Exercise $3.(40\%)$ The two bars of the figure $($of cross section $10mm10mm)$ are initially separated by a gap of $0\mathrm{.}15$ mm $.$ The initial temperature of $30$ C is raised to $100$ C FOR THE ALUMINUM BAR AB ONLY. $.*$ there is NO external force applied but the two reactions at A and C $($you cannot "Invent" one extra external load, the ONLY external input is the temperature change, and reactions at A and $B$ are the only forces$).$ YOU MUST $($in order, complete work$)$:

$($i$)$ Sketch a complete Free Body diagram to compute the force reactions at the rigid supports A and $C$ because of the temperature change.

$($ii$)$ Write a complete equilibrium equation based on $($i$)$

$($iii$)$ Sketch a complete schematics of deformations

$($iv$)$ Write a complete compatibility equation based on $($iii$).$ It must include the given material$/$dimensions data, just symbols is not enough. REQUIRED TO COMPUTE REACTIONS VALUES after $($iv$)($If done, it will not count for grade$)$

bar BC: steel

${}_{Al}=23{10}^{-6}\frac{1}{C}$

$E_{Al}=70$GPa

$E_{St}=200$GPa

$q,F_A+F_B=0$