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:

Or

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 in, l = 10 in, and w = 100 lbf/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

u(a +x) w, lbf/in