Question $1)(50$p$)$: Consider a simply supported reinforced concrete beam. The load$-$carrying capacity is:

$A_s{F}_y(d-\frac{a}{2})$

where variables $A_s,F_y,$ and $(d-\frac{a}{2})$ are all random variables. The statistics for these parameters are given in the table below. The mean moment due to the total load is known to be $130k-ft,$ and the CoV is $13$ percent. The load effect due to the total load is normally distributed. Calculate the probability of failure of the beam via Hasofer$-$Lind reliability index method $($matrix procedure$).$

$\backslash$table$[[$Parameter$,$Mean,CoV $(\%),$Distribution$],[A_s,5\mathrm{.}5i{n}^2,6,$Lognormal$],[F_y,50$ksi,$13,$Lognormal$],[(d-\frac{a}{2}),12in,15,$Normal$]]$