$(30$pts$)$ A bar with a square cross section, made of a cold$-$drawn steel $(S_{ut}=45k),$ is subject to an a tensile $(45001$b$)$ and compressive $(-5000$ lb $)$ loads. If the cross section is $2$ in $2$ in$,$ compute the following,

a$.(5$ pts $)$ Compute the fully corrected endurance limit$.$

b$.(5$ pts$)$ Compute the mean $({}_m)$ and alternating $({}_a)$ components of stress.

c$.(5$ pts$)$ Estimate equivalent completely reversed stress using modified Goodman criteria.

d$.(5$ pts$)$ What is the fatigue factor of safety $(n_f)?$ Is infinite life expected?

e$.(10$ pts$)$ How many stress cycles are expected until fatigue failure?