## 4 Parametric Study  
### Initial Imperfection Effect  
In this section, the effect of the amplitude of the initial imperfection \(\omega_{\text{m0}}\) on the lateral buckling configuration and the typical lateral buckling behavior is analyzed. The pipeline parameters, pipeline properties, soil properties, and soil parameters are kept constant, as shown in Table 1.  
The buckling amplitude development and bending stress distribution of the pipeline with the variance of \(\omega_{\text{m0}}/L_{0}\) are shown in Figure 6. As the initial imperfection ratio increases, the global buckling amplitude of the pipeline also increases. From the buckling \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline Elastic modulus \(E\) & 206 & GPa \\ External diameter \(D\) & 0.65 & m \\ Wall thickness \(t\) & 0.0227 & m \\ Steel density \(\rho\) & 7850 & kg/m\({}^{3}\) \\ Moment of inertia \(I\) & \(1.509\times 10^{-3}\) & m\({}^{4}\) \\ Coefficient of thermal & & \\ expansion \(\alpha\) & \(1.1\times 10^{-5}\) & \({}^{\circ}\)C\({}^{-1}\) \\ Axial friction coefficient \(\mu_{\text{A}}\) & 0.5 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters. Figure 5: Comparison of buckling development of pipelines in different studies: (**a**) Comparison of the buckling distributions; (**b**) comparison of the buckling maximum to \(\lambda^{2}\)[26; 27]. deformation shown in Figure 6a, the initial imperfection ratio can greatly increase the buckling amplitude of the pipeline. The pipeline still deflects in the imperfection center as \(\omega_{\text{m0}}\) increases. Therefore, the imperfection is the weak part of the pipeline, and stress concentration would occur at the imperfection. In Figure 6b, the maximum bending stress is referred as \(\sigma_{\text{Mm}}\). When the initial imperfection exists, the maximum axial compressive stress \(\sigma_{\text{Mm}}\) increases significantly with increasing initial imperfection ratio. With the increase of the initial imperfection, the maximum bending stress value moves in the positive direction of \(L_{\text{S}}\). This is because as the imperfection locates closer to the center of the pipeline, the stress is more likely to be affected by the initial imperfection amplitude. Moreover, the feeding-in amplitude increases with the increase of the imperfection during the buckling of the fixed-end pipeline. \(\lambda^{2}\) is expressed as a dimensionless form of the effective axial compressive force, and the values of \(N\) and \(\lambda^{2}\) can be obtained by solving the transcendental equations to obtain their relationship with the pipeline virtual anchor segment length \(L_{\text{S}}\). During the lateral buckling of the pipeline, both \(\lambda^{2}\) and the pull force \(N\) by the pipeline feeding-in will change with the initial imperfection \(\omega_{\text{m0}}\). They are related to the pipe virtual anchor segment length \(L_{\text{S}}\). As can be seen from Figure 7a, as \(\omega_{\text{m0}}\) increases, the \(N\) values will also decrease nonlinearly. This means that the bearing capacity of the pipeline decreases as the initial imperfection ratio increases. When the initial imperfection ratio remains unchanged, the bearing capacity increases from \(L_{\text{S}}/L_{\text{0}}=3\) to \(L_{\text{S}}/L_{\text{0}}=8\) and the corresponding tensile force  
Figure 6: Effect of the initial imperfection on the buckling configuration: (**a**) Influence of \(\omega_{\text{m0}}\) on deformation; (**b**) influence of \(\omega_{\text{m0}}\) on bending stress. \(N\) decreases by 0.21 MN. The change in tension is more drastic, and its influence cannot be ignored. It can be clearly seen that local stress vibration occurs with the decrease of \(L_{\text{S}}\). And with increasing tension of the pipeline, the lateral buckling of the pipeline maintains the feed. When the pipeline is perfect, the local stress vibration is not obvious. However, when the initial imperfection is considered, the vibration becomes obvious.