고체역학 (Beer) — Shearing Stresses in Beams: Problem 6.20

6.20 A timber beam $AB$ of length $L$ and rectangular cross section carries a uniformly distributed load $w$ and is supported as shown. (a) Show that the ratio $\tau_m / \sigma_m$ of the maximum values of the shearing and normal stresses in the beam is equal to $2h/L$, where $h$ and $L$ are, respectively, the depth and the length of the beam. (b) Determine the depth $h$ and the width $b$ of the beam, knowing that $L = 5\text{ m}$, $w = 8\text{ kN/m}$, $\tau_m = 1.08\text{ MPa}$, and $\sigma_m = 12\text{ MPa}$.