Exercise $3$: Combine loading $(20$ points$)$

A cylindrical bar of diameter $20$ mm and length $100$ mm $($Figure l$)$ is subjected to a combined load $\backslash (\backslash$mathrm$\{$P$\}=8\mathrm{.}0\backslash )$ kN and $\backslash (\backslash$mathrm$\{$T$\}=30\backslash$mathrm$\{$~N$\}.\backslash$mathrm$\{$m$\}\backslash ).$ The material of the bar has a Young's modulus of $200$ GPa and a yield strength of $280$ MPa. We neglect the weight of the beam. Consider the cut section at $\backslash (\backslash$mathrm$\{$x$\}=0\backslash ),$ calculate the stress tensor, and verify whether plastic deformation Figure $1$ at these two points using Von Mises criterion:

a$.$ Point O at the center of the cross$-$section. $(10$ points$)$

b$.$ Point A on the top of the cross$-$section $(10$ points$)$