$(20$ points$)$ 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 between $0$ and $1800in-lbf.$ 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 $K_{fs}=1\mathrm{.}7.$ For a design factor of $2\mathrm{.}5,$ determine the minimal acceptable diameter of section BC using the Goodman criterion.