고체역학 (Gere) — Stresses in Beams (Basic Topics): Problem 5.7-1

5.7-1 A tapered cantilever beam $AB$ of length $L$ has square cross sections and supports a concentrated load $P$ at the free end (see figure part a). The width and height of the beam vary linearly from $h_A$ at the free end to $h_B$ at the fixed end. Determine the distance $x$ from the free end $A$ to the cross section of maximum bending stress if $h_B = 3h_A$. (a) What is the magnitude $\sigma_{max}$ 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 load $P$ is now applied as a uniform load of intensity $q = P/L$ over the entire beam, $A$ is restrained by a roller support, and $B$ is a sliding support (see figure part b). What is the ratio of the maximum stress to the stress at the location of maximum moment?