Question $3$: A simply supported beam with a span of $6$ m is subjected to a non$-$uniform loading and end moment as shown. The beam has the cross$-$section given and is made from a titanium alloy having an allowable normal stress of ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=500$MPa, allowable shear stress of ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=85$MPa, and elastic modulus of $E=120$GPa.

$($a$)$ sketch the deflected shape of the beam and then use the method of integration to determine the maximum deflection of the beam.

$($b$)$ does the beam satisfy deflection requirements given that the maximum allowable deflection is equal to $\frac{L}{360}($where L is the span of the beam$).$