$120$

E$){}_x=x^2+y,{}_y-y^2$;$,{}_z=-x+b_y+z$

At peint $(3,1,5)$

a$)$ Find the stress $x^{\text{'}},y^{\text{'}},z^{\text{'}}$

$2xy=2yz=2xz=0$

$l_1=1,m_2=\frac{1}{2},n_2=\sqrt[\phantom{2}]{\frac{3}{2}},n_3=\frac{1}{2},m_3-\sqrt[\phantom{2}]{\frac{3}{2}}$

$\left[\begin{array}{ccccccc}1& 0& 0& 112& -\sqrt[\phantom{2}]{312}& +\sqrt[\phantom{2}]{\frac{3}{2}}& \frac{1}{2}\end{array}\right]$

$0$

$0$

$0\left[\begin{array}{ccccccc}10& 0& 0& -4& 0& 0& 8\end{array}\right]$

$0$

$0\left[\begin{array}{ccccccc}1& 0& 0& 112& \sqrt[\phantom{2}]{312}& -\sqrt[\phantom{2}]{312}& \frac{1}{2}\end{array}\right]$

$0$

$0$

$[{}^i]=\left[\begin{array}{cccccc}10& 0& 0& 5& 3\sqrt[\phantom{2}]{3}& -1\end{array}\right]$

$0$

Find priniple stresses $($HW$3)$