고체역학 (Gere) — Torsion: Problem 3.11-2

3.11-2 A solid circular bar having diameter \(d\) is to be replaced by a rectangular tube having cross-sectional dimensions \(d \times 2d\) to the median line of the cross section (see figure). Determine the required thickness \(t_{\text{min}}\) of the tube so that the maximum shear stress in the tube will not exceed the maximum shear stress in the solid bar.