Multi-spring analysis method for horizontal bearing deformation of large-diameter foundation of offshore wind turbine

Abstract

The invention provides a multi-spring analysis method for horizontal bearing deformation of a large-diameter foundation of an offshore wind turbine. The multi-spring analysis method comprises the following steps that the geometric dimension and material attributes of the large-diameter single-pile foundation are determined; determining mechanical parameters of a soil body; bending deformation of a pile body is regarded as an Euler-Bernoulli beam problem, and the pile body is discretized through a finite element method; deriving an element stiffness matrix on the basis of an Euler-Bernoulli beam theory; a transverse soil spring is arranged at a node, and the tangent rigidity is calculated by adopting a hyperbolic p-y curve; a rotary spring is arranged to simulate pile-soil interface rotation damping, and a nonlinear m-theta curve is adopted; a pile bottom horizontal shearing force spring and a pile bottom bending moment spring are arranged, and a hyperbolic curve type relation is adopted; and assembling a global stiffness equation, and carrying out iterative solution by using Newton-Raphson. According to the method, the nonlinear interaction of the four soil springs is comprehensively considered, high-precision pile foundation response prediction is achieved, and the limitation that only p-y springs are considered in a traditional method is overcome; and node concentrated load or node distributed load can be freely applied, so that the calculation efficiency and the engineering applicability are remarkably improved.

Description

Technical Field

[0001] This invention belongs to the field of marine engineering technology, specifically relating to a multi-spring analysis method for horizontal bearing deformation of large-diameter offshore wind turbine foundations. Background Technology

[0002] In nearshore waters, large-diameter monopile foundations have become the mainstream form of offshore wind turbine foundations due to their simple structure and convenient installation. Currently, large-diameter monopile foundations for offshore wind turbines are typically hollow steel pipe piles with a diameter of 5-10 m, which will bear huge horizontal overturning loads during service.

[0003] Currently, the API standard-recommended Py curve method is still widely used in engineering for pile foundation horizontal bearing capacity analysis. The API method simplifies the soil around the pile as a series of independent horizontal soil springs. However, numerous studies have shown that this simplified method has significant shortcomings: First, the Py curve recommended by the standard is significantly softer and has a lower horizontal ultimate bearing capacity compared to the actual pile-soil reaction curve; second, the traditional method is designed for small-diameter flexible long piles and does not consider the soil resistance caused by the rigid rotation of the pile (m-θ effect); third, it does not fully consider the horizontal shear force (Hv) and bending moment (M) at the pile bottom. b -θ b The contribution of springs is not considered; finally, the coupling effect between springs is ignored, which leads to conservative calculation results and waste of engineering costs.

[0004] While existing finite element analysis methods offer high accuracy, they suffer from complex modeling, time-consuming calculations, and a lack of systematic consideration of the combined effects of various soil springs, hindering their rapid application in engineering design. Currently available commercial software that allows multi-spring coupling analysis (such as Abaqus, Monople Designer, and OpenSees) suffers from operational complexity, high theoretical requirements, and incompatibility with most engineering design software, thus restricting the development of refined design for the horizontal bearing deformation of large-diameter single piles. Summary of the Invention

[0005] The main objective of this invention is to provide a multi-spring analysis method for horizontal load-bearing deformation of large-diameter foundations for offshore wind turbines, addressing the aforementioned problems.

[0006] Therefore, the above-mentioned objective of the present invention is achieved through the following technical solution:

[0007] A multi-spring analysis method for horizontal load-bearing deformation of large-diameter offshore wind turbine foundations includes the following steps:

[0008] S1. Determine the geometric dimensions of the large-diameter monopile foundation and the material properties of the steel pipe piles;

[0009] S2. Determine the soil mechanical parameters and the soil layer distribution within the pile foundation embedment depth range;

[0010] S3. Treat the bending deformation of a large-diameter single pile under horizontal load as the deformation problem of Euler-Bernoulli beam on a nonlinear foundation reaction model. Discretize the pile body using the finite element method, divide it into several beam elements, and introduce soil springs at the nodes to obtain the overall stiffness matrix.

[0011] S4. Based on the Euler-Bernoulli assumption, the element stiffness matrix is ​​derived for a single beam element using Hermite shape functions.

[0012] S5. Set transverse soil springs at the nodes, use hyperbolic py curves to calculate the horizontal reaction force and tangential stiffness of the soil, and assemble the local spring stiffness matrix.

[0013] S6. Set a rotational spring at the node to simulate the rotational damping of the pile-soil interface, calculate the tangential stiffness using a nonlinear m-θ curve, and assemble the local spring stiffness matrix.

[0014] S7. Install horizontal shear spring H at the pile bottom node. b -v and pile bottom bending moment spring M b -θ b The soil resistance and tangent stiffness are calculated using a hyperbolic relationship, and a local spring stiffness matrix is ​​assembled.

[0015] S8. Assemble the global stiffness equation, consider the nonlinearity of the soil spring, and solve it using the Newton-Raphson iterative method.

[0016] While adopting the above technical solutions, the present invention may also adopt or combine the following technical solutions:

[0017] As a preferred embodiment of the present invention: in step S3, the overall stiffness matrix is...

[0018] ;

[0019] In the formula, It is the overall stiffness matrix; It is the stiffness matrix of the beam element; It is a py spring with a local spring stiffness matrix; It is the m-θ spring of the local spring stiffness matrix; It is the stiffness matrix of the concentrated spring at the pile bottom.

[0020] As a preferred embodiment of the present invention: in step S4, the element stiffness matrix is...

[0021] ;

[0022] In the formula, It is the length of the beam element; It refers to the bending stiffness of the pile cross section.

[0023] As a preferred embodiment of the present invention: In step S5, the local spring stiffness matrix of the transverse soil spring is...

[0024] ;

[0025] ;

[0026] ;

[0027] In the formula, p is the horizontal reaction force of the soil; y is the horizontal displacement of the pile node; It is the ultimate limit value of soil reaction force; It refers to the stiffness or slope of the Py curve; It is the initial stiffness of the Py curve.

[0028] As a preferred embodiment of the present invention: In step S6, the local spring stiffness matrix of the rotary spring is...

[0029] ;

[0030] ;

[0031] ;

[0032] In the formula, m is the bending moment exerted by the soil on the pile per unit length; It is the rotation angle of the pile section; It is the limit value of the turning bending moment; It refers to the stiffness or slope of the m-θ curve; It is the initial stiffness of the m-θ curve.

[0033] As a preferred technical solution of the present invention: In step S7, the horizontal shear spring H b -v and pile bottom bending moment spring M b -θ b The local spring stiffness matrix is,

[0034] ;

[0035] In the formula, It is the initial stiffness of the horizontal shear spring at the pile bottom; It is the initial stiffness of the pile bottom bending moment spring.

[0036] As a preferred embodiment of the present invention: In step S8, the global stiffness equation is:

[0037] ;

[0038] ;

[0039] In the formula, It is the global residual; It is the external load vector; It is the internal force vector of the beam element; It is the vector of the internal force of the spring; It is the displacement increment; It is the overall tangential stiffness.

[0040] As a preferred embodiment of the present invention, the method further includes a force and moment balance verification step, which verifies the correctness of the calculation results by calculating the residual force and residual bending moment of the system.

[0041] As a preferred technical solution of the present invention, the method further includes a contribution analysis step of each spring, which quantifies the relative contribution of each spring in resisting external loads through normalization processing.

[0042] As a preferred technical solution of the present invention: when the nodal spring represents a distributed load, the equivalent stiffness and internal forces are calculated using Gaussian integral, wherein:

[0043] The tangential stiffness and internal force of the py distributed spring are linearized using Gaussian integration and expressed as:

[0044] ;

[0045] ;

[0046] The tangential stiffness and internal forces of the m–θ distributed spring are linearized using Gaussian integration and expressed as:

[0047] ;

[0048] ;

[0049] In the formula, It is the tangential stiffness of the m–θ distributed spring; is the m–θ distributed spring internal force; B is the matrix that maps nodal displacements to the second derivative (curvature).

[0050] The pile bottom spring is always treated as a concentrated load at the node.

[0051] Compared with the prior art, the present invention has the following beneficial effects:

[0052] 1) The method of this invention is based on the Euler-Bernoulli beam theory and systematically considers four types of soil springs (horizontal py, pile side m-θ, pile bottom Hv, pile bottom M). b -θ bThe comprehensive effect of the method overcomes the limitation of traditional methods that only consider the py spring, and significantly improves the calculation accuracy.

[0053] 2) This invention uses the finite element method to solve beam elements, which is more accurate than the traditional Winkler foundation beam differential method. At the same time, the element nodes can input both concentrated forces and distributed forces. The former is consistent with the beam-spring finite element method in Abaqus finite element method, and the latter is consistent with the traditional Winkler method. It can replace software such as LPILE, PileLAT, and SACS for horizontal bearing design calculation of large-diameter single pile foundations. Therefore, this invention is efficient and convenient to use.

[0054] 3) This invention adopts a modular design, and the activation state of each spring can be controlled independently, which facilitates the analysis of the influence of different boundary conditions and soil parameters, and provides a flexible tool for optimization design;

[0055] 4) This invention is not limited to large-diameter monopiles for offshore wind power, but is also applicable to offshore wind power suction bucket foundations, jacket wind turbine foundations, etc., and can be used for pile foundation dynamic response analysis, which enhances the engineering practicality of the method. Attached Figure Description

[0056] Figure 1 This is a schematic diagram of the stress on a large-diameter monopile foundation provided by the present invention.

[0057] Figure 2 This is a schematic diagram of the Euler-Bernoulli beam four-spring method used in the calculations of this invention.

[0058] Figure 3 This is a schematic diagram of the hyperbolic curve used in the soil spring of this invention.

[0059] Figure 4 This is a comparison chart of the calculation results of this invention and the finite element calculation results. Detailed Implementation

[0060] The present invention will now be described in further detail with reference to the accompanying drawings and specific embodiments.

[0061] A multi-spring analysis method for horizontal load-bearing deformation of large-diameter offshore wind turbine foundations, specifically including the following steps:

[0062] S1. Determine the geometric dimensions and material properties of the large-diameter monopile foundation. The geometric dimensions of the large-diameter monopile foundation include the outer diameter D, wall thickness t, cantilever height e, and embedment depth L. The material properties of the steel pipe pile include the elastic modulus E. p Poisson's ratio v;

[0063] S2. Determine the soil mechanical parameters and the soil layer distribution within the pile foundation embedment depth range. Soil mechanical parameters include the undrained shear strength s of the soil. uSoil elastic modulus E s ;

[0064] S3, When a large-diameter monopile is subjected to a horizontal load (such as...) Figure 1 The bending deformation of the pile (as shown) is considered as a deformation problem of the Euler-Bernoulli beam in a nonlinear foundation reaction model. The pile is discretized using the finite element method, divided into several beam elements, and soil springs (such as...) are introduced at the nodes. Figure 2 As shown), the global stiffness matrix is ​​obtained:

[0065] ;

[0066] ;

[0067] In the formula, It is the overall stiffness matrix; It is the stiffness matrix of the beam element; It is a py spring with a local spring stiffness matrix; It is the m-θ spring of the local spring stiffness matrix; It is the stiffness matrix of the concentrated springs at the pile bottom; This refers to the lateral displacement of the pile; For the elastic modulus and moment of inertia of the pile; The soil reaction force is a nonlinear function. The pile body is discretized using the finite element method, divided into several beam elements, and soil springs are introduced at the nodes to obtain the overall stiffness equation.

[0068] S4. The pile is a homogeneous, linearly elastic beam. Based on the Euler-Bernoulli assumption, the beam undergoes bending, but the cross-section remains perpendicular to the neutral axis. Shear deformation is neglected. For a single beam element, the length is... Beam deflection The Hermite-shaped function is:

[0069] ;

[0070] In the formula, It is a standard form function; It is along the coordinates of the beam element; and These are the displacement and rotation degrees of freedom of the element nodes, respectively.

[0071] Among them, the standard form function is:

[0072] ;

[0073] In the formula, The internal energy work (or elastic strain energy) is:

[0074] ;

[0075] Expressing v using shape functions yields the element stiffness matrix:

[0076] ;

[0077] In the formula, B is the matrix that maps nodal displacements to their second derivatives (curvature), where curvature... , Node degrees of freedom .

[0078] After integration, the local stiffness matrix is:

[0079] ;

[0080] In the formula, It is the length of the beam element; It represents the bending stiffness of the pile cross section, and this matrix indicates the bending stiffness relationship of the linear beam element.

[0081] S5. Install transverse soil springs at the nodes, using a hyperbolic Py curve (e.g., ...). Figure 3 (As shown) Calculate the horizontal reaction force and tangential stiffness of the soil, assemble the local spring stiffness matrix, and calculate using the hyperbolic Py curve according to the following formula:

[0082] ;

[0083] Tangential stiffness is expressed as:

[0084] ;

[0085] At this point, the system has 2N degrees of freedom, where N is the number of nodes. After introducing spring stiffness into each node, the local stiffness matrix is ​​expanded as follows:

[0086] ;

[0087] In the formula, p is the horizontal reaction force of the soil; y is the horizontal displacement of the pile node; It is the ultimate limit value of soil reaction force; It refers to the stiffness or slope of the Py curve; It is the initial stiffness of the Py curve.

[0088] S6. A rotational spring is installed at the node to simulate the rotational damping of the pile-soil interface. The tangential stiffness is calculated using a nonlinear m-θ curve, and a local spring stiffness matrix is ​​assembled. This matrix represents the resisting bending moment generated by the soil side friction around the pile axis during rotation. Its nonlinear relationship is as follows:

[0089] ;

[0090] Tangential stiffness is expressed as:

[0091] ;

[0092] In the formula, and These are the initial stiffness and ultimate bearing capacity of the m-θ spring, respectively. An additional rotational degree-of-freedom stiffness term is added to each node, resulting in a total rotational stiffness of:

[0093] ;

[0094] The updated element stiffness is:

[0095] ;

[0096] In the formula, m is the bending moment exerted by the soil on the pile per unit length; It is the rotation angle of the pile section; It is the limit value of the turning bending moment; It refers to the stiffness or slope of the m-θ curve; It is the initial stiffness of the m-θ curve.

[0097] S7. Install horizontal shear spring H at the pile bottom node. b -v and pile bottom bending moment spring M b -θ b The soil resistance and tangent stiffness are calculated using a hyperbolic relationship, and a local spring stiffness matrix is ​​assembled. When considering the pile bottom H... b and M b When the spring is in motion, the horizontal spring H b -v、M b -θ b Applying the displacement and rotation degrees of freedom at the pile bottom nodes respectively, the overall stiffness matrix becomes:

[0098] ;

[0099] ;

[0100] In the formula, It is the initial stiffness of the horizontal shear spring at the pile bottom; It is the initial stiffness of the pile bottom bending moment spring.

[0101] The pile bottom spring adopts a hyperbolic relationship:

[0102] ;

[0103] ;

[0104] The tangential stiffness is:

[0105] ;

[0106] ;

[0107] In the formula, and , and These are the horizontal shear springs at the pile bottom, H. b -v and pile bottom bending moment spring M b -θ b The initial stiffness and ultimate load values.

[0108] In steps S5-S7, the activation status of the four types of soil springs is independently controlled by switch variables to realize the analysis of different combined working conditions.

[0109] S8. After assembly, the global stiffness equation is obtained:

[0110] ;

[0111] In the formula, Let be the vector of the degrees of freedom of the nodes; Let be the external load vector. Since the soil spring is nonlinear, its stiffness matrix depends on the current displacement; therefore, an iterative solution (Newton–Raphson) is used. Let the internal force vector of the beam element be: (For a linear elastic beam, the internal forces are linear with displacement.) = The internal force vector contributed by the spring is... (Nonlinear). The global residual is:

[0112] ;

[0113] Solving for the increment :

[0114] ;

[0115] In the formula, It is the global residual; It is the external load vector; It is the internal force vector of the beam element; It is the vector of the internal force of the spring; It is the displacement increment; It is the overall tangential stiffness.

[0116] The overall tangential stiffness is consistent with the global stiffness mentioned above:

[0117] ;

[0118] Initial stiffness of transverse soil springs and ultimate bearing capacity Initial stiffness of the rotational spring and ultimate bearing capacity Initial stiffness of the horizontal shear spring at the pile bottom and ultimate bearing capacity Initial stiffness of the pile bottom bending moment spring and ultimate bearing capacity The solution is determined based on the soil properties and the boundary conditions at the bottom of the pile, or it is derived theoretically.

[0119] The method also includes a force and moment balance verification step, which verifies the correctness of the calculation results by calculating the residual force and residual bending moment of the system.

[0120] The method also includes a contribution analysis step for each spring, which quantifies the relative contribution of each spring in resisting external loads through normalization.

[0121] The above form applies to concentrated forces at nodes. When a node spring represents a distributed load, i.e., the py spring and m–θ spring are input with distributed loads of kN / m and kN·m / m respectively, the current local displacement needs to be applied at the Gauss point. and its derivative The nonlinear reaction density and tangential stiffness density are calculated, and then the equivalent stiffness and equivalent internal forces of the elements are obtained by shape function integration. These are then assembled into the global system, and the nonlinear equations are solved using Newton-Raphson.

[0122] When the nodal spring represents a distributed load, Gaussian integrals are used to calculate the equivalent stiffness and internal forces.

[0123] The tangential stiffness and internal force of the py distributed spring are linearized using Gaussian integration and expressed as:

[0124] ;

[0125] ;

[0126] In the formula, Calculate at the Gauss point based on the current v(x).

[0127] The tangential stiffness and internal forces of the m–θ distributed spring are linearized using Gaussian integration and expressed as:

[0128] ;

[0129] ;

[0130] In the formula, It is the tangential stiffness of the m–θ distributed spring; is the m–θ distributed spring internal force; B is the matrix that maps nodal displacements to the second derivative (curvature).

[0131] The above integration was performed using Gaussian pointwise numerical integration. We then proceeded to assemble the local element stiffness matrix and the global stiffness matrix. The pile bottom springs remained unaffected, as they were always nodal concentrated loads.

[0132] During the solution process, a Newton-Raphson iteration is performed for each load step:

[0133] Calculate using the current u at each Gauss point in the cell. , ;

[0134] calculate , , , ;

[0135] Obtaining unit contribution by performing numerical integration , , , ;

[0136] Assemble the global K, Find Δu and update it;

[0137] Convergence judgment (residuals and increments); if convergence is not achieved, return to updating stiffness.

[0138] Steps S5, S6 and S7 are not in any particular order, but step S5 is mandatory. Steps S6 and S7 are determined based on the actual conditions of the pile body and pile end.

[0139] Example: A large-diameter monopile foundation for an offshore wind power project has a diameter of D=4 m, a wall thickness of t=0.00635+D / 100, a pile embedment depth of L=32 m, a pile top horizontal load eccentricity of e=8 m, and a pile elastic modulus E. p =70 GPa;

[0140] The seabed foundation soil is soft clay with an undrained shear strength of s. u =10 kPa, soil elastic modulus E s =200s u =2000 kPa;

[0141] 1) First, establish a beam-spring finite element model in Abaqus: Discretize the pile into 20 beam elements, and configure py, m-θ, pile bottom Hv, and pile bottom M within the embedment depth. b -θ b Four types of nonlinear soil springs;

[0142] Setting parameters for a soil spring: py spring: =4800 kN / m, =7680 kN; m-θ spring: =144000 kN·m / rad, =6400 kN·m; Hv spring at pile bottom: =10000 kN / m, =60 kN; pile bottom M b -θ b spring: =1,000,000 kN·m / rad, =1500 kN·m.

[0143] Based on the initial stiffness and ultimate bearing capacity, input the load-displacement curves of each soil spring into Abaqus to define the connector properties. Input the half-load-displacement curves for the first and last beam element nodes below the burial depth.

[0144] Calculate the load-displacement curve at the pile top, and select and activate each spring stiffness to obtain the horizontal load displacement response, such as... Figure 4 As shown;

[0145] 2) Using Matlab programming, the pile was discretized into n beam elements. The Newton-Raphson iterative method was used for nonlinear solution to obtain the pile response. The results were then compared with those obtained using Abaqus finite element analysis. Figure 4 As shown, the analysis results of this invention under each example condition are completely consistent with the Abaqus finite element results; and the force and moment balance verification shows that the residual force is less than 1×10⁻⁻⁻⁶. 7 kN, residual bending moment less than 1×10⁻ 9 kN·m, to prove the correctness of the calculation results.

[0146] This invention allows for free input of concentrated force at nodes (P=p×L). e The load values ​​at the first and last nodes below the burial depth should be halved when inputting the concentrated load (kN) or the distributed load (p, kN / m). However, when inputting the distributed load, no additional processing is required because Gaussian point-by-point integration is used. Moreover, the pile segment division density n has no impact on the settlement result if the numerical accuracy is met, which reflects the rationality and applicability of the present invention.

[0147] The technical solution of the present invention has been described in conjunction with the specific experimental procedures shown in the accompanying drawings. However, the scope of protection of the present invention is not limited to these specific embodiments. Without departing from the principles of the present invention, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions resulting from such changes or substitutions will all fall within the scope of protection of the present invention.

Claims

1. A multi-spring analysis method for horizontal load-bearing deformation of large-diameter offshore wind turbine foundations, characterized in that, Includes the following steps: S1. Determine the geometric dimensions of the large-diameter monopile foundation and the material properties of the steel pipe piles; S2. Determine the soil mechanical parameters and the soil layer distribution within the pile foundation embedment depth range; S3. Treat the bending deformation of a large-diameter single pile under horizontal load as the deformation problem of Euler-Bernoulli beam on a nonlinear foundation reaction model. Discretize the pile body using the finite element method, divide it into several beam elements, and introduce soil springs at the nodes to obtain the overall stiffness matrix. S4. Based on the Euler-Bernoulli assumption, the element stiffness matrix is ​​derived for a single beam element using Hermite shape functions. S5. Set transverse soil springs at the nodes, use hyperbolic py curves to calculate the horizontal reaction force and tangential stiffness of the soil, and assemble the local spring stiffness matrix. S6. Set a rotational spring at the node to simulate the rotational damping of the pile-soil interface, calculate the tangential stiffness using a nonlinear m-θ curve, and assemble the local spring stiffness matrix. S7. Install horizontal shear spring H at the pile bottom node. b -v and pile bottom bending moment spring M b -θ b The soil resistance and tangent stiffness are calculated using a hyperbolic relationship, and a local spring stiffness matrix is ​​assembled. S8. Assemble the global stiffness equation, consider the nonlinearity of the soil spring, and solve it using the Newton-Raphson iterative method.

2. The method according to claim 1, characterized in that: In step S3, the global stiffness matrix is: ； In the formula, It is the overall stiffness matrix; It is the stiffness matrix of the beam element; It is a py spring with a local spring stiffness matrix; It is the m-θ spring of the local spring stiffness matrix; It is the stiffness matrix of the concentrated spring at the pile bottom.

3. The method according to claim 1, characterized in that: In step S4, the element stiffness matrix is ； In the formula, It is the length of the beam element; It refers to the bending stiffness of the pile cross section.

4. The method according to claim 1, characterized in that: In step S5, the local spring stiffness matrix of the transverse soil spring is: ； ； ； In the formula, p is the horizontal reaction force of the soil; y is the horizontal displacement of the pile node; It is the ultimate limit value of soil reaction force; It refers to the stiffness or slope of the Py curve; It is the initial stiffness of the Py curve.

5. The method according to claim 1, characterized in that: In step S6, the local spring stiffness matrix of the rotary spring is: ； ； ； In the formula, m is the bending moment exerted by the soil on the pile per unit length; It is the rotation angle of the pile section; It is the limit value of the turning bending moment; It refers to the stiffness or slope of the m-θ curve; It is the initial stiffness of the m-θ curve.

6. The method according to claim 1, characterized in that: In step S7, the horizontal shear spring H b -v and pile bottom bending moment spring M b -θ b The local spring stiffness matrix is, ； In the formula, It is the initial stiffness of the horizontal shear spring at the pile bottom; It is the initial stiffness of the pile bottom bending moment spring.

7. The method according to claim 1, characterized in that: In step S8, the global stiffness equation is: ； ； In the formula, It is the global residual; It is the external load vector; It is the internal force vector of the beam element; It is the vector of the internal force of the spring; It is the displacement increment; It is the overall tangential stiffness.

8. The method according to claim 1, characterized in that: The method also includes a force and moment balance verification step, which verifies the correctness of the calculation results by calculating the residual force and residual bending moment of the system.

9. The method according to claim 1, characterized in that: The method also includes a spring contribution analysis step, which quantifies the relative contribution of each spring in resisting external loads through normalization.

10. The method according to claim 1, characterized in that: When the nodal spring represents a distributed load, Gaussian integrals are used to calculate the equivalent stiffness and internal forces, where: The tangential stiffness and internal force of the py distributed spring are linearized using Gaussian integration and expressed as: ； ； In the formula, It is the tangential stiffness of the py-distributed spring; It represents the internal force of the spring; N is the standard shape function. These are the coordinates of the beam element nodes. The tangential stiffness and internal forces of the m–θ distributed spring are linearized using Gaussian integration and expressed as: ； ； In the formula, It is the tangential stiffness of the m–θ distributed spring; is the m–θ distributed spring internal force; B is the matrix that maps nodal displacements to second derivatives / curvatures; The pile bottom spring is always treated as a concentrated load at the node.

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