EXAMPLE $1\mathrm{.}6$ Three$-$Dimensional Stress in a Hub

A steel shaft is to be force fitted into a fixed$-$ended cast$-$iron hub. The

shaft is subjected to a bending moment $M,$ a torque $T,$ and a vertical

force $P,$ Fig. $1\mathrm{.}20$a$.$ Suppose that at a point $Q$ in the hub, the stress field is

as shown in Fig. $1\mathrm{.}20$b$,$ represented by the matrix

$\left[\begin{array}{ccc}-19& -4\mathrm{.}7& 6\mathrm{.}45\\ -4\mathrm{.}7& 4\mathrm{.}6& 11\mathrm{.}8\\ 6\mathrm{.}45& 11\mathrm{.}8& -8\mathrm{.}3\end{array}\right]$MPa

Determine the principal stresses and their orientation with respect to

the original coordinate system.