$3.$ A critical stress element has a nominal bending stress cycling between $-20$ kpsi $($compression$)$ and $80$ kpsi $($tension$).$ There are no axial, torsion or transverse shear stresses acting on the element. There is a bending fatigue stress concentration factor of $\backslash (\backslash$mathrm$\{$K$\}\_\{\backslash$mathrm$\{$f$\}\}=1\mathrm{.}5\backslash )$ that needs to be applied to the bending stresses. The completely adjusted endurance limit for the material is $\backslash (\backslash$mathrm$\{$Se$\}=40\backslash$mathrm$\{$kpsi$\}\backslash ).$ Material properties are $\backslash (\backslash$mathrm$\{$Sy$\}=100\backslash$mathrm$\{$kpsi$\}\backslash )$ and Sut $=140$ kpsi $.$

$($a$)$ Determine the fatigue factor of safety using the Goodman criteria.

$($b$)$ Estimate the fatigue life in cycles using the Goodman criteria. $($Go through the process of estimating fatigue life even if the safety factor in $($a$)$ is $1\mathrm{.}0$ or greater to show me you know how to do it$.)$