Select the lightest W$12$ section $($A$992$ steel$)$ for the beam shown below.

The member is laterally braced only at supports $R$ and $S.$

Disregard shear strength and deflection requirements.

$P=$ factored concentrated axial load $=30$kips

$F=$ factored concentrated transverse load $=40$kips

$W=$ factored uniform distributed load $=1\frac{k}{f}t(It$ includes the self$-$weight $of$ the beam$)$

If you used a table or graph for answering a question, indicate the corresponding reference and write the number of the equation used to calculate that value. Justify the use of that equation.

Vertical and axial loads are factored

Distributed load includes beam self$-$weight