Problem $2.$ A simply supported shaft is shown in the following figure. A constant magnitude transverse load of $400$ lb is applied as the shaft rotates subjected to a time varying torque that varies from $-500$ to $500$ lb $.$in$.$ Using ASME distortional energy elliptical theory, find the diameter of the shaft required to obtain a safety factor of $2$ in fatigue loading for an infinite life if the shaft is steel with $S_{ut}=108k,$ and $S_y=62k.$ Assume no stress concentration and $50\%$ reliability.

$(30$ marks$)$