## 2 Mathematical Model and Assumptions  
Due to the incomplete structure of the pipeline and the initial imperfection, pipelines are more prone to lateral buckling with the decrease of critical temperature difference. As shown in Figure 1, when full-contact repeat imperfection is considered, the imperfection profile can be expressed as:  
\[\omega_{0}(x)=\frac{\omega_{\mathrm{m0}}}{2}\mathrm{sin}(\lambda_{\mathrm{e}} x)\ 0\leq x\leq L_{0} \tag{1}\]  
where \(\lambda_{\mathrm{e}}=\pi/L_{0}\); \(\omega_{\mathrm{m0}}\) is the amplitude of imperfection and \(L_{0}\) is the half wavelength of imperfection. For submarine pipelines, there is a temperature difference \(\Delta\)T between the fluid flowing inside the pipe and the environment. If the pipe can expand, the axial compressive strain caused to thermal expansion will accumulate at the axial resistance between the pipe and the seabed. Therefore, within the online elastic response range, the axial compressive force \(P_{0}\) corresponding to the axial strain should be written as:  
\[P_{0}=AEx\Delta\mathrm{T} \tag{2}\]  
where \(E\) is the elastic modulus, \(A\) is the cross-sectional area of the pipeline, \(\alpha\) is the linear thermal expansion coefficient. The lateral buckling behavior of the pipeline is simulated using the linear beam theory. The equation controlling the lateral imperfection of the beam is [22]:  
\[EI\frac{d^{4}\omega}{dx^{4}}+P\frac{d^{2}\omega}{dx^{2}}=-F_{A}-P\frac{d^{2} \omega_{0}}{dx^{2}} \tag{3}\]  
where \(P\) is the effective axial compression force at the center of the buckle, \(F_{A}\) is the axial soil resistance, \(\omega\) is pipeline buckling amplitude. In this study, the sea bed is assumed to be fully rigid. A new parameter \(\lambda\) can then be introduced as follows:  
\[\lambda^{2}=\frac{P}{EI} \tag{4}\]  
The general solution of the equation can be written as:  
\[\omega(x)=A_{1}+A_{2}x+A_{3}\cos\lambda x+A_{4}\sin\lambda x-\frac{F_{A}}{2 \lambda^{2}EI}x^{2}+\frac{\omega_{\mathrm{m0}}\lambda^{2}}{2(\lambda_{\mathrm{ e}}^{2}-\lambda^{2})}\sin\lambda_{\mathrm{e}}x \tag{5}\]  
The pipeline at the anchor point (\(x=0\)) of the buckling amplitude is inextensible, unshearable. So, the lateral deflection and slope must be zero. \[\left\{ \begin{array}{l}\omega(0)=0\\ \\ \frac{d\omega}{dx}(0)=0\end{array} \right. \tag{6}\]  
Figure 1: Modes of the initial imperfection. At the other end of the pipeline, buckling amplitude, shearing force, and bending moment are zero at \(x=L_{\text{S}}\). \[\left\{ \begin{array}{l}\omega(L_{\text{S}})=0\\ \\ \frac{d\omega}{dx}(L_{\text{S}})=0\\ \\ \frac{d^{2}\omega}{dx^{2}}(L_{\text{S}})=0\\ \end{array} \right. \tag{7}\]  
Combining Equations (5)-(7), the solved equation coefficients are:  
\[\left\{ \begin{array}{l}\text{A}_{1}=\frac{F_{A}}{\lambda^{2}EI}\\ \text{A}_{2}=-\frac{F_{A}L_{\text{S}}}{\lambda^{2}EI(1-\lambda\cos\lambda L_{ \text{S}})}+\frac{\omega_{\text{m}}\omega_{\text{D}}\lambda_{\text{S}}(\lambda \cos\lambda L_{\text{S}}-\cos\lambda_{\text{S}}L_{\text{S}})}{2(\lambda_{ \text{e}}^{2}-\lambda^{2})(1-\lambda\cos\lambda L_{\text{S}})}\\ \text{A}_{3}=-\frac{F_{A}}{\lambda^{2}EI}\\ \text{A}_{4}=-\frac{F_{A}L_{\text{S}}}{\lambda^{2}EI(1-\lambda\cos\lambda L_{ \text{S}})}+\frac{\omega_{\text{m}}\omega_{\text{D}}\lambda_{\text{S}}(2\cos \lambda L_{\text{S}}-1-\cos\lambda_{\text{S}}L_{\text{S}})}{2(\lambda_{\text{ e}}^{2}-\lambda^{2})(1-\lambda\cos\lambda L_{\text{S}})}\end{array} \right. \tag{8}\]  
The buckling part of the pipe is regarded as a separate short pipe and divided into three parts, namely the buckle section (\(0<x<L_{0}+\Delta L\)), the sliding section (\(L_{0}+\Delta L<x<L_{\text{S}}\)), and the anchor section (\(x>L_{\text{S}}\)). As shown in Figure 2, the central part of the pipeline will move horizontally during lateral buckling. Then, when the pipeline enters the buckling section, the effective axial compression force of the pipe drops, pulling more pipes into the buckling.