A steel bar ABC having two different dimensions is held between rigid supports, as shown in the Figure Q$3($a$).$ The modulus of elasticity is $200$ GPa $,$ and the coefficient of thermal expansion is $11\mathrm{.}7\frac{{10}^{-6}}{}C.$ The bar is subjected to a uniform temperature increase of $50C.$

$($a$)$ Determine the following quantities:

i$.$ the compressive force $P$ in the bar;

ii$.$ the maximum compressive stress ${}_B$ at point $B$;

iii. the displacement ${}_B$ of point $B.$

$($CO$2$: PO$1-19$ Marks$)$

Figure Q$3($a$)$

$($b$)$ If the rigid support at A is replaced by an elastic support having spring constant $k=50M\frac{N}{m},$ as shown in Figure $3($b$).$ Determine the following quantities:

i$.$ the compressive force $P$ in the bar;

ii$.$ the maximum compressive stress ${}_B$ at point $B$;

$($CO$2$: PO$1-6$ Marks$)$

Figure Q$3($b$)$