# Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method
## 3 Results and Discussion  
The plate, subject to the loading and boundary conditions illustrated in Figure 1, was analyzed with \(l_{1}\) = 1 m, \(l_{3}\) = 2 m, and a plate thickness of \(h\) = 0.1 m, where the applied load is defined as \(P\) (\(x_{1}\), \(x_{3}\)) = 10,000 N/m\({}^{2}\). Solutions were obtained for three different material configurations: homogeneous isotropic, laminated composite, and curved laminated composite. For the composite case, the matrix material was assigned [PERSON]'s modulus of \(E_{1}\) = 20 GPa and Poisson's ratio \(\
u_{1}\) = 0.3, while the reinforcement material was assigned \(E_{2}\) = 200 GPa and \(\
u_{2}\)=0.3. The volume fractions of both components were set to \(\eta_{1}\) = \(\eta_{2}\) = 0.5. In the homogeneous isotropic case, the material properties of the matrix were used. The deflection values and the corresponding relative errors (\(e_{r}\)) for the homogeneous isotropic plate at the coordinates \(x_{1}\) = \(l_{1}/2\), \(x_{3}\) = \(l_{3}/2\), obtained from the literature [30], the ANSYS model, and the MATLAB code developed in this study, are presented in Table 1. It should be noted that in Ref. [30], a direct numerical value for the deflection is not provided; instead, an analytical formula is given for homogeneous isotropic rectangular plates under uniform loading. In this study, the deflection value was computed using that formula and then compared with the result obtained from the developed MATLAB code. The analytical formula for the maximum deflection, wmax, for a homogeneous isotropic plate under uniform loading can be expressed as follows:  
\[w_{max}=\ \frac{0.0101p_{0}a^{4}}{D} \tag{30}\]  
where a is the plate's span in the \(x_{1}\) direction, and \(D\) is the flexural rigidity of the plate defined as  
\[D=\frac{Et^{3}}{12(1-v^{2})} \tag{31}\]  
This formula was used to compute the deflection value for comparison with the MATLAB code result. The ANSYS simulations were conducted using SHELL181 elements for both homogeneous isotropic and laminated composite plates. For the composite plate, the ACP (Pre) module was used to define the layered configuration. In both models, a mesh with 800 elements and 861 nodes was used to ensure consistency in comparison. In addition, a convergence check was performed by increasing the number of Fourier terms from 5 to 10 and 15, and the difference between 10 and 15 terms was found to be less than 0.001%. Therefore, the number of terms was not increased further, and the result presented in Table 1 corresponds to the case with five terms. Figure 3 shows the deflection values of the homogeneous isotropic plate obtained from the ANSYS program. \begin{table}
\begin{tabular}{c c c} \hline \multicolumn{3}{c}{**Midpoint Deflection (mm)**} \\ \hline
[30] & **ANSYS** & **MATLAB** \\ \hline \(-0.05515\) & \(-0.05696\) & \(-0.05536\) \\  & \(e_{r}=2.81\%\) & \(e_{r}=0.38\%\) \\ \hline \end{tabular}
\end{table} 
Table 1: Midpoint deflection values for an isotropic plate. Figure 3: Deflection contour plot for homogeneous isotropic plate (ANSYS). The moment values \(M_{11}\) and \(M_{33}\) for the homogeneous isotropic case, obtained from the literature [30] and the MATLAB code developed in this study, are presented in Table 2. The values presented in this table were evaluated at the center of the plate (\(x_{1}=l_{1}/2\), \(x_{3}=l_{3}/2\)). Similarly, the stress values \(\sigma_{11}\) and \(\sigma_{33}\) obtained from the ANSYS model for the homogeneous isotropic case are compared with those obtained from the present study in Table 3. 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 values of homogeneous isotropic rectangular plates, obtained by the Navier method based on the Kirchhoff-Love plate theory, are found to be in good agreement with the results presented by [PERSON] [30] and those obtained from the ANSYS model.