P$2$:

An unnotched solid beam with a square cross$-$section is made of

SAE $4142$ steel $(450$ HB $)-$ See Table $9\mathrm{.}1($Appendix$).$ The beam is

loaded in cyclic bending between $M_{\text{m}\text{i}\text{n}}=0$ and $M_{\text{m}\text{a}\text{x}}=(20)kN.$

m $.$ The expected service life is $130,000$ cycles. Determine length

$a$ of the cross$-$section if the safety factor on life is equal to $18.$

Assume that Goodman's rule applies to this problem.