## 5 Test Results and Evaluation
### The Amplitude of Shape Imperfections in the Circumferential Direction
Here, \(w_{rms}\) in panel (j) is the amplitude of shape imperfections, i.e.,  
\[w_{rms}=\sqrt{\frac{1}{9}\times\sum_{n=2}^{9}w_{n}^{2}} \tag{5}\]  
with  
\(w_{n}\): each amplitude  
As a result, we see the correlation between \(w_{2}/t\) and \(w_{rms}/t\) and KDF from Figure 10. When the dimensionless shape imperfections, i.e., the amplitude of the shape imperfections divided by \(t\) increases in value, the KDF decreases. In other words, the KDF decreases as the shape imperfections become larger. The intercepts of the linear regression lines, KDF\({}_{0}\), obtained by the least square method are all between 0.5 and 0.65. This means KDF\({}_{0}\), i.e., the KDF, is between 50 and 65% even when there are no shape imperfections in the circumferential direction. For the KDF, it is between 50 and 65%, a possible cause could be the variation in the thickness of the cylinder. Figure 10: _Cont._  
The relationship between KDF, shape imperf, and offset was investigated through ANOVA (analysis of variance) and multiple regression analysis (Equation (6)). Qualitative variables with offset \(\delta_{j}\) were represented by dummy variables (0 or 1). \[KDF=\sum_{n=2}^{10}a_{n}(\frac{w_{n}}{t})+b_{0}(\delta_{0})+b_{90}(\delta_{90})+ b_{180}(\delta_{180})+b_{270}(\delta_{270})+KDF_{0} \tag{6}\]  
with  
\(a_{n}\): slope of \(w_{n}/t\) (\(n\) = 2-10)  
\(n\): amplitude in wave numbers  
\(\delta_{j}\): \( \begin{cases}0(\text{offset}\
eq j\text{deg})\\ 1(\text{offset}=j\text{deg})\end{cases} \)  
\(j\): deg of offset (\(j=0,90,180,270\) deg)  
Since the CFRP cylinders were offset during the buckling tests, the data with offsets at each angle should be considered as being distinct from the data without offsets. Additionally, due to the point load, the first and second tests of no offsets were not applied to the exact same location. This ensured the independence of the data. Since the second and third tests of no offsets were continuous compressive static tests, the third test of no offsets was not included in the ANOVA. \(w_{rms}/t\) includes \(w_{2}/t\) to \(w_{10}/t\), so \(w_{rms}/t\) was investigated separately. The \(p\)-value represents the probability that the data fit the regression equation by coincidence. When the \(p\)-value is small, it indicates that the occurrence of such randomness is unlikely, and the regression equation is of significance. A \(p\)-value below 5% is statistically significant, and a \(p\)-value below 1% indicates high statistical significance. Small \(p\)-values are hoped for to prove hypotheses with limited amounts of data [27]. In Table 7, the \(p\)-value of \(w_{rms}/t\) is 5.85 \(\times\)\(10^{-09}\), which is much smaller than 1%, indicating high statistical significance. The \(p\)-values of the offsets are larger than 5%, indicating no statistical significance. The slope (a) of \(w_{rms}/t\) is \(-0.166\). This result was obtained using simple linear regression analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \(\mathbf{KDF_{0}}\) & \(\bolds{w_{rms}/t}\) \\ \hline \(p\)-value & \(3.04\times 10^{-73}\) & \(5.85\times 10^{-09}\) \\ slope (a) & \(0.557\) & \(-0.166\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: \(p\)-value of regression analyses of \(w_{rms}/t\). Figure 10: KDF versus \(w_{2}/t\)-\(w_{10}/t\) and \(w_{rms}/t\). Table 8 shows the results of multiple regression analyses of \(w_{2}/t\)-\(w_{10}/t\). Here, the \(p\)-values of \(w_{4}/t\), \(w_{6}/t\), and \(w_{8}/t\) are extremely small, and the slope (a) is too large. For example, when \(w_{i}/t\) changes, KDF approaches 0. The \(w_{8}/t\) of slope (a) is 0.308, and the slope (a) \(\times\) (range of \(w_{8}/t\)) = 0.308 \(\times\) 1.3 = 0.4. Furthermore, the KDF is \(0.665+0.4=1.065\), which is greater than 100%. Regarding the cause, since each variable affects one another, it can be speculated that multi-collinearity exists with \(w_{2}/t\)-\(w_{10}/t\). The multi-collinearity is the occurrence of high intercorrelations among two or more independent variables in a multiple regression model.