## Finite Element Model
### Displacement and Strain Representation  
In terms of Assumption 3 in Section 2.1, both the concrete slab and the steel beam have the same transverse displacement \(w^{(e)}\) and normal rotation \(\theta^{(e)}\) at each node of the element, but they could have different axial displacements. Consequently, longitudinal slip \(s^{(e)}\) exists at the interface between the concrete slab and the steel beam; however, interface slip \(s^{(e)}\) is relative to the axial displacement \(\left[u_{c}^{(e)},u_{s}^{(e)}\right]\) and the rotation \(\theta^{(e)}\). Therefore, the composite beam element model has 4 degrees of freedom at each node, i.e., \(\left[u_{c}^{(e)},u_{s}^{(e)},w^{(e)},\theta^{(e)}\right]\). The displacement field of the composite beam element model is depicted in Figure 2.  
Based on Figure 2, the expression of the nodal displacement vector of the composite beam element model can be presented as:  
\[\mathbf{a}^{(e)}=\left[u_{cl}^{(e)},u_{sl}^{(e)},w_{l}^{(e)},\theta_{l}^{(e)},u_{ cr}^{(e)},u_{sr}^{(e)},w_{r}^{(e)},\theta_{r}^{(e)}\right]^{T} \tag{2}\]  
in which the subscripts \(l\) and \(r\) represent the left node and the right node of the composite beam element, respectively. Figure 1: Springs for shear connectors. Figure 2: Displacement field of the composite beam element model. As shown in Figure 2, the expression of the axial displacement \(\overline{u}_{c}^{(e)}(x,z)\) at any point on the cross-section of the concrete slab can be written as  
\[\overline{u}_{c}^{(e)}(x,z)=u_{c}^{(e)}(x)-z_{c}\cdot\theta^{(e)}(x) \tag{3}\]  
Similarly, the expression of the axial displacement \(\overline{u}_{s}^{(e)}(x,z)\) at any point on the cross-section of the steel beam can be written as  
\[\overline{u}_{s}^{(e)}(x,z)=u_{s}^{(e)}(x)-z_{s}\cdot\theta^{(e)}(x) \tag{4}\]  
At the same time, the interface slip \(s^{(e)}\) between the concrete slab and the steel beam of the composite beam element model can be expressed as  
\[s^{(e)}=\left(\overline{u}_{c}^{(e)}\right)_{z_{c}=\frac{-b_{c}}{2}}-\left( \overline{u}_{s}^{(e)}\right)_{z_{s}=\frac{b_{s}}{2}}=u_{c}^{(e)}-u_{s}^{(e)}+ \theta^{(e)}\cdot d \tag{5}\]  
where \(d=(h_{c}+h_{s})/2\). Therefore, the interface slip of the left node \(s_{l}^{(e)}\) and the interface slip of the right node \(s_{r}^{(e)}\) can be expressed as  
\[s_{l}^{(e)}=u_{cl}^{(e)}-u_{sl}^{(e)}+\theta_{l}^{(e)}\cdot d=[1,-1,0,d,0,0,0, 0]\cdot\mathbf{a}^{(e)}=\left[\mathbf{B}_{kl}^{(e)}\right]^{T}\cdot\mathbf{a}^{(e)} \tag{6}\]  
\[s_{r}^{(e)}=u_{cr}^{(e)}-u_{sr}^{(e)}+\theta_{r}^{(e)}\cdot d=[0,0,0,0,1,-1,0, d]\cdot\mathbf{a}^{(e)}=\left[\mathbf{B}_{kr}^{(e)}\right]^{T}\cdot\mathbf{a}^{(e)} \tag{7}\]  
where \(\left[\mathbf{B}_{kl}^{(e)}\right]^{T}\) and \(\left[\mathbf{B}_{kr}^{(e)}\right]^{T}\) are the slip-displacement matrices. Generally, small deflection theory is employed, and then the strain-displacement relationships on the cross-section of the concrete slab and the steel beam can be written as,  
\[\overline{\epsilon}_{c}^{(e)}(x)=\frac{d\overline{u}_{c}^{(e)}}{dx}=\frac{du _{s}^{(e)}}{dx}-z_{c}\frac{d\theta^{(e)}}{dx}=\frac{du_{c}^{(e)}}{dx}-z_{c} \frac{d^{2}w^{(e)}}{dx^{2}} \tag{8}\]  
\[\mathbf{a}^{(e)}=\left[u_{cl}^{(e)},u_{sl}^{(e)},w_{l}^{(e)},\theta_{l}^{(e)},u_{ cr}^{(e)},u_{sr}^{(e)},w_{r}^{(e)},\theta_{r}^{(e)}\right]^{T} \tag{2}\]0  
In view of Assumption 1 in Section 2.1, it is reasonable to consider the two components of composite beams separately, as a Euler-Bernoulli beam. Employing the Euler-Bernoulli beam theory, it is reasonable to neglect the cross-sectional shear deformation. In this paper, the axial displacement \(u_{c}^{(e)}\) or \(u_{s}^{(e)}\) is interpolated by using the linear Lagrangian function, and the transversal displacement \(w\) is interpolated by using the cubic Hermitian interpolation function.