Method for analyzing horizontal vibration dynamic response of piles in liquefied soil based on pasternak foundation model

Abstract

Based on the liquefied soil in the Pasternak foundation model pile-pile horizontal vibration dynamic response analysis method, the harmonic excitation model based on the Pasternak foundation model is established, the fluid control equation of liquefied layer is established according to the influence of liquefied soil on the active pile I, and the dynamic balance equation of the active pile I in the liquefied soil and the non-liquefied soil is created; Further create the dynamic balance equation of the passive pile II in the liquefied soil and the non-liquefied soil; Based on the dynamic balance equation of the active pile I and the passive pile II, the pile-pile interaction factor is obtained; The Pasternak foundation model used in the application considers the shearing effect of the soil around the pile, simulates the restraining action of the soil around the pile on the pile body, and is suitable for the horizontal vibration dynamic response problem of the pile foundation under the action of the harmonic load; The liquefied soil is equivalent to the fluid, and the non-liquefied soil is equivalent to the Pasternak foundation considering the influence of the shearing deformation of the foundation, and according to the joint action of the multi-directional coupling load on the pile top, the horizontal vibration dynamic response of the pile-pile in the liquefied soil and the non-liquefied soil is obtained; It has the advantages of fast response, small error and high accuracy.

Description

Technical Field

[0001] The present invention belongs to the technical field of pile foundation dynamic response, and in particular relates to a method for analyzing pile-pile horizontal vibration dynamic response in liquefied soil based on a Pasternak foundation model. Background Art

[0002] In recent years, the phenomenon of building damage caused by soil liquefaction during earthquakes has received increasing attention. Investigations and studies of pile foundation damage in liquefied soil have revealed that pile foundations in liquefied soil often fail due to earthquakes, resulting in severe loss of life and property. Consequently, a growing number of researchers have investigated the seismic performance of pile foundations in liquefied soil. Previous studies have mostly considered liquefied soil as a solid foundation, using strength reduction methods to investigate its performance. However, liquefied soil, a process of solid-to-liquid transformation, exhibits mechanistic differences, resulting in unsatisfactory results and limited application in practical engineering projects. This method for analyzing the horizontal dynamic response of liquefied soil pile foundations based on the Pasternak foundation model is the first to be applied to the study of pile foundation vibration. Similar patent applications have not been found in published patents or non-patent literature domestically or internationally. Summary of the Invention

[0003] To overcome the deficiencies of the above-mentioned prior art, the present invention aims to provide a method for analyzing the dynamic response of pile-pile horizontal vibration in liquefied soil based on the Pasternak foundation model. Based on the Pasternak foundation model, a dynamic equation for the horizontal vibration of a single pile in layered soil under pile-soil coupling is established under axial force load. The pile foundation is simplified to an Euler beam model, and the completely liquefied soil is regarded as a fluid. The fluid dynamics governing equation is used to simulate the liquefied soil portion. Each layer of non-liquefied soil is equivalent to a Pasternak foundation. A mechanical model that considers the shear effect of the soil around the non-liquefied pile is proposed. This method analyzes the horizontal dynamic response of a pile foundation in layered soil under pile-soil coupling under the second-order effect of axial pressure. This method is applicable to the dynamic response of piles under axial force horizontal harmonic excitation, and features fast response, good accuracy, and low error. It can provide theoretical guidance and reference for practical engineering.

[0004] To achieve the above object, the present invention adopts the following technical solutions:

[0005] The analysis method of pile-pile horizontal vibration dynamic response in liquefied soil based on the Pasternak foundation model specifically includes the following steps:

[0006] Step 1: Establish a simplified calculation model for pile-pile horizontal harmonic excitation in liquefied soil based on the Pasternak foundation model: divide it into active pile I and passive pile II. The top of active pile I is subjected to force, and the adjacent pile affected by active pile I is passive pile II; apply a horizontal harmonic excitation force Q0e to the top of active pile I.iωt 、M0e iωt , where Q0 and M0 are the amplitudes of the exciting force, N0 is the axial force, and ω is the circular frequency of the exciting force. t is time; in addition, the thickness, stiffness coefficient, damping coefficient and foundation shear coefficient of the jth layer of soil are h j 、 and The pile length is divided into the pile length L1 for liquefied soil layer and the pile length L2 for non-liquefied soil layer, and the pile diameter is d;

[0007] Step 2: Based on the model of step 1, simulate the effective fluid such as the liquefied soil layer, and establish the fluid control equation of the liquefied layer according to the hydraulic fluid wave theory;

[0008] Step 3: Based on the liquefied layer fluid control equation established in step 2, a dynamic equilibrium equation of the active pile 1 in the liquefied soil is established;

[0009] Step 4: Based on the dynamic equilibrium equation of the active pile 1 in liquefied soil established in step 3, and according to the pile body continuity condition at the interface between liquefied soil and non-liquefied soil, the dynamic equilibrium equation of the active pile 1 in non-liquefied soil is established; based on the Euler beam theory, the pile top impedance of the active pile 1 is further determined;

[0010] Step 5: Based on the established dynamic equilibrium equations of the active pile I in liquefied soil and non-liquefied soil, the dynamic equilibrium equation of the passive pile II in liquefied soil is established by considering the influence of the vibration of the active pile I on the passive pile II.

[0011] Step 6: Based on the dynamic equilibrium equation of the passive pile II in liquefied soil, and in accordance with the continuous contact condition between the fluid and the pile, the dynamic equilibrium equation of the passive pile II in non-liquefied soil is created;

[0012] Step 7: Based on the dynamic equilibrium equations of the active pile I and passive pile II, and according to the pile-pile interaction principle, the pile-pile interaction factor is obtained.

[0013] The simplified calculation model is based on the following basic assumptions:

[0014] 1) The pile body of active pile I is simplified to a circular uniform cross-section, homogeneous Euler beam;

[0015] 2) The soil around the pile is divided into n layers along the longitudinal direction of the pile body, and each layer of soil is simplified to the Pasternak foundation model to describe the pile-soil interaction;

[0016] 3) All parts of the pile-soil model system meet the small deformation condition, and the pile-soil interface is in full contact with no relative sliding;

[0017] 4) Only horizontal displacement occurs at the pile top, and the pile bottom is constrained by the fixed end.

[0018] The fluid motion control equation of the liquefied soil layer established in step 2 is specifically:

[0019]

[0020] Wherein: is a fluid velocity potential function;

[0021] The liquefied soil layer needs to meet the following boundary conditions when moving:

[0022] a) When the bottom of the liquefied soil layer z = h1, its vertical velocity is zero, that is:

[0023]

[0024] b) Ignoring the influence of gravity waves on the fluid, on the surface of the liquefied soil layer:

[0025]

[0026] c) The fluid is stationary at the radial infinity:

[0027]

[0028] d) The contact continuity condition of the fluid and the pile body is:

[0029]

[0030] In the formula: is the horizontal displacement of the pile section in water, and r0 is the radius of the pile body.

[0031] The specific method of step 3 is:

[0032] According to the final steady-state vibration of the pile body in the liquefied soil layer, the fluid velocity potential function can be expressed as:

[0033]

[0034] Further, by using separation of variables, the fluid velocity potential function can be set as:

[0035]

[0036] Substituting formula (7) into formula (1) and considering the boundary conditions (2)-(5), the solution of the fluid velocity potential function is:

[0037]

[0038] In the formula, K1(iα n r) is a first-order Bessel function, A is an undetermined coefficient, and α n =(2n-1)πi / (2h1),

[0039] According to the Bernouli equation, the hydrodynamic pressure acting on the active pile I can be further calculated as:

[0040]

[0041] Where, ρ l is the density of liquefied soil;

[0042] Substituting formula (9) into formula (8) yields:

[0043]

[0044] According to the Euler-Bernouli beam theory, the dynamic equilibrium equation of the active pile I unit in liquefied soil is obtained as follows (0≤z≤h1):

[0045]

[0046] Where, E p , I p 、m p are the elastic modulus, section moment of inertia and mass per unit length of the pile, respectively, and N0 is the axial force acting on the pile top;

[0047] The potential function of pile horizontal displacement and fluid motion velocity can be expressed as:

[0048]

[0049] Where, is the horizontal displacement amplitude of the pile in the liquefied soil layer;

[0050] Substituting formula (12) into formula (11) can further obtain the following equation:

[0051]

[0052] Where M p =E p I p ,

[0053] Obviously, Equation (13) is a fourth-order linear non-homogeneous differential equation with constant coefficients. Its corresponding solution consists of a general solution and a special solution. The displacement homogeneous general solution of the equation is:

[0054]

[0055] Where, D1, D2, D3, and D4 are unknown coefficients;

[0056] The non-homogeneous solution of formula (13) is set as:

[0057]

[0058] Substituting formula (15) into formula (13) yields:

[0059]

[0060] Therefore, the general solution of equation (13) is:

[0061]

[0062] The specific method of step 4 is:

[0063] 4.1) Establish the dynamic equilibrium equation of active pile I in non-liquefied soil

[0064] According to the contact continuity condition between the fluid and the pile, Equations (8) and (17) are substituted into Equation (4) to obtain:

[0065]

[0066] Where:

[0067] Multiply both sides of equation (18) by ch(α n z), and then integrate it over the interval [0,h1], we can get:

[0068]

[0069] Where:

[0070]

[0071] Then the solution for the horizontal displacement of the pile in liquefied soil is:

[0072]

[0073] In the formula: eta1=κS1, eta2=κS2, eta3=κS3, eta4=κS4,

[0074] According to Euler beam theory, the relationship between pile body rotation angle, bending moment, shear force and pile body displacement is obtained, which is further simplified as follows:

[0075]

[0076] T1(z) is expressed as:

[0077] T1(z)=[t1(z) t2(z) t3(z)t4(z)] (22)

[0078] Where:

[0079]

[0080] From this, we can deduce that the relationship between the horizontal displacement at the top and bottom of the pile section in the liquefied soil layer and the rotation angle, bending moment, and shear force is:

[0081]

[0082] Then, the dynamic equilibrium equation of the pile unit in the jth (j is the number of soil layers, j = 2, ..., m, ..., n) layer of the non-liquefied soil layer is obtained as follows:

[0083]

[0084] Where: is the horizontal displacement of the mass point of the pile body in the jth section; is the shear stiffness of the foundation soil around the j-th layer of piles; B0 = 0.9 (1.5d + 0.5) is the calculated width of the pile;

[0085] for and It is determined according to the following formula:

[0086]

[0087] Where, is the shear wave velocity of the soil around the pile; and are the elastic modulus, density, damping coefficient and Poisson's ratio of the soil around the pile; is the dimensionless frequency; is the shear layer thickness of the j-th layer of foundation soil,

[0088] Furthermore, the horizontal displacement of the pile can be expressed as

[0089]

[0090] Where, is the horizontal displacement amplitude of the pile body mass point at the jth section in non-liquefied soil;

[0091] set up Substituting equation (27) into equation (24) we can further obtain the following equation:

[0092]

[0093] Where,

[0094] Then the general displacement solution of equation (28) is:

[0095]

[0096] Where, is the coefficient to be determined;

[0097] 4.2) Determine the pile top impedance of active pile I based on Euler beam theory

[0098] According to Euler beam theory, the relationship between the rotation angle, bending moment, shear force and pile displacement of the j-th segment of pile in non-liquefied soil can be further obtained, which can be simplified as follows:

[0099]

[0100] F j (z) can be expressed as:

[0101] F j (z)=[f1(z) f2(z) f3(z) f4(z)] (31)

[0102] Where:

[0103]

[0104] The relationship between the horizontal displacement at the top and bottom of the pile section in the non-liquefied soil layer and the rotation angle, bending moment, and shear force can be deduced from the combined equation (23):

[0105]

[0106] Where, [F] = [F n (h n )][F n-1 (h n-1 )]…[F3(h3)][F2(h2)][T1(h1)][T1(0)] -1 ;

[0107] Further considering the pile bottom as a fixed end constraint condition, then:

[0108]

[0109] Combining equations (32) and (33), we can obtain:

[0110]

[0111] Where,

[0112] The pile top horizontal impedance K can be further obtained h , swing impedance K r and horizontal-swing coupling impedance K hr They are:

[0113]

[0114] The specific method of step 5 is:

[0115] According to the Euler-Bernouli beam theory, the dynamic equilibrium equation of the passive pile II in liquefied soil can be obtained as follows (0≤z≤h1):

[0116]

[0117] Further simplifying equation (38) yields:

[0118]

[0119] Among them, M p =E p I p ,

[0120] The corresponding solution of equation (39) consists of two parts: the general solution and the special solution. The general solution of the equation is:

[0121]

[0122] in, D II1 、D II2 、D II3 、D II4 is the coefficient to be determined;

[0123] The non-homogeneous special solution of the equation can be set as:

[0124]

[0125] Substituting equation (41) into equation (39) yields

[0126]

[0127] In summary, the general solution of equation (39) is

[0128]

[0129] The specific method of step 6 is:

[0130] According to the continuity condition of passive pile II at the interface between liquefied soil and non-liquefied soil, we can get

[0131]

[0132] in,

[0133] Trigonometric function cos(α nz) is an orthogonal function system on the interval [-h1, 0], i.e.

[0134]

[0135] Multiplying both sides of equation (44) by cos(α n z) and integrating over the interval [0, h1], we have:

[0136]

[0137] where,

[0138] Substituting equation (46) into equation (43), the analytical solution of the passive pile II horizontal displacement in liquefiable soil layer can be obtained as:

[0139]

[0140] where, η1= κS II1 , η2= κS II2 , η3= κS II3 , η4= κS II4 ,

[0141] According to the Euler beam theory, the relationship expressions of the passive pile II pile body rotation angle, bending moment, shear force and displacement can be further obtained as:

[0142]

[0143] where, T II1 (z) = [t II1 (z) t II2 (z) t II3 (z) t II4 (z)],

[0144] The relationship between the upper and lower horizontal displacement, rotation angle, bending moment and shear force of the passive pile II pile section in liquefiable soil layer is:

[0145]

[0146] The horizontal displacement of each unit of the passive pile II pile body can be expressed as Further according to the dynamic interaction between the pile and soil in the non-liquefiable soil, the dynamic equilibrium equation of the passive pile II in the jth (j = 2,..., m,..., n) section in the non-liquefiable soil can be simplified as:

[0147]

[0148] Further simplifying equation (50) yields:

[0149]

[0150] in,

[0151] The general solution of the homogeneous equation corresponding to equation (51) is:

[0152]

[0153] in, is the coefficient to be determined;

[0154] The special solution of equation (51) is:

[0155]

[0156] in, is the coefficient to be determined;

[0157] The complete solution of equation (51) can be expressed as:

[0158]

[0159] The specific method of step 7 is:

[0160] According to Euler beam theory, the relationship between pile body rotation angle, bending moment, shear force and pile body horizontal displacement is:

[0161]

[0162] in,

[0163] Combining equations (54) and (55), the relationship between the displacement, rotation angle, internal force and the undetermined coefficient of the jth section of passive pile II can be obtained, which can be simplified as follows:

[0164]

[0165] in,

[0166] F bj (z)=[f b1j (z) f b2j (z) f b3j (z) f b4j (z)];

[0167]

[0168] The relationship between the horizontal displacement, rotation angle, bending moment and shear force of the upper and lower ends of the jth segment of the passive pile II in the local coordinate system in the non-liquefied soil is obtained as follows:

[0169]

[0170] Considering the continuity of the soil layer at the interface in the non-liquefied soil, the relationship between the jth segment and the j+1th segment of the cross section of the pile in the non-liquefied soil is obtained as follows:

[0171] The horizontal displacement, rotation angle, bending moment and shear force of the pile body are continuous, that is:

[0172]

[0173] The relationship between the horizontal displacement, rotation angle, bending moment and shear force of the pile bottom and the interface between the liquefied soil and the non-liquefied soil in the non-liquefied soil is obtained by simultaneously solving equations (57) and (58) as follows:

[0174]

[0175] In the formula: [FA] = [F aj (h n )][F aj (0)] -1 [F a(j-1) (h j-1 )][F a(j-1) (0)] -1 …[F a2 (h2)][F a2 (0)] -1 ;

[0176] [FB] = [F bj (h n )][F bj (0)] -1 [F b(j-1) (h j-1 )][F b(j-1) (0)] -1 …[F b2 (h2)][F b2 (0)] -1 ;

[0177] Considering the continuity of the horizontal displacement, rotation angle, bending moment and shear force of the passive pile II in the local coordinate system at the interface between the liquefied soil and the non-liquefied soil, the following equation is obtained:

[0178]

[0179] Based on the superposition principle, the expression relationship between the displacement and internal force of the top and bottom of the passive pile II is obtained as follows:

[0180]

[0181] Among them, [TA]=[FA][T II1 (h1)][T II1 (0)] -1 ;[TB]=[FB][T1(h1)][T1(0)] -1 ;

[0182] The transfer matrices TA and TB in equation (61) are further decomposed into 2×2 sub-matrices, namely:

[0183]

[0184] According to the top constraint condition of passive pile II, that is,

[0185]

[0186] Substituting equation (64) into equation (63) and combining it with the expression of the top impedance of active pile I, we can obtain:

[0187]

[0188] Where [α(S,θ)] = -[TA 11 ] -1 [TB 11 +TB 12 K S ];

[0189] make The horizontal and rocking dynamic interaction factors of adjacent piles in liquefied soil are obtained as follows:

[0190]

[0191] Compared with the prior art, the present invention has the following beneficial effects:

[0192] 1. The present invention is based on the Pasternak foundation liquefied soil pile foundation horizontal dynamic response analysis method. The Pasternak foundation model it adopts can better simulate the constraint effect of the soil around the pile on the pile body according to the shear effect of the soil around the pile, and can be applied to the horizontal vibration dynamic response problem of the pile foundation under simple harmonic load;

[0193] 2. The present invention equates liquefied soil to a fluid and non-liquefied soil to a Pasternak foundation that takes into account the influence of foundation shear deformation. Based on the combined action of multi-directional coupled loads at the pile top, the pile-to-pile horizontal vibration dynamic response in liquefied soil-non-liquefied soil is obtained. The dynamic response of the pile group is solved by changing the parameters according to the method of the present invention without the need for repeated modeling.

[0194] 3. The method of the present invention changes the parameters to solve the dynamic response of pile groups, which only takes about 30 seconds, greatly saving calculation time. The analytical solution derived by this technical solution is a closed-form solution, which can provide more rigorous technical theoretical guidance and reference for the horizontal vibration design of pile groups in liquefied soil.

[0195] The pile-pile horizontal vibration dynamic response analysis method in liquefied soil based on the Pasternak foundation model is the first time it has been applied to the research field of pile foundation vibration technology. There is no record of it in the patents and non-patent literature published domestically and abroad. BRIEF DESCRIPTION OF THE DRAWINGS

[0196] Figure 1 This is a simplified model diagram of pile-pile horizontal harmonic excitation in liquefied soil based on the Pasternak foundation model. DETAILED DESCRIPTION

[0197] The present invention will be further described in detail below in conjunction with the embodiments of the present invention.

[0198] See also Figure 1 A method for analyzing the dynamic response of pile-pile horizontal vibration in liquefied soil based on the Pasternak foundation model comprises the following steps:

[0199] Step 1: Establish a simplified calculation model based on the following basic assumptions:

[0200] 1) The pile body of active pile I is simplified to a circular uniform cross-section, homogeneous Euler beam;

[0201] 2) The soil around the pile is divided into n layers along the longitudinal direction of the pile body, and each layer of soil is simplified to the Pasternak foundation model to describe the pile-soil interaction;

[0202] 3) All parts of the pile-soil model system meet the small deformation condition, and the pile-soil interface is in full contact with no relative sliding;

[0203] 4) Only horizontal displacement occurs at the pile top, and the pile bottom is constrained by the fixed end;

[0204] A simplified calculation model for pile-pile horizontal harmonic excitation in liquefied soil based on the Pasternak foundation model is established: the model is divided into active pile I and passive pile II. The active pile is subjected to force at the top. Since the active pile I is subjected to force at the top, the vibration affects the surrounding soil and then affects the adjacent piles. Therefore, the adjacent pile affected by the active pile I is the passive pile II. A horizontal harmonic excitation force Q0e is applied to the top of the active pile I. iωt 、M0e iωt , where Q0 and M0 are the amplitudes of the exciting force, N0 is the axial force, and π is the circular frequency of the exciting force. t is time; in addition, the thickness, stiffness coefficient, damping coefficient and foundation shear coefficient of the jth layer of soil are hj 、 and The pile length is divided into the pile length L1 for liquefied soil layer, the pile length L2 for non-liquefied soil layer, and the pile diameter is d.

[0205] Step 2: Based on the model in step 1, establish and solve the liquefied layer fluid control equation;

[0206] The liquefied soil layer is simulated by using an effective fluid. Based on the hydraulic fluid wave theory, the governing equation of the fluid motion of the liquefied soil layer in the cylindrical coordinate system is established:

[0207]

[0208] in: is the fluid velocity potential function;

[0209] The liquefied soil layer must meet the following boundary conditions when moving:

[0210] a) When z = h1 at the bottom of the liquefied soil layer, its vertical velocity is zero, that is:

[0211]

[0212] b) Ignoring the influence of gravity waves on the fluid, on the surface of the liquefied soil layer:

[0213]

[0214] c) The fluid is stationary at radial infinity:

[0215]

[0216] d) The contact continuity condition between the fluid and the pile is:

[0217]

[0218] Where: is the horizontal displacement of the pile section in water, and r0 is the radius of the pile body.

[0219] Step 3: Based on the liquefied layer fluid control equation established in step 2, the dynamic equilibrium equation of the active pile I in the liquefied soil is established:

[0220] According to the fact that the pile body is finally in steady-state vibration in the liquefied soil layer, the fluid velocity potential function can be expressed as:

[0221]

[0222] Further using separation of variables, the fluid velocity potential function can be set as:

[0223]

[0224] Substituting equation (7) into equation (1) and considering boundary conditions (2) to (5), the solution of the fluid velocity potential function is:

[0225]

[0226] In the formula, K1(iα n r) is a first-order Bessel function, A is the coefficient to be determined, α n =(2n-1)πi / (2h1),

[0227] According to the Bernouli equation, the hydrodynamic pressure acting on the active pile I can be further calculated as:

[0228]

[0229] Where, ρ l is the density of liquefied soil;

[0230] Substituting formula (9) into formula (8) yields:

[0231]

[0232] According to the Euler-Bernouli beam theory, the dynamic equilibrium equation of the active pile I unit in liquefied soil is obtained as follows (0≤z≤h1):

[0233]

[0234] Where, E p , I p 、m p are the elastic modulus, section moment of inertia and mass per unit length of the pile, respectively, and N0 is the axial force acting on the pile top;

[0235] The potential function of pile horizontal displacement and fluid motion velocity can be expressed as:

[0236]

[0237] Where, is the horizontal displacement amplitude of the pile in the liquefied soil layer;

[0238] Substituting formula (12) into formula (11) can further obtain the following equation:

[0239]

[0240] Where M p =E p I p ,

[0241] Obviously, Equation (13) is a fourth-order linear non-homogeneous differential equation with constant coefficients. Its corresponding solution consists of a general solution and a special solution. The displacement homogeneous general solution of the equation is:

[0242]

[0243] Where, D1, D2, D3, and D4 are unknown coefficients;

[0244] The non-homogeneous solution of formula (13) is set as:

[0245]

[0246] Substituting formula (15) into formula (13) yields:

[0247]

[0248] Therefore, the general solution of equation (13) is:

[0249]

[0250] Step 4: Based on the dynamic equilibrium equation of the active pile 1 in liquefied soil established in step 3, and according to the pile body continuity condition at the interface between liquefied soil and non-liquefied soil, the dynamic equilibrium equation of the active pile 1 in non-liquefied soil is established; based on the Euler beam theory, the pile top impedance of the active pile 1 is further determined:

[0251] 4.1) Establish the dynamic equilibrium equation of active pile I in non-liquefied soil

[0252] According to the contact continuity condition between the fluid and the pile, Equations (8) and (17) are substituted into Equation (4) to obtain:

[0253]

[0254] Where:

[0255] Multiply both sides of equation (18) by ch(α n z), and then integrate it over the interval [0,h1], we can get:

[0256]

[0257] Where:

[0258]

[0259]

[0260] Then the solution for the horizontal displacement of the pile in liquefied soil is:

[0261]

[0262] In the formula: eta1=κS1, eta2=κS2, eta3=κS3, eta4=κS4,

[0263] According to Euler beam theory, the relationship between pile body rotation angle, bending moment, shear force and pile body displacement is obtained, which is further simplified as follows:

[0264]

[0265] T1(z) is expressed as:

[0266] T1(z)=[t1(z) t2(z) t3(z) t4(z)] (22)

[0267] Where:

[0268]

[0269]

[0270] From this, we can deduce that the relationship between the horizontal displacement at the top and bottom of the pile section in the liquefied soil layer and the rotation angle, bending moment, and shear force is:

[0271]

[0272] Then, the dynamic equilibrium equation of the pile unit in the jth (j is the number of soil layers, j = 2, ..., m, ..., n) layer of the non-liquefied soil layer is obtained as follows:

[0273]

[0274] Where: is the horizontal displacement of the mass point of the pile body in the jth section; is the shear stiffness of the foundation soil around the j-th layer of piles; B0 = 0.9 (1.5d + 0.5) is the calculated width of the pile;

[0275] for and It is determined according to the following formula:

[0276]

[0277] Where, is the shear wave velocity of the soil around the pile; and are the elastic modulus, density, damping coefficient and Poisson's ratio of the soil around the pile; is the dimensionless frequency; is the shear layer thickness of the j-th layer of foundation soil,

[0278] Furthermore, the horizontal displacement of the pile can be expressed as

[0279]

[0280] Where, is the horizontal displacement amplitude of the pile body mass point at the jth section in non-liquefied soil;

[0281] set up Substituting equation (27) into equation (24) we can further obtain the following equation:

[0282]

[0283] Where,

[0284] Then the general displacement solution of equation (28) is:

[0285]

[0286] Where, is the coefficient to be determined;

[0287] 4.2) Determine the pile top impedance of active pile I based on Euler beam theory

[0288] According to Euler beam theory, the relationship between the rotation angle, bending moment, shear force and pile displacement of the j-th segment of pile in non-liquefied soil can be further obtained, which can be simplified as follows:

[0289]

[0290] F j (z) can be expressed as:

[0291] F j (z)=[f1(z) f2(z) f3(z) f4(z)] (31)

[0292] Where:

[0293]

[0294] The relationship between the horizontal displacement at the top and bottom of the pile section in the non-liquefied soil layer and the rotation angle, bending moment, and shear force can be deduced from the combined equation (23):

[0295]

[0296] Where, [F] = [F n (h n )][F n-1 (h n-1)]…[F3(h3)][F2(h2)][T1(h1)][T1(0)] -1 ;

[0297] Further considering the pile bottom as a fixed end constraint condition, then:

[0298]

[0299] Combining equations (32) and (33), we can obtain:

[0300]

[0301] Where,

[0302] The pile top horizontal impedance K can be further obtained h , swing impedance K r and horizontal-swing coupling impedance K hr They are:

[0303]

[0304] Step 5: Based on the established dynamic equilibrium equations of the active pile I in liquefied soil and non-liquefied soil, and considering the influence of the vibration of the active pile I on the passive pile II, the dynamic equilibrium equation of the passive pile II in liquefied soil is established:

[0305] According to the Euler-Bernouli beam theory, the dynamic equilibrium equation of the passive pile II unit in liquefied soil can be obtained as follows (0≤z≤h1):

[0306]

[0307] Further simplifying equation (38) yields:

[0308]

[0309] Among them, M p =E p I p ,

[0310] The corresponding solution of equation (39) consists of two parts: the general solution and the special solution. The general solution of the equation is:

[0311]

[0312] in, D II1 、D II2 、D II3 、D II4 is the coefficient to be determined;

[0313] The non-homogeneous special solution of the equation can be set as:

[0314]

[0315] Substituting equation (41) into equation (39) yields

[0316]

[0317] In summary, the general solution of equation (39) is

[0318]

[0319] Step 6: Based on the dynamic equilibrium equation of the passive pile II in liquefied soil, and in accordance with the continuous contact condition between the fluid and the pile, the dynamic equilibrium equation of the passive pile II in non-liquefied soil is created;

[0320] According to the continuous contact condition between the fluid and the pile body, we can get

[0321]

[0322] in,

[0323] Trigonometric function cos(α n z) is an orthogonal function system on the interval [-h1,0], that is:

[0324]

[0325] Multiply both sides of equation (44) by cos(α n z), and integrating it over the interval [0,h1], we can get:

[0326]

[0327] in,

[0328] Substituting Equation (46) into Equation (43), the analytical solution for the horizontal displacement of passive pile II in liquefied soil layer can be obtained as:

[0329]

[0330] Where η1 = κS II1 ,η2=κS II2 ,η3=κS II3 ,η4=κS II4 ,

[0331] According to Euler beam theory, the relationship between the rotation angle, bending moment, shear force and displacement of passive pile II can be further obtained as follows:

[0332]

[0333] Among them, T II1 (z)=[t II1 (z)t II2 (z)t II3 (z)t II4 (z)],

[0334] The relationship between the horizontal displacement, rotation angle, bending moment and shear force at the upper and lower ends of the passive pile II segment in the liquefied soil layer is:

[0335]

[0336] The horizontal displacement of each unit of the passive pile II can be expressed as According to the dynamic interaction between the pile and the soil in non-liquefied soil, the dynamic equilibrium equation of the passive pile II in the jth (j = 2, ..., m, ..., n) segment in non-liquefied soil can be simplified as follows:

[0337]

[0338] Further simplifying equation (50) yields:

[0339]

[0340] in,

[0341] The general solution of the homogeneous equation corresponding to equation (51) is:

[0342]

[0343] in, is the coefficient to be determined;

[0344] The special solution of equation (51) is:

[0345]

[0346] in, is the coefficient to be determined;

[0347] The complete solution of equation (51) can be expressed as:

[0348]

[0349] Step 7: Based on the dynamic equilibrium equations of the active pile I and passive pile II, and according to the pile-pile interaction principle, the pile-pile interaction factor is obtained:

[0350] For Euler beams, the relationship between pile rotation angle, bending moment, shear force and pile horizontal displacement is:

[0351]

[0352] in,

[0353]

[0354] Combining equations (54) and (55), the relationship between the displacement, rotation angle, internal force and the undetermined coefficient of the jth section of passive pile II can be obtained, which can be simplified as follows:

[0355]

[0356] in,

[0357] F bj (z)=[f b1j (z)f b2j (z)f b3j (z)f b4j (z)];

[0358]

[0359] In the local coordinate system, the relationship between the horizontal displacement, rotation angle, bending moment and shear force at the upper and lower ends of the passive pile II section j in non-liquefied soil can be obtained as follows:

[0360]

[0361] Considering the continuity of the soil layer boundary in non-liquefied soil, the cross section between the jth and j+1th sections of the pile body in the non-liquefied soil layer is:

[0362] The horizontal displacement, rotation angle, bending moment and shear force of the pile body are continuous, that is:

[0363]

[0364] The relationship between the pile bottom and the liquefied soil-non-liquefied soil interface, the horizontal displacement, rotation angle, bending moment, and shear force of the pile body in non-liquefied soil can be obtained by combining equations (57) and (58):

[0365]

[0366] In the formula: [FA]=[F aj (h n )][F aj (0)] -1 [F a(j-1) (hj-1 )][F a(j-1) (0)] -1 …[F a2 (h2)][F a2 (0)] -1 ;

[0367] [FB]=[F bj (h n )][F bj (0)] -1 [F b(j-1) (h j-1 )][F b(j-1) (0)] -1 …[F b2 (h2)][F b2 (0)] -1 ;

[0368] In the local coordinate system, the horizontal displacement, rotation, bending moment and shear force of the passive pile II at the interface between liquefied soil and non-liquefied soil are considered to be continuous, that is:

[0369]

[0370] Based on the superposition principle, the displacement internal force expression relationship between the top and bottom of passive pile II can be obtained as follows:

[0371]

[0372] Among them, [TA]=[FA][T II1 (h1)][T II1 (0)] -1 ;[TB]=[FB][T1(h1)][T1(0)] -1 ;

[0373] The transfer matrices TA and TB in equation (61) are further decomposed into 2×2 sub-matrices, namely:

[0374]

[0375]

[0376] According to the top constraint condition of passive pile II, that is,

[0377]

[0378] Substituting equation (64) into equation (63) and combining it with the expression of the top impedance of active pile I, we can obtain:

[0379]

[0380] Where [α(S,θ)] = -[TA 11 ] -1 [TB 11 +TB 12 K S ];

[0381] make The horizontal and rocking dynamic interaction factors of adjacent piles in liquefied soil are obtained as follows:

[0382]

[0383] Example:

[0384] In order to explore the influence of soil stratification characteristics, soil shear effect, liquefied soil depth, liquefied soil density, pile top axial force on the horizontal dynamic response of pile foundation, liquefied soil and non-liquefied soil are divided into three layers; unless otherwise specified, the pile length H = 15m, the pile radius r = 0.5m, and the pile body density ρ p =2500kg / m 3 , pile elastic modulus E p =2.55×10 10 Pa; fluid density in liquefied soil ρ w =1000kg / m 3 The angle between the two piles is 0°. The buried depth and thickness of each soil layer and the soil layer parameter values ​​are listed in Table 1.

[0385] Table 1 Depth and thickness of each soil layer

[0386]

[0387]

[0388] According to the method of the present invention, the above parameters are substituted into the solution to solve the horizontal dynamic response of the wedge pile top. Compared with the existing results, it is found that the agreement is good, thus verifying the rationality of the present model. At the same time, the above solution process is converted into a code calculation, which is time-saving, highly accurate, and easy to change the parameters for solution. It can be applied to the analysis of the horizontal dynamic response of piles in liquefied soil.

[0389] In summary, the present invention is based on the Pasternak foundation liquefied soil pile foundation horizontal dynamic response analysis method. First, the adopted Pasternak foundation model 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 the horizontal vibration dynamic response of the pile foundation under the action of simple harmonic load; secondly, the liquefied soil is equivalent to a fluid, and the non-liquefied soil is equivalent to a Pasternak foundation considering the influence of foundation shear deformation, and considering the joint action of the multi-directional coupling load at the pile top, the pile-pile horizontal vibration dynamic response in liquefied soil-non-liquefied soil is derived. The method of the present invention can solve the dynamic response of the pile group by arbitrarily changing the parameters, and it only takes about 30 seconds. There is no need for repeated modeling, which greatly saves calculation time. The analytical solution derived by using this technical solution is a closed-form solution, which can provide more rigorous technical theoretical guidance and reference for the horizontal vibration design of pile groups in liquefied soil.

Claims

1. A method for analyzing the dynamic response of pile-pile horizontal vibration in liquefied soil based on the Pasternak foundation model, characterized by: The specific steps include: Step 1: Establish a simplified calculation model for pile-pile horizontal harmonic excitation in liquefied soil based on the Pasternak foundation model: the model is divided into active piles I and passive piles II. The active pile I is subjected to the top force, and the adjacent piles affected by the active pile I are passive piles II. Apply a horizontal harmonic exciting force Q0e to the top of active pile I iωt 、M0e iωt , where Q0 and M0 are the amplitudes of the exciting force, N0 is the axial force, and ω is the circular frequency of the exciting force. t is time; In addition, the thickness, stiffness coefficient, damping coefficient and foundation shear coefficient of the jth layer of soil are h j 、 and The pile length is divided into the pile length L1 for liquefied soil layer and the pile length L2 for non-liquefied soil layer, and the pile diameter is d; Step 2: Based on the model of step 1, simulate the effective fluid such as the liquefied soil layer, and establish the fluid control equation of the liquefied layer according to the hydraulic fluid wave theory; Step 3: Based on the liquefied layer fluid control equation established in step 2, a dynamic equilibrium equation of the active pile 1 in the liquefied soil is established; Step 4: Based on the dynamic equilibrium equation of the active pile 1 in liquefied soil established in step 3, and according to the pile body continuity condition at the interface between liquefied soil and non-liquefied soil, the dynamic equilibrium equation of the active pile 1 in non-liquefied soil is established; based on the Euler beam theory, the pile top impedance of the active pile 1 is further determined; Step 5: Based on the established dynamic equilibrium equations of the active pile I in liquefied soil and non-liquefied soil, the dynamic equilibrium equation of the passive pile II in liquefied soil is established by considering the influence of the vibration of the active pile I on the passive pile II. Step 6: Based on the dynamic equilibrium equation of the passive pile II in liquefied soil, and in accordance with the continuous contact condition between the fluid and the pile, the dynamic equilibrium equation of the passive pile II in non-liquefied soil is created; Step 7: Based on the dynamic equilibrium equations of the active pile I and passive pile II, and according to the pile-pile interaction principle, the pile-pile interaction factor is obtained.

2. The method for analyzing pile-pile horizontal vibration dynamic response in liquefied soil based on the Pasternak foundation model according to claim 1 is characterized in that: The simplified calculation model is based on the following basic assumptions: 1) The pile body of active pile I is simplified to a circular uniform cross-section, homogeneous Euler beam; 2) The soil around the pile is divided into n layers along the longitudinal direction of the pile body, and each layer of soil is simplified to the Pasternak foundation model to describe the pile-soil interaction; 3) All parts of the pile-soil model system meet the small deformation condition, and the pile-soil interface is in full contact with no relative sliding; 4) Only horizontal displacement occurs at the pile top, and the pile bottom is constrained by the fixed end.

3. The method for analyzing pile-pile horizontal vibration dynamic response in liquefied soil based on the Pasternak foundation model according to claim 1 is characterized in that: The control equation of fluid motion in the liquefied soil layer established in step 2 is specifically: in: is the fluid velocity potential function; The liquefied soil layer must meet the following boundary conditions when moving: a) When z = h1 at the bottom of the liquefied soil layer, its vertical velocity is zero, that is: b) Ignoring the influence of gravity waves on the fluid, on the surface of the liquefied soil layer: c) The fluid is stationary at radial infinity: d) The contact continuity condition between the fluid and the pile is: Where: is the horizontal displacement of the pile section in water, and r0 is the radius of the pile body.

4. The method for analyzing pile-pile horizontal vibration dynamic response in liquefied soil based on the Pasternak foundation model according to claim 3 is characterized in that: The specific method of step 3 is: According to the fact that the pile body is finally in steady-state vibration in the liquefied soil layer, the fluid velocity potential function can be expressed as: Further using separation of variables, the fluid velocity potential function can be set as: Substituting equation (7) into equation (1) and considering boundary conditions (2) to (5), the solution of the fluid velocity potential function is: In the formula, K1(iα n r) is a first-order Bessel function, A is the coefficient to be determined, α n =(2n-1)πi / (2h1), j=1,2,3... According to the Bernouli equation, the hydrodynamic pressure acting on the active pile I can be further calculated as: Where, ρ l is the density of liquefied soil; Substituting formula (9) into formula (8) yields: According to the Euler-Bernouli beam theory, the dynamic equilibrium equation of the active pile I unit in liquefied soil is obtained as follows (0≤z≤h1): Where, E p , I p 、m p are the elastic modulus, section moment of inertia and mass per unit length of the pile, respectively, and N0 is the axial force acting on the pile top; The potential function of pile horizontal displacement and fluid motion velocity can be expressed as: Where, is the horizontal displacement amplitude of the pile in the liquefied soil layer; Substituting formula (12) into formula (11) can further obtain the following equation: Where M p =E p I p , Obviously, Equation (13) is a fourth-order linear non-homogeneous differential equation with constant coefficients. Its corresponding solution consists of a general solution and a special solution. The displacement homogeneous general solution of the equation is: Where, D1, D2, D3, and D4 are unknown coefficients; The non-homogeneous solution of formula (13) is set as: Substituting formula (15) into formula (13) yields: Therefore, the general solution of equation (13) is:

5. The method for analyzing pile-pile horizontal vibration dynamic response in liquefied soil based on the Pasternak foundation model according to claim 4 is characterized in that: The specific method of step 4 is: 4.1) Establish the dynamic equilibrium equation of active pile I in non-liquefied soil According to the contact continuity condition between the fluid and the pile, Equations (8) and (17) are substituted into Equation (4) to obtain: Where: Multiply both sides of equation (18) by ch(α n z), and then integrate it over the interval [0,h1], we can get: Where: Then the solution for the horizontal displacement of the pile in liquefied soil is: In the formula: η1=κS1, η2=κS2, η3=κS3, η4=κS4, According to Euler beam theory, the relationship between pile body rotation angle, bending moment, shear force and pile body displacement is obtained, which is further simplified as follows: T1(z) is expressed as: T1(z)=[t1(z) t2(z) t3(z) t4(z)] (22) Where: From this, we can deduce that the relationship between the horizontal displacement at the top and bottom of the pile section in the liquefied soil layer and the rotation angle, bending moment, and shear force is: Then, the dynamic equilibrium equation of the pile unit in the jth (j is the number of soil layers, j = 2, ..., m, ..., n) layer of the non-liquefied soil layer is obtained as follows: Where: is the horizontal displacement of the mass point of the pile body in the jth section; is the shear stiffness of the foundation soil around the j-th layer of piles; B0 = 0.9 (1.5d + 0.5) is the calculated width of the pile; for and It is determined according to the following formula: Where, is the shear wave velocity of the soil around the pile; and are the elastic modulus, density, damping coefficient and Poisson's ratio of the soil around the pile; is the dimensionless frequency; is the shear layer thickness of the j-th layer of foundation soil, Furthermore, the horizontal displacement of the pile can be expressed as Where, is the horizontal displacement amplitude of the pile body mass point at the jth section in non-liquefied soil; set up Substituting equation (27) into equation (24) we can further obtain the following equation: Where, Then the general displacement solution of equation (28) is: Where, is the coefficient to be determined; 4.2) Determine the pile top impedance of active pile I based on Euler beam theory According to Euler beam theory, the relationship between the rotation angle, bending moment, shear force and pile displacement of the j-th segment of pile in non-liquefied soil can be further obtained, which can be simplified as follows: F j (z) can be expressed as: F j (z)=[f1(z) f2(z) f3(z) f4(z)] (31) Where: The relationship between the horizontal displacement at the top and bottom of the pile section in the non-liquefied soil layer and the rotation angle, bending moment, and shear force can be deduced from the combined equation (23): Where, [F] = [F n (h n )][F n-1 (h n-1 )]…[F3(h3)][F2(h2)][T1(h1)][T1(0)] -1 ; Further considering the pile bottom as a fixed end constraint condition, then: Combining equations (32) and (33), we can obtain: Where, The pile top horizontal impedance K can be further obtained h , swing impedance K r and horizontal-swing coupling impedance K hr They are:

6. The method for analyzing pile-pile horizontal vibration dynamic response in liquefied soil based on the Pasternak foundation model according to claim 5 is characterized in that: The specific method of step 5 is: According to the Euler-Bernouli beam theory, the dynamic equilibrium equation of the passive pile II in liquefied soil can be obtained as follows (0≤z≤h1): Further simplifying equation (38) yields: Among them, M p =E p I p , The corresponding solution of equation (39) consists of two parts: the general solution and the special solution. The general solution of the equation is: in, D II1 、D II2 、D II3 、D II4 is the coefficient to be determined; The non-homogeneous special solution of the equation can be set as: Substituting equation (41) into equation (39) yields In summary, the general solution of equation (39) is 7. The method for analyzing pile-pile horizontal vibration dynamic response in liquefied soil based on the Pasternak foundation model according to claim 6, characterized in that: The specific method of step 6 is: According to the continuity condition of passive pile II at the interface between liquefied soil and non-liquefied soil, we can get in, Trigonometric function cos(α n z) is an orthogonal function system on the interval [-h1,0], that is: Multiply both sides of equation (44) by cos(α n z), and integrating it over the interval [0,h1], we can get: in, Substituting Equation (46) into Equation (43), the analytical solution for the horizontal displacement of passive pile II in liquefied soil layer can be obtained as: where, η1 = κS II1 , η2 = κS II2 , η3 = κS II3 , η4 = κS II4 , According to Euler beam theory, the relationship between the rotation angle, bending moment, shear force and displacement of passive pile II can be further obtained as follows: Among them, T II1 (z) = [t II1 (z) t II2 (z) t II3 (z) t II4 (z)], The relationship between the horizontal displacement, rotation angle, bending moment and shear force at the upper and lower ends of the passive pile II segment in the liquefied soil layer is: The horizontal displacement of each unit of the passive pile II can be expressed as According to the dynamic interaction between pile and soil in non-liquefied soil, the dynamic equilibrium equation of passive pile II in section j (j = 2, ..., m, ..., n) in non-liquefied soil can be simplified as follows: Further simplifying equation (50) yields: in, The general solution of the homogeneous equation corresponding to equation (51) is: in, is the coefficient to be determined; The specific solution of equation (51) is: in, is the coefficient to be determined; The complete solution of equation (51) can be expressed as:

8. The method for analyzing pile-pile horizontal vibration dynamic response in liquefied soil based on the Pasternak foundation model according to claim 7, characterized in that: The specific method of step 7 is: According to Euler beam theory, the relationship between pile body rotation angle, bending moment, shear force and pile body horizontal displacement is: in, Combining equations (54) and (55), the relationship between the displacement, rotation angle, internal force and the undetermined coefficient of the jth section of passive pile II can be obtained, which can be simplified as follows: in, F bj (z)=[f b1j (z) f b2j (z) f b3j (z) f b4j (z)]; In the local coordinate system, the relationship between the horizontal displacement, rotation angle, bending moment and shear force at the upper and lower ends of the passive pile II section j in non-liquefied soil can be obtained as follows: Considering the continuity of the soil layer boundary in non-liquefied soil, the horizontal displacement, rotation angle, bending moment and shear force of the pile body at the sections j and j+1 of the non-liquefied soil layer are continuous, that is: The relationship between the pile bottom and the liquefied soil-non-liquefied soil interface, the horizontal displacement, rotation angle, bending moment, and shear force of the pile body in non-liquefied soil can be obtained by combining equations (57) and (58): Where: [FA] = [F aj (h n )][F aj (0)] -1 [F a(j-1) (h j-1 )][F a(j-1) (0)] -1 …[F a2 (h2)][F a2 (0)] -1 ; [FB] = [F bj (h n )][F bj (0)] -1 [F b(j-1) (h j-1 )][F b(j-1) (0)] -1 …[F b2 (h2)][F b2 (0)] -1 ; In the local coordinate system, the horizontal displacement, rotation, bending moment and shear force of the passive pile II at the interface between liquefied soil and non-liquefied soil are considered to be continuous, that is: Based on the superposition principle, the displacement internal force expression relationship between the top and bottom of passive pile II can be obtained as follows: Among them, [TA]=[FA][T II1 (h1)][T II1 (0)] -1 ;[TB]=[FB][T1(h1)][T1(0)] -1 ; The transfer matrices TA and TB in equation (61) are further decomposed into 2×2 sub-matrices, namely: According to the top constraint condition of passive pile II, that is, Substituting equation (64) into equation (63) and combining it with the expression of the top impedance of active pile I, we can obtain: Where [α(S,θ)] = -[TA 11 ] -1 [TB 11 +TB 12 K S ]; make The horizontal and rocking dynamic interaction factors of adjacent piles in liquefied soil are obtained as follows:

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