$6-57$ A schematic of a clutch$-$testing machine is shown. The steel shaft rotates at a constant speed $.$ Anaxial load is applied to the shaft and is cycled from zero to $P.$ The torque $T$ induced by the clutch face onto the shaft is given by

$T=\frac{fP(D+d)}{4}$

where $D$ and $d$ are defined in the figure and $f$ is the coefficient of friction of the clutch face. The shaft is machined with $S_y=120k$ and $S_{ut}=145k.$ The theoretical stress$-$concentration factors for the fillet are $3\mathrm{.}0$ and $1\mathrm{.}8$ for the axial and torsionabloading, respectively.

Assume the load variation $P$ is synchronous with shaft rotation. With $f=0\mathrm{.}3,$ find the maximum allowable load $P$ such that the shaft will survive a minimum of ${10}^6$ cycles with a factor of safety of $3.$ Use the Goodman criterion Check for yielding.

Proble