Pile foundation seismic response analysis method and system suitable for liquefaction site

Abstract

The invention provides a pile foundation seismic response analysis method and system suitable for a liquefaction site, and the method comprises the steps: building a control equation of a liquefied soil layer and a control equation of a non-liquefied soil layer based on a potential flow theory and a Biot dynamic consolidation theory, building a balance equation of pile foundations in the liquefied soil layer and the non-liquefied soil layer based on an Euler-Bernoulli beam theory, and carrying out the seismic response analysis of the pile foundations in the liquefied soil layer and the non-liquefied soil layer. An SH wave is introduced to analyze the seismic response of the pile foundation in the whole frequency domain; the lateral response of the pile foundation at the interface of the liquefied soil layer and the non-liquefied soil layer is analyzed through a transfer matrix method, the continuity of a dynamic response vector is ensured, and therefore the safety and reliability of pile foundation design are effectively improved.

Description

Technical Field

[0001] The present invention belongs to the technical field of geotechnical engineering, and relates to a method and system for analyzing the seismic response of pile foundations applicable to liquefied sites. Background Art

[0002] As an important form of deep foundation, pile foundations are widely used in projects such as buildings, factories, bridges, seaport terminals, oil production platforms, and nuclear power plants. Because pile foundations have characteristics such as large bearing capacity, good stability, and small settlement value, they can better adapt to various geological conditions and load conditions. Under seismic action, the pile foundations in liquefied soil layers will produce complex dynamic responses. Under seismic action, the pile foundations in liquefied soil layers will not only cause the pile foundations to become unstable and damaged due to the reduction of the stiffness of the liquefied soil, but also the lateral expansion of the soil and the inertial force of the superstructure caused by the earthquake will jointly cause excessive bending stress in the pile body and result in strength failure. Therefore, it is crucial to accurately analyze and predict the dynamic response of pile foundations.

[0003] In the prior art, due to the lack of effective theoretical analysis means and calculation methods, it often relies on static models or simplified dynamic models, and these methods usually ignore the nonlinear properties of the soil and the interaction between soil layers. In addition, many methods fail to effectively consider the complex behavior of liquefied soil layers under dynamic loads, so it may lead to underestimation or overestimation of the performance of pile foundations in practical engineering applications. Summary of the Invention

[0004] Aiming at the problems existing in the prior art, the present invention provides a method and system for analyzing the seismic response of pile foundations applicable to liquefied sites, thereby effectively improving the safety and reliability of pile foundation design.

[0005] The present invention is realized through the following technical solutions:

[0006] A method for analyzing the seismic response of pile foundations applicable to liquefied sites includes:

[0007] Determine the boundary conditions based on the characteristics of pile-soil interaction and the physical characteristics of the boundary;

[0008] Establish the control equation of the liquefied soil layer based on the potential energy flow theory, and obtain the dynamic pressure acting on the unit pile length of the pile foundation by the liquefied soil layer according to the boundary conditions;

[0009] Based on Biot's dynamic consolidation theory, establish the control equation of the non-liquefied soil layer, and obtain the lateral impedance coefficient of the non-liquefied soil layer according to the boundary conditions;

[0010] Based on Euler-Bernoulli beam theory, establish the equilibrium equations of the pile foundation in the liquefied soil layer and the non-liquefied soil layer respectively through the dynamic pressure acting on the pile foundation by the liquefied soil layer and the lateral impedance coefficient of the non-liquefied soil layer;

[0011] Based on the equilibrium equation, the dynamic responses of the pile foundation in liquefied soil layers and non-liquefied soil layers are coupled to obtain the lateral response of the pile foundation.

[0012] Through the lateral response of the pile foundation, under the action of SH waves, using the transfer matrix method, through the correlation of pile-soil interaction, elastic vibration and rigid body motion coupling, the seismic response of the pile foundation in the liquefied site is obtained.

[0013] Preferably, the control equation of the liquefied soil layer is established based on the potential energy flow theory, and the specific expression is:

[0014]

[0015] In the formula, k = ω / C represents the compression wave number, ω represents the excitation frequency, represents the compression wave velocity propagating through the liquefied soil layer, ρ l represents the density of the liquefied soil, λ l represents the Lame modulus of the liquefied soil layer, and φ is the potential function of the liquefied soil layer.

[0016] Preferably, the dynamic pressure acting on the unit pile length of the pile foundation in the liquefied soil layer is expressed as:

[0017]

[0018] In the formula: m = 1, 2, 3…, where m is the number of eigenfunctions considered in the solution, ρ is the density of the liquefied soil layer, H1 represents the depth of the liquefied soil layer, ω represents the excitation frequency, I n and K n are the first and second coefficients of the nth-order modified Bessel function of the first kind respectively, p is the dynamic pressure at the interface between the pile foundation and the liquefied soil layer, A lm is the undetermined coefficient established according to the boundary conditions, a lm represents the coefficient related to the Fourier series of the liquefied soil layer, k represents the compression wave number, K1 represents the second coefficient of the nth-order modified Bessel function, d represents the diameter of the pile foundation, z represents the depth of the pile foundation in the liquefied soil layer, z is the depth of the pile foundation in the liquefied soil layer, and i = √-1.

[0019] Preferably, based on the Biot dynamic consolidation theory, the control equation of the non-liquefied soil layer is established, and according to the boundary conditions, the lateral impedance coefficient of the non-liquefied soil layer is obtained. The specific process is as follows:

[0020] Based on Biot's dynamic consolidation theory, the control equations of non-liquefied soil layers are established. The Helmholtz decomposition theorem and Fourier series expansion are used to decouple the control equations of non-liquefied soil layers. Then, according to the boundary conditions, the potential function and Fourier coefficients are determined to obtain the displacement and stress of the soil mass. The lateral resistance is obtained by integrating the stress acting on the pile body by the soil mass, and the lateral impedance coefficient of the non-liquefied soil layer can be obtained by dividing the lateral resistance by the pile body displacement.

[0021] Preferably, the expression of the control equation of the non-liquefied soil layer is:

[0022]

[0023] In the formula: and are the radial and tangential vibration displacements of the soil skeleton respectively, and are the radial and tangential vibration displacements of the pore fluid related to the soil skeleton respectively; the Laplace operator is G * =G(1 + 2iβ s ) represents the complex Lame modulus of the non-liquefied soil layer, G represents the shear modulus of the soil mass, β s is the hysteretic damping, ρ s and ρ f are the densities of soil particles and pore fluid respectively, is the fluid density for partial derivative, ρ is the density of the non-liquefied soil layer, ω represents the excitation frequency, e represents the expansion of the solid matrix, α represents the proportion of pore water with free acceleration in the soil mass, where λ s represents the Lame modulus of the non-liquefied soil layer, i = √-1.

[0024] Preferably, the expression of the lateral resistance coefficient of the non-liquefied soil layer is:

[0025]

[0026] In the formula: G * =G(1 + 2iβ s ) represents the complex Lame modulus of the non-liquefied soil layer, is the Fourier coefficient of the pile vibration displacement, are the Fourier coefficients of the effective normal stress vibration, shear stress vibration and pore pressure vibration in the non-liquefied soil layer respectively, ω represents the excitation frequency, d is the diameter of the pile foundation, and r is the radius of the pile foundation.

[0027] Preferably, the equilibrium equations of the pile foundation in the liquefied soil layer and the non-liquefied soil layer are specifically expressed as:

[0028]

[0029] In the formula: represents the wave number of the pile foundation, E p , I p , ρ p and A p respectively represent the elastic modulus, moment of inertia, density of the pile foundation, and cross-sectional area of the pile foundation. is the lateral resistance coefficient of the non-liquefied soil layer, G * = G(1 + 2iβ s ) represents the complex Lamé modulus of the non-liquefied soil layer. is the displacement of the pile foundation in the liquefied soil layer. is the displacement of the pile foundation in the non-liquefied soil layer, a lm , a sm are respectively the coefficients related to the Fourier series in the liquefied soil layer and the non-liquefied soil layer, and d is the diameter of the pile foundation.

[0030] Preferably, the dynamic responses of the pile foundation in the liquefied soil layer and the non-liquefied soil layer are coupled to obtain the lateral response of the pile foundation.

[0031] Based on the lateral impedance coefficient obtained in the non-liquefied soil layer and the Euler-Bernoulli beam theory, the equilibrium equation of the pile foundation in the non-liquefied soil layer is established, and the relationship between the dynamic response of the pile foundation in the non-liquefied soil layer, the vector of undetermined coefficients, and the transformation matrix is represented in matrix form.

[0032] Based on the dynamic pressure acting on the pile foundation in the liquefied soil layer and the Euler-Bernoulli beam theory, the equilibrium equation of the pile foundation in the liquefied soil layer is established, and the relationship between the dynamic response of the pile foundation in the liquefied soil layer, the vector of undetermined coefficients, and the transformation matrix is represented in matrix form.

[0033] Based on the transfer matrix method, the correlation between the vectors of undetermined coefficients in the liquefied soil layer and the non-liquefied soil layer is represented, and then the dynamic response of the pile foundation in the liquefied soil layer and the dynamic response in the non-liquefied soil layer are coupled to obtain the lateral displacement field of the pile foundation.

[0034] Preferably, the seismic response of the pile foundation in the liquefied site is taken, specifically:

[0035] The dynamic response generated by elastic vibration is obtained through the lateral response of the pile foundation under the action of the SH wave transmitted from the bedrock under the action of pile-soil interaction, and the dynamic response generated by rigid body motion is obtained for the soil layer without piles under the action of the SH wave. By using the continuity condition between adjacent pile segments, the transfer matrix method is used to represent the correlation between the vectors of undetermined coefficients of elastic vibration and rigid body motion, and then these two dynamic responses are coupled to obtain the seismic response of the pile foundation.

[0036] A seismic response analysis system for pile foundations applicable to liquefied sites includes

[0037] A boundary condition acquisition module, which is used to determine boundary conditions based on the characteristics of pile-soil interaction and the physical characteristics of the boundary;

[0038] A dynamic pressure acquisition module, which is used to establish a control equation for the liquefied soil layer based on the potential energy flow theory, and obtain the dynamic pressure acting on the unit pile length of the pile foundation according to the boundary conditions;

[0039] A lateral impedance coefficient acquisition module, which is used to establish a control equation for the non-liquefied soil layer based on the Biot dynamic consolidation theory, and obtain the lateral impedance coefficient of the non-liquefied soil layer according to the boundary conditions;

[0040] An equilibrium equation establishment module, which is used to establish equilibrium equations for the pile foundation in the liquefied soil layer and the non-liquefied soil layer respectively based on the Euler-Bernoulli beam theory, through the dynamic pressure acting on the pile foundation by the liquefied soil layer and the lateral impedance coefficient of the non-liquefied soil layer;

[0041] A lateral response acquisition module, which couples the dynamic responses of the pile foundation in the liquefied soil layer and the non-liquefied soil layer respectively based on the equilibrium equation to obtain the lateral response of the pile foundation;

[0042] An earthquake response module, through the lateral response of the pile foundation, under the action of SH waves, obtains the earthquake response of the pile foundation in the liquefied site through the coupling result of pile-soil interaction, elastic vibration and rigid body motion.

[0043] Compared with the prior art, the present invention has the following beneficial technical effects:

[0044] In a method for analyzing the seismic response of a pile foundation applicable to liquefiable sites provided by the present invention, the control equation of the liquefied soil layer is derived from the potential flow theory, and the variable separation method is adopted to obtain the expression of the potential function of the liquefied soil layer, and then the dynamic pressure of the pile foundation in the liquefied soil layer is obtained; according to Biot's poroelastic medium consolidation theory, the control equation of the non-liquefied soil layer is established, and then according to Helmholtz's velocity decomposition theorem, the decoupled control equation of the non-liquefied soil layer is expressed in matrix form, and the variable separation method is adopted to obtain the expression of the potential function of the non-liquefied soil layer, and then the lateral resistance coefficient of the non-liquefied soil layer is obtained; using Euler-Bernoulli beam theory, the equilibrium equation of the pile embedded in the liquefied soil layer and the non-liquefied soil layer is established; in order to make the dynamic response vector of the pile foundation must show continuity, the transfer matrix method is used to obtain the correlation between the undetermined coefficient vectors of the liquefied soil layer and the non-liquefied soil layer; the seismic response of the pile is determined through the pile-soil interaction and the lateral displacement field of the non-liquefied soil layer. In the present invention, the liquefied soil layer is regarded as a viscous compressible fluid, and the dynamic behavior of the liquefied soil layer can be described by the potential flow theory; the dynamic behavior of the non-liquefied soil layer is based on Biot's dynamic consolidation theory, considering the motion equilibrium and seepage equilibrium of the soil skeleton and pore fluid. The dynamic response of the pile foundation in the liquefied soil layer and the non-liquefied soil layer can be analyzed by Euler-Bernoulli beam theory; the lateral response of the pile foundation at the interface between the liquefied soil layer and the non-liquefied soil layer is analyzed by the transfer matrix method, ensuring the continuity of the dynamic response vector; the seismic response analysis considers the SH wave propagating from the rigid bedrock and the pile-soil interaction. In summary, the present invention provides a new calculation method for analyzing the dynamic response of pile foundations in liquefiable sites, thereby effectively improving the safety and reliability of pile foundation design.

[0045] Further, according to the characteristics of the liquefied soil layer, by introducing the hysteretic damping ratio, the Lame modulus is in the form of a complex number, successfully replacing the real number form in the traditional linear elastic model;

[0046] Further, based on Helmholtz's theorem, four potential functions are introduced to decouple the control equation of the non-liquefied soil layer, and the Fourier series expansion technique is introduced to conduct a detailed dynamic response analysis of the displacement, stress and pore water pressure of the pile foundation and the surrounding soil layer; by expressing the relationships of various variables in matrix form, the complex dynamic response calculation process is made more standardized, thereby improving the solution efficiency. Description of the Drawings

[0047] Figure 1 is the core step flow chart corresponding to the method of the present invention in the embodiment;

[0048] Figure 2 Pile-soil interaction model under seismic action; Detailed Embodiments

[0049] The following will further elaborate on the present invention in conjunction with specific embodiments, which are explanations of the present invention rather than limitations.

[0050] In order to enable those skilled in the art to better understand the solution of the present invention, the following will clearly and completely describe the technical solutions in the embodiments of the present invention with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the scope of protection of the present invention.

[0051] 1. Please refer to Figure 1 , in the specific embodiment of the present invention, a pile foundation seismic response analysis method applicable to liquefied sites is provided, including the following steps:

[0052] Determine the boundary conditions based on the characteristics of pile-soil interaction and the physical characteristics of the boundary;

[0053] Regard the liquefied soil layer as a viscous compressible fluid, establish the control equation of the liquefied soil layer based on the potential flow theory, and finally obtain the dynamic pressure acting on the unit pile length of the pile foundation according to the boundary conditions;

[0054] In the liquefied soil layer, the pile foundation is mainly affected by dynamic pressure. Therefore, based on the dynamic pressure acting on the pile foundation in the liquefied soil layer and the Euler-Bernoulli beam theory, establish the equilibrium equation of the pile foundation in the liquefied soil layer, and finally represent the relationship between the dynamic response of the pile foundation in the liquefied soil layer, the vector of undetermined coefficients, and the transformation matrix in the form of a matrix;

[0055] Based on Biot's dynamic consolidation theory, establish the control equation of the non-liquefied soil layer, decouple the control equation of the non-liquefied soil layer by using the Helmholtz decomposition theorem and Fourier series expansion, and then determine the potential function and Fourier coefficients according to the boundary conditions to obtain the displacement and stress of the soil. Integrate the stress acting on the pile body by the soil to obtain the lateral resistance, and finally divide the lateral resistance by the pile body displacement to obtain the lateral impedance coefficient of the non-liquefied soil layer;

[0056] In the non-liquefied soil layer, based on the lateral impedance coefficient obtained in the non-liquefied soil layer and the Euler-Bernoulli beam theory, establish the equilibrium equation of the pile foundation in the non-liquefied soil layer, and finally represent the relationship between the dynamic response of the pile foundation in the non-liquefied soil layer, the vector of undetermined coefficients, and the transformation matrix in the form of a matrix;

[0057] In order to make the dynamic response between the two pile segments show continuity, it is necessary to represent the correlation between the vectors of undetermined coefficients in the liquefied soil layer and the non-liquefied soil layer based on the transfer matrix method. Finally, couple the dynamic response of the pile foundation in the liquefied soil layer with the dynamic response in the non-liquefied soil layer to obtain the lateral response of the pile foundation;

[0058] The lateral response of the pile foundation obtained from the pile-soil interaction will obtain the dynamic response generated by elastic vibration under the action of SH waves transmitted from the bedrock. The soil layer without piles will obtain the dynamic response generated by rigid body motion under the action of SH waves. In order to show the continuity of these two dynamic responses, the transfer matrix method is used to represent the correlation between the undetermined coefficient vectors of elastic vibration and rigid body motion. Finally, by coupling these two dynamic responses, the seismic response of the pile foundation can be obtained.

[0059] The specific process is as follows:

[0060] 2. Determine the boundary conditions based on the characteristics of the pile-soil interaction and the physical characteristics of the boundary. Specifically:

[0061] Due to the neglect of the gravity effect, the surface of the liquefied soil layer is free, and the potential function at the interface between the liquefied layer and the non-liquefied layer remains unchanged. This interface is equivalent to the bottom of the liquefied layer:

[0062]

[0063] In the formula: φ is the potential function of the liquefied soil layer, and z is the depth of the pile foundation in the liquefied soil layer;

[0064] (2) The liquefied layer and the non-liquefied layer at infinity are stationary:

[0065]

[0066] In the formula: and are the radial and tangential vibration displacements of the soil skeleton respectively, and are the radial and tangential vibration displacements of the pore fluid related to the soil skeleton respectively;

[0067] (3) The displacement of the non-liquefied soil layer at its interface with the rigid bedrock is zero:

[0068]

[0069] In the formula: are the radial and tangential vibration displacements of the soil skeleton respectively, are the radial and tangential vibration displacements of the pore fluid related to the soil skeleton respectively. H1 refers to the depth of the liquefied soil layer, and H2 refers to the depth of the non-liquefied soil layer;

[0070] (4) The potential function and the dynamic pressure p at the interface between the pile and the liquefied soil layer satisfy:

[0071]

[0072] where: \(i = \sqrt{-1}\), \(\rho_{l}\) represents the density of liquefied soil, represents the lateral displacement of the pile foundation in the liquefied soil layer.

[0073] (5) The interface between the pile and the non-liquefied soil layer satisfies the conditions of perfect contact and impermeability:

[0074]

[0075] where: and are the radial and tangential vibration displacements of the soil skeleton respectively, \(u\) represents the lateral displacement of the pile foundation in the non-liquefied soil layer, and are the radial and tangential vibration displacements of the pore fluid related to the soil skeleton respectively;

[0076] 2.1 Governing equations of liquefied soil layer

[0077] The liquefied soil layer is regarded as a viscous compressible fluid, and the governing equations of the liquefied soil layer for fluid motion in cylindrical coordinates are derived based on the potential flow theory as follows:

[0078]

[0079] where, \(k=\omega / C\) represents the compression wave number, \(\omega\) represents the excitation frequency, represents the propagation speed of the compression wave in the liquefied soil layer, \(\rho\) l represents the density of liquefied soil, \(\lambda\) l represents the Lame modulus of liquefied soil. For a linear elastic medium, the Lame modulus is usually a real number, and \(\varphi\) is the potential function of the liquefied soil layer.

[0080] However, for a liquefied soil layer similar to a viscous fluid, its Lame modulus must be expressed in complex form.

[0081] Therefore, the hysteretic damping ratio \(\zeta\) is adopted, and its complex corresponding is used to replace the real Lame modulus and is defined as follows:

[0082]

[0083] where, \(\eta = \omega H_{1} / C\) is the dimensionless frequency, \(i=\sqrt{-1}\). This equivalent of the viscous fluid dissipation behavior simplifies the actual situation, but it provides a direct and effective method to simulate the energy loss caused by internal dissipation.

[0084] Substitute Equation (1) into Equation (6), and adopt the method of separation of variables, the general solution of the dynamic pressure in the liquefied soil layer can be obtained as follows:

[0085] \(\varphi(r,\theta,z)=R\) l (r)\(\Theta\) l (\(\theta\))\(Z\)l (z) (8)

[0086] Among them, the separated variable Θ l , Z l , R l can be expressed as:

[0087] Θ l ″ + n 2 Θ l =0 (9a)

[0088]

[0089]

[0090] Using the solution of formula (9), the potential function expression of the liquefied soil layer can be obtained:

[0091]

[0092] Among them, I n and K n are the first and second kind coefficients of the n-th order modified Bessel function respectively, A lm , B lm , C l , D l , E l and F l are undetermined coefficients established according to the boundary conditions. From the boundary conditions of the liquefied soil layer at infinity, the surface and the bottom in formula (1 - 2) and m = 1, 2, 3…, m represents the number of characteristic functions considered in the solution. In addition, the dynamic behavior of the pile foundation during an earthquake is axisymmetric along it, so D l =0 n = 1, and the potential function of the liquefied soil layer can be expressed as:

[0093]

[0094] Substituting the dynamic pressure expression in formula (4) into formula (11), the dynamic pressure expression acting on the pile foundation per unit pile length of the liquefied soil layer is:

[0095]

[0096] 2.2 Governing equations of non-liquefied soil layers

[0097] The governing equations of non-liquefied soil layers include the motion equilibrium of the soil skeleton and pore fluid and the seepage equilibrium, and are established based on Biot's porous elastic dynamic consolidation theory:

[0098]

[0099]

[0100] where the Laplace operator is are the radial and tangential vibration displacements of the soil skeleton, are the radial and tangential vibration displacements of the pore fluid related to the soil skeleton, respectively. G * = G(1 + 2iβ s ) represents the complex Lamé modulus of the non-liquefied soil layer, β s is the hysteretic damping, λ s is the Lamé modulus of the non-liquefied soil layer, G is the shear modulus of the soil, e represents the expansion of the solid matrix, α represents the proportion of pore water with free acceleration in the soil, and i = √-1. The compression coefficient is where K s , K f represent the bulk moduli of soil particles and pore fluid, respectively, n s represents the soil porosity. The density of the non-liquefied soil layer is expressed as ρ = (1 - n s )ρ s + n s ρ f , where ρ s and ρ f are the densities of soil particles and pore fluid, respectively; the permeability coefficient is expressed as k d , and g represents the acceleration due to gravity.

[0101] According to the Helmholtz velocity decomposition theorem, four potential functions are introduced to decouple the above control equations of the non-liquefied soil layer:

[0102]

[0103] Substituting Equation (14) into Equation (13), the control equations can be expressed in matrix form:

[0104]

[0105] where m = 1, 2, 3…, m represents the number of eigenfunctions considered in the solution, which satisfies the boundary conditions in Equation (3).

[0106] To make the potential functions have non-zero solutions, the determinant of the matrix in Equation (15) must be equal to zero. Based on this condition, the following expressions can be obtained, which provide the decoupling formulas for the above potential functions:

[0107]

[0108] where β1, β2, and β3 are attenuation coefficients, equal to the wave numbers of the linear elastic medium:

[0109]

[0110] Among them, the calculation coefficients of the attenuation coefficients d1 and d2 can be expressed as follows:

[0111]

[0112] By using the method of separation of variables, combining the symmetry of the pile foundation movement and the boundary conditions in formula (2-3), the expression of the above potential function can be obtained:

[0113]

[0114] ψ sm,fm (r,θ,z) = A sm3,fm3 K1(β m3 r)sinθ(19b)

[0115] The displacements, stresses, pore pressures of non-liquefied soil and the displacements of the pile can all be expanded into Fourier series with variable z values. The displacements of the pile can also be expanded as:

[0116]

[0117] Among them, α represents the proportion of pore water with free acceleration in the soil mass, is the Fourier coefficient of the vibration displacement of the soil mass; are respectively the Fourier coefficients of the effective normal stress vibration, shear stress vibration and pore pressure vibration; is the Fourier coefficient of the pile vibration displacement. Substituting Eqs. (14) and (19) into Eq. (20), the expression of the Fourier coefficient can be obtained:

[0118]

[0119] Among them, α m1 , α m2 , α m3 are proportional parameters related to the characteristics of the non-liquefied soil layer, β m1 , β m2 , β m3 represent the Fourier coefficients of the attenuation coefficients and can be determined according to the control equation in the matrix form shown in formula (15):

[0121]

[0122] Among them, a sm represents the coefficient related to the Fourier series in the non-liquefied soil layer, α represents the proportion of pore water with free acceleration in the soil mass, A sm1 , A sm2 A sm3The Fourier coefficients of the undetermined coefficients established according to the boundary conditions, n s represents the porosity of the soil mass. According to the governing equation in formula (13), the expression of the sum of the Fourier coefficients in the non-liquefied soil layer can also be obtained Expression of:

[0123]

[0124] where f m1 ~f m8 The expression of can be written as:

[0125]

[0126]

[0127] f m3 =-2K1(β m3 r)-β m3 rK0(β m3 r)(24c)

[0128] f m4 =2K1(β m1 r)+β m1 rK0(β m1 r)(24d)

[0129] f m5 =2K1(β m2 r)+β m2 rK0(β m2 r)(24e)

[0130]

[0131] where K0 represents the second-kind coefficient of the nth-order modified Bessel function.

[0132] The lateral resistance coefficient of the non-liquefied soil layer can be expressed by the following formula:

[0133]

[0134] 2.3 Equilibrium equation of the pile

[0135] Using the Euler-Bernoulli beam theory, the equilibrium equations of the pile segments in the liquefied soil layer and the non-liquefied soil layer can be established as follows:

[0136]

[0137] where represents the wave number of the pile foundation, E p , I p , ρ p and Ap represent the elastic modulus, moment of inertia, density, and cross-sectional area of the pile, respectively. The solution of the above equilibrium equation includes the general solution and the particular solution, and can be written as:

[0138]

[0139] where and are undetermined coefficients, corresponding to the general solutions of the pile segments in the liquefied soil layer and the non-liquefied soil layer, respectively.

[0140] 2.3.1 Liquefied Soil Layer

[0141] Substitute equations (4) and (11) into equation (27), and the equilibrium equation of the pile in the liquefied soil layer can be obtained:

[0142]

[0143] Multiply both sides of equation (28) by cos(a lm z) and integrate over the interval [-H1, 0]. Considering the orthogonality of the Fourier cosine series, the expression of the undetermined coefficient A lm can be obtained:

[0144]

[0145] where the integration constants e 1m , e 2m , e 3m and e 4m can be expressed as follows:

[0146]

[0147] Substitute equations (29 - 30) into equation (27), and the solution of the equilibrium equation of the pile in the liquefied soil layer can be rewritten as:

[0148]

[0149] Then, the lateral displacement slope angle bending moment and shear force of the pile segment in the liquefied soil layer and the relationship with the undetermined coefficient are expressed in matrix form as follows:

[0150]

[0151] The above matrix form relationship can be abbreviated as follows:

[0152] R l (z) = Τ l (z)U l(34)

[0153] Among them, the vector represents the dynamic response of the pile segment in the liquefied soil layer, and the vector represents the undetermined coefficients, and Τ l (z) is the transformation matrix in the liquefied soil layer.

[0154] 2.3.2 Non-liquefied soil layer

[0155] According to the Biot porous elastic medium consolidation theory, the control equations of the non-liquefied soil layer are established. Then, based on the Helmholtz velocity decomposition theorem, the decoupled control equations of the non-liquefied soil layer are expressed in matrix form. Then, using the method of separation of variables, the expression of the potential function of the non-liquefied soil layer is obtained, and further the lateral resistance coefficient of the non-liquefied soil layer is obtained;

[0156] Substituting formula (20) into formula (27), the equilibrium equation of the pile in the non-liquefied soil layer can be rewritten as:

[0157]

[0158] Similarly, by considering the orthogonality of the Fourier cosine series, the expression of the Fourier coefficients can be obtained

[0159]

[0160] Among them, the integral constants e1′ m , e2′ m , e3′ m , e4′ m are expressed as follows:

[0161]

[0162] Substituting formula (36) into formula (27), the solution of the equilibrium equation of the pile in the non-liquefied soil layer can be expressed as follows:

[0163]

[0164] Similar to formula (33), the lateral displacement slope angle bending moment shearing force of the pile segment in the non-liquefied soil layer, as well as the dynamic response of the undetermined coefficients and the undetermined coefficient vector are represented. The relationship between them is expressed in matrix form as follows:

[0165]

[0166] The above relationship can be simplified to:

[0167] R s Τ(z) = s U(z) s (40)

[0168] where Τ s Τ(z) is the transformation matrix of the non - liquefied soil layer.

[0169] 3.1 Lateral Response and Seismic Response of Pile Foundations

[0170] Between two adjacent pile segments, that is, at the interface between the liquefied and non - liquefied soil layers which are coplanar, the dynamic response vectors must exhibit continuity. Therefore, the transfer matrix method can be used to obtain the correlation between these undetermined coefficient vectors U l and U s :

[0171]

[0172] In addition, the pile tip is fixed on the rigid bedrock, satisfying the hinge condition, while the pile head can translate and rotate freely. Therefore, the undetermined coefficient vector U s , can be obtained by the following formula:

[0173]

[0174] where the vector represents the boundary conditions at the pile tip and the pile head.

[0175] The incident SH - wave propagating from the rigid bedrock is simulated as a harmonic motion to analyze the seismic response of the pile in the entire frequency domain. For simplicity, the spatial variations of the free - field soil displacement in the radial and circumferential directions are ignored. Therefore, the lateral displacement field of the non - liquefied soil layer can be expressed as follows:

[0176]

[0177] where is the amplitude at the rigid bedrock, is the complex wave number of the SH - wave vertically propagating in the non - liquefied soil layer.

[0178] The seismic response of the pile is determined by the scattered wave provided by the pile - soil interaction and the lateral displacement field of the non - liquefied soil layer. Therefore, the equilibrium equation in Equation (27) can be rewritten as:

[0179]

[0180] The expression of the Fourier coefficient can be obtained according to its orthogonality:

[0181]

[0182] where,

[0183] Substituting Equation (45) into Equation (44), the vibration displacement of the pile in the non-liquefied soil layer can be written as:

[0184]

[0185] where

[0186] The seismic response of the pile consists of two parts: the dynamic response generated by elastic vibration and the dynamic response generated by rigid body motion.

[0187] The seismic response vector R s (z) and the undetermined coefficient vectors U representing elastic vibration and rigid body motion respectively s and The relationship between them can be expressed in matrix form as follows:

[0188]

[0189] By using the continuity conditions between adjacent pile segments, the correlation between the undetermined coefficient vectors U l and U s can be obtained:

[0190]

[0191] During an earthquake, the correlation between the above undetermined coefficient vectors U s and U g can be obtained through the following formula:

[0192]

[0193] where the vector represents the boundary conditions under seismic action.

[0194] In the method for analyzing the seismic response of pile foundations applicable to liquefied sites provided by the present invention, the liquefied soil layer is regarded as a viscous compressible fluid, and the dynamic behavior of the liquefied soil layer can be described by potential flow theory; the dynamic behavior of the non-liquefied soil layer is based on Biot's dynamic consolidation theory, considering the motion equilibrium and seepage equilibrium of the soil skeleton and pore fluid. The dynamic response of the pile foundation in the liquefied soil layer and the non-liquefied soil layer can be analyzed by Euler-Bernoulli beam theory; the lateral response of the pile foundation at the interface between the liquefied soil layer and the non-liquefied soil layer is analyzed by the transfer matrix method to ensure the continuity of the dynamic response vector; the seismic response analysis considers the SH wave propagating from the rigid bedrock and the pile-soil interaction.

[0195] The above are only the preferred embodiments of the present invention, and do not impose any form of limitation on the present invention; any ordinary technical personnel in the industry can smoothly implement the present invention as shown in the accompanying drawings of the specification and as described above; however, any minor changes, modifications, and equivalent variations made by those skilled in the art within the scope of the technical solution of the present invention by using the technical content disclosed above are all equivalent embodiments of the present invention; at the same time, any equivalent changes, modifications, and evolutions made to the above embodiments based on the substantial technology of the present invention still fall within the protection scope of the technical solution of the present invention.

Claims

1. A seismic response analysis method for pile foundations applicable to liquefied sites, characterized in that, including determining boundary conditions based on the characteristics of pile-soil interaction and the physical properties of the boundary; establishing the control equation of the liquefied soil layer based on the potential energy flow theory, and obtaining the dynamic pressure acting on the unit pile length of the pile foundation by the liquefied soil layer according to the boundary conditions; establishing the control equation of the non-liquefied soil layer based on Biot's dynamic consolidation theory, and obtaining the lateral impedance coefficient of the non-liquefied soil layer according to the boundary conditions; based on Euler-Bernoulli beam theory, respectively establishing the equilibrium equations of the pile foundation in the liquefied soil layer and the non-liquefied soil layer through the dynamic pressure acting on the pile foundation by the liquefied soil layer and the lateral impedance coefficient of the non-liquefied soil layer; coupling the dynamic responses of the pile foundation in the liquefied soil layer and the non-liquefied soil layer based on the equilibrium equations to obtain the lateral response of the pile foundation; through the lateral response of the pile foundation, under the action of SH waves, using the transfer matrix method, and through the correlation of pile-soil interaction, elastic vibration and rigid body motion coupling, obtaining the seismic response of the pile foundation in the liquefied site.

2. The seismic response analysis method of a pile foundation applicable to liquefied sites according to claim 1, wherein establishing the control equation of the liquefied soil layer based on the potential energy flow theory, and the specific expression is: where k = ω / C represents the compression wave number, ω represents the excitation frequency, represents the compression wave velocity propagating through the liquefied soil layer, ρ l represents the density of the liquefied soil, λ l represents the Lame modulus of the liquefied soil layer, φ is the potential function of the liquefied soil layer, and r represents the radius of the pile foundation.

3. A method for analyzing the seismic response of a pile foundation applicable to liquefied sites according to claim 1, wherein the dynamic pressure acting on the unit pile length of the pile foundation by the liquefied soil layer, and the expression is: In the formula: where m is the number of eigenfunctions considered in the solution, ρ represents the density of the liquefied soil layer, a lm represents the coefficient related to the Fourier series of the liquefied soil layer, H1 represents the depth of the liquefied soil layer, ω represents the excitation frequency, I n and K n are the first and second kind coefficients of the n-th order modified Bessel function respectively, p is the dynamic pressure at the interface between the pile foundation and the liquefied soil layer, A lm is the undetermined coefficient established according to the boundary conditions, k represents the compression wave number, K1 represents the second kind coefficient of the n-th order modified Bessel function, d represents the pile foundation diameter, z represents the depth of the pile foundation in the liquefied soil layer, 4. A method for analyzing the seismic response of a pile foundation applicable to liquefiable sites according to claim 1, characterized in that, establishing the control equation of the non-liquefied soil layer based on Biot's dynamic consolidation theory, and obtaining the lateral impedance coefficient of the non-liquefied soil layer according to the boundary conditions. The specific process is: establishing the control equation of the non-liquefied soil layer based on Biot's dynamic consolidation theory, decoupling the control equation of the non-liquefied soil layer by using the Helmholtz decomposition theorem and Fourier series expansion, then determining the potential function and Fourier coefficients according to the boundary conditions, obtaining the displacement and stress of the soil, and obtaining the lateral resistance by integrating the stress acting on the pile body by the soil, and dividing the lateral resistance by the pile body displacement to obtain the lateral impedance coefficient of the non-liquefied soil layer.

5. A method for analyzing the seismic response of a pile foundation applicable to liquefiable sites according to claim 4, characterized in that, the control equation of the non-liquefied soil layer, and the expression is: Wherein: and are respectively the radial and tangential vibration displacements of the soil skeleton, and are respectively the radial and tangential vibration displacements of the pore fluid related to the soil skeleton; The Laplace operator is G * = G(1 + 2iβ s ) represents the complex Lame modulus of the non-liquefied soil layer, G represents the shear modulus of the soil, and β s is the hysteretic damping, ρ s and ρ f are the densities of soil particles and pore fluid respectively, is the density of the pore fluid for taking the partial derivative; ρ is the density of the non-liquefied soil layer, ω represents the excitation frequency, e represents the expansion of the solid matrix, and α represents the proportion of pore water with free acceleration in the soil mass. where λ s represents the Lame modulus of the non-liquefied soil layer.

6. The seismic response analysis method of a pile foundation applicable to a liquefied site according to claim 1, wherein The lateral resistance coefficient of the non-liquefied soil layer The expression is: Where: G * = G(1 + 2iβ s ) represents the complex Lamé modulus of the non-liquefied soil layer, is the Fourier coefficient of the pile vibration displacement, are the Fourier coefficients of the effective normal stress vibration, shear stress vibration, and pore pressure vibration in the non-liquefied soil layer respectively, ω represents the excitation frequency, d is the diameter of the pile foundation, and r is the radius of the pile foundation.

7. A method for analyzing the seismic response of a pile foundation applicable to liquefiable sites according to claim 1, characterized in that, the equilibrium equations of the pile foundation in the liquefied soil layer and the non-liquefied soil layer, and the expression is: In the formula: represents the wave number of the pile foundation, E p , I p , ρ p and A p respectively represent the elastic modulus, moment of inertia, density of the pile foundation, and cross-sectional area of the pile foundation. is the lateral resistance coefficient of the non-liquefied soil layer, G * = G(1 + 2iβ s ) represents the complex Lamé modulus of the non-liquefied soil layer. is the displacement of the pile foundation in the liquefied soil layer. is the displacement of the pile foundation in the non-liquefied soil layer, a lm , a sm are the coefficients related to the Fourier series in the liquefied soil layer and the non-liquefied soil layer respectively. d is the diameter of the pile foundation, and z is the depth of the pile foundation in the soil layer.

8. A seismic response analysis method for pile foundations applicable to liquefied sites according to claim 1, characterized in that, coupling the dynamic responses of the pile foundation in the liquefied soil layer and the non-liquefied soil layer to obtain the lateral response of the pile foundation; establishing the equilibrium equation of the pile foundation in the non-liquefied soil layer based on the lateral impedance coefficient obtained in the non-liquefied soil layer and Euler-Bernoulli beam theory, and expressing the relationship between the dynamic response, the vector of undetermined coefficients and the transformation matrix of the pile foundation in the non-liquefied soil layer in matrix form; establishing the equilibrium equation of the pile foundation in the liquefied soil layer based on the dynamic pressure acting on the pile foundation by the liquefied soil layer and Euler-Bernoulli beam theory, and expressing the relationship between the dynamic response, the vector of undetermined coefficients and the transformation matrix of the pile foundation in the liquefied soil layer in matrix form; representing the correlation between the vectors of undetermined coefficients in the liquefied soil layer and the non-liquefied soil layer based on the transfer matrix method, and then coupling the dynamic response of the pile foundation in the liquefied soil layer and the dynamic response in the non-liquefied soil layer to obtain the lateral displacement field of the pile foundation.

9. A method for analyzing the seismic response of a pile foundation applicable to liquefied sites according to claim 1, characterized in that, obtaining the seismic response of the pile foundation in the liquefied site, specifically: The dynamic response generated by elastic vibration is obtained under the action of SH waves transmitted from bedrock under the action of pile - soil interaction through the lateral response of the pile foundation. The dynamic response generated by rigid body motion is obtained for the soil layer without piles under the action of SH waves. By using the continuity conditions between adjacent pile segments, the transfer matrix method is adopted to represent the correlation between the vectors of undetermined coefficients of elastic vibration and rigid body motion, and then these two dynamic responses are coupled to obtain the seismic response of the pile foundation.

10. A pile foundation seismic response analysis system applicable to liquefied sites, characterized in that, Including, a boundary condition acquisition module for determining boundary conditions based on the characteristics of pile - soil interaction and the physical characteristics of the boundary; a dynamic pressure acquisition module for establishing the control equation of the liquefied soil layer based on the potential energy flow theory and obtaining the dynamic pressure acting on the unit pile length of the pile foundation according to the boundary conditions; a lateral impedance coefficient acquisition module for establishing the control equation of the non - liquefied soil layer based on Biot's dynamic consolidation theory and obtaining the lateral impedance coefficient of the non - liquefied soil layer according to the boundary conditions; an equilibrium equation establishment module for establishing the equilibrium equations of the pile foundation in the liquefied soil layer and the non - liquefied soil layer respectively through the dynamic pressure acting on the pile foundation by the liquefied soil layer and the lateral impedance coefficient of the non - liquefied soil layer based on Euler - Bernoulli beam theory; a lateral response acquisition module for coupling the dynamic responses of the pile foundation in the liquefied soil layer and the non - liquefied soil layer respectively based on the equilibrium equations to obtain the lateral response of the pile foundation; a seismic response module for obtaining the seismic response of the pile foundation in the liquefied site through the lateral response of the pile foundation under the action of SH waves, which is the result of the coupling of pile - soil interaction, elastic vibration and rigid body motion.

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