Problem $3(35$ Points$)$

A rotating shaft of $20$ mm diameter is simply supported.

The shaft is loaded with a transverse load of $10$ kN as shown in the figure. The shaft is made from AISI $1095$ hot$-$rolled steel. The surface has been machined. The shaft operate at temperature $\backslash ($ T$=450^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).$ Consider a reliability factor of $\backslash (95\backslash \%\backslash ).$

Determine

$($a$)$ Calculate the reaction forces $\backslash ($ R$\_\{1\}\backslash )$ and $\backslash ($ R$\_\{2\}\backslash )(2$ points$)$

$($b$)$ Draw the shear force and bending moment diagrams and determine the maximum bending moment and shear force. $(6$ points$)$

$($c$)$ Determine the critical location of the shaft and the maximum effective stresses. $(3$ points$)$

$($d$)$ Calculate the safety factor against yielding. Does the shaft undergo local yielding? $(2$ points$)$

$($e$)$ Determined the endurance limit$,$ adjusted as necessary with Marin factors. $(12$ points$)$

$($f$)$ Calculate the fatigue factor of safety based on achieving infinite life. $(2$ points$)$

$($g$)$ If the fatigue factor of safety is less than $1($hint: it should be for this problem$),$ then estimate the life of the part in number of rotations $($hint: Consider the ultimate strength of material at $\backslash ($ T$=450^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).(8$ points$)$