The two span beam ABCDE shown in Fig. $4$ has a hinge support at A and roller supports at B

and D$.$ The span BD carries a distributed load, which varies linearly from $0$ kN$/$m at B to

$30$ kN$/$m at D$.$ There is a concentrated moment of $50$ kNm at point C$.$ The flexural rigidity of

EI is constant throughout the beam.

$($i$)$ What are the boundary conditions?

$($ii$)$ Determine the bending moment equation for the beam using step functions in

terms of the reactions at A and B$.$

$($iii$)$ Determine the equation for the deflection of the beam in terms of the reaction at

A$.$

$($iv$)$ Find the reactions $($unit kN$)$ at A$,$ B and D$.$

$($v$)$ Determine the bending moment $($unit kNm$)$ at point E$.$