Question $3$

A solid round bar with a diameter of $50$ mm has a machined groove that is cut to a diameter of $45$ mm with a radius of $2\mathrm{.}5$ mm $.$ The bar does not rotate. The bar is cyclically loaded with a repeated bending load that causes the bending moment at the groove to fluctuate between $0$ and $3000$ N$.$m$.$ The bar is hot rolled AISI $1095.$

$($i$)$ Determine the factor of safety for fatigue based on infinite life using the modified Goodman criterion.

$($ii$)$ Compare the fatigue and static strength factors of safety.