$4\mathrm{.}11.$ Derive the local finite element stiffness matrix for a beam with combined transverse loading and axial force.

The stiffness matrix for axial force acting on a rod or beam was derived as an example of elementary elasticity in Prob. $2\mathrm{.}9$ using the variational function, and the derivation was repeated in Prob. $2\mathrm{.}21$ using a direct approach. In this application the body force will be omitted and replaced by axial forces $N_1$ and $N_2$ acting at joints $($nodes$)1$ and $2.$ The node loadings and corresponding displacements are shown in Fig. $4-11.$ The stiffness matrix for axial forces from Prob. $2\mathrm{.}9$ is repeated here:

$\frac{AE}{L}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\{[u_1],[u_2]\}=\{[N_1],[N_2]\}$

where $u_1$ and $u_2$ are the node displacements. The complete local stiffness matrix is obtained by combining Eq$.($a$)$ above and $Eq.f)$ of Prob. $4\mathrm{.}4$:

$\{[N_1],[R_1],[M_1],[N_2],[R_2],[M_2]\}=\left[\begin{array}{cccccc}{C}_1& 0& 0& -C_1& 0& 0\\ 0& 12C_2& 6C_{2}L& 0& -12C_2& 6C_{2}L\\ 0& 6C_{2}L& 4C_2{L}^2& 0& -6C_{2}L& 2C_2{L}^2\\ -C_1& 0& 0& C_1& 0& 0\\ 0& -12C_2& -6C_{2}L& 0& 12C_2& -6C_{2}L\\ 0& 6C_{2}L& 2C_2{L}^2& 0& -6C_{2}L& 4C_2{L}^2\end{array}\right]\{[u_1],[v_1],[{}_1],[u_2],[v_2],[{}_2]\}$

where

$C_1=A\frac{E}{L},$ and $,C_2=E\frac{\text{I}}{L^3}.$

$($c$)$

The axial and transverse deformations are uncoupled for beams when the local axis of the beam coincides with the global axis.

$4\mathrm{.}25.$ The fixed$-$fixed beam of Fig. $4-22$ has an axial force applied at $2\frac{L}{3}.$ Use the finite element derived in Prob. $4\mathrm{.}11$ to compute the axial reactions.

$**$solve problem $4\mathrm{.}25$ only, $4\mathrm{.}11$ is for reference as its mentioned in $4\mathrm{.}25.$