Hints:

You must solve the problem iteratively. $1)$ Guess a diameter; $2)$ Solve for the factor of safety for yield and fatigue; $3)$ Compare to required factor of safety; $4)$ Make shaft larger or smaller and resolve.

$2$ design iterations are acceptable for the problem.

Problem:

Bearing reactions $R_1$ and $R_2$ are exerted on the shaft shown in the figure, which rotates at $950$ rev$/$min and supports an $8-$kip bending force. Use a $1095$ HR steel. The surfaces are machined.

Specify a diameter $d$ using a design factor of $n_d=1\mathrm{.}6$ for a life of $10$ hr $.$