A steel shaft is to be designed so that it is supported by roller bearings. The basic geometry is shown in the figure from Prob. 4€“45, with l = 300 mm, a = 100 mm, and F = 3 kN. The allowable slope at the bearings is 0.001 mm/mm without bearing life penalty. For a design factor of 1.28, what uniform-diameter shaft will support the load without penalty? Determine the maximum deflection of the shaft.
From Prob. 4€“45, 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,

1/4 1/4 32Fb(12-b?) 32 Fa2) dk