## Finite Element Model
### Displacement and Strain Representation
(1) Function of axial displacement  
The axial displacement \(u_{c}^{(e)}\) or \(u_{s}^{(e)}\) (as shown in Figure 2) at any point along the length of the composite beam element can be expressed in a general form by adopting the dimensionless coordinate system as  
\[u(\xi)^{(e)}=\left[N(\xi)_{1}^{(e)},N(\xi)_{2}^{(e)}\right]\left[ \begin{array} []{c}u_{l}^{(e)}\\ u_{r}^{(e)}\end{array} \right] \tag{10}\]  
in which the shape function vector \(\left[N(\xi)_{1}^{(e)},N(\xi)_{2}^{(e)}\right]=[1-\xi/2,1+\xi/2]\). Therefore, the axial displacement \(u_{c}^{(e)}\) or \(u_{s}^{(e)}\) can be written as  
\[\left\{ \begin{array}{l}u(\xi)_{c}^{(e)}=\left[N(\xi)_{1}^{(e)},N(\xi)_{2}^ {(e)}\right]\left[u_{cl}^{(e)},u_{cr}^{(e)}\right]^{T}\\ u(\xi)_{s}^{(e)}=\left[N(\xi)_{1}^{(e)},N(\xi)_{2}^{(e)}\right]\left[u_{sl}^{( e)},u_{sr}^{(e)}\right]^{T}\end{array} \right. \tag{11}\]The axial strain-displacement relationship can be represented as  
\[\frac{du^{(e)}(\xi)}{d\xi}=\left[\left(\frac{dN_{1}}{d\xi}\right)^{(e)},\left( \frac{dN_{2}}{d\xi}\right)^{(e)}\right]\left[ \begin{array}{c}u_{l}^{(e)}\\ u_{r}^{(e)}\end{array} \right]=\left[-\frac{1}{2},\frac{1}{2}\right]\left[ \begin{array}{c}u_{l}^{(e)}\\ u_{r}^{(e)}\end{array} \right] \tag{12}\]  
and the relationship between absolute coordinates and relative coordinates is  
\[\left\{ \begin{array}{c}x=\frac{L^{(e)}}{2}(1+\xi)\\ dx=\frac{L^{(e)}}{2}d\xi\end{array} \right. \tag{13}\]  
Thus, the axial strain-displacement relationship for the concrete slab and the steel beam in absolute coordinates can be given as  
\[\left\{ \begin{array}{c}\frac{du_{r}^{(e)}(x)}{dx}=\left[-\frac{1}{L^{(e)}}, \frac{1}{L^{(e)}}\right]\left[\begin{array}{c}u_{cl}^{(e)}\\ u_{cr}^{(e)}\end{array}\right]\\ \frac{du_{r}^{(e)}(x)}{dx}=\left[-\frac{1}{L^{(e)}},\frac{1}{L^{(e)}}\right] \left[\begin{array}{c}u_{sl}^{(e)}\\ u_{sr}^{(e)}\end{array}\right]\end{array} \right. \tag{14}\]  
(2) Function of deflection  
The deflection \(w^{(e)}(x)\) (as shown in Figure 2) at any point along the length of the composite beam element can be written by utilizing the dimensionless coordinate system as  
\[w^{(e)}(\xi)=\left[N_{bi}^{(e)}(\xi)\right]^{T}\cdot\mathbf{a}_{b}^{(e)} \tag{15}\]  
in which  
\[\mathbf{a}_{b}^{(e)}=\left[w_{l}^{(e)},\theta_{l}^{(e)},w_{r}^{(e)},\theta_{r}^{( e)}\right]^{T} \tag{16}\]  
\[N_{bi}^{(e)}(\xi)=\left\{ \begin{array}{c}N_{3}^{(e)}(\xi)=(2-3\xi+\xi^{3})/4 \\ \overline{N}_{3}^{(e)}(\xi)=(1-\xi-\xi^{2}+\xi^{3})/4\\ N_{4}^{(e)}(\xi)=(2+3\xi-\xi^{3})/4\\ \overline{N}_{4}^{(e)}(\xi)=(-1-\xi+\xi^{2}+\xi^{3})/4\end{array} \right. \tag{17}\]  
and the relationship between absolute coordinates and relative coordinates is  
\[\left(\frac{d^{2}w}{dx^{2}}\right)^{(e)}=\frac{4}{\left(L^{(e)}\right)^{2}} \left(\frac{d^{2}w}{d\xi^{2}}\right)^{(e)} \tag{18}\]  
Thus, the curvature at any point along the length of the composite element can be written by using the coordinate transformation as  
\[ \begin{array}{c}\frac{d^{2}w^{(e)}}{dx^{2}}=\frac{4}{\left(L^{(e)}\right)^{ 2}}\left[\frac{d^{2}N_{3}^{(e)}}{dx^{2}},\frac{d^{2}\overline{N}_{3}^{(e)}}{ dx^{2}}.\frac{L^{(e)}}{2},\frac{d^{2}\overline{N}_{4}^{(e)}}{dx^{2}}.\frac{L^{(e)}}{ dx^{2}}.\frac{L^{(e)}}{2}\right]\cdot\mathbf{a}_{b}^{(e)}=\\ \left[\frac{6\xi}{\left(L^{(e)}\right)^{2}},\frac{-1+3\xi}{L^{(e)}},\frac{-6 \xi}{\left(L^{(e)}\right)^{2}}.\frac{1+3\xi}{L^{(e)}}\right]\cdot\mathbf{a}_{b}^{ (e)}=\left[\mathbf{B}_{1}^{(e)},B_{2}^{(e)},B_{3}^{(e)},B_{4}^{(e)}\right]\cdot \mathbf{a}_{b}^{(e)}=\left[\mathbf{B}^{(e)}\right]^{T}\cdot\mathbf{a}_{b}^{(e)}\end{array}  \tag{19}\]  
where \(\left[\mathbf{B}^{(e)}\right]^{T}\) is the longitudinal curvature-displacement matrix. Thus, the axial strain at any point of cross-section of the concrete slab \(\overline{\varepsilon}_{c}^{(e)}\) can be written by substituting Equations (14) and (19) into Equation (8) as  
\[