Given a T$-$frame with width $\backslash ($ a$+2$ a $\backslash )$ and height $\backslash ($ b $\backslash ).$ A vertical force $\backslash ($ P $\backslash )$ is acting at the tip in $\backslash ($ D $\backslash ).$ All beams in the frame are linear elastic and can be considered as Timoshenko beams. Solve the following problems by the principle of virtual work.

$1)$ Determine the vertical displacement $\backslash ($ w$\_\{$D$\}\backslash )$ at the location of the force. $(5$p$)$

$2)$ Determine the rotation $\backslash (\backslash$theta$\_\{$C$\}\backslash )$ at the joint $\backslash ($ C $\backslash ).(5$p$)$

$3)$ Determine the vertical displacement $\backslash ($ w$\_\{$B$\}\backslash )$ at the left support in $\backslash ($ B $\backslash ).(5$p$)$