Stress values for an $x($horizontal$)-y($vertical$)$ axis system are shown in the three figures below.

Fig. HW$3-0$

Fig. HW$3-1$

Fig. HW$3-2$

Fig. HW$3-3$

For the plane stress state shown in Figure HW$3-1,$ use the analytical in$-$plane rotation$-$of$-$axes

solutions $($e$.$g$.,$ Handout $2)$ to determine $($and show$)$ the following:

a$.$ the $3-$D stress matrix for the state of stress relative to the $x-y-z$ axes shown, above left.

b$.$ the stress element on which the $3$ principal stresses act, showing its orientation $($angle$)$ relative to the

original $x-y$ axes

c$.$ the stress element on which the maximum shear stress in the $x-y$ plane acts $($including the normal stresses

acting on the element$),$ showing the element's orientation relative to the original $x-y$ axes.

d$.$ The value of the maximum shear stress in $3-$D