$.($ten points$)$A $15m$ long beam is subjected to a uniformly$-$distributed load, w$.$ The built$-$up beam section consists

of two standard steel W$-$sections bolted together as shown.

$($a$)$ Calculate the maximum moment, $M_{\text{m}\text{a}\text{x}},$ that can be applied to the cross$-$section if the allowable

bending stress is ${}_{\text{a}\text{l}\text{l}\text{o}\text{w}}=200$MPa.

$($b$)$ Calculate the maximum shear force, $V_{\text{m}\text{a}\text{x}},$ that can be applied to the cross$-$section if the maximum

allowable shear force per bolt is $13\mathrm{.}5kN$ and the bolts are uniformly spaced at $175mm$ along the

length of the beam.

$($c$)$ Based on the results from parts $($a$)$ and $($b$),$ what is the maximum allowable load, $w,$ for the beam?