## 1 Introduction  
Submarine pipeline running at high temperature and high pressure will trigger global thermal expansion. When the axial pressure caused by the thermal expansion and the pressure increase is greater than the critical axial pressure of the pipeline, local buckling, fracture, fatigue, or excessive displacement of the pipeline would occur [1]. During the manufacturing or laying of submarine pipelines, imperfection would occur, which may develop into weak surfaces when the pipeline buckles. Therefore, it is necessary to study the lateral buckling behavior of submarine pipelines containing imperfection. To date, considerable efforts have been made to improve the analytical understanding of the thermal buckling behavior of defective pipelines. [PERSON] [2] first derived the analytical solution for both lateral and upheaval buckling of ideal submarine pipeline embedded on ideal rigid foundation, and deduced the corresponding critical axial force, buckling displacement amplitude, and bending moment based on the assumption of buckle mode. [PERSON] and [PERSON] [3] conducted both theoretical analysis and experimental study on the axial buckling of pipelines under high temperature and high pressure. Three different initial geometric imperfections were investigated to obtain the critical flexion load of the submarine pipeline. In addition, [PERSON] et al. [4] analyzed the influence of the geometry of initial imperfection on the lateral buckling of submarine pipelines, and obtained the critical axial force of the lateral buckling of pipeline laying under different initial imperfection. [PERSON] et al. [5]. derived the analytical solution of the critical buckling load considering the initial geometric imperfection size, imperfection waveform, and geometric instraightness (\(\omega_{\text{m0}}/L_{0}\)) of the initial imperfection. Furthermore, [PERSON] et al. [6] established an analytical solution for local buckling in unburied pipelines based on a nonlinear pipe-soil interaction model, and found that the snap-through required higher temperature differences between the pipes and the environment. [PERSON] et al. [7] and [PERSON] et al. [8] established an analytical solution for the high-order lateral overall buckling of the pipeline considering the single-arch symmetric initial imperfection and the double-arch antisymmetric initial imperfection. For the analytical model of pipeline, pipeline has been modeled as moving beams placed on a rigid [9] or flexible [10] seabed. [PERSON] et al. [10] proposed the lateral buckling control equation of elastic rods with bending stiffness on buckling foundations, and since then, which provides the basis for most theoretical studies on the thermal buckling of submarine pipelines. Based on the Euler-Bernoulli beam theory, [PERSON] [11] and [PERSON] et al. [12] studied the buckling of the beam under different types of thermal loads. [PERSON] et al. [13] studied the thermal buckling of the pipe as a refined beam model, and analyzed the lateral shear deformation, volume fraction, and internal radius of the pipeline on the critical buckling. [PERSON] et al. [14] analyzed the thermal buckling behavior of the pipeline at low temperature based on a viscoelastic plate model. [PERSON] [15] discussed the formulation for the nonlinear buckling problem of buckling loads and posterior buckling states. In recent years, scholars have found that when the pipeline releases the internal axial force through buckling deformation, a virtual anchor point (VAS) will form at the end of the slide segment [16; 17]. The virtual anchor point divides the pipeline into multiple independent buckling short pipes. All existing analytical solutions for lateral buckling assume that the pipeline have infinite length, which indicates that the axial force at the end of the sliding segment is fully constrained. The axial force at the virtual anchor points between multiple buckles is less than the fully constrained axial force, and does not satisfy the basic assumption of the classical analytical solution.