Gere & Goodno — Torsion: Problem 3.4-15

3.4-15 A uniformly tapered aluminum-alloy tube $AB$ with a circular cross section and length $L$ is shown in the figure. The outside diameters at the ends are $d_A$ and $d_B = 2d_A$. A hollow section of length $L/2$ and constant thickness $t = d_A/10$ is cast into the tube and extends from $B$ halfway toward $A$. (a) Find the angle of twist $\phi$ of the tube when it is subjected to torques $T$ acting at the ends. Use numerical values: $d_A = 65\text{ mm}$, $L = 1.2\text{ m}$, $G = 27\text{ GPa}$, and $T = 4.5\text{ kN}\cdot\text{m}$. (b) Repeat part (a) if the hollow section has constant diameter $d_A$ (see figure part b). Derive a formula for the angle of twist $\phi$ of the tube when it is subjected to torques $T$ acting at the ends.