A $400$ mm diameter solid propeller shaft has an output of $16$ MW$.$ When the forward speed of the vessel is $8\mathrm{.}33$ m$/$s$,$ the speed of the propeller is $120$ rpm$.$ The propeller has a mass of $12$ tons and overhangs the stern$-$tube by $2$ m$.$

Calculate:

$($a$)$ The max. shear stress due to the torque produced.

$($b$)$ The total compressive stresses due to the shaft thrust and bending.

$($c$)$ Construct Mohr's circle indicating the lines of action on the shaft of the stresses calculated above.

$($d$)$ For an element on the surface of the shaft find:

$($i$)$ The principle stresses and the planes on which they act.

$($ii$)$ The max. shear stresses and the planes on which they act.