A solid shaft is mounted on smooth ball bearings at A and B and is made of an AISI $1035$ CD steel that has a minimum yield strength of $S_y=460$MPa. The shaft is carrying pulleys C and D with the shown belt tensions acting on the pulleys. The magnitudes and directions of the support forces, the resultant transverse forces and torque loads on the shaft are given in the $3$D FBD of the belt drive system shown. Assuming a design factor of $2\mathrm{.}0$ and considering stresses due to bending and torsion:

a$)(15$ points$)$ identify the most critical section for design $($draw clear xy and xz plane shear force, bending moment and torque diagrams$)($Choose origin at A$,$ y $-$axis vertical and x $-$axis along the shaft$)$

b$)(15$ points$)$ determine the minimum shaft diameter to avoid yielding using: $($i$)$ Distortion Energy $($DE$)$ theory, and $($ii$)$ Maximum shear stress $($MSS$)$ theory