# Study of a Layered Plate Grider Element of Composite Materials and Its Applications
## 3 LPGE Composite Action
### Girder Element Constraint Matrix  
In the girder element parallel to the \(x\)-axis, every node has six degrees of freedom as:  
\[\delta_{\text{Lx}i}=\left[u_{\text{Lx}i}\quad v_{\text{Lx}i}\quad w_{\text{Lx} i}\quad\theta_{\text{Lx}xi}\quad\theta_{\text{Lx}yi}\quad\theta_{\text{Lx}zi} \right]^{\text{T}} \tag{21}\]  
in which the superscript \"Lx\" represents the girder parallel to the \(x\)-axis.  
However, in the LPGE, the degree of freedom \(\theta_{zi}\) related to the girder element \(\theta_{\text{Lx}zi}\) is the virtual degree of freedom. To create the deformation coordination conditions between the plate element and the girder element parallel to the \(x\)-axis, \(\theta_{\text{Lx}zi}\) can be obtained through the rotation along the tangent of the boundary within the bottom surface of the plate, which is given as:  
\[\theta_{\text{Lx}z}=-\frac{\partial u}{\partial y} \tag{22}\]  
It is assumed that the displacements and deformations of the steel deck and girder at the interface satisfy the following conditions:  
1. The normal line at the base on the steel plate crossing the junction point of the girder and plate before deformation is in a straight line, and the straight line remains unchanged after deformation;
2. Before deformation, the deck including the cross-sectional area of the boundary and the section of the girder element passing through the point (including this normal line) is defined in a plane. It remains in the same plane after deformation.  
It is noted that the displacements of the girder nodes and the steel plate nodes exhibit the following relationship according to the deformation coordination conditions for steel plates and girder elements above:  
\[\begin{array}{l}u_{\text{Lx}}^{\text{t}}=\left.u^{\text{b}}\right|_{\eta=0} \quad v_{\text{Lx}}^{\text{t}}=\left.v^{\text{b}}\right|_{\eta=0}\quad\quad \quad\quad\quad w_{\text{Lx}}^{\text{t}}=\left.w^{\text{b}}\right|_{\eta=0}\\ \theta_{\text{Lx}}=\left.\theta_{x}\right|_{\eta=0}\quad\theta_{\text{Lx}y}= \left.\theta_{y}\right|_{\eta=0}\quad\theta_{\text{Lx}z}=\left.\theta_{z} \right|_{\eta=0}=-\frac{\partial u}{\partial y}\right|_{\eta=0}\end{array} \tag{23}\]  
in which the superscript \"t\" represents the top of the girder, while the superscript \"b\" represents the bottom of the steel plate.  
Figure 3: LPGE chart.  
Based on the above hypothesis, the displacements of the girder nodes can be expressed as:  
\[\delta_{\rm Lx}=\begin{bmatrix}0&\mathbf{L}_{\rm Lx1}&\mathbf{L}_{\rm Lx2}&0\\ \mathbf{L}_{\rm Lx1}&0&0&\mathbf{L}_{\rm Lx2}\end{bmatrix}\begin{split} \delta_{1}\\ \delta_{2}\\ \delta_{3}\\ \delta_{4}\end{split}=L_{\rm Lx}\delta \tag{24}\]  
in which  
\[\mathbf{L}_{\rm Lx1}=\begin{bmatrix}\frac{1}{2}&0&0&-\frac{h_{\rm Lx}}{2}&0&0\\ 0&\frac{1}{2}&0&0&-\frac{h_{\rm Lx}}{2}&0\\ 0&0&\frac{1}{2}&0&0&0\\ 0&0&0&\frac{1}{2}&0&0\\ 0&0&0&0&\frac{1}{2}&0\\ -\frac{1}{2b}&0&0&0&0\end{bmatrix} \tag{25}\]  
\[\mathbf{L}_{\rm Lx2}=\begin{bmatrix}\frac{1}{2}&0&0&-\frac{h_{\rm Lx}}{2}&0&0\\ 0&\frac{1}{2}&0&0&-\frac{h_{\rm Lx}}{2}&0\\ 0&0&\frac{1}{2}&0&0&0\\ 0&0&0&\frac{1}{2}&0&0\\ 0&0&0&0&\frac{1}{2}&0\\ \frac{1}{2b}&0&0&0&0\end{bmatrix} \tag{26}\]  
in which \(L_{\rm Lx}\) denotes the constraint matrix for the girder parallel to the \(x\)-axis and the layered plate, and \(h_{\rm Lx}\) denotes the distance from the girder centroid to the girder top.  
Similarly, the constraint matrix for the girder parallel to the \(y\)-axis and the layered plate is obtained as follows:  
\[\mathbf{L}_{\rm Ly}=\begin{bmatrix}0&0&\mathbf{L}_{\rm Ly1}&\mathbf{L}_{\rm Ly2}\\ \mathbf{L}_{\rm Ly2}&\mathbf{L}_{\rm Ly1}&0&0\end{bmatrix} \tag{27}\]  
in which the superscript \"Ly\" represents the girder parallel to the \(y\)-axis.  
While considering the composite action of the girder and the plate, the girder element stiffness matrix is converted as follows:  
\[\mathbf{K}_{\rm Lx}^{*}=\mathbf{L}_{\rm Lx}^{\rm T}\mathbf{K}_{\rm Lx}\mathbf{L}_{\rm Lx} \tag{28}\]  
\[\mathbf{K}_{\rm Ly}^{*}=\mathbf{L}_{\rm Ly}^{\rm T}\mathbf{K}_{\rm Ly}\mathbf{L}_{\rm Ly} \tag{29}\]