$2.(9$ marks$)$ A simply supported beam is subjected to a uniformly distributed load with density $\backslash ($ w $\backslash )$ as shown in Fig. $($a$).$ The cross$-$section of the beam is shown in Fig. $($b$).$ The maximum allowable shear stress for the material of the beam is $10$ MPa $.($i$)$ Determine the maximun allowable value of $\backslash ($ w $\backslash )$ that could be sustained by the beam; $($ii$)$ determine the value of the minimum transverse shear stress developed in the beam and where the minimum shear stress occurs.

$($a$)$

$($b$)$