A solid round bar with diameter of $50$ mm has a groove cut to a diameter of $45$ mm$,$ with a

radius of $2\mathrm{.}5$ mm$.$ The bar is not rotating. The bar is loaded with a repeated bending load

that causes the bending moment at the groove to fluctuate between $0$ and $2825$ Nm$.$ The

bar is hot$-$rolled AISI $1095,$ but the groove has been machined. Determine the factor of

safety for fatigue based on infinite life and the factor of safety for yielding $($a$)$ Using

Modified Goodman; $($b$)$ ASME Elliptic.