Components of the stress tensor $\sigma$ at a point, with respect to the
$(x, y, z)$ coordinate system, are given as
$\begin{bmatrix} 18 & 24 & 0 \\ 24 & 32 & 0 \\ 0 & 0 & -20 \end{bmatrix}$ (MPa).
a) Find the principal stresses $\sigma_i$ and the unit vectors $e_i$ along the principal
directions. Arrange $\sigma_i$ such that $\sigma_1 \ge \sigma_2 \ge \sigma_3$. Further, choose the senses of $e_i$
such that $e_1 \cdot e_2 \times e_3 = +1$.
b) Find the hydrostatic and deviatoric parts of $\sigma$.
(c) Find the maximum shear stress $\sigma_{s\text{ max}}$ and the normal to the planes on
which $\sigma_{s\text{ max}}$ acts. Express the normal in terms of the unit vectors $(i, j, k)$.
