고체역학 (Gere) — Stresses in Beams (Basic Topics): Problem 5.7-4

5.7-4 The spokes in a large flywheel are modeled as beams fixed at one end and loaded by a force \(P\) and a couple \(M_0\) at the other (see figure). The cross sections of the spokes are elliptical with major and minor axes (height and width, respectively) having the lengths shown in the figure part a. The cross-sectional dimensions vary linearly from end A to end B. Considering only the effects of bending due to the loads \(P\) and \(M_0\), determine the following quantities. (a) The largest bending stress \(\sigma_A\) at end A. (b) The largest bending stress \(\sigma_B\) at end B. (c) The distance \(x\) to the cross section of maximum bending stress. (d) The magnitude \(\sigma_{max}\) of the maximum bending stress. (e) Repeat part d if a uniform load \(q(x) = 10P/3L\) is added to loadings \(P\) and \(M_0\), as shown in the figure part b. Given data: - Length \(L = 1.25 \text{ m}\) - At end A (free end): \(h_A = 90 \text{ mm}\), \(b_A = 60 \text{ mm}\) - At end B (fixed end): \(h_B = 120 \text{ mm}\), \(b_B = 80 \text{ mm}\) - Load \(P = 12 \text{ kN}\) (downward) - Moment \(M_0 = 10 \text{ kN}\cdot\text{m}\) (clockwise at end A)