Problem $3$:

$(a,b)$ The plane structures shown consist of rigid weightless bars connected by linear springs, each of stiffness $k.$ Degrees of freedom are horizontal translations $u_i$ and small rotations ${}_i$ for $i=1,2,$ as shown. Vertical motion and out$-$of$-$plane displacements are not allowed. In each case determine the $4$ by $4$ structure stiffness matrix in terms of $k$ and $b.$