Optional for problem practice: For the given $3$D stress state, the non$-$zero stresses are given:

${}_{}=150$MPa

${}_{yy}=70$MPa

${}_{zz}=-80$MPa

${}_{xy}=-45$MPa

a$.$ Sketch an appropriate $2$D state of stress $($clearly mark axes$)$ showing a logical plane

b$.$ Write the $3$D stress tensor $($matrix$)$

c$.$ Draw the $3$D Mohr's circle to scale

d$.$ Determine the ordered principal stresses

e$.$ Calculate the absolute maximum shear stress

f$.$ Determine ${}_{x^{\text{'}}{x}^{\text{'}}}$ and ${}_{x^{\text{'}}{y}^{\text{'}}}$ for the $x^{\text{'}}$ axis located $0\mathrm{.}7$ rad counterclockwise from the $x$ axis $($original

orientation$)$