# Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method
## 2 Materials and Methods
\[\text{M}_{ij}=\int_{-h/2}^{h/2}\sigma_{ij}x_{2}dx_{2}\ \ \ \text{i},\text{j}=1,2,3. \tag{18}\]  
Taking into account the condition \(\sigma_{22}=0\) and integrating Equation (16) through the thickness of the plate,  
\[N_{12,1}+N_{23,3}=P(x_{1},x_{3}) \tag{19}\]  
is obtained, where \(P(x_{1},x_{3})\) is the distributed vertical load on the plate surface. The moments obtained from Equation (18),  
\[M_{11,1}+M_{13,3}-N_{12}=0 \tag{20}\]  
\[M_{13,1}+M_{33,3}-N_{23}=0 \tag{21}\]  
are obtained. Differentiating Equation (20) with respect to \(x_{1}\) and Equation (21) with respect to \(x_{3}\) and substituting the resultant equations in Equation (19) gives the following equilibrium equation for a plate:  
\[M_{11,11}+2M_{13,13}+M_{33,33}=P(x_{1},x_{3}) \tag{22}\]  
where, from (18),  
\[ \begin{bmatrix}M_{11}\\ M_{33}\\ M_{13}\end{bmatrix} = \begin{bmatrix}A_{11}&A_{13}&0\\ A_{13}&A_{33}&0\\ 0&0&A_{55}\end{bmatrix} . \begin{bmatrix}-\frac{b^{3}}{12}w_{,11}\\ -\frac{b^{3}}{12}w_{,33}\\ -\frac{b^{3}}{6}w_{,13}\end{bmatrix}  \tag{23}\]  
By substituting Equation (23) into Equation (22), the governing equation of the plate becomes  
\[ \begin{array}{c}-\frac{b^{3}}{12}\Big{[}A_{11}(x_{1})w_{,11}+A_{13}(x_{1})w_ {,33}\Big{]}_{,11}-\frac{b^{3}}{3}\big{[}A_{55}(x_{1})w_{,13}\big{]}_{,13}-\\ -\frac{b^{3}}{12}[A_{13}(x_{1})w_{,11}+A_{33}(x_{1})w_{,33}]_{,33}=P(x_{1},x_{ 3})\end{array}  \tag{24}\]  
Accordingly, the plate equilibrium equation is obtained. The boundary conditions for the simply supported plate on all four edges are  
\[x_{1}=0,1_{1};\ \ w=0,\ \ w_{,11}=0\ \ \ x_{3}=0,1_{3};\ \ w=0,\ \ w_{,33}=0. \tag{25}\]  
The deflection expression that satisfies the boundary conditions of Equation (24) is given as  
\[w(x_{1},x_{3})=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}w_{mn}\,\sin\frac{m\pi x _{1}}{l_{1}}\sin\frac{n\pi x_{3}}{l_{3}} \tag{26}\]  
According to the Navier method, the distributed load \(P(x_{1},x_{3})\) is expressed in the form of a series [30] as  
\[P(x_{1},x_{3})=\sum_{m=1}^  
Substituting Equations (26) and (27) into Equation (24),  
\[ \begin{array}{ll}\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}w_{mn} \frac{h^{3}}{12}&\left[\left(A_{11,11}\left(\frac{m\pi}{l_{1}}\right)^{2}-A_{11 }\left(\frac{m\pi}{l_{1}}\right)^{4}+A_{13,11}\left(\frac{n\pi}{l_{3}}\right)^ {2}\right.\right.\\ &\left.\left.-2A_{13}\left(\frac{m\pi}{l_{1}}\right)^{2}\left(\frac{n\pi}{l_{3} }\right)^{2}-4A_{55}\left(\frac{m\pi}{l_{1}}\right)^{2}\left(\frac{n\pi}{l_{3} }\right)^{2}\right.\\ &\left.-A_{33}\left(\frac{n\pi}{l_{3}}\right)^{4}\right)sin\frac{m\pi x_{1}}{l _{1}}sin\frac{n\pi x_{3}}{l_{3}}\\ &\left.+\left(2A_{11,1}\left(\frac{m\pi}{l_{1}}\right)^{3}+2A_{13,1}\left( \frac{m\pi}{l_{1}}\right)\left(\frac{n\pi}{l_{3}}\right)^{2}\right.\\ &\left.+4A_{55,1}\left(\frac{m\pi}{l_{1}}\right)\left(\frac{n\pi}{l_{3}}\right) ^{2}\right)cos\frac{m\pi x_{1}}{l_{1}}sin\frac{n\pi x_{3}}{l_{3}}\\ &=\sum\limits_{m=1}^{\infty}\sum\limits_{n=1}^{\infty}P_{mn}\sin\frac{m\pi x_ {1}}{l_{1}}sin\frac{m\pi x_{3}}{l_{3}}\.\\ &\left.m,n=1,3,5\ldots\right.\end{array}  \tag{29}\]  
Here, the coefficients \(w_{mn}\) are determined for each pair of \(m\) and \(n\) by utilizing the known values of \(P_{mn}\). In this study, only the expression in Equation (28) is used for the case of a uniformly distributed load. For other loading conditions, new numerical results can be obtained from Equation (29) by employing the expressions given by [PERSON] [30]. Equation (29) is first solved for \(w_{mn}\), and the obtained coefficients are then used in Equation (26) to calculate the deflection at the desired points. Consequently, the moment and stress values follow from Equation (23) and Equation (11), respectively. For a homogeneous isotropic plate, Equation (29) has constant coefficients; thus, the \(w_{mn}\) coefficients can be easily determined as functions of \(P_{mn}\). However, in the case of a composite plate, the dependence of these coefficients on \(\mathrm{x_{1}}\) transforms the governing equation into a variable-coefficient form.