A method for simulating buckling of subsea pipelines based on the pipeline element method

Abstract

This invention provides a method for simulating buckling of subsea pipelines based on the pipeline element method, addressing the problems of low computational efficiency, complex modeling, and low accuracy in existing methods. Based on Euler-Bernoulli beam theory, this invention models the pipeline as a series of efficient pipeline elements. Internal pressure is converted into equivalent temperature rise, and the effective thermal expansion force generated by the effective temperature rise is applied to both ends of the pipeline element to construct the total potential energy equation. The second derivative of the total potential energy equation is performed to obtain the tangent stiffness matrix of the pipeline element. An updated Lagrange method is used for coordinate transformation. The secant relationship is used to calculate the pipeline element and soil resistance. A Newton-Raphson incremental-iterative numerical program is established to perform nonlinear analysis of the pipeline, progressively solving for the deformation and stress of the pipeline under complex loads. This invention significantly reduces the number of elements, improves computational efficiency, and accurately simulates nonlinear pipe-soil interactions. It is applicable to various pipelines, including short pipes, long pipes, and corroded pipes, as well as complex working conditions such as high temperature and high pressure.

Description

Technical Field

[0001] This invention relates to the field of pipeline buckling simulation technology, and in particular to a method for simulating the buckling of subsea pipelines based on the pipeline element method. Background Technology

[0002] In the process of marine resource development, subsea pipelines serve as the core infrastructure for oil and gas transportation, and their safety and stability directly determine development efficiency. As development moves towards deeper and more open seas, pipeline lengths increase, structures become more complex, and they are often exposed to high-temperature and high-pressure environments, making buckling problems increasingly prominent. Therefore, buckling analysis has become a key aspect of pipeline design and operation and maintenance.

[0003] Existing methods for analyzing buckling of subsea pipelines are mainly divided into three categories:

[0004] Test method: It can intuitively show the buckling response under high temperature and high pressure, but it is significantly limited by the test cycle and site, and the performance difference between the scaled model and the actual pipeline (scale effect) is difficult to quantify, which leads to limited applicability of the results.

[0005] Analytical solution: Although it can accurately predict the working condition of subsea pipelines under simple working conditions, it is only suitable for performance prediction under simple working conditions. In complex sea conditions, a large number of simplification assumptions need to be introduced. In particular, it cannot accurately describe the nonlinear pipe-soil interaction and is difficult to meet the engineering accuracy requirements.

[0006] Numerical methods: Numerical methods can accurately predict the buckling response of pipelines through modeling, and can also consider nonlinear pipe-soil interactions through finite element method, SPI element method, or discrete spring model. However, these methods have significant shortcomings in terms of computational efficiency, modeling complexity, and analytical accuracy. For example, while the finite element method can accurately simulate the buckling behavior of pipelines, it is computationally intensive, especially when the pipeline is long, requiring a large amount of computational resources and time. The discrete spring element method simulates soil resistance by adding discrete springs between pipeline elements, which simplifies the modeling process to some extent, but still requires a large number of spring elements to accurately simulate the nonlinear characteristics of the soil, resulting in low computational efficiency. Although the PSI element method is relatively simple in modeling and calculation, its accuracy and applicability are limited, making it difficult to accurately capture the buckling behavior of pipelines under complex conditions.

[0007] In summary, existing methods struggle to balance computational efficiency, modeling complexity, and analytical accuracy. Therefore, there is an urgent need to provide novel buckling analysis methods that can overcome these bottlenecks. Summary of the Invention

[0008] To address the shortcomings of existing technologies, this invention provides a method for simulating buckling of subsea pipelines based on the pipeline element method. This invention can achieve a synergistic improvement in computational efficiency, analytical accuracy, and applicability to operating conditions, thus meeting the needs of practical engineering for buckling analysis of subsea pipelines.

[0009] The technical solution of this invention is: a method for simulating buckling of subsea pipelines based on the pipeline element method, comprising the following steps:

[0010] S1) Based on the Euler-Bernoulli beam theory, the pipeline is modeled as a series of efficient pipeline elements; and the axial, lateral and vertical deformation of each pipeline element, as well as the energy absorbed by the soil around the pipeline element, are calculated.

[0011] S2) Apply internal pressure to the pipe unit, convert the internal pressure into an equivalent temperature rise and calculate the effective temperature rise of the pipe unit. Apply the effective thermal expansion force generated by the effective temperature rise to both ends of the pipe unit, and then integrate the effective thermal expansion force into the overall stiffness matrix of the pipe unit, while constructing the total potential energy equation.

[0012] S3) Perform second-order derivative calculation on the total potential energy equation to obtain the tangent stiffness matrix of the pipe element, predict the deformation of the pipe element under incremental load, and obtain the nodal displacement of the pipe element under each incremental load.

[0013] S4) The coordinate transformation is performed using the updated Lagrange method to obtain the stiffness matrix in the global coordinate system;

[0014] S5) Calculate the pipe unit and soil resistance using secant relationships;

[0015] S6) Establish a Newton-Raphson incremental-iterative numerical program to perform nonlinear analysis on the pipeline, set convergence criteria for the displacement increment and load mass of the pipeline element, and solve the deformation and stress of the pipeline under complex loads step by step.

[0016] Preferably, in step S1), the formulas for calculating the axial, lateral, and vertical deformations of each pipe unit are as follows:

[0017] ; (1)

[0018] ; (2)

[0019] ; (3)

[0020] ; (4)

[0021] In the formula, , , These represent the axial, lateral, and vertical deformation displacements of the pipeline unit, respectively. Indicates the axial position of the pipe; It indicates torsional deformation.

[0022] Indicates the length of the pipe unit; , These represent the axial deformation displacements at the left and right ends of the pipe unit, respectively. , These represent the lateral deformation displacements at the left and right ends of the pipe unit, respectively. , These represent the vertical deformation displacements at the left and right ends of the pipe unit, respectively. and , and as well as and These represent the rotation angles around the x-axis, z-axis, and y-axis of the left and right ends of the pipeline unit, respectively.

[0023] Preferably, in step S1), the energy absorbed by the soil is calculated using the Gauss-Legendal integral method, i.e.:

[0024] ; (5)

[0025] In the formula, This represents the energy absorbed by the soil surrounding the pipeline unit; , , These represent the energy absorbed by the pipeline unit in the vertical, horizontal, and axial directions, respectively.

[0026] Preferably, in step S2), the internal pressure is converted into an equivalent temperature rise and the effective temperature rise of the pipeline is calculated, i.e.:

[0027] ; (6)

[0028] In the formula, Indicates the effective temperature difference; Indicates the actual temperature rise; This indicates the equivalent temperature rise caused by the internal pressure of the pipe; This indicates the change in internal pressure. , , , These represent the diameter, wall thickness, Poisson's ratio, and Young's modulus of the pipe unit, respectively. The coefficient of thermal expansion of a material.

[0029] Preferably, in step S2), the total potential energy equation is expressed as:

[0030] ; (7)

[0031] In the formula, This represents the total potential energy of the unit; Work done on external loads; This refers to the strain energy of the pipe unit itself; Energy absorbed by the soil surrounding the pipeline; The energy generated for effective temperature rise.

[0032] Preferably, in step S3), the second derivative of the total potential energy equation is performed to derive the tangent stiffness matrix of the element, predict the deformation of the pipeline under incremental load, and obtain the nodal displacement of the pipeline under each incremental load, i.e.:

[0033] ; (8)

[0034] ; (9)

[0035] In the formula, The pipe unit represents the first The total potential energy equation for all degrees of freedom; , These represent the displacements corresponding to m and n degrees of freedom, respectively. This represents the load corresponding to the m-th degree of freedom; This represents the tangent stiffness matrix of the pipe element; These represent the linear stiffness matrix, geometric stiffness matrix, effective thermal expansion stiffness matrix, and axial, lateral, and vertical soil stiffness matrices of the pipeline element, respectively.

[0036] Preferably, in step S4), the stiffness matrix in the global coordinate system... Represented as:

[0037] ; (10)

[0038] In the formula, Indicates the first Stiffness matrix of each pipe element; Here is the tangent stiffness matrix for the pipe element; This is the transformation matrix; This represents the total number of pipe units.

[0039] Preferably, in step S5), the secant relationship is used to calculate the unit and the soil resistance, that is:

[0040] ; (11)

[0041] In the formula, Indicates total resistance; For the resistance of the pipeline unit itself; The resistance generated by the soil.

[0042] Preferably, in step S6), a convergence criterion is set for the displacement increment and load mass of the pipeline element, and the deformation and stress of the pipeline under complex loads are solved step by step, i.e.:

[0043] ;(12)

[0044] ; (13)

[0045] ;(14)

[0046] In the formula, , They represent the first The external load vector, pipe element resistance, initial unbalanced force, initial unbalanced displacement, and overall displacement vector of the next iteration; Indicates the allowable error; This indicates the transpose operation.

[0047] The beneficial effects of this invention are as follows:

[0048] 1. This invention reduces modeling and computation time by modeling the pipeline as a series of efficient pipeline elements, which significantly reduces modeling and computation time. Compared with traditional methods (such as Abaqus) and discrete spring element methods, the number of elements required is greatly reduced, and computational resources and time are significantly reduced.

[0049] 2. This invention can efficiently simulate complex working conditions. When dealing with complex load conditions, this invention can efficiently control the initial defects, damage and soil resistance of pipelines, avoiding the cumbersome process that requires a lot of computational resources and time in traditional methods.

[0050] 3. This invention accurately simulates nonlinear pipe-soil interaction. By considering the nonlinear interaction between the pipe and the soil, this invention can accurately simulate the buckling behavior of the pipe under complex conditions; the simulation accuracy of pipe buckling behavior is high.

[0051] 4. This invention converts thermal expansion and internal pressure into equivalent temperature rise, which is then applied to the pipe unit in the form of internal force. This allows for accurate simulation of the buckling behavior of pipes under high temperature and high pressure conditions, thus improving the accuracy of the analysis.

[0052] 5. This invention is applicable not only to two-dimensional and three-dimensional analysis, but also to different types of pipes (short pipes, long pipes, corroded pipes) and complex engineering conditions (such as high temperature and high pressure, nonlinear soil-pipe interaction, sleepers). Attached Figure Description

[0053] Figure 1This is a flowchart illustrating the method of Embodiment 1 of the present invention;

[0054] Figure 2 This is a schematic diagram of a single pipe unit in the pipe unit method of Embodiment 2 of the present invention;

[0055] Figure 3 This is a flowchart of the two-dimensional pipe buckling analysis in Embodiment 2 of the present invention;

[0056] Figure 4 This is a comparison chart of the buckling amplitude of the pipeline calculated using the method of the present invention and the analytical solution at different temperatures in Embodiment 3 of the present invention;

[0057] Figure 5 The graph shows a comparison of the relationship between the lateral displacement of the monitoring point and the load coefficient calculated by the method of this invention and the ABAQUS finite element method, respectively. Detailed Implementation

[0058] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings:

[0059] Example 1

[0060] like Figure 1 As shown, this embodiment provides a method for simulating buckling of subsea pipelines based on the pipeline element method, including the following steps:

[0061] S1) Based on the Euler-Bernoulli beam theory, the pipeline is modeled as a series of efficient pipeline elements; and the axial, lateral and vertical deformation of each pipeline element, as well as the energy absorbed by the soil around the pipeline element, are calculated.

[0062] In this embodiment, the formulas for calculating the axial, lateral, and vertical deformations of each pipe unit are as follows:

[0063] ; (1)

[0064] ; (2)

[0065] ; (3)

[0066] ; (4)

[0067] In the formula, , , These represent the axial, lateral, and vertical deformation displacements of the pipeline unit, respectively. Indicates the axial position of the pipe; Indicates torsional deformation; Indicates the length of the pipe unit; , These represent the axial deformation displacements at the left and right ends of the pipe unit, respectively. , These represent the lateral deformation displacements at the left and right ends of the pipe unit, respectively. , These represent the vertical deformation displacements at the left and right ends of the pipe unit, respectively. and , and as well as and These represent the rotation angles around the x-axis, z-axis, and y-axis of the left and right ends of the pipeline unit, respectively.

[0068] This embodiment calculates the energy absorbed by the soil using the Gauss-Legendal integral method, integrating the continuously distributed soil resistance into the stiffness matrix of the pipe element, thus avoiding the cumbersome modeling required by traditional methods that necessitates a large number of discrete spring elements. This embodiment further differentiates the energy generated by soil resistance into lateral, vertical, and axial directions, i.e.:

[0069] ; (5)

[0070] In the formula, This represents the energy absorbed by the soil surrounding the pipeline unit; , , These represent the energy absorbed by the pipeline unit in the vertical, horizontal, and axial directions, respectively.

[0071] S2) Apply internal pressure to the pipe unit and convert the internal pressure into an equivalent temperature rise. Calculate the effective temperature rise of the pipe unit. Apply the effective thermal expansion force generated by the effective temperature rise to both ends of the pipe unit in the form of an axial force. Then integrate the effective thermal expansion force into the overall stiffness matrix of the pipe unit (refer to equation (9)). At the same time, construct the total potential energy equation; that is:

[0072] ; (6)

[0073] In the formula, Indicates the effective temperature difference; Indicates the actual temperature rise; This indicates the equivalent temperature rise caused by the internal pressure of the pipe; This indicates the change in internal pressure. , , , These represent the diameter, wall thickness, Poisson's ratio, and Young's modulus of the pipe unit, respectively. The coefficient of thermal expansion of a material.

[0074] The total potential energy equation is expressed as follows:

[0075] ; (7)

[0076] In the formula, This represents the total potential energy of the unit; Work done on external loads; This refers to the strain energy of the pipe unit itself; Energy absorbed by the soil surrounding the pipeline; The energy generated for effective temperature rise.

[0077] in,

[0078] ; (7-1)

[0079]

[0080] ; (7-6)

[0081] ; (7-7)

[0082] ; (7-8)

[0083] In the formula, , and These represent the shear stresses, which are the normal stresses along the pipe axis on the pipe cross-section. , , , These represent the strain on the cross-section of the pipe; and , and , and , and as well as and 1 and 2 represent the axial load, bending moment about the y-axis, bending moment about the z-axis, shear force in the y-axis and z-axis directions, and moment of inertia of the pipe section about the y-axis and z-axis, respectively; 1 and 2 represent the left and right ends of the pipe element, respectively. The length of the pipe unit; Indicates the cross-sectional area; , , These represent the axial, lateral, and vertical deformation displacements of the pipeline unit, respectively. Let be the angle of rotation of the pipe element around the x-axis.

[0084] S3) Perform second-order derivative calculations on the total potential energy equation to obtain the tangent stiffness matrix of the pipe element. Using the Newton-Raphson iteration method, solve the updated stiffness equation in each incremental load step to obtain the nodal displacements of the pipe under the current load step. This allows for accurate prediction of the pipe's deformation under incremental loads.

[0085] ; (8)

[0086] ; (9)

[0087] In the formula, The pipe unit represents the first The total potential energy equation for all degrees of freedom; , Respectively represent the land and Displacement corresponding to each degree of freedom; Indicates the first The load corresponding to each degree of freedom; This represents the tangent stiffness matrix of the pipe element; These represent the linear stiffness matrix, geometric stiffness matrix, effective thermal expansion stiffness matrix, and axial, lateral, and vertical soil stiffness matrices of the pipeline element, respectively.

[0088] S4) The coordinate transformation is performed using the updated Lagrange method to obtain the stiffness matrix in the global coordinate system; the stiffness matrix in the global coordinate system Represented as:

[0089] ; (10)

[0090] In the formula, Indicates the first Stiffness matrix of each pipe element; Here is the tangent stiffness matrix for the pipe element; This is the transformation matrix; This represents the total number of pipe units.

[0091] S5) Calculate the pipe element and soil resistance using secant relationships; calculate the element and soil resistance using secant relationships, i.e.:

[0092] ; (11)

[0093] In the formula, Indicates total resistance; For the resistance of the pipeline unit itself; The resistance generated by the soil.

[0094] (11-1)

[0095] (11-2)

[0096] In the formula, , These represent the self-resistance at the left end of the pipe unit in the x, y, and z directions, respectively. These represent the self-bending moments at the left end of the pipe element in the x, y, and z directions, respectively. These represent the self-resistance at the right end of the pipe element in the x, y, and z directions, respectively. These represent the self-bending moments at the right end of the pipe element in the x, y, and z directions, respectively. These represent the resistance generated at the left end of the element by the soil in the x, y, and z directions, respectively. These represent the bending moments generated at the left end of the element by the soil in the x, y, and z directions, respectively. These represent the resistance generated at the right end of the element by the soil in the x, y, and z directions, respectively. These represent the bending moments generated at the right end of the element by the soil in the x, y, and z directions, respectively.

[0097] S6) Establish a Newton-Raphson incremental-iterative numerical program to perform nonlinear analysis on the pipeline, focusing on the displacement increment of the pipeline elements. and load increment A convergence criterion is set. If the convergence criterion is met, the next iteration is performed; if the convergence criterion is not met, the current iteration is repeated to gradually solve for the deformation and stress of the pipeline under complex loads.

[0098] ;(12)

[0099] ; (13)

[0100] ;(14)

[0101] In the formula, , They represent the first The external load vector, pipe element resistance, initial unbalanced force, initial unbalanced displacement, and overall displacement vector of the next iteration; Indicates the allowable error; This indicates the transpose operation.

[0102] Example 2

[0103] like Figure 2 As shown, taking the two-dimensional pipe buckling response analysis as an example, this embodiment models the pipe as a series of efficient pipe elements, considering factors such as the nonlinear interaction between the pipe and the soil, thermal expansion, and internal pressure, significantly improving computational efficiency and analysis accuracy. Figure 3As shown, the specific implementation process is as follows:

[0104] 1. Initialize parameters: Based on the pipe's geometry, material properties, and boundary conditions, input the pipe element parameters, including the pipe's length, diameter, wall thickness, elastic modulus, Poisson's ratio, and coefficient of thermal expansion; set the initial load to an unbalanced force to initiate the iteration process;

[0105] 2. Update effective temperature rise: Calculate the effective temperature rise of the pipeline based on its thermal expansion characteristics and current temperature conditions, and then obtain the effective thermal expansion matrix;

[0106] 3. Update the soil stiffness matrix: Based on the nonlinear soil resistance curve, calculate the soil stiffness matrix at different Gaussian points using the Gauss-Legend integral method to obtain the soil stiffness matrix. Then, based on the nodal displacements, update the tangent value of the soil-concentrated pipe interaction curve to ensure that the nonlinear characteristics of soil resistance are accurately reflected in the analysis.

[0107] 4. Obtain the element tangential stiffness matrix and convert it to the global stiffness matrix: Calculate the element tangential stiffness matrix based on the pipe's geometric parameters, material properties, and effective temperature rise, and use a transformation matrix to convert the element tangential stiffness matrix into the global stiffness matrix;

[0108] 5. Apply boundary conditions: Apply displacement and force boundary conditions according to the constraints and load types;

[0109] 6. Calculate unbalanced forces: Calculate the unbalanced forces at the nodes based on the applied loads and their element resistances;

[0110] 7. Calculate the unit and soil resistance based on the secant relationship;

[0111] 8. Check convergence: If the node displacement increment is less than the allowable error, then convergence is achieved and the result is obtained; otherwise, update the node displacement and repeat the process steps.

[0112] Table 1 shows the results obtained by reproducing Li Xueyou's 2020 using the method provided by this invention in this embodiment. The table analyzes and compares the differences between the analysis results of short pipes with different lengths and different numbers of pipe units and the theoretical solutions.

[0113]

[0114] As can be seen from Table 1, the critical buckling temperature of the pipeline can be accurately captured by using only four pipeline elements to simulate the submarine pipeline, with a maximum relative error of only 0.08%. This means that the method provided by the present invention can reduce the number of elements to be modeled while maintaining sufficient accuracy, thus effectively improving computational efficiency.

[0115] Example 3

[0116] This embodiment considers a 1500-meter-long subsea pipeline with defect ratios of 0.007 and 0.01, and takes into account nonlinear soil resistance. The buckling amplitude of the pipeline at different temperatures is calculated using the method of this invention. Figure 4 As shown, the buckling amplitude gradually increases with increasing temperature, and the increase also gradually increases with increasing temperature. The simulation results of this method are consistent with those of Taylor (1986). i The analytical solutions are basically consistent, indicating that the method has sufficient accuracy in analyzing the buckling response of long tubes considering nonlinear tube-soil interaction.

[0117] Example 4

[0118] Consider a 20-meter-long subsea pipeline, with soil of stiffness 10 kN / m² laid on top and bottom. The left end of the pipeline is fixed, while the right end is subjected to a series of complex loads (800 kN axial force, 60 kN vertical shear force, 80 kN horizontal shear force, 50 kN vertical moment, and 60 kN horizontal moment). By monitoring the displacement and load at the right end, the results of this method are compared with the results of Abaqus modeling. Figure 5 As shown, to achieve similar simulation accuracy, the traditional discrete spring method requires 80 elements, while the pipe finite element method only requires 20 elements. This demonstrates the superior computational efficiency and applicability of the method described in this invention for short pipe analysis.

[0119] The embodiments and descriptions above are merely illustrative of the principles and preferred embodiments of the present invention. Various changes and modifications may be made to the present invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed.

Claims

1. A method for simulating buckling of subsea pipelines based on the pipeline element method, characterized in that, Includes the following steps: S1) Based on the Euler-Bernoulli beam theory, the pipeline is modeled as a series of efficient pipeline elements; and the axial, lateral and vertical deformation of each pipeline element, as well as the energy absorbed by the soil around the pipeline element, are calculated. S2) Apply internal pressure to the pipe unit, convert the internal pressure into an equivalent temperature rise and calculate the effective temperature rise of the pipe unit. Apply the effective thermal expansion force generated by the effective temperature rise to both ends of the pipe unit, and then integrate the effective thermal expansion force into the overall stiffness matrix of the pipe unit, while constructing the total potential energy equation. S3) Perform second-order derivative calculation on the total potential energy equation to obtain the tangent stiffness matrix of the pipe element, predict the deformation of the pipe element under incremental load, and obtain the nodal displacement of the pipe element under each incremental load. S4) The coordinate transformation is performed using the updated Lagrange method to obtain the stiffness matrix in the global coordinate system; S5) Calculate the pipe unit and soil resistance using secant relationships; S6) Establish a Newton-Raphson incremental-iterative numerical program to perform nonlinear analysis on the pipeline, set convergence criteria for the displacement increment and load mass of the pipeline element, and solve the deformation and stress of the pipeline under complex loads step by step. In step S1), the formulas for calculating the axial, lateral, and vertical deformations of each pipe unit are as follows: ； （1） ； （2） ； （3） ； （4） In the formula, , , These represent the axial, lateral, and vertical deformation displacements of the pipeline unit, respectively. Indicates the axial position of the pipe; Indicates torsional deformation; Indicates the length of the pipe unit; , These represent the axial deformation displacements at the left and right ends of the pipe unit, respectively. , These represent the lateral deformation displacements at the left and right ends of the pipe unit, respectively. , These represent the vertical deformation displacements at the left and right ends of the pipe unit, respectively. and , and as well as and These represent the rotation angles around the x-axis, z-axis, and y-axis of the left and right ends of the pipe unit, respectively. In step S2), the internal pressure is converted into an equivalent temperature rise and the effective temperature rise of the pipeline is calculated, that is: ； （6） In the formula, Indicates the effective temperature difference; Indicates the actual temperature rise; This indicates the equivalent temperature rise caused by the internal pressure of the pipe; This indicates the change in internal pressure. , , , These represent the diameter, wall thickness, Poisson's ratio, and Young's modulus of the pipe unit, respectively. The coefficient of thermal expansion of the material; In step S2), the total potential energy equation is expressed as: ； （7） In the formula, Represents the total potential energy; Work done on external loads; This refers to the strain energy of the pipe unit itself; Energy absorbed by the soil surrounding the pipeline; The energy generated for effective temperature rise.

2. The method for simulating buckling of subsea pipelines based on the pipeline element method according to claim 1, characterized in that: In step S1), the energy absorbed by the soil is calculated using the Gauss-Legendal integral method, i.e.: ； （5） In the formula, This represents the energy absorbed by the soil surrounding the pipeline unit; , , These represent the energy absorbed by the pipeline unit in the vertical, horizontal, and axial directions, respectively.

3. The method for simulating buckling of subsea pipelines based on the pipeline element method according to claim 1, characterized in that: In step S3), the second derivative of the total potential energy equation is calculated to derive the tangent stiffness matrix of the element, predict the deformation of the pipe under incremental load, and obtain the nodal displacement of the pipe under each incremental load, i.e.: ； （8） ； （9） In the formula, Represents the first pipe unit The total potential energy equation for all degrees of freedom; , These represent the displacements corresponding to m and n degrees of freedom, respectively. This represents the load corresponding to the m-th degree of freedom; This represents the tangent stiffness matrix of the pipe element; These represent the linear stiffness matrix, geometric stiffness matrix, effective thermal expansion stiffness matrix, and axial, lateral, and vertical soil stiffness matrices of the pipeline element, respectively.

4. The method for simulating buckling of subsea pipelines based on the pipeline element method according to claim 3, characterized in that: In step S4), the stiffness matrix in the global coordinate system Represented as: ； （10） In the formula, Indicates the first Stiffness matrix of each pipe element; Here is the tangent stiffness matrix for the pipe element; This is the transformation matrix; This represents the total number of pipe units.

5. The method for simulating buckling of subsea pipelines based on the pipeline element method according to claim 4, characterized in that: In step S5), the secant relationship is used to calculate the unit and the soil resistance, that is: ； （11） In the formula, Indicates total resistance; For the resistance of the pipeline unit itself; The resistance generated by the soil.

6. The method for simulating buckling of subsea pipelines based on the pipeline element method according to claim 5, characterized in that: In step S6), convergence criteria are set for the displacement increment and load mass of the pipeline element, and the deformation and stress of the pipeline under complex loads are solved step by step, i.e.: ； （12） ； （13） ； （14） In the formula, , They represent the first The external load vector, pipe element resistance, initial unbalanced force, initial unbalanced displacement, and overall displacement vector of the next iteration; Indicates the allowable error; This indicates the transpose operation.

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