고체역학 (Beer) — Shearing Stresses in Beams: Problem 6.19

6.19 A timber beam \(AB\) of length \(L\) and rectangular cross section carries a single concentrated load \(P\) at its midpoint \(C\). (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 \(h/2L\), 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 \(L = 2 \text{ m}\), \(P = 40 \text{ kN}\), \(\tau_m = 960 \text{ kPa}\), and \(\sigma_m = 12 \text{ MPa}\).