## 5 Test Results and Evaluation
### Evaluation of the Amplitude of Shape Imperfections in the Axial Direction  
After the Fourier transform in the circumferential direction (Equation (1)), ten values each of the \(w_{0,j}\) (diameter expansion and contraction on the elevation position \(j\)), \(w_{R,1,j}\), and \(w_{1,1,j}\) (rigid body movement at elevation position j) in the axial direction were Fourier transformed using Equations (8)-(10). \[w_{n,j}=W_{R,n,j}\cos n\theta_{i}+W_{I,n,j}\sin n\theta_{i} \tag{8}\]  
\[W_{R,n,j}  
\[c_{I,n,p}=\sum_{j=1}^{10}w_{I,n,j}(\frac{\sin p\pi z_{j}}{l}) \tag{14}\]  
with  
\(w_{R,n,j}\): coefficient of the real part of circumferential-direction Fourier transform at each height. \(w_{I,n,j}\): coefficient of the imaginary part of circumferential-direction Fourier transform at each height. The shape of the cylinder in the axial direction is obtained after the second Fourier transform (Figure 11). Because the KDF on the \(y\)-axis is a dimensionless quantity, the Fourier coefficients for each shape are divided by \(t\) in order to make the transformation dimensionless. In this figure, the shapes 4 and 5 are both related to \(b_{I,0,1}\), i.e., the drum shape appears when \(b_{I,0,1}>0\), while the barrel shape appears when \(b_{I,0,1}<0\). The relationship between the KDF and shape imperfections 1 to 5 are shown in Figure 12. Here, there are correlations between the KDF and shapes 1, 2, 3, and 4, wherein the KDF decreases as the dimensionless values of the Fourier coefficients of shapes 1 and 3 divided by \(t\) increase, while that of shapes 2 and 4 decreases. In other words, shapes 2 and 4 increase the buckling load. The dimensionless value in the case of shape 5 tended to fall slightly. The intercepts of the linear regression KDF\({}_{0}\) values or KDF were between 45 and 60%. The results of the multiple regression analyses and the \(p\)-values obtained by ANOVA are shown in Table 12 (Equation (15)). Here, shape 4 and shape 5 were combined for ANOVA and multiple regression analysis. \[KDF=a_{1}(\frac{\sqrt{b_{R,1,1}^{2}-c_{R,1,1}^{2}}}{t})+a_{2}(\frac{b_{R,0,1}} {t})+a_{3}(\frac{\sqrt{b_{I,1,1}^{2}-c_{I,1,1}^{2}}}{t})+a_{4,5}(\frac{b_{I,0, 1}}{t})+b_{0}(\delta_{0})+\cdots+b_{270}(\delta_{270})+KDF_{0} \tag{15}\]  
The \(p\)-values of shape 1 are smaller than 5%, indicating statistical significance. The \(p\)-value of shape 3 is much smaller than 1%, indicating high statistical significance. The slopes (a) of shapes 1 and 3 are all negative; these shape imperfections lower the buckling load. The \(p\)-values of shape 2 and shapes 4 and 5 are over 5%, indicating statistical insignificance. The shape imperfections of shape 2 and shapes 4 and 5 have no effect on the buckling load. The \(p\)-values of offsets are larger than 5%, indicating no statistical significance. Figure 11: Imperfections in the shape of the thin-walled cylinder in the axial direction. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Shape 1** & **Shape 2** & **Shape 3** & **Shape 4, 5** \\  & **Incline** & **Trapezoid** & **Bow** & **Drum, Barrel** \\ \hline \(p\)-value & 0.0469 & 0.0861 & 0.000193 & 0.177 \\ slope (\(a\)) & \(-\)0.0447 & 0.0979 & \(-\)0.0252 & 0.0225 \\ \hline  & Offset 0\({}^{\circ}\) & Offset 90\({}^{\circ}\) & Offset 180\({}^{\circ}\) & Offset 270\({}^{\circ}\) \\ \hline \(p\)-value & 0.763 & 0.916 & 0.817 & 0.801 \\ slope (\(b\)) & \(-\)0.00484 & 0.00169 & \(-\)0.00372 & \(-\)0.00405 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: \(p\)-value of regression analyses. Figure 12: KDF versus the shape in the axial direction of (**a**) shape 1; (**b**) shape 2; (**c**) shape 3; (**d**) shape 4; and (**e**) shape 5. Effect on KDF by Relationship Between Load Offset and Direction of Maximum (\(-\)Minimum) Amplitude of Local Shape Imperfections  
The local shape imperfections of the out-plane leading to a decrease in KDF were predicted. In particular, when axial compressive load is concentrated at locations of large local shape imperfections in the out-plane deformation, the buckling load was assumed to decrease.