# Finite Elements for Higher Order Steel-Concrete Composite Beams
## 2 Recall of the Analytical Model
### Balance Conditions  
The weak form of the equilibrium conditions is derived by means of the virtual work theorem by considering the displacement field previously introduced and the constitutive laws (9)-(11) (Appendix A). By grouping the generalized displacements in the vector,  
\[\mathbf{d}^{T}=\left[\begin{array}{cccccc}w_{c0}&\Phi_{c}&f_{c}&w_{s0}&\Phi_{s}& f_{sv}&f_{sh}&v_{0}\end{array}\right] \tag{16}\]  
the following equilibrium conditions in weak form can be obtained:  
\[\int_{0}^{L}[\mathbf{K}\ \mathcal{D}\mathbf{d}-\overline{\mathbf{n}}]\cdot\mathcal{D}\mathbf{ \hat{d}}\ dz=\int_{0}^{L}\mathbf{p}\cdot\mathbf{\hat{d}}\ dz+\left.\left(\mathbf{P}_{ \alpha}\cdot\mathbf{\hat{d}}\right)\right|_{\alpha=0,L}\qquad\forall\mathbf{\hat{d}} \ \
eq\mathbf{0} \tag{17}\]  
where \(\mathbf{\hat{d}}\) is the variation of the displacement field and \(\mathcal{D}\) is a formal linear differential operator defined as  
\[\mathcal{D}\mathbf{d}(z)=\left[\begin{array}{c}\mathbf{d}\\ \mathbf{d}^{\prime}\end{array}\right] \tag{18}\]In Equation (17),  
\[K(z)=\begin{bmatrix}K_{1,1}&K_{1,2}\\ K_{2,1}&K_{2,2}\end{bmatrix}=\begin{bmatrix}G_{e}J_{e}+\rho\ A_{ce}&-\rho\ A_{cs}& \mathbf{0}&\mathbf{0}&\mathbf{0}&G_{e}\ L_{c}\\ -\rho\ A_{sc}&G_{s}J_{s}+\rho\ A_{ss}&\mathbf{0}&\mathbf{0}&\mathbf{0}&G_{s}\ L_{s}\\ \mathbf{0}&\mathbf{0}&0&\mathbf{0}&\mathbf{0}&0\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&E_{e}\ I_{e}+E_{r}\ I_{r}&\mathbf{0 }&\mathbf{0}\\ \mathbf{0}&\mathbf{0}&\mathbf{0}&\mathbf{0}&E_{s}\ I_{s}&\mathbf{0}\\ G_{e}\ L_{c}^{T}&G_{s}\ L_{s}^{T}&0&\mathbf{0}&\mathbf{0}&\mathbf{0}&G_{e}\ A_{c}+G_{s}\ m_{s}\end{bmatrix} \tag{19}\]  
is a matrix that contains inertia terms of the cross section of the beam components and those due to the interface connection (Appendix A). Furthermore,  
\[\boldsymbol{p}^{T}(z)=\left[\begin{array}{cccccc}q_{cz}&m_{cx}&\omega_{cy} &q_{sz}&m_{sx}&\omega_{sy}&\omega_{sx}&q_{y}\end{array}\right] \tag{20}\]  
\[\boldsymbol{P}_{\kappa}^{T}=\left[\begin{array}{cccccc}N_{cza}&M_{cxa}&W_{ cya}&N_{sza}&M_{sxa}&W_{sya}&W_{sxa}&Q_{ya}\end{array}\right] \tag{21}\]  
are the vectors grouping resultants of the external forces along the beam axis and at the beam end cross sections. In detail, the above vectors contain longitudinal forces (\(q\) and \(N\)), bending moments (\(m\) and \(M\)), bi-ments (\(\omega\) and \(W\)), and vertical loads (\(q\) and \(Q\)). Finally, in Equation (17),  
\[\bar{\boldsymbol{n}}(z,t)=\begin{bmatrix}\bar{\boldsymbol{n}}_{1}\\ \bar{\boldsymbol{n}}_{2}\end{bmatrix}=\begin{bmatrix}\mathbf{0}\\ \mathbf{0}\\ E_{c}I_{c}\bar{e}_{c}+E_{r}I_{r}\bar{\boldsymbol{r}}_{r}\\ E_{s}I_{s}\bar{\boldsymbol{n}}_{s}\\ 0\end{bmatrix} \tag{22}\]  
is the vector grouping resultants of stresses due to the restrained stress-independent strains. By integrating parts and considering the fundamental theorem of variational calculus, the following strong form of the equilibrium conditions can be obtained:  
\[-K_{2,2}d^{\prime\prime}+(K_{1,2}-K_{2,1})d^{\prime}+K_{1,1}d=\boldsymbol{p} -\bar{\boldsymbol{n}}_{2}^{\prime} \tag{23}\]  
with the relevant boundary conditions  
\[\left[-K_{2,1}d-K_{2,2}d^{\prime}+\bar{\boldsymbol{n}}_{2}-\boldsymbol{P}_{0} \right]\cdot\hat{\boldsymbol{d}}\big{|}_{0}=\mathbf{0} \forall\hat{\boldsymbol{d}} \tag{24}\]  
\[\left[K_{2,1}d+K_{2,2}\ d^{\prime}-\bar{\boldsymbol{n}}_{2}-P_{L}\right]\cdot \hat{\boldsymbol{d}}\big{|}_{L}=\mathbf{0} \forall\hat{\boldsymbol{d}} \tag{25}\]