고체역학 (Gere) — Torsion: Problem 3.11-14

3.11-14 A long, thin-walled tapered tube \(AB\) with a circular cross section (see figure) is subjected to a torque \(T\). The tube has length \(L\) and constant wall thickness \(t\). The diameter to the median lines of the cross sections at the ends \(A\) and \(B\) are \(d_A\) and \(d_B\), respectively. Derive the following formula for the angle of twist of the tube: \[ \phi = \frac{2TL}{\pi Gt} \left( \frac{d_A + d_B}{d_A^2 d_B^2} \right) \] *Hint:* If the angle of taper is small, you may obtain approximate results by applying the formulas for a thin-walled prismatic tube to a differential element of the tapered tube and then integrating along the axis of the tube.