Chapter 4, Problem 45P

The designer of a shaft usually has a slope constraint imposed by the bearings used. This limit will be denoted as ξ. If the shaft shown in the figure is to have a uniform diameter d except in the locality of the bearing mounting, it can be approximated as a uniform beam with simple supports. Show that the minimum diameters to meet the slope constraints at the left and right bearings are, respectively,

$d_L={\left|\frac{32Fb\left(l^2-b^2\right)}{3\pi El\xi }\right|}^{1/4}{d}_R={\left|\frac{32Fb\left(l^2-a^2\right)}{3\pi El\xi }\right|}^{1/4}$

Problem 4–45