Question $1$:

Three forces are applied to the bar as shown in the figure below.

$($a$)$ Draw the stress element for a point $a$ on the surface of the bar showing all stress values on the element.

$($b$)$ Sketch a neat and fully labelled Mohr's circle for the element in $($a$)$ and determine the principal stresses and the maximum shear stress.

$($c$)$ Determine the factor of safety with respect to yield according to $($i$)$ the maximum$-$distortionenergy criterion, $($ii$)$ the maximum$-$shearing$-$stress criterion.