The bar AB$,$ fixed at both ends, has a cross$-$sectional area of $1200$ mm$2$ and is made of a steel which is assumed to be elastic, perfectly plastic with Young$$s modulus E $=200$ GPa and yield strength Sy $=250$ MPa.

If a force F of $520$ kN is applied and then removed, determine the residual stress in the bar and the permanent displacement of Point C$.$

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