oblom $2(12/35$ points$)$

At a machined rotating shaft shoulder, the small diameter $d=30mm,$ the large diameter $D=44mm,$ and the fillet radius is $r=3mm.$ The shaft is subjected to a repeated bending moment of $125N-m,$ at an operating temperature of $400C.$ The shaft has been machined from AISI$-1050$ Cold Drawn material $(S_{ut}=690$MPa;$S_y=580$MPa $).$ The reliability goal is $99\%.$

a$)$ Estimate the endurance limit of the shaft.

b$)$ Determine the fatigue factor$-$of$-$safety using the Modified Goodman Criteria.

c$)$ If the shaft has an initial tensile stress at the shaft shoulder of $20$ MPa $,$ introduced during manufacturing, what would be the new factor$-$of$-$safety against fatigue failure using the Modified Goodman Criteria?