Question $3$: A finite element analysis is to be carried out for the plane structure shown in the following figure which contains both frame and bar elements. Nodes $1$ and $2($element $1),$ and nodes $2$ and $3($element $2)$ are connected by frame elements whose bending rigidity $($EI$)$ and extensional rigidity $($EA$)$ are given by ${10}^{6}N*m^2$ and $5{10}^{7}N,$ respectively whereas nodes $2$ and $4($element $3)$ are connected by a bar element whose extensional rigidity $(EA)$ is given by $2{10}^{6}N.$ A vertical downward load of $5$ kN and an anticlockwise moment of $7\mathrm{.}5kN.m$ are applied at node $2$ as shown.

Write the condensed equations related to the primary variables in a matrix form, using the minimum number of elements. Consider the frame element as a combination of a cubic approximation Euler Bernoulli's element and a linear bar element. Also, consider a linear element for the bar.

$(6$ points$)$