Consider the spring assemblage with the elements and nodes numbered as

shown in the figure below. The spring constants are $k_1=2000l\frac{b}{i}n.,k_2=$

$1000l\frac{b}{i}n.$ and $k_3=3000l\frac{b}{i}n.$

$($a$)$ Assemble the global stiffness matrix $K$ using the direct stiffness method.

$($b$)$ Consider the case where nodes $2$ and $4$ are held fixed, and the other two

nodes are subjected to forces $F_1=1000lb$ and $F_3=-250lb.$ Evaluate the

displacements of nodes $1$ and $3,$ and the reaction forces at nodes $2$ and $4.$