## 4 Reference Case Study
In this case, we have included the results obtained without strain and for the three values of strain selected: \(10^{-7}\), \(10^{-5}\) and \(10^{-3}\). The ratio f\({}_{\text{ANP}}\)/f\({}_{\text{FEM}}\) as a function of the strain for the first nine modes of the reference case has been plotted in Figure 5.  
Observing the graphs presented in Figure 5, in the range of low levels of strain the ratio is virtually unity, which means that for that strain range the behaviour is plate-like. Its value decreases when the strain increases, being very low with large strains. The behaviour changes to increase the imposed strain, moving away from the plate behaviour. Figures 4 and 5 are useful to identify the strain values where the solution separates from the classic solutions. It can be observed in Figure 5 how for values of strain below \(10^{-6}\) the plate behaviour from Equation (1) is valid. In this zone, we can say that pure bending behaviour is dominant, and the strain is not significant. On the other side, from Figure 4 it is observed that for values of strain  
\begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & & & \multicolumn{6}{c}{**STRAIN**} \\ \cline{3-10}  & & \multicolumn{2}{c}{**0**} & \multicolumn{2}{c}{**10\({}^{-7}\)**} & \multicolumn{2}{c}{**10\({}^{-5}\)**} & \multicolumn{2}{c}{**10\({}^{-3}\)**} \\ \cline{3-10}  & \multirow{2}{*}{\(\mathbf{f_{ANP}}\)} & \multirow{2}{*}{\(\mathbf{f_{FEM}}\)} & \(\mathbf{f_{ANP}}\) & \multirow{2}{*}{\(\mathbf{f_{FEM}}\)} & \(\mathbf{f_{ANP}}\) & \multirow{2}{*}{\(\mathbf{f_{FEM}}\)} & \(\mathbf{f_{ANP}}\) & \(\mathbf{f_{FEM}}\) & \(\mathbf{f_{ANP}}\) & \(\mathbf{f_{FEM}}\) & \(\mathbf{f_{ANP}}\) \\  & & & & & & & & & & \\ \hline Mode 1 & 10.179 & 10.179 & 1.000 & 10.310 & 0.987 & 19.038 & 0.535 & 151.958 & 0.067 \\ Mode 2 & 21.184 & 21.185 & 1.000 & 21.338 & 0.993 & 32.984 & 0.642 & 242.399 & 0.087 \\ Mode 3 & 3  
above \(10^{-3}\), the results obtained are close to those predicted with the membrane theory Equation (2). We can say that is the zone with pure membrane behaviour where the stiffness of the plate is negligible. Among these extremes, a transition zone is observed where neither of those solutions are valid. Another interesting parameter that can be evaluated is the ratio between the natural frequency of the modes and the frequency of the first mode (\(\mathrm{f_{n}/f_{1}}\)). This has been a useful parameter to evaluate the presence of pre-strain with experimental data [28]. If the natural modes of vibration are calculated analytically by Equation (1) or Equation (2), and these are referred to the first mode dividing by its value, the results are constants that can be calculated with the relations of the \(\alpha\) constants for plate and membrane cases respectively (Table 3). In order to evaluate separately the response of the first nine FEM vibration modes, the ratio \(\mathrm{f_{n}/f_{1}}\) has been plotted in Figure 6 for the calculated strain values. The same tendency is observed in all the graphs of Figure 6. For low strains, the numerical ratio tends to the ratio obtained with the plate behaviour, and for higher strains to the membrane behaviour. This is repeated in all the graphed modes, which present horizontal asymptotes in the region of lower and higher strains, whose value coincides with the relationships between the analytical frequencies of the plate and membrane cases. The transition zone is observed clearly. Figure 6: _Cont._