Rod $\backslash ($ A B $\backslash )$ has a solid circular cross$-$section with radius $\backslash ($ R$=5\backslash$mathrm$\{$~mm$\}\backslash ).$ It is fixed at its ends, subjected to an axial load $\backslash ($ P $\backslash )$ as shown in the following figure and made of an elastic perfectly plastic material having $\backslash (\backslash$sigma $\backslash$mathrm$\{$y$\}=420\backslash$mathrm$\{$MPa$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$E$\}=70\backslash )$ GPa. If a force $\backslash ($ P$=50\backslash$mathrm$\{$kN$\}\backslash )$ is applied to the rod at noint $\backslash ($ C $\backslash )$ and then removed

a$)$ Determine the residual stress in each segment.

b$)$ Find the permanent deformation at point $\backslash ($ C $\backslash ).$

c$)$ Determine the smallest load $\backslash ($ P $\backslash )$ needed to cause segment CB to yield.