고체역학 (Beer) — Torsion: Problem 3.150

3.150 A hollow cylindrical shaft of length \(L\), mean radius \(c_m\), and uniform thickness \(t\) is subjected to a torque of magnitude \(T\). Consider, on the one hand, the values of the average shearing stress \(\tau_{ave}\) and the angle of twist \(\phi\) obtained from the elastic torsion formulas developed in Secs. 3.1C and 3.2 and, on the other hand, the corresponding values obtained from the formulas developed in Sec. 3.10 for thin-walled shafts. (a) Show that the relative error introduced by using the thin-walled-shaft formulas rather than the elastic torsion formulas is the same for \(\tau_{ave}\) and \(\phi\) and that the relative error is positive and proportional to the ratio \((t/c_m)^2\). (b) Compare the percent error corresponding to values of the ratio \(t/c_m\) of 0.1, 0.2, and 0.4.