고체역학 (Gere) — Torsion: Problem 3.4-10

3.4-10 A tapered bar \(AB\) with a solid circular cross section is twisted by torques \(T\) (see figure). The diameter of the bar varies linearly from \(d_A\) at the left-hand end to \(d_B\) at the right-hand end. (a) Confirm that the angle of twist of the tapered bar is \(\phi = \frac{32TL}{3\pi G(d_B - d_A)} \left( \frac{1}{d_A^3} - \frac{1}{d_B^3} \right)\) (b) For what ratio \(d_B/d_A\) will the angle of twist of the tapered bar be one-half the angle of twist of a prismatic bar of diameter \(d_A\)? (The prismatic bar is made of the same material, has the same length, and is subjected to the same torque as the tapered bar.)