As shown in Figure $1,$ the beam ABC is loaded with a uniform distributed load $($UDL$)$ of $w$ in the segment of $AB$ and $BC.$ Support $A$ is a pin, and Support $C$ is a roller. The beam has a flexural rigidity of $3EI$ in the segment of $AB$ and $El$ in the segment of $BC.$

$($a$)$ Draw the shear force diagram $($SFD$)$ and the bending moment diagram $($BMD$)$; Determine the expression of bending moment, $M(x),$ in terms of $w,$ L and x ;

$(10$ marks$)$

$($b$)$ Use direct integration method to determine the deflection curve of the beam, $v(x),$ in terms of $EI,w,L$ and $x.$ Then, determine the deflection at Joint B$,$ and the slope at Joint A;

$(40$ marks$)$

$($c$)$ Use moment area method to determine the deflection at Joint B$,$ the slope at Joint A$.$ Note that the flexural rigidity is different between $AB$ and $BC$;

$(30$ marks$)$

$($d$)$ If the segment AB has the flexural rigidity of EI$,$ how much should the UDL $w$ be reduced to maintain the same value of deflection at Joint B $($that was obtained in Q$1$b$)?$ How much should the UDL $w$ be reduced to maintain the same value of slope at Joint A $($that was obtained in Q$1$b$)?[$Hint: apply the method of superposition$]$ Please discuss the strengthening effect of segment AB $($from El to $3$ EI $)$ on the beam's deformations.

$(20$ marks$)$