## 2 Mathematical Model and Assumptions
Based on the profile of axial compressive force shown in Figure 3, the distribution of axial compressive force \(P(x)\) is written as:  
\[P(x)=P-N+F_{A}x\ (0\leq x\leq L_{\text{S}}) \tag{9}\]  
where \(N\) is the tension force corresponding to the beam tension. The axial friction force can be expressed as:  
\[F_{A}=\mu_{A}q \tag{10}\]  
where \(\mu_{A}\) is the axial friction coefficient between pipeline and seabed, and \(q\) is the submerged weight per unit length of the pipeline. Combining Equations (1), (3) and (19), the axial force balance is obtained:  
\[A\mathbb{E}\alpha\mathbb{T}=EI\lambda^{2}-N+\mu_{A}qL_{\text{S}} \tag{11}\]  
[PERSON] [23] and [PERSON] [24] proposed a beam model considering the linear elastic isotropic response of beam under compressive axial force \(P_{0}\). If the pipeline end remains fixed in space and the axial compression load exceeds the critical buckling load, lateral buckling of the pipeline occurs. During lateral buckling, the buckling area gradually  
Figure 2: Configuration and loads distribution. expands along the pipeline length, and the pipeline experiences planar feeding-in during lateral buckling. As shown in Figure 4, the effective compression load at the support of the pipeline reduces the tension generated along the buckling, reducing the initial axial compression load and thus reducing the support reaction [25]. Assuming that the pipeline is a limited long beam, the bending of the beam is allowed while the two ends remain fixed. Based on this assumption, the pipeline buckling shows a midplane stretching phenomenon when the effective axial stress exceeds the buckling stress. In the range of second-order effects, the midplane stretch due to buckling is defined as the feeding-in segment length and can be expressed as:  
\[L_{1}=\frac{1}{2}\int_{0}^{L_{\mathrm{S}}}\frac{d\omega}{dx}dx \tag{12}\]  
Based on Equation (12), the corresponding tensile force \(N\) at the beam supports can be given by [PERSON]'s law [25]:  
\[N=\frac{EA}{L}L_{1} \tag{13}\]  
For the eigenvalue problem of beam vibration under temperature load, there is the following eigenvalue equation:  
\[(P_{0}+4\omega_{\mathrm{L}}{}^{2})\sin k_{1}\sin\lambda k_{2}=0 \tag{14}\]  
where \(\omega_{\mathrm{L}}\) is the linear natural frequency and \(k_{1}\) and \(k_{2}\) are two constants given by:  
\[\left\{ \begin{array}{l}k_{1}=\sqrt{\frac{1}{2}(P_{0}+\sqrt{P_{0}{}^{2}+4 \omega_{\mathrm{L}}{}^{2}})}\\ k_{2}=\sqrt{\frac{1}{2}(-P_{0}+\sqrt{P_{0}{}^{2}+4\omega_{\mathrm{L}}{}^{2}}) }\end{array} \right. \tag{15}\]  
Since \(\omega_{\mathrm{L}}\) must be positive, Equation (14) yields \(k_{1}\) = n\(\pi\) and the linear natural frequency of the first mode is given by:  
Figure 4: A Schematic of the bending deformation of the end fixed-fixed beam under the compression condition. Figure 3: Axial compression distribution of pipe lateral buckling. \[\omega_{\rm L}^{2}=\pi^{2}(\Lambda\rm E\alpha\Gamma-\pi^{2}) \tag{16}\]  
For the fixed-fixed beam, the linear natural frequency of the first mode can be obtained by solving the following transcendental equation:  
\[2k_{1}k_{2}(\cos k_{1}\cos\lambda k_{2}-1)+(k_{1}-k_{2})(k_{1}+k_{2})\sin k_{1} \sin\lambda k_{2}=0 \tag{17}\]  
Considering the bending stress distribution along the pipeline, the moment \(M\) along the bending pipeline is:  
\[M=EI(\frac{d\omega^{2}}{d^{2}x}-\frac{d\omega_{0}{}^{2}}{d^{2}x})\;(0\leq x \leq L_{\rm S}) \tag{18}\]  
It is concluded that the corresponding bending stress \(\sigma_{\rm m}\) along the bending pipe is:  
\[\sigma_{\rm m}=\frac{MD}{2I} \tag{19}\]  
where \(D\) is the pipeline external diameter.