$2\mathrm{.}4-7$ A tube structure is acted on by loads at $B$ and $D,$ as shown in the figure. The tubes are joined using two flange plates at $C,$ which are bolted together using six $0\mathrm{.}5-$in$.$ diameter bolts.

$($a$)$ Derive formulas for the reactions $R_A$ and $R_E$ at the ends of the bar.

$($b$)$ Determine the axial displacements ${}_B,{}_c,$ and ${}_D$ at points $B,C,$ and $D,$ respectively.