Problem $1(30$ points$).$ A gear reduction unit uses the countershaft shown in the figure below. 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 that the shaft is running at a constant speed.

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

$($c$)$ Draw the shear$-$force and bending$-$moment diagrams for the shaft. Make one set for the horizontal plane and another 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.

$($f$)$ Find the factors of safety against yielding at the critical location according to MSS and DE theories. The shaft is made of steel with a yield strength of $50$ ksi $.$

$($g$)$ Repeat part $($f$)$ graphically by plotting the yield envelopes using MATLAB for both theories. The MATLAB scripts posted on D$2$L may be used $($make necessary modifications$),$ or feel free to write your own codes. No need to attach the codes.