Beer & Johnston — Shearing Stresses in Beams: Problem 6.60

6.60 Consider the cantilever beam AB discussed in Sec. 6.5 and the portion ACKJ of the beam that is located to the left of the transverse section CC′ and above the horizontal plane JK, where K is a point at a distance \( y < y_Y \) above the neutral axis (Fig. P6.60). (a) Recalling that \( \sigma_x = \sigma_Y \) between C and E and \( \sigma_x = (\sigma_Y / y_Y)y \) between E and K, show that the magnitude of the horizontal shearing force H exerted on the lower face of the portion of beam ACKJ is \[ H = \frac{1}{2}b\sigma_Y \left( 2c - y_Y - \frac{y^2}{y_Y} \right) \] (b) Observing that the shearing stress at K is \[ \tau_{xy} = \lim_{\Delta A \to 0} \frac{\Delta H}{\Delta A} = \lim_{\Delta x \to 0} \frac{1}{b} \frac{\Delta H}{\Delta x} = \frac{1}{b} \frac{\partial H}{\partial x} \] and recalling that \( y_Y \) is a function of x defined by Eq. (6.14), derive Eq. (6.15).