고체역학 (Gere) — Torsion: Problem 3.5-13

3.5-13 Two circular aluminum pipes of equal length \(L = 610 \text{ mm}\) are loaded by torsional moments \(T\) (see figure). Pipe 1 has outside and inside diameters \(d_2 = 76 \text{ mm}\) and \(d_1 = 64 \text{ mm}\), respectively. Pipe 2 has a constant outer diameter of \(d_2\) along its entire length \(L\) and an inner diameter of \(d_1\) but has an increased inner diameter of \(d_3 = 67 \text{ mm}\) over the middle third. Assume that \(E = 72 \text{ GPa}\), \(\nu = 0.33\), and allowable shear stress \(\tau_a = 45 \text{ MPa}\). (a) Find the maximum acceptable torques that can be applied to Pipe 1; repeat for Pipe 2. (b) If the maximum twist \(\phi\) of Pipe 2 cannot exceed \(5/4\) of that of Pipe 1, what is the maximum acceptable length of the middle segment? Assume both pipes have total length \(L\) and the same applied torque \(T\). (c) Find the new value of inner diameter \(d_3\) of Pipe 2 if the maximum torque carried by Pipe 2 is to be \(7/8\) of that for Pipe 1. (d) If the maximum normal strain in each pipe is known to be \(\epsilon_{\text{max}} = 811 \times 10^{-6}\), what is the applied torque on each pipe? Also, what is the maximum twist of each pipe? Use the original properties and dimensions.