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The rolating solid steel shaft is simply supported by bearings at points $B$ and $C$ and is driven by a gear $($not slhown$)$ which meshes with the spur gear at $D,$ which has a $150-$ mm pilch diameter. The force $F$ from the drive gear aces at a pressure angle of $20.$ The sloat transmits a torque to point A of $T_A=340N.$ mi$.$ For the shaft, $S_{ur}=560$MPa and $S_y=420$MPa, and a folly corrected endurance limit of $S_B=191$MPa is assumed Let $K_2=2\mathrm{.}2$ and $K_F=1\mathrm{.}9.$ Using a factor of safety of $2\mathrm{.}5$ determine the minimum allowable diameier of the $250-$mm section of the sluaft based on $($a$)$ a static yield analysis using the distortion energy theory and $($b$)$ a faigue$-$follure analysis.