A mirror polished test specimen with material properties specified by Point $A$ in the Figure $1$ below is subjected to a purely alternating bending stress equal to $30$ ksi $.$ The fatigue strength fraction, f $,$ equals $0\mathrm{.}9$ for these test conditions. $(15$ points$)$

a$)$ Sketch the $S-N$ curve in Figure $2$ on your right

b$)$ Label the endurance limit for this material in Figure $2$

c$)$ Graphically estimate the life for the test specimen subjected to a purely alternating stress equal to $30$ ksi $.$

d$)$ Graphically estimate the life for an as$-$forged test specimen $(k_a=0\mathrm{.}75)$ of the same material subjected to a purely alternating stress equal to $20$ ksi $.$

Show your work for c $)$ and d$)$ on the graph in Figure $2$ below.

Stress$-$Life Diagram

Figure $2$

$($purely reversing bending$)$

Answer: c$)$ Life$=$

d$)$ Life $=q,$ cycles