고체역학 (Gere) — Torsion: Problem 3.4-7

3.4-7 A solid steel shaft $ABCDE$ turns freely in bearings at points $A$ and $E$. The shaft is driven by the gear at $C$, which applies a torque $T_2 = 440 \text{ N}\cdot\text{m}$. Gears at $B$ and $D$ are driven by the shaft and have resisting torques $T_1 = 270 \text{ N}\cdot\text{m}$ and $T_3 = 170 \text{ N}\cdot\text{m}$, respectively. Segments $BC$ and $CD$ have lengths $L_{BC} = 500 \text{ mm}$ and $L_{CD} = 380 \text{ mm}$ and the shear modulus $G = 80 \text{ GPa}$. Determine the minimum required diameter ($d$) of the shaft if the allowable shear stress $\tau_a = 40 \text{ MPa}$. Also calculate the angle of twist between gears $B$ and $D$.