# 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  
The displacement at any point of the plate is controlled by the in-plane displacement, out-of-plane displacement, and the angular displacement of the bottom surface of the plate, expressed as:  
\[ \begin{array}{c}u(\zeta,\eta,z)=u_{0}(\zeta,\eta)-2\theta_{x}(\zeta,\eta)\\ v(\zeta,\eta,z)=v_{0}(\zeta,\eta)-2\theta_{y}(\zeta,\eta)\\ w(\zeta,\eta,z)=w_{0}(\zeta,\eta)\\ \theta_{x}(\zeta,\eta,z)=\theta_{x}(\zeta,\eta)\\ \theta_{y}(\zeta,\eta,z)=\theta_{y}(\zeta,\eta)\end{array}  \tag{3}\]  
Therefore, the strain field of any point on the plate is:  
\[\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}_{0}+2\boldsymbol{\kappa} \tag{4}\]  
in which \(\boldsymbol{\varepsilon}_{0}\) and \(\boldsymbol{\kappa}\) are the strain field on the bottom plate and field curvature for the plate, respectively. They are given as:  
\[\boldsymbol{\varepsilon}_{0}= \begin{bmatrix}\boldsymbol{\varepsilon}_{x0}& \boldsymbol{\varepsilon}_{y0}&\gamma_{xy0}\end{bmatrix} ^{\mathrm{T}}= \begin{bmatrix}\frac{\partial\boldsymbol{u}_{0}}{\partial x}&\frac{\partial \boldsymbol{v}_{0}}{\partial y}&\frac{\partial\boldsymbol{u}_{0}}{\partial y} +\frac{\partial\boldsymbol{v}_{0}}{\partial x}\end{bmatrix} ^{\mathrm{T}} \tag{5}\]  
When the displacement and rotation angle are interpolated independently, the field curvature of the plate should be expressed as:  
\[\boldsymbol{\kappa}= \begin{bmatrix}\boldsymbol{\kappa}_{x}&\boldsymbol{ \kappa}_{y}&\boldsymbol{\kappa}_{xy}\end{bmatrix} ^{\mathrm{T}}= \begin{bmatrix} -\frac{\partial\boldsymbol{\varepsilon}_{x}}{\partial x}&-\frac{\partial \boldsymbol{\varepsilon}_{y}}{\partial y}&-\frac{\partial\boldsymbol{ \varepsilon}_{x}}{\partial y}&\frac{\partial\boldsymbol{\varepsilon}_{y}}{ \partial x}\end{bmatrix} ^{\mathrm{T}} \tag{6}\]  
Substituting Equation (3) into Equations (5) and (6) gives:  
\[\boldsymbol{\varepsilon}=\boldsymbol{B}\delta \tag{7}\]  
where  
\[\boldsymbol{B}= \begin{bmatrix}\boldsymbol{B}_{1}&\boldsymbol{B}_{2}& \boldsymbol{B}_{3}&\boldsymbol{B}_{4}\end{bmatrix}  \tag{8}\]  
\[\boldsymbol{B}_{i}= \begin{bmatrix}\boldsymbol{B}_{i}^{(\mathrm{m})}& \boldsymbol{B}_{i}^{(\mathrm{b})}\end{bmatrix}  \tag{9}\]  
\[\boldsymbol{B}_{i}^{(\mathrm{m})}=\frac{1}{4 ab} \begin{bmatrix}b\zeta_{i}(1+ \eta_{i}\eta)&0\\ 0&a\eta_{i}(1+\zeta_{i}\xi)\\ a\eta_{i}(1+\zeta_{i}\xi)&b\zeta_{i}(1+\eta_{i}\eta)\end{bmatrix}  \tag{10}\]  
\[\boldsymbol{B}_{i}^{(\mathrm{b})}=\frac{-z}{4 ab} \begin{bmatrix}0&b\zeta_{i}(1+ \eta_{i}\eta)&0&0\\ 0&0&a\eta_{i}(1+\zeta_{i}\xi)&0\\ 0&a\eta_{i}(1+\zeta_{i}\xi)&b\zeta_{i}(1+\eta_{i}\eta)&0\end{bmatrix}  \tag{11}\]  
in which \(\boldsymbol{B}\), \(\boldsymbol{B}_{i}^{(\mathrm{m})}\), and \(\boldsymbol{B}_{i}^{(\mathrm{b})}\) are the strain matrix, film strain matrix, and bending strain matrix for the plate, respectively. Since the independent nodes are located at the bottom surface instead of the middle surface of the steel plate, \(\boldsymbol{B}_{i}^{(\mathrm{m})}\) and \(\boldsymbol{B}_{i}^{(\mathrm{b})}\) are not strictly a film strain matrix and bending strain matrix. The bending shear strain is:  
\[\boldsymbol{\gamma}= \begin{cases}\frac{\partial w}{\partial x}-\theta_{x}\\ \frac{\partial w}{\partial y}-\theta_{y}\end{cases} =\boldsymbol{B}^{\mathrm{s}}\delta \tag{12}\]  
\[\boldsymbol{B}^{\mathrm{s}}=[\boldsymbol{B}_{1}^{\mathrm{s}}\quad\boldsymbol{ B}_{2}^{\mathrm{s}}\quad\boldsymbol{B}_{3}^{\mathrm{s}}\quad\boldsymbol{B}_{4}^{ \mathrm{s}}] \tag{13}\]  
where  
\[\boldsymbol{B}_{i}^{\mathrm{s}}=\frac{1}{4 ab} \begin{bmatrix}0&0&b\zeta_{i}(1+ \eta_{i}\eta)&-ab(1+\zeta_{i}\xi)(1+\eta_{i}\eta)&0&0\\ 0&0&a\eta_{i}(1+\zeta_{i}\xi)&0&-ab(1+\zeta_{i}\xi)(1+\eta_{i}\eta)&0\end{bmatrix}  \tag{14}\]in which \(\mathbf{B}^{\rm s}\) is the bending shear strain matrix for the plate.