Q$4$

A solid titanium alloy right$-$angled rod $\backslash ($ A B C $\backslash )($Fig$.$ Q$4)$ is fixed into a wall at one end $\backslash (($A$)\backslash )$ and free at the other end $\backslash (($C$)\backslash )$ as shown. A vertical load of $800$ N $($in the negative $\backslash ($ z $\backslash )-$direction$)$ and a horizontal load of $500$ N $($in the positive $\backslash ($ x $\backslash )-$direction$)$ is applied at the free end as shown. The rod has a constant diameter of $1\mathrm{.}5$ cm $.$ The titanium alloy comprising the rod has a tensile yield strength of $1000$ MPa $.$ The point $\backslash ($ A $\backslash )$ is on the top of the outside surface of the rod and located as indicated in Fig. Q$4.$

$($a$)$ Determine the full $3$D stress tensor representing the state of stress at point $\backslash ($ A $\backslash )$ with respect to the $\backslash ($ x$,$ y$,$ z $\backslash )$ axes shown in Fig. Q$4.$

$[15]$

$($b$)$ Determine the $3$D principal stresses $\backslash (\backslash$left$(\backslash$sigma$\_\{1\},\backslash$sigma$\_\{2\}\backslash$right$.\backslash )$ and $\backslash (\backslash$left$.\backslash$sigma$\_\{3\}\backslash$right$)\backslash )$ and the maximum shear stress in the $\backslash ($ x $\backslash )-\backslash ($ y $\backslash )$ plane at point $\backslash ($ A $\backslash ).$

$($c$)$ Holding the $800$ N load constant, how much can the $500$ N load be increased before yield occurs at point $\backslash ($ A $\backslash )$ according to the Von Mises yield criterion?

$[8]$