## 4 Reference Case Study
If the ratios \(\mathrm{f_{ANM}}\)/\(\mathrm{f_{WAH}}\) and \(\mathrm{f_{ANP}}\)/\(\mathrm{f_{WAH}}\) are calculated for all the strains of the Wah case, the results obtained are very similar to the ratios \(\mathrm{f_{ANM}}\)/\(\mathrm{f_{FEM}}\) and \(\mathrm{f_{ANP}}\)/\(\mathrm{f_{FEM}}\). If the Wah points are plotted in Figures 4 and 5, these points are superimposed with the FEM results on the corresponding graphs. The main limitation of Wah solution is that the vibration modes can only be calculated for a short range of strain. This range does not cover the whole transition zone between the plate-to-membrane behaviour, as it is shown in Figure 7. These limitations are solved using finite element method, with which all modes of vibration and at any range of pre-strain can be calculated. In order to complete this study, calculations have been repeated for different thicknesses: 10 mm, 1 mm (reference case), 0.1 mm, 0.01 mm, 0.001 mm, for the same level of strains, diameter and material properties that the reference case study (steel 1 m diameter). The ratios \(\mathrm{f_{ANM}}\)/\(\mathrm{f_{FEM}}\) and \(\mathrm{f_{ANP}}\)/\(\mathrm{f_{FEM}}\) have been plotted in Figure 8 for the first five modes. Figure 7: Wah points superimposed on graphics of \(\mathrm{f_{ANM}}\)/\(\mathrm{f_{FEM}}\) (**a**) and \(\mathrm{f_{ANP}}\)/\(\mathrm{f_{FEM}}\) (**b**) ratios. In the graphs of Figure 8, the same tendency is observed for all the studied cases: for small strains, the plate behaviour is fulfilled and for large strains the membrane behaviour. There is a transition zone between these behaviours, and it moves to smaller strain values for smaller thickness/diameter ratios. It can be observed that the curves obtained follow the same shape displaced an order of magnitude \(10^{2}\) in strain, what is in concordance with the fact that the ratio between the plate and membrane solution is dependent on a ratio (h/d)/\(\epsilon^{1/2}\), considering only the geometric parameter and removing the effect of the material properties. It is remarkable that, in certain ratios of thickness/diameter simulations, the membrane behaviour is observed with very low tension applied; those are cases hypersensitive to the applied tension. The \(\mathrm{f_{n}/f_{1}}\) ratio has been plotted in Figure 9 for the FEM results of the first five modes for the different values of thickness/diameter (h/d) calculated: \(10^{-2}\), \(10^{-3}\), \(10^{-4}\), \(10^{-5}\), and \(10^{-6}\). As for the case previously studied, the transition zone is moved to lower strains or stresses applied with smaller values of thickness/diameter ratio. In addition to this theoretical analysis of the problem, it is of great relevance to evaluate the consequences of this behaviour in practical situations. Based on the data shown in Figure 8, we can calculate, for every case, the values of strain where the exact solutions (plate and membrane) present a certain error. In Figure 10 the strain for a 2% and a 5% error is represented for the different geometrical cases and strains studied using logarithmic scale in both axes. The error has been calculated as ((f\({}_{\rm ANP}\)-f\({}_{\rm FEM}\))/f\({}_{\rm ANP}\))\(\cdot\)100 and ((f\({}_{\rm ANM}\)-f\({}_{\rm FEM}\))/f\({}_{\rm ANM}\))\(\cdot\)100 for plate and membrane cases respectively. The discrete solutions obtained by Wah have also been plotted as reference. As it has been commented, these solutions do not cover the entire transition zone between plate and membrane behaviour. In this figure, we can identify a zone with a clear plate behaviour for strains below a certain value, another zone with pure membrane behaviour and an intermediate zone where the transition between both solutions must be considered. In a real situation, we must consider that the material is not infinitely elastic.