Problem $2$: Frame Analysis $(18$ Points$)$

A framed structure containing one column and one beam is modeled using two $1-$d frame elements $($combination of truss and beam elements$).$ Two versions of the frame are considered: $($i$)$ a fixed $($moment$)$ connection between beam & column; $($ii$)$ a pinned $($shear$)$ connection between beam & column. The geometry of the structure is shown below.

Fix connected frame

Pin connected frame.

The base of the left column is fixed to the ground, and the right support of the beam is pinned. The local stiffness matrix of each frame element expressed in the local $x^{\text{'}}-y^{\text{'}}$ coordinate system $(x^{\text{'}}$ aligned with the member$)$ is:

$k^e=\left[\begin{array}{cccccc}60& 0& 0& -60& 0& 0\\ 0& 72& 180& 0& -72& 180\\ 0& 180& 600& 0& -180& 300\\ -60& 0& 0& 60& 0& 0\\ 0& -72& -180& 0& 72& -180\\ 0& 180& 300& 0& -180& 600\end{array}\right]\{[u_{x^{\text{'}}}^1],[u_{y^{\text{'}}}^1],[{}^1],[u_{x^{\text{'}}}^2],[u_{y^{\text{'}}}^2],[{}^2]\}$

Determine the following:

$($a$)$ Redraw each frame and clearly number the free degrees of freedom $($DOFs$)$

$($b$)$ Compute the global stiffness matrix $K$ for both structures. Report the matrix after rows and columns from restrained degrees of freedom have been removed. $($Hint: apply transformations so that the global degrees of freedom numbered in $($a$)$ are expressed in the global $x-y$ system$)$

$($c$)$ According to undergraduate mechanics, label each frame as unstable, statically determinate, or statically indeterminate to the $n-$th degree $($provide $n).$

Hint: consider why these frames are different, how the connection affects the expected continuity of variables, and how that should affect the free and restrained DOFs of the model.