Part A $-$ Stress Transformation

Q$1.($a$)$ Draw $3$d state of stress at a point in material using a $3$D$-$ element. Show all components of the stress on three mutually perpendicular planes.

$($b$)$ Drive general equations for plain stress transformation assuming an anti$-$clockwise rotation $$ of the element around the martial point subjected to stresses ${}_x,{}_{xy},{}_y$ as shown.

$($c$)$ What are principal stresses and principal planes? and what is the value of shear stresses bon principal plane?

$($d$)$ What is the mutual angle of inclination $\frac{b}{w}$

$(1)$ principal planes and $(2)$ principal plane and shear plane. Justify your answer using equations.