Method for determining the rigidity of the top of a pile in a double-layer foundation under horizontal load

Abstract

This invention discloses an analytical method for determining the horizontal static stiffness of end-bearing piles in double-layered foundations. By obtaining the mechanical parameters of the pile and the double-layered soil, the method establishes the soil governing equations and solves them using the method of separation of variables. Combined with multi-layered boundary conditions, analytical expressions for the soil displacement field and distributed soil resistance are derived. Furthermore, a piecewise pile deflection equation is constructed, and by substituting the soil resistance and using orthogonal expansion of characteristic functions, a piecewise expression for the pile displacement is obtained. By comprehensively considering various boundary and continuity conditions at the pile top, pile bottom, and soil layer interfaces, undetermined coefficients are solved to obtain a complete analytical expression for the pile displacement response, and based on this, the horizontal static stiffness matrix at the pile top is assembled. This method overcomes the limitations of existing methods in adapting to layered foundations, avoids complex numerical calculations, and features rigorous logic, clear expression, and high computational efficiency, providing a direct and reliable theoretical tool for the design and analysis of pile foundations in layered foundations.

Description

Technical Field

[0001] This invention relates to the field of pile foundation technology, and in particular to a method for determining the top stiffness of a horizontally loaded end-bearing pile in a double-layer foundation. Background Technology

[0002] Pile foundations, due to their advantages such as high bearing capacity, good durability, small settlement, and high strength, have been widely used in various engineering constructions in recent years. Driven by practical needs in fields such as earthquake engineering, bridge engineering, and marine engineering, many researchers have analyzed the dynamic response of pile foundations in soil. In the analysis of the bearing characteristics of pile foundations under horizontal static loads, the pile top stiffness of the pile-soil system is a fundamental component, requiring the establishment of realistic mathematical models to describe the interaction between the pile and the surrounding soil. Current research mostly focuses on homogeneous foundations; however, in actual engineering, due to factors such as crustal movement and geological deposition, natural foundations are primarily heterogeneous and often possess natural layered characteristics. Currently, there are relatively few mathematical models capable of describing the horizontal static stress response characteristics of end-bearing piles in layered foundations.

[0003] Existing models describing the horizontal static load response characteristics of end-bearing piles in layered foundations are typically based on boundary element integration or finite element methods, which involve complex solution or calculation processes. Therefore, current research lacks analytical methods that can accurately describe the bearing characteristics of end-bearing piles in layered foundations under horizontal static loads. Summary of the Invention

[0004] In view of this, the purpose of this invention is to provide a method for determining the pile top stiffness of horizontally loaded end-bearing piles in a double-layer foundation.

[0005] To achieve the above-mentioned technical objectives, the technical solution adopted by this invention is as follows: An analytical method for determining the horizontal static stiffness of a mid-end-bearing pile in a double-layer foundation includes: Obtain the geometric and material parameters of the pile body of the end-bearing pile in the double-layer foundation, as well as the continuous medium mechanical parameters of the upper soil layer and the continuous medium mechanical parameters of the lower soil layer. The governing equations of the double-layer foundation soil under horizontal static load are established. The potential function is introduced and the separation of variables method is used for solution. Combining the boundary condition of soil displacement attenuation at infinity, the continuous condition of soil layer interface displacement and stress, and the pile-soil interface displacement coordination condition, the analytical expression of the displacement field of the double-layer soil and the analytical expression of the distributed soil resistance acting on the pile are derived. The segmented pile body deflection control equation for end-bearing piles in a double-layer foundation is constructed. The analytical expression of the distributed soil resistance is substituted into the equation, and the characteristic function is orthogonally expanded using the pile-soil interface displacement compatibility condition to obtain the segmented expression of the pile body horizontal displacement containing undetermined coefficients. Based on the load boundary conditions of unit displacement or unit rotation at the pile top, the fixed end constraint conditions at the pile bottom, and the continuity conditions of pile displacement and internal forces at the soil layer interface, a set of boundary condition and continuity condition equations is formed. The undetermined coefficients are solved to obtain the complete analytical expression of the pile displacement response. Based on the analytical expression of pile displacement response, the reaction force at the pile top under a unit horizontal displacement and the reaction force under a unit rotation angle are calculated respectively, and then the horizontal static stiffness matrix of the end-bearing pile in the double-layer foundation is obtained.

[0006] In some embodiments, the pile's geometric and material parameters include pile length, pile radius, pile elastic modulus, and pile section moment of inertia. The continuous medium mechanical parameters of the upper soil layer and the lower soil layer include the thickness of each soil layer, the elastic modulus of each soil layer, the shear modulus of each soil layer, and the Poisson's ratio of each soil layer.

[0007] In some embodiments, the governing equations for the two-layer foundation soil under horizontal static load are established and expressed by formula (1), which is as follows: ; In formula (1), The soil compression coefficient is . , Poisson's ratio of the soil; This represents the radial displacement of the soil. The circumferential displacement of the soil is given by the following: Represents the upper soil layer. Represents the lower soil layer. To find the sign of the partial derivative, For radial independent variable, It is the vertical independent variable.

[0008] In some embodiments, the analytical expression for the displacement field of the two-layer soil is derived and expressed by formula (2), which is as follows: ; In formula (2), The second kind of zeroth-order modified Bessel function For the second kind of first-order modified Bessel function, For radial functions containing modified Bessel functions of the second kind, For a cyclic function containing a modified Bessel function of the second kind, where, , , As the first calculation variable, , The coefficient of the first variable, The first characteristic function, As the second calculation variable, , These are the eigenvalues ​​of the first characteristic function. , , It is a circumferential corner; in, Equation (3) is expressed as follows: ; In formula (3), Let be the first characteristic function of the upper soil layer. This is the first characteristic function of the underlying soil. This refers to the total thickness of the soil layer, which is also the length of the pile. This refers to the thickness of the upper soil layer. This refers to the thickness of the lower soil layer; The analytical expression for the distributed soil resistance acting on the pile body is expressed by formula (4), which is as follows: ; In formula (4), As the third calculation variable, , soil layer shear modulus, Where is the pile radius.

[0009] In some embodiments, the deflection control equation of the segmented pile body of the end-bearing pile in a double-layer foundation is expressed by formula (5), which is as follows: ; In formula (5), The elastic modulus of the pile body. The moment of inertia of the pile section, This refers to the displacement of the upper pile segment. This refers to the displacement of the lower pile segment.

[0010] In some embodiments, a piecewise expression for the horizontal displacement of the pile body containing undetermined coefficients is obtained, which is represented by formula (6), as follows: ; In formula (6), The first undetermined coefficient, The second undetermined coefficient, The third undetermined coefficient, The fourth undetermined coefficient, The fifth undetermined coefficient, The sixth undetermined coefficient, The seventh undetermined coefficient. The eighth undetermined coefficient, As the fourth calculation variable, .

[0011] In some embodiments, a system of boundary and continuity condition equations is formed, and the undetermined coefficients are solved to obtain a complete analytical expression for the pile displacement response, including: Based on the unit displacement or unit rotation condition at the pile top, the fixed end condition at the pile bottom, and the continuity conditions of pile displacement, rotation, bending moment, and shear force at the soil layer interface, a system of linear equations with respect to the first to the eighth undetermined coefficients is constructed. Its matrix form is expressed by formula (7), which is as follows: ; In formula (7), The first variable matrix, For the second variable matrix, This is the third variable matrix; in, The complete expression is represented by formula (8), which is as follows: ; The complete expression is represented by formula (9), which is as follows: ; In formula (9), for The first characteristic function of the upper soil layer at time, for The first characteristic function of the underlying soil at that time; The first characteristic function pair of the upper soil layer Find the first derivative. , The first characteristic function pair of the lower soil layer Find the first derivative. ; The first characteristic function pair of the upper soil layer Find the second derivative. , The first characteristic function pair of the lower soil layer Find the second derivative. ; The first characteristic function pair of the upper soil layer Find the third derivative. , The first characteristic function pair of the lower soil layer Find the third derivative. ; The complete expression for the pile top subjected to a unit horizontal displacement and a fixed rotation angle is expressed by formula (10), which is as follows: ; The complete expression for the pile top subjected to unit rotation and fixed horizontal displacement is expressed by formula (11), which is as follows: .

[0012] In some embodiments, Equation (12) is expressed as follows: ; The complete expression is represented by formula (13), which is as follows: ; The complete expression is represented by formula (14), which is as follows: ; From formula (12) to formula (13), This is the fourth variable matrix. As the fifth calculation variable, The characteristic function required to satisfy the orthogonality condition, It is the integral variable.

[0013] In some embodiments, forming a system of boundary and continuity condition equations and solving for undetermined coefficients further includes: A system of linear equations with the first variable coefficients is constructed using the pile-soil interface displacement compatibility condition. Its matrix form is expressed by formula (13), which is as follows: ; Equation (13) is converted into matrix form and expressed by Equation (14), which is as follows: .

[0014] In some embodiments, the horizontal static stiffness matrix of the assembled end-bearing pile in a double-layer foundation is expressed by formula (15), which is as follows: ; The parameters of formula (15) are expressed by formula (16), which is as follows: ; In formulas (15) and (16), This is the first component of the pile top stiffness. This is the second component of the pile top stiffness. This is the third component of the pile top stiffness. This is the fourth component of the pile top stiffness. For the bending moment of the pile body, , For pile shear force, , Is to take The horizontal displacement of the pile top when it is 0. Is to take The pile top rotation angle when it is 0 for The pile top bending moment at that time for Shear force at the top of the pile.

[0015] By adopting the above technical solution, the present invention has the following beneficial effects compared with the prior art: Unlike existing technologies, the above technical solution obtains the mechanical parameters of the pile and the double-layer soil, establishes the soil control equations, and solves them using the method of separation of variables. Combined with multi-layer boundary conditions, it derives analytical expressions for the soil displacement field and distributed soil resistance. Furthermore, it constructs a piecewise pile deflection equation, substitutes the soil resistance, and uses orthogonal expansion of characteristic functions to obtain piecewise expressions for pile displacement. By comprehensively considering various boundary and continuity conditions at the pile top, pile bottom, and soil layer interfaces, it solves for undetermined coefficients, obtaining a complete analytical expression for the pile displacement response, and assembling the horizontal static stiffness matrix at the pile top. Based on rigorous continuum mechanics and analytical mathematics, it achieves an accurate description of the mechanical properties of end-bearing piles in double-layer heterogeneous foundations under horizontal static loads, directly obtaining the closed-form expression of the stiffness matrix through pure analytical derivation. This overcomes the limitations of existing methods in adapting to layered foundations, avoids complex numerical calculations, and features rigorous logic, clear expression, and high computational efficiency, providing a direct and reliable theoretical tool for the design and analysis of pile foundations in layered foundations. Attached Figure Description

[0016] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0017] Figure 1 This is a flowchart for calculating the pile top stiffness of horizontally loaded end-bearing piles in a double-layer foundation; Figure 2 It is a theoretical model of a double-layer foundation with end-bearing piles under horizontal static load. Figure 3This is a schematic diagram showing the change of the first component of the stiffness at the top of a horizontally loaded end-bearing pile in a double-layer foundation. Figure 4 This is a schematic diagram showing the variation of the second component of the stiffness at the top of a horizontally loaded end-bearing pile in a double-layer foundation. Figure 5 This is a schematic diagram showing the variation of the fourth component of the stiffness at the top of a horizontally loaded end-bearing pile in a double-layer foundation. Detailed Implementation

[0018] The present invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be particularly noted that the following embodiments are for illustrative purposes only and do not limit the scope of the invention. Similarly, the following embodiments are only some, not all, embodiments of the present invention, and all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] Please see Figures 1 to 5 This embodiment provides an analytical method for determining the horizontal static stiffness of the mid-end bearing pile in a double-layer foundation, including: Obtain the geometric and material parameters of the pile body of the end-bearing pile in the double-layer foundation, as well as the continuous medium mechanical parameters of the upper soil layer and the continuous medium mechanical parameters of the lower soil layer. The governing equations of the double-layer foundation soil under horizontal static load are established. The potential function is introduced and the separation of variables method is used for solution. Combining the boundary condition of soil displacement attenuation at infinity, the continuous condition of soil layer interface displacement and stress, and the pile-soil interface displacement coordination condition, the analytical expression of the displacement field of the double-layer soil and the analytical expression of the distributed soil resistance acting on the pile are derived. The segmented pile body deflection control equation for end-bearing piles in a double-layer foundation is constructed. The analytical expression of the distributed soil resistance is substituted into the equation, and the characteristic function is orthogonally expanded using the pile-soil interface displacement compatibility condition to obtain the segmented expression of the pile body horizontal displacement containing undetermined coefficients. Based on the load boundary conditions of unit displacement or unit rotation at the pile top, the fixed end constraint conditions at the pile bottom, and the continuity conditions of pile displacement and internal forces at the soil layer interface, a set of boundary condition and continuity condition equations is formed. The undetermined coefficients are solved to obtain the complete analytical expression of the pile displacement response. Based on the analytical expression of pile displacement response, the reaction force at the pile top under a unit horizontal displacement and the reaction force under a unit rotation angle are calculated respectively, and then the horizontal static stiffness matrix of the end-bearing pile in the double-layer foundation is obtained.

[0020] Table 1. Basic mechanical parameters of soil:

[0021] Table 2 Basic mechanical parameters of end-bearing piles:

[0022] In the calculation, the total number of eigenvalues ​​of the series Take as Convergence can then be satisfied.

[0023] In some embodiments, the pile's geometric and material parameters include pile length, pile radius, pile elastic modulus, and pile section moment of inertia. The continuous medium mechanical parameters of the upper soil layer and the lower soil layer include the thickness of each soil layer, the elastic modulus of each soil layer, the shear modulus of each soil layer, and the Poisson's ratio of each soil layer.

[0024] In some embodiments, the governing equations for the two-layer foundation soil under horizontal static load are established and expressed by formula (1), which is as follows: ; In formula (1), The soil compression coefficient is . , Poisson's ratio of the soil; This represents the radial displacement of the soil. The circumferential displacement of the soil is given by the following: Represents the upper soil layer. Represents the lower soil layer. To find the sign of the partial derivative, For radial independent variable, It is the vertical independent variable.

[0025] In some embodiments, the analytical expression for the displacement field of the two-layer soil is derived and expressed by formula (2), which is as follows: ; In formula (2), The second kind of zeroth-order modified Bessel function For the second kind of first-order modified Bessel function, For radial functions containing modified Bessel functions of the second kind, For a cyclic function containing a modified Bessel function of the second kind, where, , , As the first calculation variable, , The coefficient of the first variable, The first characteristic function, As the second calculation variable, , These are the eigenvalues ​​of the first characteristic function. , , It is a circumferential corner; in, Equation (3) is expressed as follows: ; In formula (3), Let be the first characteristic function of the upper soil layer. This is the first characteristic function of the underlying soil. This refers to the total thickness of the soil layer, which is also the length of the pile. This refers to the thickness of the upper soil layer. This refers to the thickness of the lower soil layer; The analytical expression for the distributed soil resistance acting on the pile body is expressed by formula (4), which is as follows: ; In formula (4), As the third calculation variable, , soil layer shear modulus, Where is the pile radius.

[0026] In this embodiment, the boundary conditions are: ; ; ; ; ; In some embodiments, the deflection control equation of the segmented pile body of the end-bearing pile in a double-layer foundation is expressed by formula (5), which is as follows: ; In formula (5), The elastic modulus of the pile body. The moment of inertia of the pile section, This refers to the displacement of the upper pile segment. This refers to the displacement of the lower pile segment.

[0027] In some embodiments, a piecewise expression for the horizontal displacement of the pile body containing undetermined coefficients is obtained, which is represented by formula (6), as follows: ; In formula (6), The first undetermined coefficient, The second undetermined coefficient, The third undetermined coefficient, The fourth undetermined coefficient, The fifth undetermined coefficient, The sixth undetermined coefficient, The seventh undetermined coefficient. The eighth undetermined coefficient, As the fourth calculation variable, .

[0028] In some embodiments, a system of boundary and continuity condition equations is formed, and the undetermined coefficients are solved to obtain a complete analytical expression for the pile displacement response, including: Based on the unit displacement or unit rotation condition at the pile top, the fixed end condition at the pile bottom, and the continuity conditions of pile displacement, rotation, bending moment, and shear force at the soil layer interface, a system of linear equations with respect to the first to the eighth undetermined coefficients is constructed. Its matrix form is expressed by formula (7), which is as follows: ; In formula (7), The first variable matrix, For the second variable matrix, This is the third variable matrix; in, The complete expression is represented by formula (8), which is as follows: ; The complete expression is represented by formula (9), which is as follows: ; In formula (9), for The first characteristic function of the upper soil layer at time, for The first characteristic function of the underlying soil at that time; The first characteristic function pair of the upper soil layer Find the first derivative. , The first characteristic function pair of the lower soil layer Find the first derivative. ; The first characteristic function pair of the upper soil layer Find the second derivative. , The first characteristic function pair of the lower soil layer Find the second derivative. ; The first characteristic function pair of the upper soil layer Find the third derivative. , The first characteristic function pair of the lower soil layer Find the third derivative. ; The complete expression for the pile top subjected to a unit horizontal displacement and a fixed rotation angle is expressed by formula (10), which is as follows: ; The complete expression for the pile top subjected to unit rotation and fixed horizontal displacement is expressed by formula (11), which is as follows: .

[0029] In this embodiment, the boundary conditions and the continuity conditions of the pile segment at the soil layer interface are as follows: or ; ; ; ; ; ; ; In some embodiments, Equation (12) is expressed as follows: ; The complete expression is represented by formula (13), which is as follows: ; The complete expression is represented by formula (14), which is as follows: ; From formula (12) to formula (13), This is the fourth variable matrix. As the fifth calculation variable, The characteristic function required to satisfy the orthogonality condition, It is the integral variable.

[0030] In some embodiments, forming a system of boundary and continuity condition equations and solving for undetermined coefficients further includes: A system of linear equations with the first variable coefficients is constructed using the pile-soil interface displacement compatibility condition. Its matrix form is expressed by formula (13), which is as follows: ; Equation (13) is converted into matrix form and expressed by Equation (14), which is as follows: .

[0031] In some embodiments, the horizontal static stiffness matrix of the assembled end-bearing pile in a double-layer foundation is expressed by formula (15), which is as follows: ; The parameters of formula (15) are expressed by formula (16), which is as follows: ; In formulas (15) and (16), This is the first component of the pile top stiffness. This is the second component of the pile top stiffness. This is the third component of the pile top stiffness. This is the fourth component of the pile top stiffness. For the bending moment of the pile body, , For pile shear force, , Is to take The horizontal displacement of the pile top when it is 0. Is to take The pile top rotation angle when it is 0 for The pile top bending moment at that time for Shear force at the top of the pile.

[0032] The above embodiment provides a method for determining the top stiffness of a horizontally loaded end-bearing pile in a double-layer foundation, including: 1) Determining the mechanical parameters of the end-bearing pile and the soil through on-site exploration, testing, and analysis; it should be noted that the type of pile described in this invention is a cylindrical end-bearing pile, and the soil is a double-layer elastic soil; the soil mechanical parameters include the thickness of each soil layer, the elastic modulus of each soil layer, the shear modulus of each soil layer, and the Poisson's ratio of each soil layer; the end-bearing pile mechanical parameters include the pile length, pile radius, pile elastic modulus, and pile section moment of inertia. 2) Establishing a theoretical model of the end-bearing pile in a double-layer foundation under horizontal static load, establishing the soil control equation, and solving for soil displacement and soil skin friction according to boundary conditions. 3) Establishing segmented pile control equations, and substituting the soil skin friction according to the solution results of step 2) to obtain the expression for the horizontal displacement of the pile containing undetermined coefficients. 4) Based on the derivation results of step 3), and combined with the boundary conditions, pile-soil continuity conditions, and pile segment continuity conditions at the soil layer interface, solve for the undetermined coefficients in the expression for the horizontal displacement of the pile. 5) Based on the results of steps 3) and 4), calculate the horizontal displacement at the pile top. The pile top stiffness can be obtained based on the input load, thus determining the method for determining the pile top stiffness of the horizontally loaded end-bearing pile in a double-layer foundation.

[0033] This invention employs a theoretical approach, treating the soil surrounding the pile as an elastic continuous medium. Based on the proposed theoretical model, an analytical formula for determining the pile top stiffness of a horizontally loaded end-bearing pile in a double-layer foundation is derived, demonstrating rigorous logic. By substituting relevant parameters for calculation, the resulting graph accurately describes the bearing characteristics of the end-bearing pile in a double-layer foundation under horizontal static load. This method is simple, efficient, and provides specific and clear calculation results, offering a valuable reference for the engineering design of end-bearing piles in layered foundations.

[0034] Furthermore, the functional units in the various embodiments of the present invention can be integrated into one processing unit, or each unit can exist physically separately, or two or more units can be integrated into one unit. The integrated unit can be implemented in hardware or as a software functional unit.

[0035] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of this invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) or processor to execute all or part of the steps of the methods of various embodiments of this invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), magnetic disks, or optical disks.

[0036] The above description is only a part of the embodiments of the present invention and does not limit the scope of protection of the present invention. Any equivalent device or equivalent process transformation made based on the content of the present invention specification and drawings, or direct or indirect application in other related technical fields, are similarly included within the patent protection scope of the present invention.

Claims

1. An analytical method for determining the horizontal static stiffness of an end-bearing pile in a double-layered ground, characterized in that, include: Obtain the geometric and material parameters of the pile body of the end-bearing pile in the double-layer foundation, as well as the continuous medium mechanical parameters of the upper soil layer and the continuous medium mechanical parameters of the lower soil layer. The governing equations of the double-layer foundation soil under horizontal static load are established. The potential function is introduced and the separation of variables method is used for solution. Combining the boundary condition of soil displacement attenuation at infinity, the continuous condition of soil layer interface displacement and stress, and the pile-soil interface displacement coordination condition, the analytical expression of the displacement field of the double-layer soil and the analytical expression of the distributed soil resistance acting on the pile are derived. The segmented pile body deflection control equation for end-bearing piles in a double-layer foundation is constructed. The analytical expression of the distributed soil resistance is substituted into the equation, and the characteristic function is orthogonally expanded using the pile-soil interface displacement compatibility condition to obtain the segmented expression of the pile body horizontal displacement containing undetermined coefficients. Based on the load boundary conditions of unit displacement or unit rotation at the pile top, the fixed end constraint conditions at the pile bottom, and the continuity conditions of pile displacement and internal forces at the soil layer interface, a set of boundary condition and continuity condition equations is formed. The undetermined coefficients are solved to obtain the complete analytical expression of the pile displacement response. Based on the analytical expression of the pile displacement response, the reaction force at the pile top under a unit horizontal displacement and the reaction force under a unit rotation angle are calculated respectively, and then the horizontal static stiffness matrix of the end-bearing pile in the double-layer foundation is obtained.

2. The method for analytical determination of horizontal static stiffness of end-bearing pile in double-layer ground according to claim 1, characterized in that, The geometric and material parameters of the pile include pile length, pile radius, pile elastic modulus, and pile section moment of inertia; The continuous medium mechanical parameters of the upper soil layer and the lower soil layer include the thickness of each soil layer, the elastic modulus of each soil layer, the shear modulus of each soil layer, and the Poisson's ratio of each soil layer.

3. The method for analytical determination of horizontal static stiffness of end-bearing pile in double-layer ground according to claim 1, characterized in that, The governing equations for the two-layer foundation soil under horizontal static load are established and expressed by formula (1), which is as follows: ； In formula (1), The soil compression coefficient is . , Poisson's ratio of the soil; This represents the radial displacement of the soil. The circumferential displacement of the soil is given by the following: Represents the upper soil layer. Represents the lower soil layer. To find the sign of the partial derivative, For radial independent variable, It is the vertical independent variable.

4. The analytical method for determining the horizontal static stiffness of the mid-end bearing pile in a double-layer foundation according to claim 3, characterized in that, The analytical expression for the displacement field of the two-layer soil is derived and expressed by formula (2), which is as follows: ； In formula (2), The second kind of zeroth-order modified Bessel function For the second kind of first-order modified Bessel function, For radial functions containing modified Bessel functions of the second kind, For a cyclic function containing a modified Bessel function of the second kind, where, , , As the first calculation variable, , The coefficient of the first variable, The first characteristic function, As the second calculation variable, , These are the eigenvalues ​​of the first characteristic function. , , It is a circumferential corner; in, Equation (3) is expressed by the following formula: ； In formula (3), Let be the first characteristic function of the upper soil layer. This is the first characteristic function of the underlying soil. This refers to the total thickness of the soil layer, which is also the length of the pile. This refers to the thickness of the upper soil layer. This refers to the thickness of the lower soil layer; The analytical expression for the distributed soil resistance acting on the pile body is expressed by formula (4), which is as follows: ； In formula (4), As the third calculation variable, , soil layer shear modulus, Where is the pile radius.

5. The analytical method for determining the horizontal static stiffness of the mid-end bearing pile in a double-layer foundation according to claim 4, characterized in that, The deflection control equation for segmented piles in a double-layer foundation is expressed by formula (5), which is as follows: ； In formula (5), The elastic modulus of the pile body. The moment of inertia of the pile section, This refers to the displacement of the upper pile segment. This refers to the displacement of the lower pile segment.

6. The analytical method for determining the horizontal static stiffness of the mid-end bearing pile in a double-layer foundation according to claim 5, characterized in that, The piecewise expression for the horizontal displacement of the pile body containing undetermined coefficients is obtained, which is expressed by formula (6), as follows: ； In formula (6), The first undetermined coefficient, The second undetermined coefficient, The third undetermined coefficient, The fourth undetermined coefficient, The fifth undetermined coefficient, The sixth undetermined coefficient, The seventh undetermined coefficient. The eighth undetermined coefficient, As the fourth calculation variable, .

7. The analytical method for determining the horizontal static stiffness of the mid-end bearing pile in a double-layer foundation according to claim 6, characterized in that, By forming a system of boundary and continuity condition equations and solving for the undetermined coefficients, a complete analytical expression for the pile displacement response is obtained, including: Based on the unit displacement or unit rotation condition at the pile top, the fixed end condition at the pile bottom, and the continuity conditions of pile displacement, rotation, bending moment, and shear force at the soil layer interface, a system of linear equations with respect to the first to the eighth undetermined coefficients is constructed. Its matrix form is expressed by formula (7), which is as follows: ； In formula (7), For the first variable matrix, For the second variable matrix, This is the third variable matrix; in, The complete expression is represented by formula (8), which is as follows: ； The complete expression is represented by formula (9), which is as follows: ； In formula (9), for The first characteristic function of the upper soil layer at time , for The first characteristic function of the underlying soil at that time; The first characteristic function pair of the upper soil layer Find the first derivative. , The first characteristic function pair of the lower soil layer Find the first derivative. ; The first characteristic function pair of the upper soil layer Find the second derivative. , The first characteristic function pair of the lower soil layer Find the second derivative. ; The first characteristic function pair of the upper soil layer Find the third derivative. , The first characteristic function pair of the lower soil layer Find the third derivative. ; The complete expression for the pile top subjected to a unit horizontal displacement and a fixed rotation angle is represented by formula (10), which is as follows: ； The complete expression for the pile top subjected to unit rotation and fixed horizontal displacement is represented by formula (11), which is as follows: 。 8. The analytical method for determining the horizontal static stiffness of the mid-end bearing pile in a double-layer foundation according to claim 7, characterized in that, Equation (12) is expressed as follows: ； The complete expression is represented by formula (13), which is as follows: ； The complete expression is represented by formula (14), which is as follows: ； From formula (12) to formula (13), This is the fourth variable matrix. As the fifth calculation variable, The characteristic function required to satisfy the orthogonality condition, It is the integral variable.

9. The analytical method for determining the horizontal static stiffness of the mid-end bearing pile in a double-layer foundation according to claim 8, characterized in that, The process of forming a system of boundary and continuity condition equations and solving for the undetermined coefficients also includes: A system of linear equations with the first variable coefficients is constructed using the pile-soil interface displacement compatibility condition. Its matrix form is expressed by formula (13), which is as follows: ； Equation (13) is converted into matrix form and expressed by Equation (14), which is as follows: 。 10. The analytical method for determining the horizontal static stiffness of the mid-end bearing pile in a double-layer foundation according to claim 9, characterized in that, The horizontal static stiffness matrix of the assembled end-bearing pile in the double-layer foundation is expressed by formula (15), which is as follows: ； The parameters of formula (15) are expressed by formula (16), which is as follows: ； In formulas (15) and (16), This is the first component of the pile top stiffness. This is the second component of the pile top stiffness. This is the third component of the pile top stiffness. This is the fourth component of the pile top stiffness. For the bending moment of the pile body, , For pile shear force, , Is to take The horizontal displacement of the pile top when it is 0. Is to take The pile top rotation angle when it is 0 for The pile top bending moment at that time for Shear force at the top of the pile.

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