Problem $2($Part $2-$Stresses$)$

A circular shaft of diameter $D=1\frac{1}{8}$ in $(1\mathrm{.}125$ in $.)$ is made of AISI $1045$ CD$.$ It has a notch of radius $\frac{1}{16}$ in$.$ at the midway point. The shaft is subjected to a steady torque of

$1200$ lbf $-$in$.$ The forces on the shaft in the xy and xz planes are shown in the figure below. Design for a reliability of $99\%.$

$($i$)$ Determine the moments and torques $($Ma$,$ Mm; Ta$,$ Tm$)$ at the critical location.

$($ii$)$ Determine the stresses ${}_a{}_m{}_a{}_m$ after determining the stress concentration factors.

$($iii$)$ Determine von$-$Mises stresses ${}^2$ and ${}^{\text{'}}$

$($iv$)$ Determine equivalent completely reversed stress ${}_{ar}=\frac{a^{\text{'}a}}{1-\frac{{}^{\text{'}m}}{S_{ut}}}$