고체역학 (Gere) — Stresses in Beams (Basic Topics): Problem 5.5-12

5.5-12 A railroad tie (or sleeper) is subjected to two rail loads, each of magnitude \( P = 175 \text{ kN} \), acting as shown in the figure. The reaction \( q \) of the ballast is assumed to be uniformly distributed over the length of the tie, which has cross-sectional dimensions \( b = 300 \text{ mm} \) and \( h = 250 \text{ mm} \). Calculate the maximum bending stress \( \sigma_{\max} \) in the tie due to the loads \( P \), assuming the distance \( L = 1500 \text{ mm} \) and the overhang length \( a = 500 \text{ mm} \).