고체역학 (Gere) — Stresses in Beams (Basic Topics): Problem 5.7-2

5.7-2 A tall signboard is supported by two vertical beams consisting of thin-walled, tapered circular tubes (see figure part a). For purposes of this analysis, each beam may be represented as a cantilever AB of length \( L = 8.0 \text{ m} \) subjected to a lateral load \( P = 2.4 \text{ kN} \) at the free end. The tubes have a constant thickness \( t = 10.0 \text{ mm} \) and average diameters \( d_A = 90 \text{ mm} \) and \( d_B = 270 \text{ mm} \) at ends A and B, respectively. Because the thickness is small compared to the diameters, the moment of inertia at any cross section may be obtained from the formula \( I = \pi d^3 t / 8 \) (see Case 22, Appendix E); therefore, the section modulus may be obtained from the formula \( S = \pi d^2 t / 4 \). (a) At what distance \( x \) from the free end does the maximum bending stress occur? 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 concentrated load \( P \) is applied upward at A and downward uniform load \( q(x) = 2P/L \) is applied 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?