The beam is built in at the right end and simply supported at the left end and at the midspan $($Fig$.1).$ The beam is loaded by a distributed load linearly increasing from O to w between the left end and midspan and remaining constant at the w value from the midspan to the right end $($Fig$.1).$ Derive the equation for the w value at the onset of the plastic collapse in terms of the beam cross$-$section fully plastic moment, Mp$,$ and beam length, L$.(12\mathrm{.}5\%$ of final grade$)$ The beam is built in at the right end and simply supported at the left end

and at the midspan $($Fig$.1).$ The beam is loaded by a distributed load linearly increasing from $0$

to $w$ between the left end and midspan and remaining constant at the $w$ value from the midspan to

the right end $($Fig$.1).$ Derive the equation for the $w$ value at the onset of the plastic collapse in

terms of the beam cross$-$section fully plastic moment, $M_p,$ and beam length, $L.$