고체역학 (Gere) — Stresses in Beams (Basic Topics): Problem 5.7-3

5.7-3 A tapered cantilever beam AB with rectangular cross sections is subjected to a concentrated load \( P = 220 \text{ N} \) and a couple \( M_0 = 90 \text{ N}\cdot\text{m} \) acting at the free end (see figure part a). The width \( b \) of the beam is constant and equal to \( 25 \text{ mm} \), but the height varies linearly from \( h_A = 50 \text{ mm} \) at the loaded end to \( h_B = 75 \text{ mm} \) at the support. (a) At what distance \( x \) from the free end does the maximum bending stress \( \sigma_{\text{max}} \) occur? What is the magnitude \( \sigma_{\text{max}} \) of the maximum bending stress? What is the ratio of the maximum stress to the largest stress \( \sigma_B \) at the support? (b) Repeat part a if, in addition to \( P \) and \( M_0 \), a triangular distributed load with peak intensity \( q_0 = 3P/L \) acts upward over the entire beam as shown in the figure part b. What is the ratio of the maximum stress to the stress at the location of maximum moment?