Figure $2$ shows a side view and an end view of a cantilevered rectangular beam with dimensions of $0\mathrm{.}1$ m long, $0\mathrm{.}12$ m high and $0\mathrm{.}05$ m wide. A $6$ kN point load is applied to the end of the beam as illustrated.

$($a$)$Calculate the normal and shear stresses in the X and Y directions at point A$.$

$($b$)$Calculate the principal stresses at point A$.$

$($c$)$ Calculate the orientation of the principal planes at point A $($with respect to the X axis$).$

$($d$)$Determine which principal stresses act on which principal planes.

$($e$)$Sketch the element at point A$,$ clearly showing the orientation of the principal planes and the magnitude and direction of the principal stresses.

$($f$)$ Calculate the maximum and minimum shear stresses at point A$.$

$($g$)$ What are the values of the normal stresses in the planes containing the maximum and minimum shear stresses at point A$?$