Question $1$:

The stress tensor ${}_{ij}$ at a point in a body is given by

${}_{ij}=\left[\begin{array}{ccc}0& 0& 0\\ 0& 300& 100\sqrt[\phantom{2}]{3}\\ 0& 100\sqrt[\phantom{2}]{3}& 100\end{array}\right](stress$ units$)$

Find:

$($a$)$ the magnitude of the stress vector on an area element whose unit

normal vector is given by $n=(\frac{1}{2},\frac{1}{2},\frac{1}{\sqrt[\phantom{2}]{2}}).$

$($b$)$ The orientation of the principal axes of stresses.

$($c$)$ The magnitudes of the principal stresses.

$($d$)$ The magnitude of the octahedral stresses.

$($e$)$ The magnitudes of the maximum shear stresses.