A method for analyzing horizontal dynamic response of a single pile under Rayleigh waves

Abstract

The embodiment of the present invention discloses a method for analyzing the horizontal dynamic response of a single pile under the action of Rayleigh waves, which includes: S1, creating a free field lateral displacement equation generated by any soil layer under the action of Rayleigh waves; S2, creating a dynamic equilibrium equation of a pile body unit under the action of Rayleigh waves; S3, based on the horizontal displacement of the pile body, simplifying the dynamic equilibrium equation of the pile body unit under the action of Rayleigh waves to obtain and solve the corresponding simplified equation; S4, based on the Euler beam theory, determining the relationship between the pile body rotation angle, bending moment, shear force and the horizontal displacement of the pile body to form a corresponding relationship equation; S5, solving the relationship equation to obtain the corresponding analytical result of the horizontal displacement of the pile body. The present invention considers the horizontal vibration force analysis of the pile foundation under the axial force of the pile top in the layered soil, which can better simulate the constraint effect of the soil around the pile on the pile body, and can be applied to the problem of the dynamic response of the horizontal vibration of the pile foundation under the action of the earthquake Rayleigh wave.

Description

Technical Field

[0001] The invention relates to the technical field of pile foundation vibration analysis, and in particular to a method for analyzing horizontal dynamic response of a single pile under the action of Rayleigh waves. Background Art

[0002] In recent years, pile foundations have been commonly damaged in major earthquakes. When an earthquake occurs, the pile foundation will vibrate due to the seismic waves in the foundation soil, and the vibration of the pile will also change the movement of the foundation soil around the pile. When studying the dynamic response of pile foundations under earthquakes, the interaction between pile and soil motion is a key factor.

[0003] Therefore, more and more scholars have conducted relevant research on the horizontal dynamic response of pile foundations under seismic waves. The study of the dynamic response of piles under lateral seismic loads is usually divided into numerical solutions and analytical solutions. Numerical solutions require a lot of computer resources and time, and cannot guarantee mathematical rigor. There are few methods in the analytical solution that can simply reveal the influence of soil shear deformation on the calculation results, and most of these studies currently only simplify the soil around the pile into a Winkler foundation during the pile foundation vibration analysis, and do not fully consider the influencing factors of its shear deformation, especially the influencing factors of the horizontal vibration force of the pile foundation under the axial force of the pile top. Summary of the invention

[0004] Based on this, in order to solve the shortcomings of the existing technology, a method for analyzing the horizontal dynamic response of a single pile under the action of Rayleigh waves is proposed.

[0005] A method for analyzing the horizontal dynamic response of a single pile under the action of Rayleigh waves, characterized in that it comprises the following steps:

[0006] S1. Create the free-field lateral displacement equation generated by any soil layer under the action of Rayleigh waves;

[0007] S2. Based on the free-field lateral displacement equation, a dynamic equilibrium equation of the pile unit under the action of Rayleigh waves is created;

[0008] S3, based on the horizontal displacement of the pile body, simplify the dynamic equilibrium equation of the pile body unit under the action of the Rayleigh wave to obtain and solve the corresponding simplified equation;

[0009] S4. Based on the Euler beam theory, determine the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement to form a corresponding relationship equation;

[0010] S5. Solve the relationship equation to obtain the corresponding analytical result of pile body horizontal displacement.

[0011] Optionally, in one embodiment, in S1, the specific steps of creating a free-field lateral displacement equation generated by any soil layer under the action of Rayleigh waves include:

[0012] According to the wave equation theory of elastic body, the free field displacement equation generated at the depth z of the jth layer of soil under the action of Rayleigh wave is obtained as follows:

[0013]

[0014] In the formula, u 1j 、u 2j 、u 3j 、u 4j are unknown complex-valued coefficients, j is less than the total number of soil layers n, V pl is the compression wave velocity of soil, V sl is the shear wave velocity of soil, c is the phase velocity of Rayleigh wave;

[0015] For the bottom nth layer of soil, since there is only a downward wave but no upward wave, the corresponding free field displacement equation is:

[0016]

[0017] In the formula, u 1n 、u 2n are unknown complex-valued coefficients.

[0018] Optionally, in one embodiment, in S2, based on the free field lateral displacement equation, the specific steps of creating a dynamic equilibrium equation of the pile unit under the action of Rayleigh waves include:

[0019] Combining the theories of Euler beam and Pasternak foundation model, the dynamic equilibrium equation of the pile unit under the action of Rayleigh wave is obtained as follows (3):

[0020]

[0021] In the formula, is the horizontal displacement of the j-th pile unit mass point, is the free field lateral displacement generated by the Rayleigh wave at the jth soil layer depth z; E p ,I p 、m p are the elastic modulus, section moment of inertia and mass per unit length of the pile respectively; and are the stiffness coefficient, damping coefficient and foundation shear stiffness of the foundation soil around the j-th pile layer; B 0 =0.9(1.5d+0.5) is the calculated width of the pile.

[0022] Optionally, in one embodiment, in S3, based on the horizontal displacement of the pile body, the specific steps of simplifying the dynamic equilibrium equation of the pile body unit under the action of the Rayleigh wave to obtain and solve the corresponding simplified equation include:

[0023] The horizontal displacement of the pile is expressed as:

[0024]

[0025] In the formula, is the horizontal displacement amplitude of the pile body particle;

[0026] Let W p =E p I p , Substituting equations (1), (2), and (4) into equation (3) respectively, we can obtain the simplified equation:

[0027]

[0028]

[0029] In the formula,

[0030] Solving differential equations (5) and (6) we can obtain:

[0031]

[0032]

[0033] In the formula,

[0034] Optionally, in one embodiment, in S4, based on the Euler beam theory, the specific steps of determining the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement to form a corresponding relationship equation include:

[0035] Based on the Euler beam theory, the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement is:

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042] In the formula,

[0043]

[0044]

[0045]

[0046]

[0047] make f 5j =2λ j χ j , f 5n =2λ n χ n ,

[0048] The unknown coefficient A in formula (9) and (10) 2_2j , B 2_2j , C 2_2j , D 2_2j , A 2_3j , B 2_3j , C 2_3j , D 2_3j , A 2_4j , B 2_4j , C 2_4j , D 2_4j , A 2_2n , B 2_2n , C 2_2n , D 2_2n , A 2_3n , B 2_3n , C 2_3n , D 2_3n , A 2_4n , B 2_4n , C 2_4n , D 2_4n According to A in formula (7) and (8), 2_1j , B 2_1j , C 2_1j , D 2_1j , A 2_1n , B 2_1n , C 2_1n , D 2_1n And further expressed as:

[0049]

[0050]

[0051] Then at the connecting section between the jth and j+1th sections of the pile body, the horizontal displacement, rotation angle, bending moment and shear force of the pile body are continuous, that is, they satisfy:

[0052]

[0053] Then, the coefficient A is obtained by combining equations (11) and (13): 2_1j , B 2_1j , C 2_1j , D 2_1j , A 2_2j , B 2_2j , C 2_2j , D 2_2j , A 2_3j , B 2_3j , C 2_3j , D 2_3j , A 2_4j , B 2_4j , C 2_4j , D 2_4j , satisfying the following matrix equations:

[0054]

[0055] Where: {T I_j}=[A 1_1j B 1_1j C 1_1j D 1_1j ] T ;

[0056]

[0057]

[0058] From formula (14), we can further obtain:

[0059]

[0060] Where:

[0061]

[0062] Based on the recursive relationship, the coefficient matrix corresponding to the mth pile section {T I_m} is represented as:

[0063]

[0064] Further consideration of the boundary conditions at the pile top and pile bottom shows that:

[0065]

[0066] Optionally, in one embodiment, in S5, the specific steps of solving the relationship equation to obtain the corresponding pile body horizontal displacement analysis result include:

[0067] Let F 5j =F 1j +F 3j ; F 6j =F 2j +F 4j , Substituting the passive pile displacement, rotation angle, bending moment and shear force expressions into formula (17) respectively, we can obtain:

[0068]

[0069]

[0070] Where: [T I_1 ]=[A 1_11 B 1_11 C 1_11 D 1_11 ] T ;

[0071]

[0072]

[0073] Substituting equation (16) into equation (18b) and combining equation (18a) to obtain the coefficient matrix According to the recursive expression (18), the pile body coefficient of each section is further obtained: Finally, the coefficient expression is substituted into equations (7) and (8) to obtain the analytical solution data of the pile body horizontal displacement. Finally, according to the relationship between the pile body bending moment, shear force and horizontal displacement, the analytical solution result of the pile body internal force is obtained.

[0074] Implementing the embodiments of the present invention will have the following beneficial effects:

[0075] The Pasternak foundation model adopted in the present invention takes into account the shear effect of the soil around the pile, and considers the horizontal vibration force analysis of the pile foundation under the axial force of the pile top, which can better simulate the constraint effect of the soil around the pile on the pile body, and can be applied to the horizontal vibration dynamic response problem of the pile foundation under the action of earthquake Rayleigh waves, and can provide theoretical guidance and reference for the seismic design of pile foundation engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0077] in:

[0078] Figure 1 A flowchart of the technical steps implemented in one embodiment;

[0079] Figure 2 A technical demonstration diagram of a model principle in an embodiment;

[0080] In the figure: A, Rayleigh wave propagation direction, B, free field displacement us(z) under the action of Rayleigh wave, C, rigid foundation. DETAILED DESCRIPTION

[0081] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0082] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as those generally understood by those skilled in the art of the technical field of the present invention. The terms used herein in the specification of the present invention are only for the purpose of describing specific embodiments and are not intended to limit the present invention. It is understood that the terms "first", "second", etc. used in the present invention can be used to describe various elements in this article, but these elements are not limited by these terms. These terms are only used to distinguish the first element from another element. For example, without departing from the scope of the present application, the first element can be referred to as the second element, and similarly, the second element can be the first element. Both the first element and the second element are elements, but they are not the same element.

[0083] In view of the pile foundation vibration analysis technology, when studying the influence of earthquake on pile foundation vibration, it is necessary to consider that the earthquake is transmitted from the source to the far field area, and the main cause of ground vibration is caused by Rayleigh waves. Therefore, it is necessary to study the horizontal dynamic response of the pile foundation under the action of Rayleigh waves, and it is necessary to give priority to the analysis of the free field displacement distribution of the foundation under the action of Rayleigh waves. Based on this, in this embodiment, a method for analyzing the horizontal dynamic response of a single pile under the action of Rayleigh waves is proposed, such as Figure 1-2 As shown, the method includes:

[0084] S1. Create the free-field lateral displacement equation generated by any soil layer under the action of Rayleigh waves;

[0085] S2. Based on the free-field lateral displacement equation, a dynamic equilibrium equation of the pile unit under the action of Rayleigh waves is created;

[0086] S3, based on the horizontal displacement of the pile body, simplify the dynamic equilibrium equation of the pile body unit under the action of the Rayleigh wave to obtain and solve the corresponding simplified equation;

[0087] S4. Based on the Euler beam theory, determine the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement to form a corresponding relationship equation;

[0088] S5. Solve the relationship equation to obtain the corresponding analytical result of pile body horizontal displacement.

[0089] Based on the above scheme, it can be known that the present invention uses the matrix transfer theory method to comprehensively describe the dynamic characteristics of a single pile in a layered foundation under the action of earthquake Rayleigh waves. First, the dynamic equilibrium equation of the pile body unit under the action of Rayleigh waves based on the free field lateral displacement equation is created, and the layered foundation in the patent model of the present invention is simplified to a Pasternak foundation, and the pile foundation is simplified to an Euler beam model. At the same time, the horizontal vibration dynamic equation of a single pile in layered soil under the action of the pile-soil coupling effect is established considering the axial force load at the top of the pile, so that it can be applied to the dynamic response problem of the pile body under the action of axial force-horizontal earthquake, which can provide theoretical guidance and reference for engineering practice.

[0090] Among them, in some specific embodiments, in S1, such as Figure 2 The model principle shown in the figure, the specific steps to create the free-field lateral displacement equation generated by any soil layer under the action of Rayleigh waves include:

[0091] According to the wave equation theory of elastic body, the free field lateral displacement equation generated at depth z of the jth layer of soil under the action of Rayleigh wave can be obtained as:

[0092]

[0093] In the formula, u 1j 、u 2j 、u 3j 、u 4j are unknown complex-valued coefficients, j is less than the total number of soil layers n, V pl is the compression wave velocity of soil, V sl is the shear wave velocity of soil, c is the phase velocity of Rayleigh wave;

[0094] For the bottom nth layer of soil, since there is only a downward wave but no upward wave, the corresponding free field displacement equation is:

[0095]

[0096] In the formula, u 1n 、u 2n is the complex-valued coefficient to be determined,

[0097] Among them, in some specific embodiments, in S2, based on the free field lateral displacement equation, the specific steps of creating the dynamic equilibrium equation of the pile body unit under the action of Rayleigh waves include:

[0098] Combining the theories of Euler beam and Pasternak foundation model, the dynamic equilibrium equation of the pile unit under the action of Rayleigh wave is obtained as follows (3):

[0099]

[0100] In the formula, is the horizontal displacement of the j-th pile unit mass point, is the free field lateral displacement generated by the Rayleigh wave at the jth soil layer depth z; E p ,I p 、m p are the elastic modulus, section moment of inertia and mass per unit length of the pile respectively; and are the stiffness coefficient, damping coefficient and foundation shear stiffness of the foundation soil around the j-th pile layer; B 0 =0.9(1.5d+0.5) is the calculated width of the pile, d is the pile diameter, N 0 is the axial force at the pile top.

[0101] Among them, in some specific embodiments, in S3, based on the horizontal displacement of the pile body, the specific steps of simplifying the dynamic equilibrium equation of the pile body unit under the action of the Rayleigh wave to obtain and solve the corresponding simplified equation include:

[0102] The horizontal displacement of the pile is expressed as:

[0103]

[0104] In the formula, is the horizontal displacement amplitude of the pile body particle;

[0105] Let W p =E p I p , Substituting equations (1), (2), and (4) into equation (3) respectively, we can obtain the simplified equation:

[0106]

[0107]

[0108] In the formula, W p is the defined coefficient,

[0109] Solving differential equations (5) and (6) we can obtain:

[0110]

[0111]

[0112] In the formula,

[0113] Among them, in some specific embodiments, in S4, based on the Euler beam theory, the specific steps of determining the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement to form a corresponding relationship equation include:

[0114] Based on the Euler beam theory, the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement is:

[0115]

[0116]

[0117]

[0118]

[0119]

[0120]

[0121] In the formula,

[0122]

[0123]

[0124]

[0125]

[0126]

[0127] make f 5j =2λ j χ j , f 5n =2λ n χ n ,

[0128] The unknown coefficient A in formula (9) and (10) 2_2j , B 2_2j , C 2_2j , D 2_2j , A 2_3j , B 2_3j , C 2_3j , D 2_3j , A 2_4j , B 2_4j , C 2_4j , D 2_4j , A 2_2n , B 2_2n , C 2_2n , D 2_2n , A 2_3n , B 2_3n , C 2_3n , D 2_3n , A 2_4n , B 2_4n , C 2_4n , D 2_4n According to A in formula (7) and (8), 2_1j , B 2_1j , C 2_1j , D 2_1j , A 2_1n , B 2_1n , C 2_1n , D 2_1n Further expressed as:

[0129]

[0130]

[0131] Then at the connecting section between the jth and j+1th sections of the pile body, the horizontal displacement, rotation angle, bending moment and shear force of the pile body are continuous, that is, they satisfy:

[0132]

[0133] Then, the coefficient A is obtained by combining equations (11) and (13): 2_1j , B 2_1j , C 2_1j , D 2_1j , A 2_2j , B 2_2j , C 2_2j , D 2_2j , A 2_3j , B 2_3j , C 2_3j , D 2_3j , A 2_4j , B 2_4j , C 2_4j , D2_4j , satisfying the following matrix equations:

[0134]

[0135] Where: {T I_j}=[A 1_1j B 1_1j C 1_1j D 1_1j ] T ;

[0136]

[0137]

[0138] From formula (14), we can further obtain:

[0139]

[0140] Where:

[0141]

[0142] Based on the recursive relationship, the coefficient matrix corresponding to the mth pile section {T I _ m} is represented as:

[0143]

[0144] Further consideration of the boundary conditions at the pile top and bottom shows that:

[0145]

[0146] Among them, in some specific embodiments, in S5, the specific steps of solving the relationship equation to obtain the corresponding pile body horizontal displacement analysis result include:

[0147] Let F 5j =F 1j +F 3j ; F 6j =F 2j +F 4j , Substituting the passive pile displacement, rotation angle, bending moment and shear force expressions into formula (17) respectively, we can obtain:

[0148]

[0149]

[0150] Where: [T I_1 ]=[A 1_11 B 1_11 C1_11 D 1_11 ]T;

[0151]

[0152]

[0153] Substituting equation (16) into equation (18b) and combining equation (18a) to obtain the coefficient matrix According to the recursive expression (18), the pile body coefficient of each section is further obtained: Finally, the coefficient expression is substituted into equations (7) and (8) to obtain the analytical solution data of the pile body horizontal displacement. Finally, according to the relationship between the pile body bending moment, shear force and horizontal displacement, the analytical solution result of the pile body internal force is obtained.

[0154] Implementing the embodiments of the present invention will have the following beneficial effects:

[0155] The present invention proposes a method for analyzing the horizontal dynamic response of pile foundation under the action of Rayleigh waves based on Pasternak foundation. The Pasternak foundation model adopted by the method takes into account the shear effect of the soil around the pile, which can better simulate the constraint effect of the soil around the pile on the pile body and can be applied to the problem of horizontal vibration dynamic response of pile foundation under the action of earthquake Rayleigh waves.

[0156] The above-mentioned embodiments only express several implementation methods of the present application, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the scope of the present application. It should be pointed out that, for a person of ordinary skill in the art, several variations and improvements can be made without departing from the concept of the present application, and these all belong to the protection scope of the present application. Therefore, the protection scope of the present application shall be subject to the attached claims.

Claims

1. A method for analyzing the horizontal dynamic response of a single pile under Rayleigh waves. It is characterized in that The steps include: S1. Create the free-field lateral displacement equation generated by any soil layer under the action of Rayleigh waves; S2. Based on the free-field lateral displacement equation, a dynamic equilibrium equation of the pile unit under the action of Rayleigh waves is created; S3, based on the horizontal displacement of the pile body, simplify the dynamic equilibrium equation of the pile body unit under the action of the Rayleigh wave to obtain and solve the corresponding simplified equation; S4. Based on the Euler beam theory, determine the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement to form a corresponding relationship equation; S5, solving the relationship equation to obtain the corresponding pile body horizontal displacement analytical result; In S1, the specific steps to create the free-field lateral displacement equation generated by any soil layer under the action of Rayleigh waves include: According to the wave equation theory of elastic body, the free field displacement equation generated at the depth z of the jth layer of soil under the action of Rayleigh wave is obtained as follows: In the formula, u 1j 、u 2j 、u 3j 、u 4j is the complex-valued coefficient to be determined, j is less than or equal to the total number of soil layers n, ω is the excitation circular frequency; Vpl is the compression wave velocity of the soil, Vsl is the shear wave velocity of the soil, and c is the Rayleigh wave phase velocity; For the bottom nth layer of soil, since there is only a downward wave but no upward wave, the corresponding free field displacement equation is: In the formula, u 1n 、u 2n is the complex-valued coefficient to be determined, In S2, based on the free-field lateral displacement equation, the specific steps of creating the dynamic equilibrium equation of the pile shaft unit under the action of Rayleigh waves include: Combining the theories of Euler beam and Pasternak foundation model, the dynamic equilibrium equation of the pile unit under the action of Rayleigh wave is obtained as follows (3): In the formula, is the horizontal displacement of the j-th pile unit mass point, is the free field lateral displacement generated by the Rayleigh wave at the jth soil layer depth z; E p ,I p 、m p are the elastic modulus, section moment of inertia and mass per unit length of the pile respectively; and are the stiffness coefficient, damping coefficient and foundation shear stiffness of the foundation soil around the j-th pile layer; B 0 =0.9(1.5d+0.5) is the calculated width of the pile, d is the pile diameter, N 0 is the axial force at the pile top.

2. According to the method for analyzing the horizontal dynamic response of a single pile under the action of Rayleigh waves according to claim 1, It is characterized in that In S3, based on the horizontal displacement of the pile body, the specific steps of simplifying the dynamic equilibrium equation of the pile body unit under the action of the Rayleigh wave to obtain and solve the corresponding simplified equation include: The horizontal displacement of the pile is expressed as: In the formula, is the horizontal displacement amplitude of the pile body particle; Let W p =E p I p , Substituting equations (1), (2), and (4) into equation (3) respectively, we can obtain the simplified equation: In the formula, W p is the defined coefficient, Solving differential equations (5) and (6) we can obtain: In the formula, 3. According to the method for analyzing the horizontal dynamic response of a single pile under the action of Rayleigh waves as described in claim 2, It is characterized in that In S4, based on the Euler beam theory, the specific steps of determining the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement to form the corresponding relationship equation include: Based on the Euler beam theory, the relationship between the pile body rotation angle, bending moment, shear force and pile body horizontal displacement is: In the formula, Let f 5j = 2λ j χ j , f 5n = 2λ n χ n , The unknown coefficient A in formula (9) and (10) 2_2j , B 2_2j , C 2_2j , D 2_2j , A 2_3j , B 2_3j , C 2_3j , D 2_3j , A 2_4j , B 2_4j , C 2_4j , D 2_4j , A 2_2n , B 2_2n , C 2_2n , D 2_2n , A 2_3n , B 2_3n , C 2_3n , D 2_3n , A 2_4n , B 2_4n , C 2_4n , D 2_4n According to A in formula (7) and (8), 2_1j , B 2_1j , C 2_1j , D 2_1j , A 2_1n , B 2_1n , C 2_1n , D 2_1n Further expressed as: Then at the connecting section between the jth and j+1th sections of the pile body, the horizontal displacement, rotation angle, bending moment and shear force of the pile body are continuous, that is, they satisfy: Then, the coefficient A is obtained by combining equations (11) and (13): 2_1j , B 2_1j , C 2_1j , D 2_1j , A 2_2j , B 2_2j , C 2_2j , D 2_2j , A 2_3j , B 2_3j , C 2_3j , D 2_3j , A 2_4j , B 2_4j , C 2_4j , D 2_4j , satisfying the following matrix equations: Where: {T I_j }=[A 1_1j B 1_1j C 1_1j D 1_1j ] T ; From formula (14), we can further obtain: Where: Based on the recursive relationship, the coefficient matrix corresponding to the mth pile section {T I_m } is represented as: Further consideration of the boundary conditions at the pile top and bottom shows that:

4. According to the method for analyzing the horizontal dynamic response of a single pile under the action of Rayleigh waves as described in claim 1, It is characterized in that In S5, the specific steps of solving the relational equation to obtain the corresponding pile body horizontal displacement analytical result include: Let F 5j =F 1j +F 3j ; F 6j =F 2j +F 4j , Substituting the passive pile displacement, rotation angle, bending moment and shear force expressions into formula (17) respectively, we can obtain: Where: [T I_1 ]=[A 1_11 B 1_11 C 1_11 D 1_11 ] T ; Substituting equation (16) into equation (18b) and combining equation (18a) to obtain the coefficient matrix According to the recursive expression (18), the pile body coefficient of each section is further obtained: Finally, the coefficient expression is substituted into equations (7) and (8) to obtain the analytical solution data of the pile body horizontal displacement. Finally, according to the relationship between the pile body bending moment, shear force and horizontal displacement, the analytical solution result of the pile body internal force is obtained.

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