고체역학 (Gere) — Torsion: Problem 3.8-15

3.8-15 The composite shaft shown in the figure is manufactured by shrink-fitting a steel sleeve over a brass core so that the two parts act as a single solid bar in torsion. The outer diameters of the two parts are \(d_1 = 40\text{ mm}\) for the brass core and \(d_2 = 50\text{ mm}\) for the steel sleeve. The shear moduli of elasticity are \(G_b = 36\text{ GPa}\) for the brass and \(G_s = 80\text{ GPa}\) for the steel. (a) Assuming that the allowable shear stresses in the brass and steel are \(\tau_b = 48\text{ MPa}\) and \(\tau_s = 80\text{ MPa}\), respectively, determine the maximum permissible torque \(T_{\text{max}}\) that may be applied to the shaft. (b) If the applied torque \(T = 2500\text{ N}\cdot\text{m}\), find the required diameter \(d_2\) so that allowable shear stress \(\tau_s\) is reached in the steel.