Method for evaluating anti-seismic performance of end bearing pile in stratified soil under action of earthquake S waves

Abstract

The invention belongs to the field of rock-soil earthquake engineering, and particularly relates to a method for evaluating the anti-seismic performance of an end bearing pile in stratified soil under the action of earthquake S waves. According to the method, a pile-soil model is established on the basis of a continuous medium theory under a linear elastic frame, and displacement of a free field and a scattered field of a soil body is decoupled through a Helmholtz decomposition method; the soil layer boundary is processed by adopting the orthogonality of a modal function characteristic function, and the motion response of the pile foundation is described by adopting a strict closed form solution. Frequency domain amplitude-frequency characteristic analysis is adopted, a motion corresponding coefficient and a motion amplification coefficient are introduced to quantitatively evaluate the influence of the soil layer thickness and the soil layer rigidity ratio on the anti-seismic performance of the end bearing pile, and theoretical support is provided for pile foundation optimization design.

Description

Technical Field

[0001] This invention belongs to the field of geotechnical earthquake engineering, specifically relating to a method for evaluating the seismic performance of end-bearing piles in stratified soil under S-wave seismic action. Background Technology

[0002] In geotechnical engineering practice, pile foundations are often used in seismic design. During earthquake propagation, in addition to the load transmitted by the superstructure, the pile foundation is also subjected to the interaction of the soil around the pile caused by the scattered waves. This interaction causes the pile to bear the bending moment distributed along the length of the soil. In actual earthquake disaster investigations, it has been found that most pile foundations fail due to bending near the ground surface.

[0003] Prior to this, calculation methods for the seismic response of end-bearing piles under S-wave loading were relatively mature. Early studies mostly employed numerical simulation methods, using finite element and boundary element methods to model the pile as a beam member and the soil as a viscoelastic half-space soil. This method is widely regarded as a reference for other assumed model analysis methods. However, this calculation method is complex and time-consuming, and it is difficult to intuitively reflect the interaction mechanism between the pile and the soil. Subsequent studies introduced more precise assumptions: simplifying the complex soil mass into a series of independent linear springs and parallel dampers, and defining the parameters of these springs, which vary with soil depth, as Winkler springs. Based on this, the medium surrounding the soil is considered as a series of uncoupled and incompressible horizontal slices, thus proposing a plane strain model. This method is simple to operate and can accurately simulate the motion response of end-bearing piles under a two-dimensional frame. However, this method is unstable, cannot obtain relevant resonant frequencies and cutoff frequencies, and cannot consider the issue of soil continuity. To further refine this theory, the Tajimi continuous medium theory was introduced, and the BDWF model was proposed. This theory no longer simply regards the pile as a one-dimensional rod but as a continuous cylinder. However, this theory still ignores the effect of the radial displacement of the soil, which makes it unable to reflect the coupling characteristics well.

[0004] In addition, much current research focuses on the impact of different pile types on pile seismic response, while the influence of soil stratification is less studied. Actual earthquake case reports show that interfaces between soil layers with significant differences in stiffness are more prone to failure. Soil heterogeneity is mainly reflected in the different physical properties of the interfaces between soil layers, and considering soil stratification is more relevant to engineering practice. Therefore, studying the influence of stratified soil properties on pile motion response under S-wave loading is an important research topic.

[0005] Therefore, there is a lack of a method in this field for analyzing mid-end-bearing piles in layered soils based on rigorous theoretical derivation. This method must first meet the requirements of accuracy in theoretical research, and then meet the needs of simplicity and convenience in engineering practice; most importantly, it must be able to provide reliable theoretical basis and technical support for pile foundation design under various complex working conditions. The establishment of this innovative method will significantly improve the seismic safety performance of major engineering structures, and has important theoretical value and engineering application prospects. Summary of the Invention

[0006] The purpose of this invention is to provide a method for evaluating the seismic performance of end-bearing piles in layered soil under S-wave seismic loading. This method establishes a pile-soil model based on continuous medium theory within a linear elastic framework. It decouples the soil's free field and scattered field displacements using the Helmholtz decomposition method, employs the orthogonality of modal function characteristic functions to handle soil layer boundaries, and uses a strictly closed-form solution to describe the pile foundation's motion response. Frequency domain amplitude-frequency characteristic analysis is used, introducing motion response coefficients and motion amplification coefficients to quantitatively evaluate the influence of soil layer thickness and soil layer stiffness ratio on the seismic performance of end-bearing piles, providing theoretical support for optimal pile foundation design.

[0007] To achieve the above-mentioned technical objectives and effects, the present invention is implemented through the following technical solution: A method for evaluating the seismic performance of mid-bearing piles in stratified soil under S-wave seismic loading includes the following steps: S1: Establishing a pile-soil system model: Based on the theory of continuous elastic dynamics, a motion response model of an end-bearing pile in stratified soil under the action of a vertical S-wave is established. The circular frequency of the seismic S-wave is ω, and the amplitude is u0. The pile-soil system includes a vertical end-bearing pile with its bottom inserted into the bedrock. The pile length is H, the diameter is d, and the elastic modulus is E. p The cross-sectional area is A p The moment of inertia of the pile cross section is I. p The mass density is ρ p The pile and soil are in complete contact with no relative sliding. The soil around the pile consists of two homogeneous layers, and each layer is an isotropic three-dimensional continuous medium. The thickness of the upper soil layer is L, and its material parameters include shear modulus G. s,1 Poisson's ratio s,1 Damping ratio β1 and density ρ s,1 The thickness of the underlying soil layer is HL, and its material parameters include shear modulus G. s,2 Poisson's ratio s,2 Damping ratio β2 and density ρ s,2 ; S2: Decoupling and solving the soil displacement field: Decoupling the total soil displacement into free field displacement and scattered field displacement, i.e. Based on the boundary conditions that the soil displacement needs to satisfy, the circumferential dynamic displacement u of the soil in a single pile is solved.θ (r,θ,z) and radial dynamic displacement u r (r,θ,z); S3: Calculate the soil friction resistance around the pile: Through stress balance analysis of the micro-element on the pile surface, combined with the soil displacement field obtained in step S2, calculate the friction resistance R(z) of the soil around the pile acting on the end-bearing pile surface. S4: Solve for the dynamic response of the pile: Using the pile boundary conditions and the frictional resistance R(z) obtained in step S3, solve for the vertical displacement w of the end-bearing pile. p (z); The boundary conditions of the pile body include zero bending moment and shear force at the ground surface, zero displacement and curvature of the bedrock, and continuous displacement, curvature, bending moment and shear force at the soil layer interface; S5: Coupled solution of pile-soil system response: Combining the complete contact boundary condition of pile and soil, the total soil displacement obtained in step S2 at the pile-soil interface is compared with the pile displacement w obtained in step S4. p (z) By combining the equations, utilizing the orthogonality of trigonometric functions, and solving for the unknown constants through integration and Fourier series expansion, the displacement response w of the end-bearing pile along depth z is obtained. p (z); S6: Quantitative assessment of seismic performance: Based on the end-bearing pile displacement response w obtained in step S5 p (z) The dynamic response characteristics of end-bearing piles are evaluated using a dual analysis method in the frequency and time domains, including dimensionless frequency parameters. Motion response coefficient I u The motion amplification factor A is used to analyze the changes in parameters of each soil layer in stratified soil; by changing the thickness of the upper soil layer and / or the stiffness ratio of the soil layer; steps S1 to S6 are repeated to calculate the changes in parameters at multiple dimensionless frequencies under different combinations of parameters. I at the location u Value and A value; plot I u And A The varying amplitude-frequency response curves are analyzed to assess the impact of changes in soil layer thickness ratio and soil layer stiffness ratio on the evaluation index I. u The influence of A and the influence of layered soil properties on the seismic performance of end-bearing piles were investigated.

[0008] Furthermore, the total soil displacement in step S2 is given by the formula... The free field displacement is determined by the formula Sure: Where i=1 and 2 represent two soil layers in a stratified soil formation, and ω is the S-wave circular frequency. For complex-valued Lamé constants, , For the complex shear wave velocity of the soil, The shear wave velocity of the soil. .

[0009] Furthermore, the boundary conditions that the soil displacement must satisfy in step S2 include: S2.1: The stress on the soil at the ground surface is zero; S2.2: The total displacement of the soil at the bedrock is u0; S2.3: Displacement is continuous at the interface between the two soil layers; S2.4: The shear stress at the interface between the two soil layers is continuous; S2.5: Displacement coordination at the pile-soil interface; Furthermore, in step S3, the frictional force exerted by the soil around the pile on the surface of the end-bearing pile is given by the formula... Sure: Where R(z) is the surface friction of the end-bearing pile; σ r τ is the radial normal stress on the pile surface. rθ Let r be the shear stress on the pile surface and r0 be the pile radius.

[0010] Furthermore, the displacement of the end-bearing pile in step S4 is given by the formula... Sure: Among them, E p For the elastic modulus of the pile, A p I is the cross-sectional area of ​​the pile. p Let ρ be the moment of inertia of the pile cross section. p The density of the pile mass.

[0011] Furthermore, the boundary conditions that the end-bearing pile displacement must satisfy in step S4 include: S4.1: The moment experienced by the end-bearing pile at the ground surface is zero; S4.2: The force on the end-bearing pile at the ground surface is zero; S4.3: The displacement of the end-bearing pile in the bedrock is u0; S4.4: The curvature of the end-bearing pile in the bedrock is zero; S4.5: The displacement of the end-bearing pile at the soil interface is continuous; S4.6: The curvature of the end-bearing pile is continuous at the soil layer interface; S4.7: The bending moment experienced by the end-bearing pile at the soil interface is continuous; S4.8: The force on the end-bearing pile at the soil interface is continuous; Furthermore, the simultaneous equations for pile-soil displacement in step S5 are derived from the formula... ,formula Sure: Furthermore, in step S6, the dimensionless frequency From the formula Determine the motion response coefficient I u From the formula The motion amplification factor A is determined by formula (9). Sure: Where ω is the angular frequency and ω1 is the dominant frequency. This refers to the displacement of the pile top of the end-bearing pile. U represents the free surface displacement, and u0 represents the displacement at the bedrock.

[0012] Furthermore, the method is applicable to the single-pile motion response of end-bearing piles, and the analytical solution of the motion response of end-bearing piles in stratified soil layers under S-wave action is derived with rigorous formula derivation.

[0013] The beneficial effects of this invention are: In traditional analysis methods, the kinematic response of end-bearing piles under S-wave action often neglects the effect of radial soil displacement, leading to the overlooking of wave coupling effects. To address this deficiency, this invention employs a rigorous closed-loop solution based on continuous medium theory, considering the soil to decouple the total soil displacement into free-field displacement and scattered-field displacement. Combined with the boundary condition of complete pile-soil contact, it accurately calculates the vertical dynamic displacement of the soil within a single pile. Furthermore, it introduces the influence of soil stratification on the pile's kinematic response, resulting in a more objective and accurate representation of the end-bearing pile's kinematic response under S-wave action. This allows for the study of the impact of stratification on the end-bearing pile's kinematic response, better serving engineering practice.

[0014] This invention establishes the dynamic equilibrium equations for a single pile by introducing the Euler-Bernoulli beam theory, thus establishing the connection between the pile and soil motion equations. This theory effectively improves the accuracy of calculations compared to previous one-dimensional rod theories. In frequency domain analysis, this invention uses dimensionless frequency parameters and motion response coefficients I... uThe motion amplification factor A directly reflects the influence of soil stratification on dynamic response, which not only improves the computational efficiency of the model, enabling it to quickly adapt to engineering application needs, but also ensures computational reliability under complex working conditions.

[0015] This invention systematically characterizes the dynamic response model of end-bearing piles based on continuous elastic dynamics theory, fully considering the physical properties of the soil surrounding the pile (such as shear modulus, Poisson's ratio, and damping ratio) and the structural properties of the pile (such as elastic modulus, cross-sectional area, and mass density). By refining the calculation of pile-soil interaction forces, it not only improves the realism of the simulation but also helps to reveal the dynamic characteristics of the end-bearing pile system. Through quantitative analysis of pile-soil interaction, this invention provides clear theoretical support for further pile foundation design and optimization.

[0016] This invention employs dimensionless frequency to study the dynamic response characteristics of end-bearing piles in layered soil. In the frequency domain, the motion response coefficients I are plotted. u The amplitude-frequency curves can be used to intuitively analyze the influence of soil layer thickness, stiffness and other properties on the dynamic response of end-bearing piles.

[0017] Of course, any product implementing this invention does not necessarily need to achieve all of the advantages described above at the same time. Attached Figure Description

[0018] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments 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.

[0019] Figure 1 This is a schematic diagram of a pile-soil model; Figure 2 A schematic diagram illustrating the effect of changing the thickness ratio of stratified soil layers on the pile top motion response; Figure 3 A schematic diagram illustrating the effect of changing the stiffness ratio of stratified soil layers on the pile top motion response. Figure 4 This is a schematic diagram illustrating the effect of pile displacement on depth under different soil layer thickness ratios. Figure 5 This is a schematic diagram of the overall steps of the present invention. Detailed Implementation

[0020] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0021] Example 1 The method for evaluating the seismic performance of end-bearing piles in stratified soil under S-wave action, as described in this embodiment, includes the following steps: S1: Based on the theory of continuous elastic dynamics, a motion response model of an end-bearing pile in stratified soil under the action of a vertical S-wave is established. The circular frequency of the seismic S-wave is ω, and the amplitude is u0. The pile-soil system includes a vertical end-bearing pile with its bottom inserted into the bedrock. The pile length is H, the diameter is d, and the elastic modulus is E. p The cross-sectional area is A p The moment of inertia of the pile cross section is I. p The mass density is ρ p The pile and soil are in complete contact with no relative sliding. The soil around the pile consists of two homogeneous layers, and each layer is an isotropic three-dimensional continuous medium. The thickness of the upper soil layer is L, and its material parameters include shear modulus G. s,1 Poisson's ratio s,1 Damping ratio β1 and density ρ s,1 The thickness of the underlying soil layer is HL, and its material parameters include shear modulus G. s,2 Poisson's ratio s,2 Damping ratio β2 and density ρ s,2 ; S2: Decouple the total soil displacement into free field displacement and scattered field displacement, i.e. Based on the boundary conditions that the soil displacement needs to satisfy, the circumferential dynamic displacement u of the soil in a single pile is solved. θ (r,θ,z) and radial dynamic displacement u r (r,θ,z); S3: The stress balance on the pile surface is established by using infinitesimal elements and the integral method is used to calculate the frictional force of the soil around the pile acting on the surface of the end-bearing pile. S4: Based on the Euler-Bernoulli beam theory, establish the dynamic equilibrium equations for a single pile. Utilize the boundary conditions that the displacement of the end-bearing pile in the soil must satisfy to establish the equations. Then, calculate the vertical displacement w of the end-bearing pile using the frictional force exerted by the surrounding soil on the pile. p (z); S5: Based on the boundary conditions of complete contact between the pile and the soil and no relative sliding, establish the connection between the overall soil displacement equation and the end-bearing pile displacement equation. In the subsequent processing of infinite series, consider the orthogonality of trigonometric functions, adopt the integral form and introduce Fourier series expansion to solve the last unknown constant of the end-bearing pile displacement equation. S6: Quantitative assessment of seismic performance: Based on the end-bearing pile displacement response w obtained in step S5 p (z) The dynamic response characteristics of end-bearing piles are evaluated using a dual analysis method in the frequency and time domains, including dimensionless frequency parameters. Motion response coefficient I u The motion amplification factor A is used to analyze the changes in parameters of each soil layer in stratified soil; by changing the thickness of the upper soil layer and / or the stiffness ratio of the soil layer; steps S1 to S6 are repeated to calculate the changes in parameters at multiple dimensionless frequencies under different combinations of parameters. I at the location u Value and A value; plot I u And A The varying amplitude-frequency response curves are analyzed to assess the impact of changes in soil layer thickness ratio and soil layer stiffness ratio on the evaluation index I. u The influence of A and the influence of layered soil properties on the seismic performance of end-bearing piles were investigated.

[0022] In this embodiment, the total soil displacement mentioned in step S2 is given by the formula The free field displacement is determined by the formula Sure: Where i=1 and 2 represent two soil layers in a stratified soil formation, and ω is the S-wave circular frequency. For complex-valued Lamé constants, , For the complex shear wave velocity of the soil, The shear wave velocity of the soil. .

[0023] In this embodiment, the boundary conditions that the soil displacement must satisfy in step S2 are as follows: S2.1: The stress on the soil at the ground surface is zero; S2.2: The total displacement of the soil at the bedrock is u0; S2.3: Displacement is continuous at the interface between the two soil layers; S2.4: The shear stress at the interface between the two soil layers is continuous; S2.5: Displacement coordination at the pile-soil interface; In this embodiment, the frictional force exerted by the soil around the pile on the surface of the end-bearing pile in step S3 is given by the formula... Sure: Where R(z) is the surface friction of the end-bearing pile; σ r τ is the radial normal stress on the pile surface. rθ Let r be the shear stress on the pile surface and r0 be the pile radius.

[0024] In this embodiment, the end-bearing pile displacement mentioned in step S4 is given by the formula Sure: Among them, E p For the elastic modulus of the pile, A p I is the cross-sectional area of ​​the pile. p Let ρ be the moment of inertia of the pile cross section. p The density of the pile mass.

[0025] In this embodiment, the boundary conditions that the end-bearing pile displacement must satisfy in step S4 are as follows: S4.1: The moment experienced by the end-bearing pile at the ground surface is zero; S4.2: The force on the end-bearing pile at the ground surface is zero; S4.3: The displacement of the end-bearing pile in the bedrock is u0; S4.4: The curvature of the end-bearing pile in the bedrock is zero; S4.5: The displacement of the end-bearing pile at the soil interface is continuous; S4.6: The curvature of the end-bearing pile is continuous at the soil layer interface; S4.7: The bending moment experienced by the end-bearing pile at the soil interface is continuous; S4.8: The force on the end-bearing pile at the soil interface is continuous; In this embodiment, the simultaneous equations of pile-soil displacement described in step S5 are derived from the formula... ,formula Sure: In this embodiment, the dimensionless frequency mentioned in step S6 From the formula Sure Where ω is the angular frequency and ω1 is the dominant frequency.

[0026] In this embodiment, the frequency domain analysis in step S6 specifically includes: (1) Motion response coefficient I u From the formula The motion amplification factor A is determined by formula (9): in, This refers to the displacement of the pile top of the end-bearing pile. U represents the free surface displacement, and u0 represents the displacement at the bedrock.

[0027] (3) Plot the motion response curve of the end-bearing pile and analyze the influence of different soil layer thickness ratios and soil layer stiffness ratios on the trend of motion response curve changes; In this embodiment, the method is applicable to the single-pile motion response of end-bearing piles, and the analytical solution of the motion response of end-bearing piles in stratified soil layers under S-wave action is obtained by rigorous formula derivation.

[0028] The preferred embodiments of the present invention disclosed above are merely illustrative of the invention. These preferred embodiments do not exhaustively describe all details, nor do they limit the invention to the specific implementations described. Clearly, many modifications and variations can be made based on the content of this specification. This specification selects and specifically describes these embodiments to better explain the principles and practical applications of the invention, thereby enabling those skilled in the art to better understand and utilize the invention. The invention is limited only by the claims and their full scope and equivalents.

Claims

1. A method for evaluating the seismic performance of an end-bearing pile in layered soil under seismic S-wave action, characterized in that, The method comprises the following steps: S1: Establishing a pile-soil system model: based on continuous elastic dynamics theory, a motion response model of an end-bearing pile in layered soil under vertical S wave is established, the circular frequency of the seismic S wave is ω, the amplitude is u0, the pile-soil system comprises a vertical end-bearing pile, the pile bottom is inserted into a bedrock, the pile length is H, the diameter is d, the elastic modulus is E p , the cross-sectional area is A p , the inertia moment of the pile cross section is I p , the mass density is ρ p , the pile soil is completely in contact and has no relative sliding, the soil around the pile is a homogeneous soil body distributed in two layers, and the soil in each soil layer is an isotropic three-dimensional continuous medium, the thickness of the upper soil body is L, and the material parameters thereof include shear modulus G s,1 , Poisson's ratio v s,1 , damping ratio β1 and density ρ s,1 , the thickness of the lower soil body is H-L, and the material parameters thereof include shear modulus G s,2 , Poisson's ratio v s,2 , damping ratio β2 and density ρ s,2 ; S2: Decoupling and solving the soil displacement field: decoupling the total displacement of the soil into free field displacement and scattered field displacement, that is , combined with the boundary conditions that the soil displacement needs to satisfy, solving the circumferential dynamic displacement u θ (r, θ, z) and the radial dynamic displacement u r (r, θ, z) of the single pile soil; S3: calculating the soil frictional resistance around the pile: through stress balance analysis of the pile surface microelement, and in combination with the soil displacement field obtained in step S2, the soil frictional resistance R(z) acting on the surface of the end-bearing pile is calculated; S4: solving the dynamic response of the pile body: using the pile body boundary conditions and the calculated friction resistance R(z) in step S3, solving the vertical displacement w of the end-bearing pile p (z); the pile body boundary conditions include zero ground surface bending moment and shear force, base rock displacement u0 and zero curvature, continuous displacement, curvature, bending moment and shear force at the soil layer interface; S5: coupling solution of pile-soil system response: combining the complete contact boundary condition of pile-soil, the total displacement of soil obtained in step S2 at the pile-soil interface is coupled with the displacement w of the pile obtained in step S4 p (z) the unknown constants are solved by using the orthogonality of trigonometric functions and through integration and Fourier series expansion to obtain the displacement response w of the end-bearing pile along the depth z p (z); S6: Quantitative evaluation of anti-seismic performance: based on the displacement response w of the end-bearing pile obtained in step S5 p (z), the dynamic response characteristics of the end-bearing pile are evaluated by using the dual analysis method of frequency domain and time domain, including the dimensionless frequency parameter , the motion response coefficient I u , and the motion amplification coefficient A u The parameter changes of each soil layer in the layered soil are analyzed by changing the thickness of the upper layer and / or the stiffness ratio of the soil layers; steps S1 to S6 are repeated to calculate the I value and A value at multiple dimensionless frequencies u at different parameter combinations; the amplitude-frequency characteristic curves of I u and A with are drawn, the influence law of the thickness ratio of the soil layers and the stiffness ratio of the soil layers on the evaluation indexes I u and A is analyzed, and the influence of the layered soil properties on the anti-seismic performance of the end-bearing pile is evaluated.

2. The method for evaluating the seismic performance of an end-bearing pile in layered soil under S-wave action according to claim 1, characterized in that: The total displacement of the soil mass in the step S2 is determined by the formula The free-field displacement is determined by the formula ​ where i = 1, 2 represent two soil layers of the layered soil respectively, ω is the circular frequency of S wave, is the complex Lame constant, , is the complex shear wave velocity of the soil, is the shear wave velocity of the soil, .

3. The method for evaluating the seismic performance of an end-bearing pile in layered soil under S-wave action according to claim 1, characterized in that: The boundary conditions to be met by the soil displacement in step S2 include: S2.1: the stress of the soil at the ground surface is zero; S2.2: the total displacement of the soil at the bedrock is u0; S2.3: the displacement is continuous at the interface between two soil layers; S2.4: the shear stress is continuous at the interface between two soil layers; S2.5: the displacement is coordinated at the pile-soil interface.

4. The method for evaluating the seismic performance of an end-bearing pile in layered soil under S-wave action according to claim 1, characterized in that: The friction force of the soil around the pile acting on the surface of the end-bearing pile in the step S3 is calculated by the formula Determined: where R(z) is the surface friction of the pile; σ r is the radial normal stress on the pile surface, τ rθ is the shear stress on the pile surface, and r0 is the pile radius.

5. The method for evaluating the seismic performance of an end-bearing pile in layered soil under S-wave action according to claim 1, characterized in that: The end bearing pile displacement in the step S4 is determined by the formula Determined: where E p is the pile elastic modulus, A p is the pile cross-sectional area, I p is the moment of inertia of the pile cross-section, and p p is the pile mass density.

6. The method for evaluating the seismic performance of an end-bearing pile in layered soil under S-wave action according to claim 1, characterized in that: The boundary conditions to be met by the end-bearing pile displacement in step S4 include: S4.1: the moment acting on the end-bearing pile at the ground surface is zero; S4.2: the force acting on the end-bearing pile at the ground surface is zero; S4.3: the movement displacement of the end-bearing pile at the bedrock is u0; S4.4: the curvature of the end-bearing pile at the bedrock is zero; S4.5: the displacement of the end-bearing pile is continuous at the interface between soil layers; S4.6: the curvature of the end-bearing pile is continuous at the interface between soil layers; S4.7: the bending moment acting on the end-bearing pile is continuous at the interface between soil layers; S4.8: the force acting on the end-bearing pile is continuous at the interface between soil layers.

7. The method for evaluating the seismic performance of an end-bearing pile in layered soil under S-wave action according to claim 1, characterized in that: The simultaneous equations of pile soil displacement in the step S5 are determined by the formula , the formula . 。 8. The method for evaluating the seismic performance of a pile with end bearing in layered soil under S-wave action according to claim 1, characterized in that: The dimensionless frequency in step S6 is determined by the formula The motion response coefficient I is determined by the formula u The motion amplification coefficient A is determined by the formula ​​ where ω is the circular frequency, and ω1 is the main frequency; is the displacement of the pile top of the end-bearing pile, is the displacement of the free-field ground surface, and u0 is the displacement at the bedrock.

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