A tall signboard is supported by two vertical beams consisting of thin-walled, tapered circular tubes [see figure]. For purposes of this analysis, each beam may be represented as a cantilever AB of length L = 8.0 m subjected to a lateral load P = 2.4 kN at the free end. The tubes have constant thickness t = 10.0 mm and average diameters dA = 90 mm and dB = 270 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 = pd3t/8 (see Case 22, Appendix D), and therefore, the section modulus may be obtained from the formula S = pd2t/4.
(a) At what distance x from the free end does the maximum bending stress occur? What is the magnitude σmax of the maximum bending stress? What is the ratio of the maximum stress to the largest stress σB at the support?
(b) Repeat (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. What is the ratio of the maximum stress to the stress at the location of maximum moment?

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P 24 kN Wind A load L=8.0 m t 10.0 mm da=90 mm dB-270 mm 2P トーーーL=8.0 m