## 6 Conclusions  
For the circumferential direction, KDF decreases as the shape imperfections increase in amplitude. The intercepts of the linear regression KDF\({}_{0}\) obtained by the least squares method all lie between 50 and 65%. Namely, the KDF is 50-65% even when there are no shape imperfections in the circumferential direction. A possible cause could be the variation in thickness of the cylinder. This issue will be considered in the future. The simple linear regression analysis for \(w_{rms}/t\) indicated with high statistical significance that \(w_{rms}/t\) can lower the buckling load. The ANOVA indicated high statistical significance; in particular, results for \(w_{2}/t\)-\(w_{10}/t\) indicate that \(w_{7}/t\) has a small effect on improving the KDF.  
Regarding the axis direction, KDF decreases as the dimensionless values of the Fourier coefficients of shapes 1 and 3 increase. On the other hand, they increase as dimensionless values for shapes 2 and 4 decrease, i.e., the shape imperfections of shapes 2 and 4 enhance the buckling load. The \(p\)-values for shapes 1, 2, 3, and 4 are all smaller than 5%, indicating statistical significance. The \(p\)-value of shape 1 is smaller than 5%, indicating statistical significance. The \(p\)-value of shape 3 is much smaller than 1%, indicating high statistical significance. The \(p\)-values of shapes 2, 4, and 5 are over 5%, indicating statistical insignificance.  
The offset direction and offset distance both affect the KDF, i.e., the KDF does not have a simple correlation with offset direction, shape imperfections, or offset distance. This conclusion is consistent with the ANOVA.  
In the evaluation of the relationship between KDF and r/t, it is evident that the KDF decreases as the r/t ratio increases. The KDFs scatter across both sides of the line; thus, this linearity cannot be used for prediction. This conclusion is consistent with previous studies [22]. Data with a KDF below 40% were obtained for \(r/t\) values between 500 and 900; this should be considered when using the buckling load. But this linearity cannot be directly used in the design.  
Figure 15: KDF versus \(r/t\). (The references for each data are by [PERSON] [8], [PERSON] [9], [PERSON] [10], [PERSON] [2], [PERSON] [11], [PERSON] [12], [PERSON] [13], [PERSON] [14], [PERSON] [15], [PERSON] [16], [PERSON] [17], [PERSON] [18], [PERSON] [19], [PERSON] [20], [PERSON] [21], [PERSON] [22], and [PERSON] [24]).  
This paper can be used for designing CFRP cylinders with FEM and can also provide references for designing thin-walled cylindrical structures. When \(r/t>500\), a KDF of 50% or less should be considered when calculating the buckling load of thin-walled cylinders. For example, it can be used to design the rocket body of an ultra-compact hybrid rocket [29].  
Conceptualization, A.T.; methodology, A.T.; software, A.T. and J.B.; validation, A.T. and J.B.; formal analysis, A.T. and J.B.; data curation, J.B.; investigation, J.B.; writing--original draft preparation, J.B.; writing--review and editing, R.K. and T.M. All authors have read and agreed to the published version of the manuscript. This work was supported by JKA and its promotion funds from KEIRIN RACE.  
Not applicable.  
The data presented in this study are available on request from the corresponding author.  
The authors declare no conflicts of interest.