고체역학 (Gere) — Torsion: Problem 3.11-6

3.11-6 A thin-walled circular tube and a solid circular bar of the same material (see figure) are subjected to torsion. The tube and bar have the same cross-sectional area and the same length. What is the ratio of the strain energy \(U_1\) in the tube to the strain energy \(U_2\) in the solid bar if the maximum shear stresses are the same in both cases? (For the tube, use the approximate theory for thin-walled bars.)