Rod $AB$ has a solid circular cross$-$section with radius $R=5\mathrm{.}5mm.$ It is fixed at its ends, subjected to an axial load P as shown in the following figure and made of an elastic perfectly plastic material having $y=420$MPa and $E=70$GPa. If a force $P=60kN$ is applied to the rod at point $C$ and then removed.

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

b$)$ Find the permanent deformation at point C $.$

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