## 4 Reference Case Study  
In this section, attention will be paid to the results corresponding to the steel membrane 1 m diameter and 1 mm thickness. The natural frequencies for the cases of plate and membrane subjected  
Figure 3: Finite element modal simulation of circular membrane. (**a**) Scheme of pre-strain; (**b**) Geometry; (**c**) Element type; (**d**) Meshing; (**e**) Some displacement solutions: modes 2, 3 and 4.  
to the deformations will be calculated using the classic theory besides the results of the simulation in the ANSYS program. Relations between these different parameters will be calculated and plotted to establish the influence of the pre-strain on the behaviour of this reference case. The analytical natural frequencies of the membrane behaviour (f\({}_{\text{ANM}}\)) have been calculated with the Equation (2) besides the plate behaviour (f\({}_{\text{ANP}}\)) which was calculated using the Equation (1). On the other hand, the finite elements model (FEM) frequencies have also been obtained (f\({}_{\text{FEM}}\)). The first nine analytical natural frequencies for membrane behaviour (f\({}_{\text{ANM}}\)), the simulation frequencies (f\({}_{\text{FEM}}\)) and the ratio between the analytical and simulation frequencies (f\({}_{\text{ANM}}\)/f\({}_{\text{FEM}}\)) are shown in Table 1 for the selected strains: \(10^{-7}\), \(10^{-5}\), and \(10^{-3}\). These values can be considered as low, medium and high pre-strain. It can be observed how the analytical and numerical results are closer when strain increases. The ratio f\({}_{\text{ANM}}\)/f\({}_{\text{FEM}}\) for the first nine modes of the reference case (h/d = \(10^{-3}\)) for different levels of pre-strain have been plotted in Figure 4.  
In Figure 4 it is observed that for small values of strain, the calculated and simulated frequencies are far away, being the ratio far from 1, so the behaviour as membrane and therefore Equation (2) are not met in that range. \begin{table}
\begin{tabular}{l c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**STRAIN**} \\ \cline{2-10}  & \multicolumn{3}{c}{**10\({}^{-7}\)**} & \multicolumn{3}{c}{**10\({}^{-5}\)**} & \multicolumn{3}{c}{**10\({}^{-3}\)**} \\ \cline{2-10}  & **f\({}_{\text{ANM}}\)** & **f\({}_{\text{FEM}}\)** & **f\({}_{\text{ANM}}\)** & **f\({}_{\text{FEM}}\)** & **f\({}_{\text{ANM}}\)** & **f\({}_{\text{FEM}}\)** & **f\({}_{\text{ANM}}\)** & **f\({}_{\text{FEM}}\)** & **f\({}_{\text{ANM}}\)** \\ \hline Mode 1 & 1.496 & 10.310 & 0.145 & 14.959 & 19.038 & 0.786 & 149.593 & 151.958 & 0.984 \\ Mode 2 & 2.385 & 21.338 & 0.112 & 23.845 & 32.984 & 0.723 & 238.452 & 242.399 & 0.984 \\ Mode 3 & 3.195 & 34.918 & 0.092 & 31.953 & 48.318 & 0.661 & 319.533 & 325.372 & 0.982 \\ Mode 4 & 3.435 & 39.797 & 0.086 & 34.349 & 53.474 & 0.642 & 343.491 & 349.922 & 0.982 \\ Mode 5 & 3.969 & 51.022 & 0.078 & 39.694 & 65.587 & 0.605 & 396.943 & 040.962 & 0.980 \\ Mode 6 & 4.366 & 60.791 & 0.072 & 43.658 & 75.669 & 0.577 & 436.582 & 445.791 & 0.979 \\ Mode 7 & 4.721 & 69.600 & 0.068 & 47.205 & 84.984 & 0.555 & 472.051 & 482.680 & 0.978 \\ Mode 8 & 5.238 & 84.469 & 0.062 & 52.376 & 100.146 & 0.523 & 523.761 & 536.317 & 0.977 \\ Mode 9 & 5.385 & 88.983 & 0.061 & 53.851 & 104.728 & 0.514 & 538.509 & 551.689 & 0.976 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ratio f\({}_{\text{ANM}}\)/f\({}_{\text{FEM}}\) for first nine modes and strain selected. Figure 4: Ratio f\({}_{\text{ANM}}\)/f\({}_{\text{FEM}}\) as a function of strain calculated for the first nine modes. The ratio increases with the imposed strain until reaching the value of \(1\) for strains close to \(10^{-2}\). For these strains, the Equation (2) and membrane behaviour are met. This result is fulfilled for the first nine vibration modes represented in Figure 4.  
Table 2 shows the first nine analytical frequencies calculated with Equation (1) for the behaviour as plate (f\({}_{\text{ANP}}\)), the numerical frequencies obtained with FEM (f\({}_{\text{FEM}}\)) and the ratio between both (f\({}_{\text{ANP}}\)/f\({}_{\text{FEM}}\)).