Question $3$: $(20$ points$)$

Figure $3$ shows a beam system consisting of two beams AB and BD$.$ Beam AB is fixed at the left end A$.$ Beam BD is supported by a roller at the right end. Beams AB and BD are connected by a hinge at point B$.$ Beams AB and BD have the same constant Young's modulus E and the same joint at point I. Point C is located in the middle of beam BD$.$ At point C$,$ a downward load P is applied. In addition, between point B and point C$,$ a distributed load downward load b$.$ Applying the energy methods $($virtual work or Castigliano's theorem$),$ find the vertical displacement ov and the slope e at point C $($ignore the deformation caused by the effects of the shear force, express your results using E$,$ I, P$,$ q$).$