Problem $3$:

To safely support a uniform load $w=30k\frac{N}{m},$ we need to determine the lightest$-$weight A $-36$ steel American Standard shape that can be used for a $6$ m $-$long beam. The beam is pin supported at both ends and has an additional support placed at the mid span of the beam, as shown in the figure below. Horizontal forces at all supports can be assumed equal to zero.

$($a$)$ Knowing that distanee $d=3m,$ determine the reactions at all supports.

$($b$)$ Draw the shear force and bending moment diagram for this beam.

$($c$)$ Knowing that yield stress for $A-36$ stecl is $250$ MPa and a factor of safety equal to $2$ is required for this design, select the lighted$-$weight American Standard shape to safely support the load.

$($d$)$ Determine the beam deflection at the midpoint between supports, $x=\frac{d}{2}$ Is this the beam maximum deflection?