Beer & Johnston — Principal Stresses Under a Given Loading: Problem 8.21

8.21 It was stated in Sec. 8.2 that the shearing stresses produced in a shaft by the transverse loads are usually much smaller than those produced by the torques. In the preceding problems their effect was ignored, and it was assumed that the maximum shearing stress in a given section occurred at point $H$ (see Fig. P8.21a) and was equal to the expression obtained in Eq. (8.5), namely, \[ \tau_H = \frac{c}{J} \sqrt{M^2 + T^2} \] Show that the maximum shearing stress at point $K$ (see Fig. P8.21b), where the effect of the shear $V$ is greatest, can be expressed as \[ \tau_K = \frac{c}{J} \sqrt{(M \cos\beta)^2 + \left(\frac{2}{3}cV + T\right)^2} \] where $\beta$ is the angle between the vectors $\mathbf{V}$ and $\mathbf{M}$. It is clear that the effect of the shear $V$ cannot be ignored when $\tau_K \ge \tau_H$. (Hint: Only the component of $\mathbf{M}$ along $\mathbf{V}$ contributes to the shearing stress at $K$.)