$2\mathrm{.}5-6$ A plastic bar ACB having two different solid circular cross sections is held between rigid supports as shown in the figure. The diameters in the left$-$ and right$-$hand parts are $50$ mm and $75$ mm $,$ respectively. The corresponding lengths are $225$ mm and $300$ mm $.$ Also, the modulus of elasticity $E$ is $6\mathrm{.}0$ GPa $,$ and the coefficient of thermal expansion $$ is $100\frac{{10}^{-6}}{}C.$ The bar is subjected to a uniform temperature increase of $30C.$

$($a$)$ Calculate the following quantities: $(1)$ the compressive force $N$ in the bar; $(2)$ the maximum compressive stress ${}_c$; and $(3)$ the displacement ${}_C$ of point $C.$

$($b$)$ Repeat part $($a$)$ if the rigid support at $A$ is replaced by an elastic support having spring constant $k=50M\frac{N}{m}($see figure part b ; assume that only the bar ACB is subject to the temperature increase$).$

PROB. $2\mathrm{.}5-6$