A structure is subjected to loading as shown in the figure. The E $($modulus of elasticity$)$ for the cross$-$section of the $$ shaped part of the structure is $4000$ ksi $($kips per square inch$),$ and the moment of inertia $($I$)$ is $1000$ in$^4.$ The stiffness of the springs is indicated in the figure.

$($a$)$ Calculate the reaction forces at points A and C$.$

$($b$)$ Calculate the vertical displacement at point D$.$

$($c$)$ If point A settles by $0\mathrm{.}5$ in before assembly, determine the reaction forces at points A and C after the assembly is completed.