Q$1)(45$ points$)$ The stresses $($in MPa$)$ with respect to $x,y,z$ coordinate system in absence of body forces are described by

${}_x=a{x}^2+3y^2-5z,{}_y=c{y}^2,$

${}_z=3x+y+bz-5,{}_{xy}=z+4xy-7,$

${}_{xz}=-3x+y+1,{}_{yz}=0$

Determine:

$($a$)$ the values of constants $a,b,$ and $c,$

$($b$)$ the stress components at point A with coordinates $(3,1,5),$

$($c$)$ the principal stresses and the related direction cosines at point A $,$

$($d$)$ the normal stress $$ and the shearing stress $$ at point A on a plane whose outward normal is oriented at angles of $40$ and $75$ with the $x$ and $y$ axes, respectively.

$($e$)$ the stress components with respect to $x^{\text{'}},y^{\text{'}},z^{\text{'}}$ at point A if$,$

$m_1=\frac{1}{2},n_1=\frac{\sqrt[\phantom{2}]{3}}{2},l_2=1,m_3=-\frac{\sqrt[\phantom{2}]{3}}{2},n_3=\frac{1}{2}$