A rotating solid shaft is made of cold$-$drawn steel. The material properties of the shaft is $S_{ut}=500$MPa, $S_y=400$MPa, a fully corrected endurance limit of $S_e=200$MPa, and a fatigue stress$-$concentration factor $K_f$ is $1\mathrm{.}44.$ If the load line slope $r=0\mathrm{.}6,$ use the diagram below to:

a$)$ Draw the Yield $($Langer$)$ line, Modified Goodman line, and the load line.

b$)$ Determine the alternating strength $S_a$ and the midrange strength $S_m$

c$)$ Find ${}_a$ and ${}_m$ if the factor of safety is $n_f=1\mathrm{.}5$

d$)$ For stresses in part c$,$ find ${}_{\text{m}\text{a}\text{x}}$ and ${}_{\text{m}\text{i}\text{n}}$