# 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
### Stiffness Matrix for Layered Plate Element
When considering shear deformation energy, the total potential energy of the plate is obtained as:  
\[\Pi^{*}=\Pi+\frac{Gt}{2K}\iint\limits_{\Omega}\left(\frac{\partial w}{\partial x }-\theta_{\rm x}\right)^{2}dxdy+\frac{Gt}{2k}\iint\limits_{\Omega}\left(\frac{ \partial w}{\partial y}-\theta_{y}\right)^{2}dxdy \tag{15}\]  
in which \(\Pi\) is the total potential energy of the plate based on the classical small deflection theory, \(G\) is the shear modulus of the plate, \(t\) is the thickness of the plate, and \(k\) is an introduced correction coefficient when the actual non-uniform distribution of shear stress along the thickness direction is considered. According to the principle of shear strain energy, \(k=6/5\) is taken. The stiffness matrix of a layered plate element with displacements and angles of rotation interpolating independently is obtained following the principle of minimum potential energy as:  
\[K_{\rm P}=K_{\rm c}+K_{\rm s} \tag{16}\]  
where  
\[K_{\rm c}=\int_{-(t_{\rm s}+t_{\rm c})}^{-t_{\rm s}}\int_{-1}^{1}\int_{-1}^{1} ab\mathbf{B}^{\rm T}\mathbf{D}_{\rm c}\mathbf{B}d\zeta d\eta dz+\frac{G_{\rm c}t_{\rm c}}{2k} \iint\limits_{\Omega}ab(\mathbf{B}^{\rm s})^{\rm T}\mathbf{B}^{\rm s}d\zeta d\eta \tag{17}\]  
\[K_{\rm s}=\int_{-t_{\rm s}}^{0}\int_{-1}^{1}\int_{-1}^{1}ab\mathbf{B}^{\rm T}\mathbf{D }_{\rm s}\mathbf{B}d\zeta d\eta dz+\frac{G_{\rm s}t_{\rm s}}{2k}\iint\limits_{ \Omega}ab(\mathbf{B}^{\rm s})^{\rm T}\mathbf{B}^{\rm s}d\zeta d\eta \tag{18}\]  
in which the subscript \"\(c\)\" and \"\(s\)\" represent the concrete plate and the steel plate, respectively. \(B\) and \(B^{\rm s}\) are shown in Equations (8) and (13), respectively. \(\mathbf{D}\) is the elastic constitutive matrix for the plate. Similarly, the stiffness matrix of the beam element is obtained as follows:  
\[K_{\rm Lx}=\int_{A_{\rm Lx}}\int_{-1}^{1}a\mathbf{B}_{\rm Lx}^{\rm T}\mathbf{D}_{\rm Lx }\mathbf{B}_{\rm Lx}d\zeta dA_{\rm Lx}+\frac{G_{\rm Lx}A_{\rm Lx}a}{2k_{\rm Lx}} \int_{-1}^{1}\left(\mathbf{B}_{\rm Lx}^{\rm s1}\right)^{\rm T}\mathbf{B}_{\rm Lx}^{ \rm s1}d\zeta+\frac{G_{\rm Lx}A_{\rm Lx}a}{2k_{\rm Lx}}\int_{-1}^{1}\left(\bm {B}_{\rm Lx}^{\rm s2}\right)^{\rm T}\mathbf{B}_{\rm Lx}^{\rm s2}d\zeta \tag{19}\]  
\[K_{\rm Ly}=\int_{A_{\rm Ly}}\int_{-1}^{1}bB_{\rm Ly}^{\rm T}\mathbf{D}_{\rm Ly}B_{ \rm Ly}d\zeta dA_{\rm Ly}+\frac{G_{\rm Ly}A_{\rm Ly}b}{2k_{\rm Ly}}\int_{-1}^{1 }\left(\mathbf{B}_{\rm Ly}^{\rm s1}\right)^{\rm T}\mathbf{B}_{\rm Ly}^{\rm s1}d\eta+ \frac{G_{\rm Ly}A_{\rm Ly}b}{2k_{\rm Ly}}\int_{-1}^{1}\left(\mathbf{B}_{\rm Ly}^{ \rm s2}\right)^{\rm T}\mathbf{B}_{\rm Ly}^{\rm s2}d\eta \tag{20}\]  
in which the subscript \"\(\rm Lx\)\" represents the girder parallel to the \(x\)-axis, while the subscript \"\(\rm Ly\)\" represents the girder parallel to the \(y\)-axis. \(K_{\rm Lx}\) and \(K_{\rm Ly}\) are the element stiffness matrix, which is expressed as the nodal displacement at the centroid of the girder end. The combined effect of the girder and the plate is not taken into account.