Problem points$).$ The $($huthever bein $AB$ shown below is subjected to a uniformly distributed lond of intensity u acting downward. The length of the beam is $l.$ The end $B$ is supporled on a linear spring of stillness $k.WI$ for the beam is constant. Delermine Lhe dellechion of the beam al $13$ using Castigliano's theorem.

$($Hint: Lsolute the beam from the spring by replaing the ellect of the spring by a vertical reaction at $B,$ mit druw the frec body dimgram for the beam. Nole that lies beam is statieally imdetermimate $($do a litule seIf$-$shimy on shalically indelerminale systoms to learn more$),$ becanse there are threse reachions: $($wo at A and oment, B$,$ but only two cquilibrimm cquations are avilable. This, liere is one redimdant remelion, and so we sty that this givon bumin is stalically indetermimale lo lhe first degree. Any one ont of the threa reachions can le regarded ts the redmedunt revetion. Ollen, we seloet the renction$($s$)$ Lhat makes the system look rather Stranget wis the redumbnt$($