# Finite Elements for Higher Order Steel-Concrete Composite Beams
## 2 Recall of the Analytical Model
### Active Stress Field  
A linear elastic behaviour is assumed for materials in both tension and compression (uncracked concrete under tensile stress). \(E_{\alpha}\), \(\
u_{\alpha}\), and \(G_{\alpha}=E_{\alpha}/2(1+\
u_{\alpha})\) are the relevant Young, Poisson, and shear moduli of materials (\(\alpha=c,s,r\)), while \(\rho\) is the stiffness per unit length of the slab-steel girder connection.  
According to the linear theory, the admissible displacement field defined by Equations (1)-(3) leads to the following relationships between stresses and overall displacements:  
\[\sigma_{cz}(x,y,z)=E_{c}\ \left(\mathbf{w}_{c}^{\prime}\mathbf{\cdot}\mathbf{a}_{c}-\bar{ \varepsilon}_{cz}\right)\ \ \ \ \tau_{cxz}(x,y,z)=G_{c}\ (\mathbf{w}_{c}\mathbf{\cdot}\mathbf{a}_{c,x}) \tag{9}\]  
\[\tau_{cyz}(x,y,z)=G_{c}\ \left(\mathbf{w}_{c}\mathbf{\cdot}\mathbf{a}_{c,y}+v_{0}^{\prime}\right) \ \ \ \sigma_{sz}(\xi_{i},z)=E_{s}\ \left(\mathbf{w}_{s}^{\prime}\mathbf{\cdot}\mathbf{a}_{s}-\bar{ \varepsilon}_{sz}\right) \tag{10}\]  
\[\tau_{s\xi_{2}}(\xi_{i},z)=G_{s}\ \left(\mathbf{w}_{s}\mathbf{\cdot}\mathbf{a}_{s,\xi}+v_{0}^{ \prime}\ \alpha_{i}\right)\ \ \ \ \sigma_{rz}(x,y,z)=E_{r}\ \left(\mathbf{w}_{c}^{\prime}\mathbf{\cdot}\mathbf{a}_{c}-\bar{ \varepsilon}_{rz}\right) \tag{11}\]  
where \(\sigma_{az}\) (\(\alpha=c,s,r\)) are the longitudinal normal stresses; \(\tau_{cxz}\) and \(\tau_{cyz}\) are the slab shear stresses on \(XZ\) and \(YZ\) planes, respectively; and \(\tau_{s\xi_{2}}\) is the steel girder shear stress in the plane of the \(i\)-th wall. In Equations (9)-(11), primes denote total derivatives with respect to \(z\), whereas commas denote partial derivatives. Furthermore, term \(\overline{\varepsilon}_{az}\) (\(\alpha=c,s,r\)) in Equations (9)-(11) are stress-independent strains that may represent thermal, concrete slab shrinkage, or wire pre-stressing; these are defined as  
\[\overline{\varepsilon}_{\alpha z}=\overline{\varepsilon}_{\alpha}\cdot\mathbf{a}_{c }=\left[\begin{array}{c}\epsilon_{\alpha z}\\ \kappa_{\alpha x}\\ \mu_{\alpha z}\end{array}\right]\cdot\mathbf{a}_{c}\qquad\quad\alpha=c,r \tag{12}\]  
\[\overline{\varepsilon}_{sz}=\overline{\varepsilon}_{s}\cdot\mathbf{a}_{s}=\left[ \begin{array}{c}\epsilon_{sz}\\ \kappa_{\alpha x}\\ \mu_{\alpha v}\\ \mu_{sh}\end{array}\right]\cdot\mathbf{a}_{s} \tag{13}\]  
where \(\epsilon\), \(\kappa\), and \(\mu\) are overall stress-independent longitudinal strains given by  
\[\overline{\varepsilon}_{\alpha}=\mathbf{I}_{\alpha}^{-1}\int_{A_{\alpha}}\widetilde {\varepsilon}_{\alpha}\mathbf{a}_{\alpha}\ dA\qquad\alpha=c,r \tag{14}\]  
\[\overline{\varepsilon}_{s}=\mathbf{I}_{s}^{-1}\sum_{i=1}^{n}t_{i}\int_{0}^{l_{i}} \widetilde{\varepsilon}_{s}\mathbf{a}_{s}\ d\widetilde{\varepsilon}_{i} \tag{15}\]  
in which \(\overline{\varepsilon}_{c}\), \(\widetilde{\varepsilon}_{r}\), and \(\widetilde{\varepsilon}_{s}\) are generic nonlinear stress-independent longitudinal strain fields in concrete, reinforcements, and structural steel, respectively. \(\mathbf{I}_{\alpha}\) and \(\mathbf{I}_{s}\) appearing in Equations (14)-(15) are inertia terms of the cross section of the beam components, as detailed in Appendix A. It is worth mentioning that active stresses of Equations (9)-(11), i.e., relevant to strains descending from the admissible displacement field according to constitutive relationships, do not satisfy the local equilibrium, which also requires additional non-vanishing stress components, called reactive stresses. The latter do not appear in the virtual work theorem expression and can be estimated by means of the local equilibrium conditions. These components are significant in the case of shear stresses, while normal stresses can be obtained with negligible errors from Equations (9)-(11). Expressions of the total shear stresses \(\tau_{cxz}\) and \(\tau_{sgz}\) are available in Appendix B.