Q$1$x$,$y$,$z coordinate system in absence of

body forces are described by

$\backslash$sigma $\_($x$)=$ax$^(2)+3$y$^(2)-5$z$,\backslash$sigma $\_($y$)=$cy$^(2),$

$\backslash$sigma $\_($z$)=3$x$+$y$+$bz$-5,\backslash$tau $\_($xy$)=$z$+4$xy$-7,$

$\backslash$tau $\_($xz$)=-3$x$+$y$+1,\backslash$tau $\_($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 $\backslash$sigma and the shearing stress $\backslash$tau at point A on a plane whose outward normal is

oriented at angles of $40\backslash$deg and $75\backslash$deg 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$\_(1)=(1)/(2),$n$\_(1)=(\backslash$sqrt$(3))/(2),$l$\_(2)=1,$m$\_(3)=-(\backslash$sqrt$(3))/(2),$n$\_(3)=(1)/(2)$