고체역학 (Gere) — Torsion: Problem 3.11-10

3.11-10 Compare the angle of twist \(\phi_1\) for a thin-walled circular tube (see figure) calculated from the approximate theory for thin-walled bars with the angle of twist \(\phi_2\) calculated from the exact theory of torsion for circular bars. (a) Express the ratio \(\phi_1 / \phi_2\) in terms of the nondimensional ratio \(\beta = r/t\). (b) Calculate the ratio of angles of twist for \(\beta = 5, 10, \text{ and } 20\). What conclusion about the accuracy of the approximate theory do you draw from these results?