P$8\mathrm{.}27$ A solid steel shaft supports loads $P_A=250N$ and $P_C=620N$ as shown in

Figure P$8\mathrm{.}27.$ Assume that $a=500mm,b=700mm,$ and $c=600mm.$ The bearing at

$B$ can be idealized as a roller support, and the bearing at $D$ can be idealized as a pin

support. If the allowable bending stress is $105$MPa, determine the minimum diameter

that can be used for the shaft.

FIGURE P$8\mathrm{.}27-$