Solved: A beam carrying a uniform load is simply supported

Chapter 3, Problem 3-14

(choose chapter or problem)

A beam carrying a uniform load is simply supported with the supports set back a distance a from the ends as shown in the figure. The bending moment at x can be found from summing moments to zero at section x:

\(\sum M=M+\frac{1}{2} w(a+x)^2-\frac{1}{2} w l x=0\)

or

\(M=\frac{w}{2}\left[l x-(a+x)^2\right]\)

where w is the loading intensity in lbf/in. The designer wishes to minimize the necessary weight of the supporting beam by choosing a setback resulting in the smallest possible maximum bending stress.

(a) If the beam is configured with \(a=2.25 \mathrm{in}, l=10 \mathrm{in}\), and \(w=100 \mathrm{lbf} / \mathrm{in}\), find the magnitude of the severest bending moment in the beam.

(b) Since the configuration in part (a) is not optimal, find the optimal setback a that will result in the lightest-weight beam.