# Study of a Layered Plate Grider Element of Composite Materials and Its Applications
## 2 Layered Plate Element with Independent Interpolation of Displacements and Rotation Angles  
### Degree of Freedom and Displacement Pattern of Element  
Besides vertical bending deformations, decks experience deformations along the bridge direction and in the cross-bridge direction; thus, simulation is conducted using the available plate elements. The local coordinate system _xoy_ is positioned at the bottom plate to facilitate the establishment of the deformation coordination conditions. Nodes 1, 2, 3, and 4 are respectively located at the four corner points of the bottom plate, rather than at the four corner points of the middle surface, as shown in Figure 2.  
In a local coordinate system, a layered plate with an independent interpolation of displacements and rotation angles has five degrees of freedom, excluding \(\theta_{z}\) (the rotation angle around the \(z\)-axis), while the girder elements in the LPGE have six degrees of freedom, including \(\theta_{z}\). When the stiffness matrix of each element in the local coordinate system is converted to the global coordinate system, \(\theta_{z}\) still needs to be involved so as to derive the uniform stiffness matrix for the plates and girder elements. Each nodal displacement of the layered plate element has six degrees of freedom, i.e., displacements \(u\), \(v\), and \(w\) and rotation angles \(\theta_{x}\), \(\theta_{y}\), and \(\theta_{z}\), where \(\theta_{z}\) is the virtual degree of freedom and physically meaningless, i.e.,  
\[\delta=\begin{bmatrix}\delta_{1}&\delta_{2}&\delta_{3}&\delta_{4}\end{bmatrix} ^{\mathsf{T}} \tag{1}\]  
in which, \(\delta_{i}=\begin{bmatrix}u_{i}&v_{i}&w_{i}&\theta_{xi}&\theta_{yi}&\theta_{zi} \end{bmatrix}^{\mathsf{T}}\), \(i=1\)-\(4\).  
Let \(\zeta=x/a\), \(\eta=y/b\), meaning the in-plane coordinates for the underside of the layered plate are described in terms of dimensionless form. As mentioned above, the plate adopts a four-noded rectangular plate element with six degrees of freedom at each node, i.e., three line displacements and three angular displacements. The line and angular displacements are interpolated independently. Like the plain strain element, the interpolation functions of the element adopt a Lagrange polynomial to calculate. Then, using a combination of Mindlin plate element and plane stress element displacement patterns, the displacement at the bottom surface can be expressed as:  
\[\mathbf{f}=\left\{\begin{array}{c}u(\zeta,\eta)\\ v(\zeta,\eta)\\ w(\zeta,\eta)\\ \theta_{x}(\zeta,\eta)\\ \theta_{y}(\zeta,\eta)\\ \theta_{z}(\zeta,\eta)\end{array}\right\}=\left\{\begin{array}{c}\sum\limits _{i=1}^{4}N_{i}u_{i}\\ \sum\limits_{i=1}^{4}N_{i}v_{i}\\ \sum\limits_{i=1}^{4}N_{i}w_{i}\\ \sum\limits_{i=1}^{4}N_{i}\theta_{xi}\\ \sum\limits_{i=1}^{4}N_{i}\theta_{yi}\\ \sum\limits_{i=1}^{4}N_{i}\theta_{zi}\end{array}\right. \tag{2}\]  
in which \(N_{i}\) is the shape function. It is simplified as: \(N_{i}=(1+\zeta_{i}\zeta)\)\((1+\eta_{i}\eta)/4\).  
As the displacements and rotation angles are interpolated independently, the displacements and rotation angles within this element are continuous, and so are the displacements and rotation angles of the adjacent elements. Thus, the layered plate element is a C\({}_{0}\)-type coordinating element.  
Figure 2: Layered plate element.