# Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method
## 3 Results and Discussion
For the laminated composite plate, the numerical values and corresponding relative errors (\(e_{r}\)) obtained from both the ANSYS model and the MATLAB code developed within the scope of this study at the coordinates \(x_{1}=l_{1}/2\) and \(x_{3}=l_{3}/2\) are presented in Table 4. Since no analytical formula is available in the literature for systems with the same material properties as those used in the laminated composite case, only the results obtained from the ANSYS model and the developed MATLAB code are compared in Tables 4 and 5. \begin{table}
\begin{tabular}{c c c c} \hline
**ANSYS** & **MATLAB** & **ANSYS** & **MATLAB** \\ \(\sigma_{11}\)**(MPa)** & \(\sigma_{11}\)**(MPa)** & \(\sigma_{33}\)**(MPa)** & \(\sigma_{33}\)**(MPa)** \\ \hline \multirow{3}{*}{0.60834} & 0.61393 & & 0.28473 \\  & & 0.27885 & \\ \cline{1-1}  & \(e_{r}=0.92\%\) & & \(e_{r}=2.11\%\) \\ \hline \end{tabular}
\end{table} 
Table 3: \(\sigma_{11}\) and \(\sigma_{33}\) stress values for the case of isotropic plate. \begin{table}
\begin{tabular}{c c c c} \hline
**[30]** & **MATLAB** & **[30]** & **MATLAB** \\ \(M_{11}\)**(kN.m)** & \(M_{11}\)**(kN.m)** & \(M_{33}\)**(kN.m)** & \(M_{33}\)**(kN.m)** \\ \hline \multirow{3}{*}{1.0250} & 1.032 & & 0.4745 \\  & & 0.4754 & \\ \cline{1-1}  & \(e_{r}=0.68\%\) & & \(e_{r}=0.18\%\) \\ \hline \end{tabular}
\end{table} 
Table 2: M\({}_{11}\) and M\({}_{33}\) moment values for the case of an isotropic plate. \begin{table}
\begin{tabular}{c c c} \hline
**ANSYS** & **MATLAB** & **ANSYS** & **MATLAB** \\ \(\sigma_{11}\)**(MPa)** & \(\sigma_{11}\)**(MPa)** & \(\sigma_{33}\)**(MPa)** & \(\sigma_{33}\)**(MPa)** \\ \hline \multirow{3}{*}{0.60834} & 0.61393 & & 0.28473 \\  & & 0.27885 & \\ \cline{1-1}  & \(e_{r}=0.92\%\) & & \(e_{r}=2.11\%\) \\ \hline \end{tabular}
\end{table} 
Table 4: Midpoint deflection values for laminated composite plate. Figure 4 illustrates the deflection distribution obtained from the ANSYS analysis of the laminated composite plate. The \(\sigma_{11}\) and \(\sigma_{33}\) stress values obtained from the ANSYS model for the laminated composite plate are compared with those obtained in this study in Table 5. The values presented in this table were evaluated at the center of the plate (\(x_{1}=l_{1}/2\), \(x_{3}=l_{3}/2\)). The deflection, moment, and stress results calculated for laminated composite rectangular plates using the Navier method based on the Kirchhoff-Love plate theory with MATLAB were found to be in close agreement with those obtained from ANSYS and reported in [30], with acceptable relative errors. In this study, the static analysis of a laminated composite material with periodic curvature in a single direction (\(x_{1}\)) was conducted by varying the \(\varepsilon\) parameter. Since the plate geometry must satisfy the condition  
\[\Lambda\ll h \tag{32}\]  
The deflection values at the plate's midpoint were computed up to the maximum allowable value of \(\varepsilon\) (i.e., \(\varepsilon\) = 0.009), and these results are presented in Table 6.  
Figure 5 illustrates the effect of varying the curvature parameter on the midpoint deflection for four different \(l_{1}/l_{3}\) ratios, given that \(E_{2}/E_{1}=10\). It can be observed that as the curvature increases, the amount of deflection increases significantly. Figure 5 illustrates the effect of varying the curvature parameter on the midpoint deflection for four different \(l_{1}/l_{3}\) ratios, given that \(E_{2}/E_{1}=10\). It can be observed that as the curvature increases, the amount of deflection increases significantly. Moreover, plates with lower \(l_{1}/l_{3}\) ratios (for example, 0.1 and 0.5) exhibit higher sensitivity to curvature because the load transfer is concentrated along the shorter edge, which amplifies the effect of curvature on bending stiffness. In contrast, for wider plates (\(l_{1}/l_{3}=2\)), the load is distributed over a larger area, and the impact of curvature on global deflection becomes negligible. Figure 4: Deflection contour plot for the laminated composite plate (ANSYS).