Problem $2)$ In the figure shown, shaft A$,$ made of AISI $1020$ hot$-$rolled steel, is welded to a fixed support and is subjected to loading by equal and opposite forces $\backslash ($ F $\backslash )$ via shaft $\backslash ($ B $\backslash ).$ A theoretical stress$-$concentration factor $\backslash ($ K$\_\{$t s$\}\backslash )$ of $1\mathrm{.}6$ is induced in the shaft by the $\backslash (\backslash$frac$\{1\}\{8\}\backslash )-$in weld fillet. The length of shaft $\backslash ($ A $\backslash )$ from the fixed support to the connection at shaft $\backslash ($ B $\backslash )$ is $2$ ft $.$ The load $\backslash ($ F $\backslash )$ cycles from $150$ to $500$ lbf $.$

$($a$)$ For shaft A$,$ find the factor of safety for infinite life using the Goodman fatigue failure criterion.

$($b$)$ Repeat part $($a$)$ using the Gerber fatigue failure criterion.