A large-diameter rigid pile foundation horizontal bearing response winkler multi-spring analysis method

Abstract

The application discloses a Winkler multi-spring analysis method for horizontal bearing response of a large-diameter rigid pile foundation. The application discretizes a pile body into a finite number of micro-sections, and simultaneously sets four kinds of nonlinear soil springs at nodes. Based on the Winkler foundation beam theory, the finite difference method is adopted to establish a pile body deflection differential equation, a virtual node is introduced, and the free boundary at the top of the pile and the spring boundary condition at the bottom of the pile are utilized to derive a node difference equation containing the contribution of all springs, and then the whole nonlinear stiffness matrix is assembled. The Newton-Raphson iteration method is adopted to solve the displacement field, and the internal force of the pile body is further calculated. The application systematically considers the comprehensive influence of the horizontal resistance of the soil around the pile, the rotational resistance and the resistance at the bottom of the pile, overcomes the limitation of traditional methods which only consider p-y the spring, and significantly improves the prediction accuracy of the horizontal bearing response of the large-diameter rigid pile. Meanwhile, the application provides a correct stiffness matrix expression, which is convenient for engineering application and program implementation.

Description

Technical Field

[0001] This invention relates to the fields of geotechnical engineering and marine engineering, specifically to a Winkler multi-spring nonlinear analysis method for analyzing the bearing deformation response of large-diameter rigid monopile foundations under horizontal loads. Background Technology

[0002] In offshore wind power, cross-sea bridges, and other projects, large-diameter monopile foundations (with diameters reaching 5-10 m) are among the most commonly used foundation types. These foundations bear enormous horizontal loads (such as wind, waves, and currents) and overturning moments during their service life, making their horizontal load-bearing deformation characteristics a core design concern.

[0003] Currently, the Py curve method recommended by the API (American Petroleum Institute) standard is widely used in the engineering field. This method simplifies the soil around the pile as a series of independent horizontal nonlinear springs (Py springs) and solves the pile deflection based on the Winkler foundation beam model. However, extensive research and engineering practice have shown that the traditional Py curve method is mainly developed for small-diameter flexible long piles, and has the following shortcomings when applied to large-diameter rigid or semi-rigid piles:

[0004] 1) Ignoring the rotational resistance of the pile side: The rotation of the pile body will generate a resisting bending moment (m-θ effect) at the pile-soil interface. For large-diameter piles, this resistance contributes significantly. Ignoring it will lead to an overestimation of displacement and an overly conservative estimation of bearing capacity.

[0005] 2) Oversimplified pile bottom boundary conditions: Traditional methods typically simplify the pile bottom to a hinged or free end, neglecting the horizontal resistance and bending moment resistance provided by the soil at the pile bottom (i.e., pile bottom H). b -v b and M b -θ b The spring effect is particularly pronounced for shallow rigid piles.

[0006] 3) Complexity of finite difference scheme: When using the finite difference method to solve the problem, it is necessary to make precise mathematical derivations to correctly consider the load at the top of the pile and the boundary conditions of the spring at the bottom of the pile. The existing literature often uses the beam spring finite element method, which is complicated and difficult to implement.

[0007] Therefore, developing a method for analyzing the horizontal bearing capacity of large-diameter rigid piles that can comprehensively consider the pile side horizontal and rotational resistance as well as the pile bottom resistance, and that is rigorously derived and easy to program, has significant practical engineering value. Summary of the Invention

[0008] The purpose of this invention is to overcome the shortcomings of the prior art and provide a Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations. Based on the traditional Py curve method, this invention systematically introduces the pile side m-θ rotational spring, the pile bottom horizontal spring, and the pile bottom bending moment spring. Through rigorous finite difference derivation, a unified multi-spring nonlinear analysis framework is established, and the correct overall stiffness matrix expression is given, thereby improving the prediction accuracy of the horizontal bearing response of large-diameter piles.

[0009] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0010] A Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations includes the following steps:

[0011] (1) Determine the parameters of the pile-soil system. Obtain the geometric dimensions of the large-diameter single pile: pile length L, outer diameter D, and wall thickness t, and calculate the moment of inertia I of the cross section. p And bending stiffness EI. Obtain the layered mechanical parameters of the soil around the pile, including soil type, unit weight, and undrained shear strength s. u The internal friction angle, etc., are used to subsequently determine the initial stiffness and ultimate bearing capacity of each soil spring;

[0012] (2) Constructing the Winkler multi-spring model. The pile body is discretized along its depth into n micro-segments, with a total of n+1 nodes, numbered 0, 1, 2, ..., n (0 is the pile top, n is the pile bottom). The length of each micro-segment is h = L / n. The following four nonlinear soil springs are set at each node to simulate the pile-soil interaction:

[0013] Pile-side horizontal spring (py spring): simulates the relationship between the horizontal resistance p of the soil and the horizontal displacement y of the pile body. The spring force acts on the horizontal degree of freedom of the node.

[0014] Pile-side rotational spring (m-θ spring): simulates the relationship between the rotational resistance m at the pile-soil interface and the pile rotation angle θ, with the spring torque acting on the rotational degree of freedom of the node;

[0015] Pile bottom horizontal spring (H) b -v b Spring): Only set at the pile bottom node n, to simulate the horizontal resistance H of the soil at the pile bottom to the pile tip. b With the horizontal displacement v at the bottom of the pile b Relationship;

[0016] Pile bottom moment spring (M b -θ b Spring): Only set at the pile bottom node n, to simulate the bending moment resistance M of the soil at the pile bottom to the pile tip. b With pile bottom rotation angle θ b Relationship;

[0017] The nonlinear constitutive relations of the above springs all adopt a hyperbolic model, and the specific expressions are as follows:

[0018] Py curve relationship: Tangential stiffness:

[0019] m-θ curve relationship: Tangential stiffness:

[0020] H b -v b Curve relationship: Tangential stiffness:

[0021] M b -θ b Curve relationship: Tangential stiffness:

[0022] in, , , , These are the initial stiffnesses of each spring; and These are nodes M. b -θ b Spring and H b -v b Spring tangential stiffness; , , , These are the ultimate bearing capacities of each spring, determined based on soil parameters and pile-soil interaction theory;

[0023] (3) Establish the finite difference scheme for the differential equation of pile deflection. Under the Winkler foundation beam model, the equilibrium differential equation for any small segment of the pile is:

[0024]

[0025] in, It is divided into soil horizontal reaction stiffness (representing the resistance provided by the Py spring); This is the rotational reaction stiffness of the node (representing the resistance provided by the m-θ spring). Its force-displacement relationship is:

[0026] ,

[0027] In the formula, and These are the initial stiffness and ultimate bearing capacity of the Py spring, respectively.

[0028] ,

[0029] In the formula, and These are the initial stiffness and ultimate bearing capacity of the m-θ spring, respectively.

[0030] Using a central difference scheme with equal spacing h, the fourth derivative of the displacement y is approximated as:

[0031]

[0032] Substitute (2) into (1), and consider the tangential stiffness of the spring at node i (m-θ). (Right now ), thus obtaining the difference equation for internal node i:

[0033]

[0034] To apply boundary conditions to the top of the pile (i=0) and the bottom of the pile (i=n), virtual nodes need to be introduced: virtual nodes -2 and -1 are set above the top of the pile, and virtual nodes n+1 and n+2 are set below the bottom of the pile.

[0035] (4) Handling the boundary conditions at the top of the pile (free end)

[0036] The pile top is free, subjected to a horizontal force H0 and a bending moment M0. Based on beam theory and the central difference method, the displacements of virtual nodes -2 and -1 can be eliminated, yielding the corrected difference equations for pile top nodes 0 and 1:

[0037] Node 0: ,

[0038] Node 1: ,

[0039] In the formula, the coefficients , , and , , , Determined by the pile top load and spring parameters, the specific expression is as follows:

[0040] , ,

[0041] , , ,

[0042] = , =

[0043] (5) Handling the boundary conditions at the pile bottom. A horizontal spring and a moment spring are installed at node n at the pile bottom. The force-displacement relationship is as follows:

[0044] , ,

[0045] in, , These are the initial tangential stiffnesses of the horizontal spring and the moment spring at the pile bottom, respectively.

[0046] Consider the difference form of pile base shear force and bending moment (involving virtual nodes n-2, n-1, n, n+1, n+2):

[0047] ,

[0048] Will Substitute the values ​​and consider the difference in the rotation angle θ, then eliminate the virtual nodes algebraically. , The corrected difference equations for pile bottom node n-1 and node n are obtained:

[0049] Node n-1: ,

[0050] Node n: ,

[0051] in, , , and , Determined by the spring parameters; when a) only the py spring and m-θ spring are considered at the pile bottom node, the coefficient is:

[0052] = , = , = ,

[0053] = , = ;

[0054] b) When both py spring and m-θ spring are considered at the pile bottom node, the coefficient is:

[0055] = , = ,

[0056] = , = ,

[0057] = ;

[0058] (6) Assemble the overall stiffness matrix. Apply the corrected difference equations to all nodes (including the internal node equations). and boundary node equations Arrange the nodes in order to form a vector {y} = [y0, y1, ..., y2] for the unknown displacement. n The system of linear algebraic equations of ᵀ:

[0059] [K({y})]{y} = {F}

[0060] Where [K] is an (n+1)×(n+1) matrix, and {F} is the load vector (including the contributions of H0 and M0). The matrix [K] is as follows:

[0061]

[0062] As can be seen, [K] is a strip-shaped sparse matrix, approximately pentagonal, but the first and last rows vary slightly due to boundary condition treatment. Meanwhile, the matrix elements contain the tangential stiffness of all four types of springs. , , , This reflects the combined effect of multiple springs.

[0063] (7) Nonlinear iterative solution. Since the spring stiffness depends on the current displacement, the equation system... It is nonlinear. The Newton-Raphson method is used to solve it.

[0064] 1. Let the initial displacement be {y} 0 = {0}, calculate the initial tangential stiffness of each spring.

[0065] 2. In the k-th iteration, based on the current displacement {y}k Update the tangential stiffness of each spring and assemble the current stiffness matrix. and internal force vector Calculate unbalanced forces .

[0066] 3. Solve the system of linear equations The displacement increment is obtained.

[0067] 4. Update displacement .

[0068] 5. Check convergence: If If the iteration fails, stop; otherwise, return to step 2.

[0069] Compared with the prior art, the present invention has the following beneficial effects:

[0070] 1. Comprehensive consideration of multiple resistance mechanisms: Based on the traditional Py spring, the pile-side m-θ rotational spring, pile-bottom horizontal spring, and pile-bottom bending moment spring are systematically introduced within the Winkler finite difference framework. This comprehensively reflects the contributions of the horizontal resistance, rotational resistance, and pile-bottom resistance of the soil on the pile side during the horizontal bearing process of the rigid pile, and the prediction accuracy is significantly improved.

[0071] 2. Rigorous boundary condition handling: For the free boundary at the pile top and the double spring boundary at the pile bottom, virtual nodes are eliminated through rigorous mathematical derivation to obtain the correct modified difference equations, which are then assembled into an overall stiffness matrix.

[0072] 3. Unified computational framework: The nonlinear contributions of the four types of springs are integrated into a single stiffness matrix, forming a clear and compact mathematical expression that is easy to implement in programming (e.g., MATLAB, Python). The matrix is ​​in a pentagonal band shape, resulting in high storage and solution efficiency.

[0073] 4. Flexibility and scalability: The activation of each spring can be independently controlled by switching variables, which facilitates the analysis of the importance of different spring mechanisms; the spring constitutive model is not limited to hyperbolic and can be easily replaced by other more complex models (such as tri-segmented line, Ramberg-Osgood model) to adapt to different soil conditions.

[0074] 5. Strong engineering applicability: This invention can be directly applied to the horizontal bearing analysis of large-diameter monopiles for offshore wind turbines and bridge pile foundations. The calculation results are comparable to those of general-purpose finite element software such as Abaqus, but the modeling is simple and the calculation is fast, making it particularly suitable for parameter analysis and optimization in the engineering design stage. Attached Figure Description

[0075] Figure 1 This is a schematic diagram of the multi-spring arrangement of a large-diameter single pile provided by the present invention;

[0076] Figure 2 This is a schematic diagram of the Winkler beam differential separation of the present invention;

[0077] Figure 3 This is a comparison diagram of the load-displacement curve calculated by this invention and the finite element method. Detailed Implementation

[0078] The present invention will be further described in detail below with reference to specific embodiments and accompanying drawings. This embodiment takes a large-diameter steel pipe monopile of an offshore wind farm as an example (e.g.) Figure 1 The present invention is used for analysis.

[0079] Example parameters:

[0080] The pile diameter D = 4 m, wall thickness t = 0.00635 + D / 100, pile depth L = 32 m, eccentricity of horizontal load at the pile top e = 8 m, and pile elastic modulus E p =70 GPa;

[0081] The soil is homogeneous soft clay with an undrained shear strength s u =10 kPa, soil elastic modulus E s =200s u =2000 kPa;

[0082] (1) Take the number of discrete segments of the pile body n=80, and the distance from the set distance h=0.5 m (e.g. Figure 2 ).

[0083] (2) Based on the soil parameters and pile diameter, calculate the initial stiffness and ultimate bearing capacity of each spring using the API specification recommended formula or theoretical solution:

[0084] Py spring: =4800 kN / m, =7680 kN;

[0085] m-θ spring: =144000 kN·m / rad, =6400 kN·m;

[0086] Hv spring at pile bottom: =10000 kN / m, =60kN;

[0087] Pile bottom M b -θ b spring: =1,000,000 kN·m / rad, =1500kN·m.

[0088] (3) According to the formula Establish the internal node equations. For the pile top node, according to the equation... Establish the corrected equations. For the pile bottom node, establish the equations based on the derived corrected pile bottom equations. Note that there are no soil springs in the cantilever section, therefore... = 0、 =0. Arrange the above equations in node order to form a matrix [K].

[0089] (4) Write a MATLAB program to implement the above algorithm. Set the convergence tolerance. The load is applied in stages, with iterations within each stage. During the iteration process, the tangential stiffness of each spring is calculated based on the current displacement (using the hyperbolic model formula), [K] is reassembled, and the displacement increment is solved until convergence.

[0090] (5) To verify the correctness of the present invention, the above parameters were input into Abaqus software to establish a beam-spring finite element model (using a connector to simulate a nonlinear spring). Comparing the pile top load-displacement curves calculated by both methods, the results show that the method of the present invention and the Abaqus results are in high agreement (e.g., ...). Figure 3 This demonstrates the accuracy and reliability of the present invention.

[0091] The above description is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the protection scope of the present invention.

Claims

1. A Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations, characterized in that, Includes the following steps: (1) Obtain the geometric dimensions and material properties of large-diameter single piles, as well as the layered mechanical parameters of the soil around the piles; (2) Discretize the pile body into n micro-segments, with a total of n+1 nodes, and set two types of nonlinear soil springs at the pile body nodes: pile side horizontal py spring and pile side rotation m-θ spring; set three types of springs at the pile bottom nodes: pile side horizontal py spring and pile bottom horizontal H spring. b -v b Spring, pile bottom bending moment M b -θ b spring; (3) Based on Winkler's foundation beam theory, establish the differential equation for pile segment deflection: In the formula, It is the bending stiffness of the pile section, i.e. ; The central difference scheme is used to discretize each node of the pile body, and virtual nodes (-2, -1, n+1, n+2) are introduced to meet the requirements of the fourth-order difference. The difference equation of node i is: In the formula, and These are the stiffnesses of the m-θ spring and the py spring at node i, respectively. (4) Eliminate virtual nodes -2 and -1 based on the free boundary conditions at the pile top (horizontal force H0, bending moment M0) to obtain the corrected difference equations for pile top nodes 0 and 1. (5) Based on the different constraint conditions of the pile bottom nodes and the composition of the spring components, the pile bottom shear force and bending moment have different difference expressions. Eliminate the virtual nodes n+1 and n+2 to obtain the corrected difference equations of the pile bottom nodes n-1 and n. (6) Combine all the nodal correction difference equations obtained in steps (4) and (5) to form a system of nonlinear algebraic equations [K({y})]{y} = {F} about the nodal horizontal displacement vector {y}, where [K] is the global stiffness matrix, whose elements are composed of the pile bending stiffness EI, the discrete length h, and the current tangential stiffness of each nonlinear spring: (7) The Newton-Raphson iterative method is used to solve the nonlinear equation system in step (6). In each iteration, the tangential stiffness of each spring is updated according to the current displacement, [K] is reassembled, and the displacement increment is solved until convergence. (8) Based on the converged node displacements, calculate the rotation angle, bending moment and shear force of each node of the pile body using the central difference scheme, and output the results.

2. The Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations according to claim 1, characterized in that, In step (2), the nonlinear constitutive relations of the four types of soil springs all adopt a hyperbolic model: Py curve relationship: Tangential stiffness: m-θ curve relationship: Tangential stiffness: H b -v b Curve relationship: Tangential stiffness: M b -θ b Curve relationship: Tangential stiffness: in, , , , These are the initial stiffnesses of each spring; and These are nodes M. b -θ b Spring and H b -v b Spring tangential stiffness; , , , These are the ultimate bearing capacities of each spring, determined based on soil parameters and pile-soil interaction theory.

3. The Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations according to claim 1, characterized in that, The corrected difference equations for pile top node 0 and node 1 in step (4) are as follows: Node 0: , Node 1: , in, , , and , , , Determined by the pile top load and spring parameters: ， ， ， ， ， = ， = The corrected difference equations for pile bottom node n-1 and node n in step (4) are as follows: Node n-1: , Node n: , The boundary conditions have been eliminated, and the pile bottom spring stiffness is implicitly included in the equation. Among them, , , and , Determined by the spring parameters; when a) only the py spring and m-θ spring are considered at the pile bottom node, the coefficient is: = ， = ， = ， = ， = ； b) When both py spring and m-θ spring are considered at the pile bottom node, the coefficient is: = ， = ， = ， = ， = 。 4. The Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations according to claim 1, characterized in that, In step (5), the overall stiffness matrix [K] is an (n+1)×(n+1) order matrix, and its non-zero elements are distributed in a pentagonal form, but the first two rows and the last two rows are adjusted due to boundary condition processing; the specific expressions of the matrix elements are given by the coefficients of the difference equations of each node, where the equation coefficients corresponding to the internal node i (2≤i≤n-2) are: ， ， The coefficients of the nodes near the pile top and the nodes near the pile bottom are determined according to the modified equation, and these coefficients include the effects of the pile side m-θ spring and the pile bottom spring.

5. The Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations according to claim 1, characterized in that, The Newton-Raphson iteration format in step (6) is as follows: ， ， Iterate until , where ε is the preset convergence tolerance.

6. The Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations according to claim 1, characterized in that, The method also includes independently controlling the activation state of four types of springs by switching variables to analyze the contribution of different spring mechanisms to the horizontal bearing capacity of the pile foundation.

7. The Winkler multi-spring analysis method for the horizontal bearing response of large-diameter rigid pile foundations according to claim 1, characterized in that, The method also includes a force and moment balance verification step: calculate the resultant force and resultant moment of all soil spring reactions on the pile body, and compare them with the external loads H0 and M0 at the pile top to verify the accuracy of the calculation results.

8. A computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps of the method according to any one of claims 1 to 7.

9. A computing device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor, when executing the program, implements the steps of the method according to any one of claims 1 to 7.

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