## 5 Conclusions  
An analytical solution is provided for the lateral buckling of the seabed pipeline based on the fixed-fixed ends beam model, considering the initial imperfection. The distribution of the external stress and displacement is determined by comparing the different extreme values of the axial compression stress and displacement amplitudes. The influence of the pipeline diameter-thick ratio effect on the buckling behavior is then analyzed. A detailed parametric analysis investigates the effects of temperature loading, imperfection amplitude, soil resistance on lateral buckling development and thermal vibration. The following conclusions can be drawn:  
1. When the virtual anchor section divides the pipeline into multiple short sections, the thermal vibration phenomenon can be found based on the fixed-fixed beam model. The thermal vibration of the beam will cause the fatigue effect when the pipe bears the thermal load or the axial load, reducing the service life of the pipe. Therefore, when studying the lateral buckling problem of the pipeline, attention should be paid to the thermal vibration effect of the pipeline. 2. With increasing initial imperfection amplitude, the displacement amplitude and bending stress of the pipe increase linearly and the maximum bending stress position moves toward the position of the imperfection. At the same time, with increasing initial imperfection, the frequency and density of the pipeline stress vibration also increases, making the stress mutation effect more obvious. Attention should be paid to the fatigue disruption caused by continuous stress mutations in the pipeline during buckling. 3. Virtual anchor segment length (VAS) and diameter-thick ratio of the pipeline have an obvious influence on the behavior of lateral buckling. Increasing the diameter-thick ratio significantly increases the bending stress of the pipeline, and the anchoring effect of increasing the tensile stress on the pipeline. This is advantageous for the pipeline stability. When lateral buckling of the pipeline occurs, there is an obvious size effect. Therefore, in the pipeline design, the size effect of the pipeline must be considered. 4. With increasing temperature difference, the buckling amplitude and bending stress increase linearly. In addition, the maximum bending stress point moves in the positive direction of \(L_{\text{S}}\), making the stress response at the pipe imperfection stronger. The corresponding tensile force also increases linearly with increasing temperature difference. Therefore, the pipeline running at high temperature is more likely to undergo lateral buckling at the initial imperfection. 5. As the soil resistance increases, the buckling amplitude decreases, and the bending stress increases accordingly. The maximum bending stress point moves toward the imperfection. With increasing soil resistance, pipe buckling is less likely to occur. Figure 16: Change in parameter \(\lambda^{2}\) during buckling under different solid resistance conditions. Therefore, it is prudent to prevent lateral buckling by increasing the effective friction coefficient, such as by digging trench lines. Conceptualization, S.W., L.L. and W.W.; methodology, S.W., W.W. and H.L. ; software, S.W., T.W. and H.L. ; validation, S.W., T.W. and H.L. ; formal analysis, S.W., L.L. and H.L. ; investigation, S.W., W.W. and H.L. ; resources, L.L. and W.W.; data curation, S.W. and T.W. ; writing--original draft preparation, S.W., L.L. and H.L. ; writing--review and editing, W.W. and H.L. ; supervision, L.L., W.W. and H.L. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China (Grant Nos. 52178371, 52168046, 52178321, 52108347), the Exploring Youth Project of Zhejiang Natural Science Foundation (LQ22E080010), the Outstanding Youth Project of Natural Science Foundation of Zhejiang Province (Grant No. LR21E080005), the China Postdoctoral Science Foundation Funded Project (Grant No.