Question $3.$ Numerical

a$)$ Draw Mohr's circle to obtain principal stresses and orientation of the principal planes for the

state of stress shown below for a material with yield strength of $240$ MPa :

b$)$ Find out the normal stresses $({}_{z^{\text{'}}},{}_{y^{\text{'}}})$ on a plane inclined at $30$ to the mentioned plane.

c$)$ Based on the values given and obtained in $($a$)$ and $($b$),$ determine First Invariant of Stress, $I_1$

d$)$ Draw another Mohr's Circle if additional compressive stress of $80$ MPa is incorporated in thiid

orthogonal direction.

e$)$ Considering the Mohr's circle in $($d$),$ evaluate whether the $3-$dimensional state of stress will

yield the material using both criteria.