## 4 Summary and Conclusions
The study shows that the enhancement coefficient \(\mathbf{k}\) can be represented as the change of boundary conditions. By reducing the \(l/R\) ratio by \(\sqrt{k}\) times, the improved enhanced model can predict the buckling value of the lining and the number of possible buckling lobes. 7. The enhanced prediction model can be used to design the inner wall thickness of the water-supply pipeline repaired by SSL. When the negative pressure is taken into account, the number of possible lobe formations decreases with the increase of lining wall thickness. If the lining thickness reaches a certain value, the required buckling pressure has exceeded the vacuum limit. In this case, negative pressure is not needed to be considered. However, considering the cost of the thin-walled stainless-steellining, the reinforcement model can be applied to obtain the optimal wall thickness, thus making the lining design economical and safe. Conceptualization, C.L. and X.Y. ; methodology, C.L. ; validation, S.T.A. and B.M. ; formal analysis, C.L. ; investigation, C.L., Y.Z. and W.X. ; resources, B.M. ; data curation, C.L. X.Y. and Y.Z. ; writing--original draft preparation, C.L. ; writing--review and editing, C.L. and X.Y. ; visualization, C.L. and Y.Z. ; supervision, S.T.A. and B.M. funding acquisition, B.M. All authors have read and agreed to the published version of the manuscript. This work was supported by the National Science Foundation of China (Project No. 51979254). This research was supported by Zhejiang Provincial Natural Science Foundation of China under Grant No. LGG20E080005. Not applicable. Data Availability Statement: Data is contained within the article. The authors are very grateful to the technical support provided by Fuzhou Water Engineering Co., Ltd., Fujian, China. The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results. Nonenclature \begin{tabular}{l l} \(a\) & Axial coefficient of thermal expansion of liner, generally \(a=1.2\times 10^{-5}\) mm/(mm \({}^{\circ}\)C); \\ \(E_{z}\) & Axial elastic modulus; \\ \(\Delta T\) & The difference between the test temperature and the installation temperature \\ \(H\) & The burial depth of the pipeline (From the crown to the surface); \\ \(R\) & The shell radius; \\ \(u,v,w\) & Displacement in the \(x,y\) and \(z\) directions; \\ \(N_{x},N_{y},N_{xy},N_{yx}\) & The normal force and shear force per unit length on the middle surface \\  & of the shell; \\ \(Q_{x},Q_{y}\) & The shear force per unit length of shell; \\ \(P\) & System internal pressure; \\ \(M_{x},M_{y},M_{xy}\) & Bending moment and torque per unit length of shell; \\ \(\varepsilon_{1},\varepsilon_{2},\gamma\) & The normal strain in the 1, 2 direction and shear strain; \\ \(\chi_{x},\chi_{y},\chi_{xy}\) & Variation in curvature per unit length of shell; \\ \(E\) & Elastic modulus of liner; \\ \(t\) & The average thickness of liner; \\ \(\mu\) & Poisson ratio; \\ n & The lobe number; \\ \(l\) & The length of pipeline; \\ \(\mathrm{k}_{i}\) & Enhancement coefficient of wall support; \\ \(\lambda\) & \(\pi R/l\). \\ \end{tabular}