While the exact distribution of the shearing stresses in a

Chapter 3, Problem 3.29

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While the exact distribution of the shearing stresses in a hollow-cylindrical shaft is as shown in Fig. P3.29a, an approximate value can be obtained for \(\tau_{\max }\) by assuming that the stresses are uniformly distributed over the area A of the cross section, as shown in Fig. P3.29b, and then further assuming that all of the elementary shearing forces act at a distance from O equal to the mean radius \(\frac{1}{2}\left(c_{1}+c_{2}\right)\) of the cross section. This approximate value is \(\tau_{0}=T / A r_{m}\), where T is the applied torque. Determine the ratio \(\tau_{\max } / \tau_{0}\) of the true value of the maximum shearing stress and its approximate value \(\tau_{0}\) for values of \(c_{1} / c_{2}\) respectively equal to 1.00, 0.95, 0.75, 0.50, and 0.