## 4 Example Analysis  
The whole process described above (including those in Appendices A and B), showed that the non-destructive method of this paper used mathematical logic rigorously. To further validate it, we compare its results with those of other methods on this section. Due to the limitation of NDT data found in literature on these kinds of shells, we compare our method within a special case in Section 2.1 (without corrosion and temperature) with another existing method. The NDT data adopted here is taken from an experiment described in [33]. To maintain consistency with the experimental model of [33], we also apply internal suction to our FE model to simulate equivalently the external pressure. The geometric parameters for the FE model (meeting thin wall hypothesis) are: the radius of shell \(r=0.05\) m and the wall thickness \(h_{0}=0.0005\) m, yielding a \(h_{0}/r\) ratio of 0.01. The mechanical properties are: the Young modulus \(E=650\) MPa, Poisson ratio \(\
u=0.4\), and density \(\rho=1150\) kg/m\({}^{3}\). A 1/8 symmetrical spherical shell model built by the ANSYS software is shown in Figure 3. The first two buckling modes obtained from the eigenvalue buckling analysis are shown in Figure 4.  
First, we calculate the value of \(p^{\ast}\) using the described method. The results for \(w\) and \(w/p\) are plotted in Figure 5.  
Within the plot, it is shown that the fitting line equation for the \(w\) and \(w/p\) points is:  
\[w+7.27385\ w/p=16.59268, \tag{27}\]  
therefore, we get \(S=7.27385\) by Equation (13), and then obtain \(p^{\ast}=0.08593\) through Equation (14). Figure 3: 1/8 symmetrical spherical shell FE model. In order to verify the practicality and accuracy of this method, we also calculate the \(p^{*}\) value adopting the other five existing methods, including one-order analytical value, numerical value, two-order analytical value, etc., which are shown in Table 1. The highest error is produced from the numerical method of [33], while the lowest error is produced from the lowest eigenvalue method. Because the error values from our method, when used correctly, are within the range of the other approaches, its accuracy can be considered acceptable in the field of engineering. Besides that, compared with other methods, there is no doubt that our method is the most practical and easiest one to operate in engineering. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method No.** & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** & **6 th** \\ \hline \multirow{3}{*}{Method name} & The lowest & \multirow{3}{*}{\begin{tabular}{c} The lowest \\ Eigenvalue method \\ \end{tabular} } & Traditional & Non-destruction & Analytical & Numerical & \multirow{3}{*}{\
\begin{tabular}{c} Experimental \\ method [33] \\ \end{tabular} } \\  & Eigenvalue method & Our method & & method [33] & & \\ \hline Critical load value \(p^{*}\) & 0.081574 & 0.08127 & 0.08593 & 0.08189 & 0.08790 & 0.07800 \\ \hline \multirow{3}{*}{Formulas} & \([K]+\lambda_{i}[S](\gamma_{i})=0\). & \multirow{3}{*}{\(p_{\sigma}=\frac{2\pi n^{2}}{\sqrt{3}\left(1-\gamma^{2}\right)}\)} & \multirow{3}{*}{\(p_{\sigma}=m\Delta\sqrt{Em}\)} & \multirow{3}{*}{\(p_{\sigma}=\frac{2\pi n^{2}}{\sqrt{3}\left(1-\gamma^{2}\right)}-\)} & \multirow{3}{*}{
\begin{tabular}{c} Ansys FEM \\ software. \\ \end{tabular} } & Non-destructive \\  & & & \(\frac{\pi n^{2}}{\left(1-\gamma^{2}\right)}\) & & Testing \\ \hline Relative Error & 0.03718 & 0.04193 & 0.10167 & 0.04988 & 0.12692 & Reference \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Critical load of our method comparing with other methods. Figure 4: Buckling modes obtained from the eigenvalue buckling analysis. Figure 5: Plot of \(w/p\) vs. \(w\) (\(R=50\) mm, \(h=0.5\) mm). From Table 1, we also note that the experimental values are the smallest relative to any other method (numerical methods of various orders, analytical methods, etc.).