$208$MAE$/5$

$($b$)$ Figure $4\mathrm{.}2$ shows a beam of length $L$ exposed to pure bending and shear forces. At nodes $1$ and $2,$ the shear forces are $F_2$ and $F_5,$ bending moments $F_3$ and $F_6,$ displacements $u_2$ and $u_5,$ and rotations $u_3$ and $u_6.$

The equation system matrix for the beam is

$\{[F_2],[F_3],[F_5],[F_6]\}=\left[\begin{array}{cccc}{k}_{11}& k_{12}& k_{13}& k_{14}\\ k_{21}& k_{22}& k_{23}& k_{24}\\ k_{31}& k_{32}& k_{33}& k_{34}\\ k_{41}& k_{42}& k_{43}& k_{44}\end{array}\right]\{[u_2],[u_3],[u_5],[u_6]\}$

Derive $k_{44}$ of the stiffness matrix for this single beam element.

$(15$ marks$)$