Problem $4.$

The rotating solid steel shaft is simply supported by bearings at points $B$ and $C$ and is driven by a gear $($not shown$)$ which meshes with the spur gear at $D,$ which has a $150-mm$ pitch diameter. The force $F$ from the drive gear acts at a pressure angle of $20.$ The shaft transmits a torque to point $A$ of $T_A=340N*m.$ The shaft is machined from steel with $S_y=420$MPa and $S_{ut}=560$MPa. Using a factor of safety of $2\mathrm{.}5,$ determine the minimum allowable diameter of the $250-mm$ section of the shaft based on $($a$)$ a static yield analysis using the distortion energy theory and $(b)$ a fatigue$-$failure analysis. Assume sharp fillet radii at the bearing shoulders for estimating stress$-$concentration factors.

Use Table $7-1$ to estimate the stress concentration factors $(K_t$ and $K_{ts})$ at the bearing shoulders

Table $7-1$

First Iteration Estimates for Stress$-$Concentration Factors $K_t$ and $K_{ts}.$

Warning: These factors are only estimates for use when actual dimensions are not yet determined. Do not use these once actual dimensions are available.

$\backslash$table$[[,$Bending,Torsional,Axial$],[$Shoulder fillet$-$sharp