The building beam shown below must carry a uniformly distributed service dead load of $1\mathrm{.}0klf($including self$-$weight$)$ along with a concentrated service live load at the midspan $($the location of this point is fixed$).$ To satisfy architectural requirements, the beam will have a triangular cross$-$section, as shown below $)=60$ksi,$f^{\text{'}}=(4$ksi.

$($a$)$ What is the nominal moment capacity of the beam?

$($b$)$ Has the longitudinal steel yielded?

$($c$)$ What is the maximum value of the concentrated service live load $(P_{LL})$ at which the beam barely satisfies the strength requirements?