# Static Analysis of Composite Plates with Periodic Curvatures in Material Using Navier Method
## 3 Results and Discussion
Practical context: In this study, \(l_{1}=1\) m was chosen so that the curvature parameter epsilon could be directly linked to typical manufacturing tolerances. Following Ref. [8], the wavelength-to-length ratio was set to \(\Lambda/l_{1}=1/16\), which reflects realistic fiber waviness patterns. With this setup, the peak waviness amplitude is approximately 0.0625 \(\times\)\(\varepsilon\) meters (about 62,500 \(\times\) epsilon microns). Therefore, epsilon = 0.001, 0.005, and 0.009 correspond to about 63 \(\upmu\)m, 313 \(\upmu\)m, and 563 \(\upmu\)m of fiber waviness over a 1 m layup, respectively. These magnitudes help practitioners assess when curvature effects become critical in thin laminated plates. The values of bending moments \(M_{11}/M_{33}\) at the midpoint of the laminated plate with periodic curvature are presented in Table 7. \begin{table}
\begin{tabular}{c c c c c} \hline \(\varepsilon\) & \(l_{1}/l_{3}\) & **0.1** & **0.5** & **1.0** & **2.0** \\ \hline
0.000 & \(-14.690\) & \(-12.710\) & \(-5.743\) & \(-0.7942\) \\ \hline
0.001 & \(-14.710\) & \(-12.720\) & \(-5.746\) & \(-0.7943\) \\ \hline
0.002 & \(-14.770\) & \(-12.770\) & \(-5.756\) & \(-0.7945\) \\ \hline
0.003 & \(-14.880\) & \(-12.840\) & \(-5.772\) & \(-0.7949\) \\ \hline
0.004 & \(-15.030\) & \(-12.950\) & \(-5.794\) & \(-0.7955\) \\ \hline
0.005 & \(-15.240\) & \(-13.100\) & \(-5.824\) & \(-0.7963\) \\ \hline
0.006 & \(-15.500\) & \(-13.280\) & \(-5.861\) & \(-0.7972\) \\ \hline
0.007 & \(-15.830\) & \(-13.510\) & \(-5.907\) & \(-0.7983\) \\ \hline
0.008 & \(-16.230\) & \(-13.790\) & \(-5.961\) & \(-0.7996\) \\ \hline
0.009 & \(-16.730\) & \(-14.130\) & \(-6.025\) & \(-0.8011\) \\ \hline \end{tabular}
\end{table} 
Table 6: Midpoint deflection (\(\times 10^{-3}\) mm) of a composite plate with periodic curvature. For \(E_{2}/E_{1}=10\) under varying \(\varepsilon\) and \(l_{1}/l_{3}\) ratios. Figure 5: Deflection w (\(\times 10^{-3}\) mm) of a composite plate with periodic curvature for \(\mathrm{E_{2}}/\mathrm{E_{1}}=10\). Figure 6 illustrates the distributions of \(M_{11}\) and \(M_{33}\) moments for \(E_{2}\)/\(E_{1}\) = 10 as the curvature parameter increases. It is observed that an increase in curvature leads to a noticeable rise in both moment components. For \(I_{1}\) = 1 m and \(I_{3}\) = 0.5 m (i.e., \(I_{1}\)/\(I_{3}\) = 2), the plate exhibits a beam-like behavior due to its short span along the \(x_{3}\) direction, despite being simply supported on all edges. This geometry slightly redistributes the bending moments, with a minor relaxation of \(M_{11}\) at the midpoint and a small increase in \(M_{33}\). However, the overall change remains negligible (below 1.5%) and does not affect the general trends or conclusions. Table 8 presents the values of the stresses \(\sigma_{11}\,^{1+b/2}\) and \(\sigma_{33}\,\,^{1+b/2}\) obtained for the laminated composite plate with periodic curvature. As illustrated in Figure 7, the increase in the curvature parameter not only amplifies the deflection but also leads to a noticeable rise in bending moments and internal stress distributions. This suggests that the impact of curvature extends beyond deflection alone, significantly influencing both the load-bearing capacity and stress safety of the plate. In addition to the observed increases in deflection and internal forces, it is noted that the structural response becomes more sensitive to curvature variations, particularly in plates with lower \(I_{1}\)/\(I_{3}\) ratios. This is attributed to the fact that geometric changes caused by periodic curvature can significantly influence the stiffness and internal force distribution of the plate. As the curvature increases, these changes become more prominent, potentially compromising structural performance and reliability in practical applications.