Problem $1(20$ points$).$ The figure below shows a hand crank. It has a circular cross section of $5$ cm

diameter and is loaded with a force $F$ in the negative $z-$direction. Determine the principal stresses and

maximum shear stress at the critical location of the hand crank. Your answers should be functions of $F.$

$($Hint: This is a combined stress problem. The system is subjected to both bending and torsion. Bending

moment is maximum at the fixed end. Consider two plane stress elements: One at the top, another one at

the neutral axis. Compute the principal stresses and maximum shear stresses for both elements, and decide

which element is critical.$)$