$7-5$ A rotating step shaft is loaded as shown, where the forces $F_A$ and $F_B$ are constant at $600$ lbf and $300$ lbf $,$ respectively, and the torque $T$ alternates from $0$ to $1800lbf*in.$ The shaft is to be considered simply supported at points $O$ and $C,$ and is made of AISI $1045$ CD steel with a fully corrected endurance limit of $S_e=40k.$ Let $K_f=2\mathrm{.}1$ and $K_{fj}=1\mathrm{.}7.$ For a design factor of $2\mathrm{.}5$ determine the minimal acceptable diameter of section $BC$ using the

$($a$)$ DE$-$Gerber criterion.

$($b$)$ DE$-$Goodman criterion.