The stress tensor in MPa for a three$-$dimensional system for some $x,y,z$ axis system is

$\left[\begin{array}{ccc}20& 40& -30\\ 40& 30& 25\\ -30& 25& -10\end{array}\right]$

$($a$)$ Confirm that one of the principal stresses is $65\mathrm{.}333$MPa, hence find the other principal stresses.

$(12$ marks$)$

$($b$)$ Determine the direction cosines corresponding to the principal stress $65\mathrm{.}333$MPa.

$(6$ marks$)$

$($c$)$ Sketch the $3$D Mohr's Circles and determine the maximum shear stress.

$(3$ marks$)$

$($d$)$ The yield stress for the material is $250$MPa and a safety factor of $2\mathrm{.}0$ is required for the design. Determine whether the component pass the design criteria according to i$)$ the Maximum Principal Stress criterion and ii$)$ the Von Mises criterion.

$(5$ marks$)$