Problem $3-72*$

A gear reduction unit uses the countershaft shown in the figure. Gear A receives power from another gear with the transmitted force $F_A$ applied at the $20$ pressure angle as shown. The power is transmitted through the shaft and delivered through gear $B$ through a transmitted force $F_B$ at the pressure angle shown.

$($a$)$ Determine the force $F_B,$ assuming the shaft is running at a constant speed.

$($b$)$ Find the magnitudes of the bearing reaction forces, assuming the bearings act as simple supports.

$($c$)$ Draw shear$-$force and bending$-$moment diagrams for the shaft. If needed, make one set for the horizontal plane and another set for the vertical plane.

$($d$)$ At the point of maximum bending moment, determine the bending stress and the torsional shear stress.

$($e$)$ At the point of maximum bending moment, determine the principal stresses and the maximum shear stress.