고체역학 (Gere) — Torsion: Problem 3.4-3

3.4-3 A stepped shaft ABCD consisting of solid circular segments is subjected to three torques, as shown in the figure. The torques have magnitudes of $3000\text{ N}\cdot\text{m}$, $2000\text{ N}\cdot\text{m}$, and $800\text{ N}\cdot\text{m}$. The length of each segment is $0.5\text{ m}$ and the diameters of the segments are $80\text{ mm}$, $60\text{ mm}$, and $40\text{ mm}$. The material is steel with shear modulus of elasticity $G = 80\text{ GPa}$. (a) Calculate the maximum shear stress $\tau_{\text{max}}$ in the shaft and the angle of twist $\phi_D$ (in degrees) at end D. (b) If each segment must have the same shear stress, find the required diameter of each segment in part (a) so that all three segments have shear stress $\tau_{\text{max}}$ from part (a). What is the resulting angle of twist at D?