$[31]$ The $2$D frame structure shown in Figure P$31$ is composed of two steel mem$-$

bers with Young's modulus $E=10{10}^6$ and cross sectional area $12in$;

the $2-$in dimension is perpendicular to the plane of loading. The connection

at node $2$ is treated as a welded joint and both supports at nodes $1$ and $3$ are

fixed. The loading is composed of a linearly distributed load with maximum

value of $20,000lb/$in and a concentrated moment at node $2$ of $1,500$lbin.

Using a model with two beam elements, determine:

$($a$)$ The global reduced stiffness matrix of the structure.

$($b$)$ The global lumped load vector.

$($c$)$ All displacement and rotation components at node $2.$

$($d$)$ All reactions at node $3.$

Figure P$31$