$1.(25$ points$)$ An L $-$shaped bracket lying in a horizontal plane supports a load of $\backslash (\backslash$mathrm$\{$P$\}=800\backslash$mathrm$\{$~N$\}\backslash ),$ as shown in the figure. The bracket is comprised of a hollow rectangular cross$-$section with thickness $\backslash (\backslash$mathrm$\{$t$\}=4\backslash$mathrm$\{$~mm$\}\backslash )$ and outer dimensions $\backslash (\backslash$mathrm$\{$b$\}=60\backslash$mathrm$\{$~mm$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$h$\}=100\backslash$mathrm$\{$~mm$\}\backslash ).$ The centerline lengths of the bracket arms are $\backslash ($ b$\_\{1\}=600\backslash$mathrm$\{$~mm$\}\backslash )$ and $\backslash ($ b$\_\{2\}=750\backslash$mathrm$\{$~mm$\}\backslash ).$

a$.)$ Determine the total normal stress at point A on the cross$-$section at the support.

b$.)$ Determine the total shear stress at point A on the cross$-$section at the support.

$2.(20$ points$)$ If the same hollow rectangular cross section presented in Question $1$ is used to construct the $2-$m long cantilever beam shown below, and is made from an elastic$-$perfectly plastic material with a yield stress capacity of $\backslash (\backslash$sigma$\_\{$y$\}=500\backslash$mathrm$\{$MPa$\}\backslash )$ :

a$.$ Determine the plastic moment capacity of the hollow cross section.

b$.$ Determine the load $\backslash (\backslash$mathbf$\{$P$\}\backslash )$ that will produce fully plastic bending in the beam.

Beam Cross Section