Given a plate that is $0\mathrm{.}12$ in thick and constructed from high carbon steel, $490$ Brinell hardness, with a machined finish. Assume $\backslash (95\backslash \%\backslash )$ reliability and an operating temperature of $\backslash (250^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash ).($Hint: to find a relationship between Brinell Hardness and ultimate tensile strength, see Chapter $3$ of the textbook$).$

a$)$ Construct the $\backslash (\backslash$log $\backslash$mathrm$\{$S$\}\backslash )-\backslash (\backslash$log $\backslash$mathrm$\{$N$\}\backslash )$ curve.

b$)$ The loading is completely reversed $(\backslash (\backslash$sigma$\_\{\backslash$operatorname$\{$mean$\}\}=0\backslash )),$ and infinite life is desired. What alternating stress can be allowed with a safety factor of $2?$

c$)$ If the plate is loaded axially with a force $\backslash ($ F $\backslash ),$ that varies from $0$ to Fo$.$

d$)$ Sketch the $\backslash (\backslash$sigma $\backslash )$ vs$.$ time plot and find the equivalent mean and alternating stresses. Write the stresses as a function of Fo$.$

e$)$ Sketch the appropriate material fatigue failure diagram. Include numerical values, units, and axle labels.

f$)$ If it is desired to have a safety factor of $3,$ what is the loading Fo such that the plate has infinite life? Assume the loads increase proportionally