positive in the direction of T.  Same as  j_C

∂U/∂T_B = - (T - T_B )(1.2a)/G(1.4)⁴J, positive in the direction of T_B. Coincides with j_B after adopting the same sign convention.

c) For T_B = 4T,

U = T²a/2GJ + 9T²(1.2a)/2G(1.4)⁴J

∂U/∂T = Ta/GJ + 9T(1.2a)/G(1.4)⁴J

This should be equal to

4j_B clockwise + j_C  cntr-clckwise =  - 4(T - T_B )(1.2a)/G(1.4)⁴J + (T - T_B )(1.2a)/G(1.4)⁴J +  Ta/GJ

              =  12T(1.2a)/G(1.4)⁴J - 3T(1.2a)/G(1.4)⁴J +  Ta/GJ = 9T(1.2a)/G(1.4)⁴J + Ta/GJ.  Yes.