Chapter 3.1, Problem 29P

Fig. P3.29

3.29 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 $\frac{1}{2}$ (c₁ + c₂) of the cross section. This approximate value is ${\tau }_0$ = T/Ar_m, w'here 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 c₁/c₂ respectively equal to 1.00, 0.95, 0.75, 0.50, and 0.