Method for determining vertical seismic response of friction type tubular pile in layered foundation by considering soil plug effect

Abstract

The invention discloses a method for determining vertical seismic response of a friction type tubular pile in a stratified foundation by considering a soil plug effect. The method comprises the following steps: S1, setting initial parameters for an iteration factor gamma; s2, calculating an attenuation function phi according to the iteration factor; s3, calculating an intermediate function kk and an intermediate function ck according to the attenuation function phi; s4, calculating the free field displacement of the soil body, and obtaining the displacement values of the pipe pile and the pile bottom virtual soil pile according to the free field displacement of the soil body, the intermediate function kk and the intermediate function ck; s5, calculating a new iteration factor; and S6, repeating the steps S2-S5 until the difference between the newest iteration factor and the iteration factor of the last round is smaller than a fixed value, and obtaining the final vertical seismic response by taking the free field displacement and the displacement value of the tubular pile and the pile bottom virtual soil pile obtained according to the newest iteration factor. The problem that the seismic response of the friction type pipe pile in layered soil is difficult to obtain is solved.

Description

Technical Field

[0001] The invention belongs to the technical field and relates to a method for determining the vertical seismic response of a friction type pipe pile in a layered foundation taking into account a soil plug effect. Background Art

[0002] Pile foundations have good seismic performance and are widely used in various engineering projects. Studying the seismic response of pile foundations is crucial for seismic design and seismic measures, and has become a research hotspot. At present, a lot of research has been done on the response of pile foundations under horizontal seismic motion. However, due to the damage to pile foundations caused by vertical ground motion observed in the Northbridge and Kobe earthquakes, the dynamic response of pile foundations under vertical seismic action has gradually received attention.

[0003] In actual engineering, there are different types of piles, such as concrete solid piles, steel pipe piles, PHC pipe piles, and PCC pipe piles. Unlike solid piles, pipe piles are usually fully or partially filled with soil plugs, so the soil plugs have a significant impact on the dynamic response of pipe piles. Some scholars have also considered the seismic response of pipe piles. However, there are few studies that consider the effect of soil plugs on the seismic response of pile foundations. In addition, pipe piles are usually used as friction piles, with the bottom of the pile supported by the bearing layer soil, and the soil around the pile is often distributed in layers due to long-term sedimentation or geological tectonic action. Compared with the end-bearing piles and homogeneous soil conditions considered in existing studies, the above-mentioned conditions are also very common in actual engineering. However, there has been no report on the seismic response of friction pipe piles in layered soils considering the soil plug effect.

[0004] For example, the paper "Study on the influence of soil plugging effect on the longitudinal dynamic characteristics of pipe piles, Wu Wenbing, Jiang Guosheng, etc." studied the longitudinal vibration problem of pipe piles in layered foundations when considering the soil plugging effect: First, according to the stratification of the soil on the side and end of the pile, the pile-soil system is divided into a finite number of micro-element segments along the longitudinal direction, and the longitudinal vibration control equation of the pile-soil system is established by using the virtual soil pile method and the additional mass method. Then, using the Laplace transform technology and the impedance function recursion method, the frequency domain response analytical solution of the pile top velocity of the longitudinal vibration of the pipe pile considering the soil plugging effect and the corresponding semi-analytical solution of the time domain response of the pile top velocity under the action of the semi-sine pulse excitation force are derived. Finally, the sensitivity range of the Voigt model parameters in the additional mass method is analyzed, the influence of the soil plugging effect on the longitudinal dynamic characteristics of the pipe pile is discussed, and the rationality of the proposed theoretical solution is verified by model tests. However, the difference from the present invention is that the literature studies the longitudinal vibration of the pipe pile, that is, the external load directly acts on the top of the pile foundation, and the dynamics propagate downward; while the present invention studies the response of the pipe pile to the longitudinal seismic wave, the seismic load acts on the soil at the bottom of the pile foundation, and the dynamics propagate upward, thus affecting the pile foundation. In comparison, the dynamic responses of the pile foundations of the two are different, and the problems caused by the seismic load are more complicated, and the analysis and solution are more difficult. Summary of the invention

[0005] To achieve the above objectives, the present invention provides a method for determining the vertical seismic response of friction pipe piles in layered soil taking into account the soil plug effect, which solves the problem that the seismic response of friction pipe piles in layered soil is difficult to obtain.

[0006] The technical solution adopted by the present invention is a method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect, comprising the following steps:

[0007] Step S1: setting initial parameters for the iteration factor γ;

[0008] Step S2: Calculate the attenuation function φ according to the iteration factor;

[0009] Step S3: Calculate the intermediate function k according to the attenuation function φ k and the intermediate function c k ;

[0010] Step S4: Calculate the soil free field displacement and calculate the soil free field displacement and the intermediate function k k and the intermediate function c k Get the displacement value of the pipe pile and the virtual soil pile at the bottom of the pile;

[0011] Step S5: Calculate a new iteration factor;

[0012] Step S6: Repeat steps S2 to S5 until the difference between the latest iteration factor and the iteration factor of the previous round is less than a fixed value, and take the free field displacement, the displacement value of the pipe pile and the virtual soil pile at the bottom obtained according to the latest iteration factor to obtain the final vertical seismic response.

[0013] Furthermore, the specific calculation formula of the attenuation function φ in step S2 is:

[0014]

[0015] Among them, K0 is the second kind of zero-order modified Bessel function, r represents the transverse coordinate of the cylindrical coordinate system, and r2 represents the outer diameter of the pile.

[0016] Furthermore, the intermediate function k in step S3 k and the intermediate function c k The calculation formulas are:

[0017]

[0018] In the formula, represents the shear modulus of the soil, is an intermediate function, r represents the lateral coordinate of the cylindrical coordinate system, and r2 represents the outer diameter of the pile;

[0019]

[0020] represents the equivalent elastic modulus of soil, v sk is Poisson's ratio.

[0021] Furthermore, in step S4, the free field displacement w fk The calculation formula is:

[0022] w fk =w g +w rk

[0023] In the formula, w rk represents the vertical displacement of the soil free field relative to the rigid bedrock, w g represents the amplitude of bedrock motion; where w rk The calculation formula is:

[0024] w rk =B k cos(λ k z)+C k sin(-λ k z)-w g , where H k-1 ≤z≤H k

[0025] In the formula, B k and C k All of them are unknown parameters. z represents the vertical coordinate of the cylindrical coordinate system. H k represents the total height of the k-layer soil layer, λ k is an intermediate function, and Where ω represents the circular excitation frequency, ρ sk is the soil density.

[0026] Furthermore, the calculation formula for the displacement of the pipe pile in step S4 is:

[0027]

[0028] Where D k and F k Unknown parameters, e is a natural constant, α pk , are all intermediate functions, and:

[0029]

[0030] In the formula, E p is the pile modulus, ρ p is the pile density, A p Indicates the cross-sectional area of ​​the pile, mk represents the mass of the soil plug, r1 represents the inner diameter of the pile, k rk represents the spring coefficient of the soil plug, c rk Represents the damping coefficient of the soil plug, in imaginary units ξ pk is an intermediate function, and

[0031] Furthermore, the calculation formula for the displacement of the virtual soil pile at the pile bottom in step S4 is:

[0032]

[0033] In the formula, α sk , are all intermediate functions, among which,

[0034]

[0035] In the formula, E sk is the Young's modulus of the kth soil layer, the intermediate function

[0036] Furthermore, the new iteration factor in step S5 is calculated by the following formula:

[0037]

[0038] In the formula, n represents the number of virtual soil pile layers, w k represents the vertical displacement of the kth layer of soil, and t is the time coefficient.

[0039] Furthermore, the fixed value in step S6 is 0.001.

[0040] The beneficial effects of the present invention are:

[0041] The present invention proposes a dynamic semi-analytical solution for friction pipe piles in layered soil under the action of earthquake P waves, based on the consideration of soil plugging effect: the pile-soil motion equation and boundary and continuity requirements are derived based on Hamilton's variational principle. The displacement of the pipe pile is coupled with the attenuation function representing the displacement of the soil around the pile, and finally solved by an iterative method. The parameter analysis of the three-layer soil with a soft soil interlayer in the embodiment shows that soil plugging and stratification have a significant impact on the pile-soil seismic response, which should be considered in relevant engineering practices. BRIEF DESCRIPTION OF THE DRAWINGS

[0042] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.

[0043] Figure 1 1 is a mechanical model diagram of a friction type pipe pile partially filled with soil plugs in a layered soil body according to an embodiment of the present invention, wherein (a) is a model diagram of the pile-soil system, (b) is a schematic diagram of the coordinate system and displacement components, and (c) is a model diagram of the interaction between the pipe pile and the soil plug.

[0044] Figure 2 It is a flow chart of the pile-soil response solution method according to an embodiment of the present invention.

[0045] Figure 3 It is the free field response diagram of the present invention at different excitation frequencies; wherein (a) is the real part diagram of the free field displacement response, (b) is the imaginary part diagram of the free field displacement response, (c) is the real part diagram of the free field vertical stress response, and (d) is the imaginary part diagram of the free field vertical stress response.

[0046] Figure 4 The present invention is a pile response diagram along the pile axis at different excitation frequencies; wherein (a) is a real part diagram of the pile displacement response, (b) is an imaginary part diagram of the pile displacement response, (c) is a real part diagram of the pile axial force response, and (d) is an imaginary part diagram of the pile axial force response.

[0047] Figure 5 It is a comparison diagram of the degradation solution of the present invention and the existing results.

[0048] Figure 6 The effect of different soil plug heights on I v and AM influence diagram; where (a) is the kinematic interaction coefficient I v (a) is a graph showing the change of motion amplification factor AM with different soil plug heights, and (b) is a graph showing the change of motion amplification factor AM with different soil plug heights.

[0049] Figure 7 is the kinematic interaction coefficient I v and the variation of the motion amplification coefficient AM under different soil interlayer moduli; where a) is the motion interaction coefficient I v (b) is the variation diagram of the motion amplification factor AM under different soil interlayer moduli, and (c) is the variation diagram of the motion amplification factor AM under different soil interlayer moduli.

[0050] Figure 8 The effect of different thickness of soft soil layers on I v and AM curves; where (a) is the kinematic interaction coefficient Iv (b) is the change diagram of the motion amplification factor AM under soft soil layers of different thicknesses. DETAILED DESCRIPTION

[0051] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0052] The embodiment of the present invention proposes a semi-analytical solution for the dynamic response of a friction-type pipe pile partially filled with soil plugs in a layered soil mass under the action of a time-harmonic earthquake P wave (vertical wave). The pipe pile is regarded as a tubular structure partially filled with soil plugs, and the soil plug is regarded as a concentrated mass attached to the pipe pile. The soil in the vertical projection area of ​​the pile is regarded as a soil column, while the soil and soil column around the pile are regarded as a multi-layer continuum. According to Hamilton's variational principle, the motion equations of the pile and the soil column and the pile-soil continuity condition are derived. The above-mentioned pile-soil system motion equations are solved using variable separation technology and iterative method, and combined with the pile-soil boundary and continuity requirements, the frequency domain semi-analytical solution of the pile-soil system to the earthquake P wave motion response is obtained. By comparing with the existing solutions, it is verified that the proposed solution is reasonable and feasible. Finally, a numerical example analysis is carried out to reveal the earthquake pile-soil interaction effect under different material / geometric parameters. The specific process is as follows:

[0053] 1. Model establishment and assumptions:

[0054] like Figure 1 As shown in (a) to (c) of the present invention, in the pile-soil system, the pipe pile is regarded as a linear elastic, circular, isotropic hollow rod with an inner radius of r1 and an outer radius of r2, which is completely embedded in the layered soil and partially filled with soil plugs; the modulus of the pipe pile is E p , Poisson's ratio is v p , the pile density is ρ p The soil around the pile is a layered continuum, each layer is homogeneous and isotropic, and the shear modulus is The equivalent elastic modulus is Among them, E sk is the Young's modulus of the kth soil layer, β 0k is the hysteresis damping ratio, ρ sk is the soil density, v sk is Poisson's ratio. There are n layers in the soil, of which the hollow part above the soil plug has k-1 layers, and the top of the pile has m layers. kIndicates the thickness of each layer, and the thickness of each layer is consistent. H1 indicates the thickness of all soil layers in the hollow part above the soil plug, H2 indicates the height of the soil plug section, and H3 indicates the height of the virtual soil pile. The rigid bedrock under the soil layer is affected by w g (t) = w g e iωt Vertical steady-state seismic excitation (w g represents the motion amplitude of the bedrock, ω represents the circular excitation frequency, and e iωt represents the time domain). It is assumed that the pile-soil system satisfies the ideal contact condition, that is, there is no relative pile-soil sliding and separation behavior, and the small deformation assumption is satisfied. The soil plug in the pile is regarded as having a mass of m k The contact behavior between the soil plug and the pile is described by the distributed Voigt model, which has a spring coefficient k rk and the damping coefficient c rk The soil at the vertical projection of the pipe pile is regarded as a soil column (virtual soil pile) with rod-like structural behavior.

[0055] Under the action of the incident P wave, the offset angle is θ, and the pile-soil system satisfies the axisymmetric condition, so its radial and circumferential displacements can be ignored. That is, its displacement field can be constructed as:

[0056]

[0057] w k (z,t)=w pk (z,t),z≤H m (3)

[0058] w k (z,t)=w sk (z,t),z≥H m (4)

[0059] In the formula, and They represent the radial, circumferential and axial components of the displacement of the kth layer of soil with time t, respectively; z represents the vertical coordinate of the cylindrical coordinate system, r represents the horizontal coordinate of the cylindrical coordinate system, and H m Indicates the height of the soil layer m above the pile end; w fk (z,t) is the vertical displacement of the soil in the free field with time t, w pk (z,t) is the vertical displacement of the pile with time t, w sk (z, t) is the vertical displacement of the virtual soil pile with time t; φ represents the attenuation function of the soil displacement field around the pile, and the boundary conditions satisfy φ(r→∞)=0 and φ(r=r2)=1. For simplicity, the following analysis will omit the time term t in all field variables.

[0060] 2. Control equations and solutions of pile-soil system:

[0061] Displacement of soil in free field (i.e. soil without piles):

[0062] The displacement of the soil free field to the earthquake P wave can be expressed as:

[0063] w fk =w g +w rk (5)

[0064] Among them, w rk It represents the vertical displacement of the free field soil relative to the rigid bedrock.

[0065] According to Hamilton's principle and collecting δw rk (δw rk Indicates w rk The coefficient of the first-order variation of rk The equation of motion is as follows:

[0066]

[0067] In the formula, H k represents the total height of the k-th soil layer, λ k , is an intermediate function.

[0068] The general solution of formula (6) can be written as:

[0069] w rk =B k cos(λ k z)+C k sin(-λ k z)-w g (H k-1 ≤z≤H k ) (7)

[0070]

[0071] w rn =0(z=H n ) (9)

[0072] w rk =w r(k+1) (z=H k ) (10)

[0073]

[0074] In the formula, B k and C k is an unknown coefficient, which is determined by combining the above boundary equation and continuity equation (i.e., Equation 7 to Equation 11).

[0075] Combining equations (7) to (11) yields the following matrix:

[0076] [M f ][X f ]=[V f ] (12)

[0077] In the formula, [M f ]、[X f ] and [V f ] are all free field coefficient matrices, where:

[0078]

[0079] [X f ] T =[B1 C1 B2 C2…B n-1 C n-1 B n C n ] (14)

[0080] [V f ] T =[0 w g 00…0000] (15)

[0081] In the formula, T represents transposition, p = n-1, a kj 、b kj 、c kj d kj are all intermediate functions, where a kj = cos(λ k H j ),

[0082] b kj = sin(-λ k H j ), Number of layers j = 1, 2, ..., n.

[0083] Soil plug displacement:

[0084] The energy generated by soil plugging can be written as:

[0085]

[0086] Among them, w ck represents the vertical displacement of the k-layer soil plug, k rk represents the spring coefficient of the soil plug, m k Indicates the quality of the soil plug.

[0087] Using Hamilton's principle, and collecting δw ck (δwck Indicates w ck The coefficients of the first-order variation of ) can be obtained:

[0088]

[0089] According to formula (17), we can get:

[0090]

[0091] Pipe pile displacement:

[0092] Assuming that the lateral displacement can be ignored, the non-zero strain (Equation 19-Equation 20) and stress components (Equation 21-23) of the pile-soil system can be obtained by the following formulas:

[0093]

[0094]

[0095] In the formula, ε zzk represents the vertical strain of the pile along the z direction, ε rzk represents the vertical strain of the soil along the z direction, σ zzk represents the stress on the pile body along the z direction, σ rrk represents the stress of the soil along the radial direction r, σ θθk represents the stress of soil along the tangential direction, σ rzk It represents the stress in the soil along the z direction.

[0096] The potential energy U and kinetic energy T of the pile-soil system can be expressed as:

[0097]

[0098] In the formula, Represents the cross-sectional area of ​​the pile.

[0099] According to Hamilton's principle, and collect the pk (δw pk Indicates w pk The coefficients of the first-order variation of ) can be obtained:

[0100]

[0101] In the formula, α pk , pk are all intermediate functions, among which:

[0102]

[0103] In the formula, c k , k k are all intermediate functions, among which:

[0104]

[0105]

[0106] Then the general solution of equation (26) and its boundary and continuity conditions are:

[0107]

[0108] w pk =w p(k+1) (z=H k ,k=1,2,...,m) (34)

[0109]

[0110] In the formula, the intermediate function D k and F k is an unknown coefficient determined by the boundary equation and continuity equation of the pile.

[0111] Combining (31) to (35), we can get the following matrix:

[0112] [M p ][X p ]=[V p ] (36)

[0113] In the formula, [M p ]、[X p ] and [V p ] are all pile foundation coefficient matrices, where:

[0114]

[0115] [X p ] T =[D1 F1 D2 F2 … D m-1 F m-1 D m F m ], (38)

[0116]

[0117] In the formula, the number of layers q = m-1; β pk are all intermediate functions, among which, β pk =(E p A p +2c k )α pk; k=1,2,...,m and j=1,2,...,m.

[0118] Displacement of pile at bottom of virtual soil:

[0119] Similarly, apply Hamilton's variational principle to the pile bottom virtual soil and collect δw sk (δw sk Indicates w sk The coefficients of the first-order variation of ) can be obtained:

[0120]

[0121] In the formula, the intermediate function ξ sk They are:

[0122]

[0123] The general solution of formula (40) and its boundary conditions and continuity conditions can be obtained as follows:

[0124]

[0125] w pm =w s(m+1) (z=H m ) (44)

[0126] w sn =w g (z=H n ) (45)

[0127] w sk =w s(k+1) (z=H k ,k=m+1,...,n) (46)

[0128]

[0129] In the formula, the intermediate function D k and F k The unknown coefficients will be solved through the boundary conditions and continuity conditions of the above-mentioned virtual soil piles.

[0130] Combining (43) to (47), we can get the following matrix:

[0131] [M s ][X s ]=[V s ] (48)

[0132] In the formula, [M s ]、[X s ] and [V s] are the coefficient matrices of virtual soil piles, among which:

[0133]

[0134] [X s ] T =[D m+1 F m+1 D m+2 F m+2 … D n-1 F n-1 D n F n ] (50)

[0135]

[0136] In the formula, β sk is an intermediate function, where

[0137] The number of layers p = n-1; k = m, m+1, ..., n.

[0138] Displacement field of soil around the pile:

[0139] Using the Hamilton variation principle of soil and collecting the coefficients of δφ (δφ represents the first-order variation of φ), we can obtain:

[0140]

[0141] In the formula, γ is the iteration factor;

[0142]

[0143] φ holds true under the following boundary conditions:

[0144] φ(r→∞)=0 (54)

[0145] φ(r=r2)=1 (55)

[0146] According to formulas (52) to (55), it can be determined that:

[0147]

[0148] Where K0 is the second kind zero-order modified Bessel function.

[0149] The interaction effect of the pile-soil system on earthquake P waves can be described by defining the kinematic interaction coefficient I v And the motion amplification factor AM is used to express it, and its expression is:

[0150]

[0151] Solution process:

[0152] According to equations (31) and (43), it can be found that the pile displacement w pk (z) and the virtual soil pile displacement w at the pile bottom sk The coupling relationship between (z) and the surrounding soil attenuation function φ can be decoupled by an iterative procedure, see Figure 2 , the specific process is:

[0153] (1) Setting an initial parameter for the iteration factor γ, for example, γ = 1;

[0154] (2) The attenuation function φ is calculated by expression (56);

[0155] (3) Coefficient k k 、c k Calculated by expressions (29) and (30) respectively;

[0156] (4) Expressions (5) and (7) can be used to calculate the free field displacement. Substituting the displacement results into expressions (31) and (43) can obtain the displacement values ​​of the pipe pile and the virtual soil pile at the pile bottom;

[0157] (5) The new iteration factor can be obtained by expression (53);

[0158] Repeat processes (2) to (5) until the difference between the latest iteration factor and the previous iteration factor is less than a certain value, which can generally be taken as 0.001.

[0159] Specific example analysis:

[0160] The embodiment of the present invention considers the three-layer soil condition with a weak interlayer, that is, k=2 represents a soft soil layer. Unless otherwise specified, Table 1 lists the pile-soil materials or geometric parameters used in the subsequent analysis. The excitation frequency of the earthquake P wave is determined by After dimensionless processing, C pn is the longitudinal wave velocity in the nth soil layer.

[0161] Table 1 Pile-soil material / geometric parameters

[0162] symbol definition Numeric unit <![CDATA[ρ p ]]> Pile density 2250 <![CDATA[kg / m 3 ]]> <![CDATA[E p ]]> Pile modulus 10 GPa L Pile length 10 m <![CDATA[r1]]> Pipe pile inner diameter 0.2 m <![CDATA[r2]]> Pipe pile outer diameter 0.35 m <![CDATA[ν p ]]> Poisson's ratio of pile 0.2 - <![CDATA[ρ sk ]]> Soil density per layer 1800 <![CDATA[kg / m 3 ]]> <![CDATA[E sk ]]> Soil modulus of each layer 10 MPa <![CDATA[ν sk ]]> Poisson's ratio of each soil layer 0.4 - <![CDATA[β 0k ]]> Hysteretic damping ratio of each soil layer 0.05 - <![CDATA[h k ]]> Thickness of each soil layer 3 m <![CDATA[H2]]> Pile core soil height 3 m <![CDATA[k rk ]]> Soil plug stiffness coefficient 10 MPa <![CDATA[c rk ]]> Soil plug damping coefficient 100 N·m / s

[0163] 3. Verification and comparison of solutions:

[0164] Figure 3 Figures (a) to (d) show the consistency of the free field response with the boundary conditions and continuity conditions at different excitation frequencies. It can be observed from the figure that the free field displacement and vertical stress distribution change with The change of is nonlinear. As the free field vertical stress increases, the change in free field vertical stress increases. At the same time, the stress at the pile top is zero, corresponding to the free field boundary condition formula (8). The displacement is continuous at the junction of the pipe pile and the virtual soil pile, which conforms to formula (10). The free field vertical stress is continuous, corresponding to formula (11).

[0165] Figure 4 (a) to (d) in the figure show the variation of pile response along the pile body at different excitation frequencies. As can be seen from the figure, the axial force at the top of the pile is zero, which corresponds to the pile boundary condition formula (32). The displacement and axial force are continuous at the junction of the pile and the virtual soil pile, which conforms to formulas (34) and (33). The axial force of the pile body is continuous, corresponding to formula (35).

[0166] Therefore, from Figure 3 and 4 It can be seen that the calculation results are consistent with the specified pipe pile boundary conditions, and to a certain extent verify the correctness of the technical solution of the present invention.

[0167] By setting the inner radius r1 = 0 and the number of soil layers k = 1, the vertical seismic response problem of the pipe pile is simplified to the kinematic response of a solid pile buried in a homogeneous soil layer under a time-harmonic seismic P wave. The corresponding simplified solution is then compared with the results in the existing literature (Liu, QJ, Deng, FJ, He, YB, 2014. Kinematic response of single piles to vertically incident P waves. Earthquake Engineering and Structural Dynamics. 43(6), 871-887.). Figure 5 It can be seen that the solution results proposed by the present invention are very consistent with the solution results of the existing literature, which can confirm the effectiveness of the solution proposed by the present invention.

[0168] Analysis of the seismic response characteristics of pipe piles:

[0169] Figure 6 Indicates the interaction coefficient I of different inner soil heights on motion v and the influence of motion amplification factor AM. Figure 6 As can be seen from (a) to (b) in Figure 1, soil plug has a significant effect on the seismic response of pile-soil, which should be considered in related engineering applications. Specifically, as the height of the soil plug increases, I vThe maximum frequencies of and AM become larger, while the peak values ​​show non-monotonic changes, which indicates that it is difficult for a pile completely filled with soil plugs to cause resonance. This may be because as the height of the soil plug increases, the mass of the soil inside the pile also increases, and the influence on the dynamic response of the pile also increases. In order to maximize the maximum difference between the displacement of the pile top and the displacement of the bedrock, a larger seismic excitation is required.

[0170] Figure 7 Shows and The curve changes with the different modulus of the soil interlayer. Figure 7 It can be seen from (a) to (b) that as the soil interlayer modulus E s2 The increase of I v The fluctuation trend of the curve increases accordingly, and the influence of the pile on the soil free field also increases accordingly. s2 As the value of AM increases, the maximum value of AM gradually increases, which reflects the increase in the resonance degree of the pile-soil system. This may be because the larger the modulus of the soil interlayer, the harder the soil layer, which amplifies the interaction between the pile and the soil. At this time, the difference between the seismic response of the soil to the pile and the seismic response of the soil in the free field also gradually increases.

[0171] Figure 8 The effect of different thickness of soft soil layers on I v and AM curve. Figure 8 It can be seen from (a) to (b) that the thicker the soft soil layer, the greater the I v The greater the fluctuation trend of the curve, the greater the influence of the pipe pile on the soil free field, and then tends to be stable. The peak value of AM gradually decreases with the increase of soil thickness, and the corresponding frequency also gradually increases. This is because as the thickness of the soft soil layer increases, the soil's restriction on the pipe pile becomes smaller, thereby reducing the displacement difference between the pile top and the bottom rigid bedrock.

[0172] The above description is only a preferred embodiment of the present invention and is not intended to limit the protection scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention are included in the protection scope of the present invention.

Claims

1. A method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect, characterized in that: The following steps are involved: Step S1: setting initial parameters for the iteration factor γ; Step S2: Calculate the attenuation function φ according to the iteration factor; Step S3: Calculate the intermediate function k according to the attenuation function φ k and the intermediate function c k ; Step S4: Calculate the soil free field displacement and calculate the soil free field displacement and the intermediate function k k and the intermediate function c k Get the displacement value of the pipe pile and the virtual soil pile at the bottom of the pile; Step S5: Calculate a new iteration factor; Step S6: Repeat steps S2 to S5 until the difference between the latest iteration factor and the iteration factor of the previous round is less than a fixed value, and take the free field displacement, the displacement value of the pipe pile and the virtual soil pile at the bottom obtained according to the latest iteration factor to obtain the final vertical seismic response.

2. A method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect according to claim 1, characterized in that: The specific calculation formula of the attenuation function φ in step S2 is: Among them, K0 is the second kind of zero-order modified Bessel function, r represents the transverse coordinate of the cylindrical coordinate system, and r2 represents the outer diameter of the pile.

3. The method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect according to claim 1, characterized in that: The intermediate function k in step S3 k and the intermediate function c k The calculation formulas are: In the formula, represents the shear modulus of the soil, is an intermediate function, r represents the lateral coordinate of the cylindrical coordinate system, and r2 represents the outer diameter of the pile; represents the equivalent elastic modulus of soil, v sk is Poisson's ratio.

4. The method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect according to claim 3, characterized in that: In step S4, the free field displacement w fk The calculation formula is: In fk =in g +in rk In the formula, w rk represents the vertical displacement of the soil free field relative to the rigid bedrock, w g represents the amplitude of bedrock motion; where w rk The calculation formula is: w rk =B k cos(λ k z)+C k sin(-λ k z)-w g , among themH k-1 ≤z≤H k In the formula, B k and C k All of them are unknown parameters. z represents the vertical coordinate of the cylindrical coordinate system. H k represents the total height of the k-layer soil layer, λ k is an intermediate function, and Where ω represents the circular excitation frequency, ρ sk is the soil density.

5. The method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect according to claim 4, characterized in that: The calculation formula of the pipe pile displacement in step S4 is: Where D k and F k Unknown parameters, e is a natural constant, α pk , are all intermediate functions, and: In the formula, E p is the pile modulus, ρ p is the pile density, A p Indicates the cross-sectional area of ​​the pile, m k represents the mass of the soil plug, r1 represents the inner diameter of the pile, k rk represents the spring coefficient of the soil plug, c rk Represents the damping coefficient of the soil plug, in imaginary units ξ pk is an intermediate function, and 6. A method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect according to claim 5, characterized in that: The calculation formula of the displacement of the virtual soil pile at the pile bottom in step S4 is: In the formula, α sk , are all intermediate functions, among which, In the formula, E sk is the Young's modulus of the kth soil layer, the intermediate function 7. A method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect according to claim 6, characterized in that: The new iteration factor in step S5 is calculated by the following formula: In the formula, n represents the number of virtual soil pile layers, w k represents the vertical displacement of the kth layer of soil, and t is the time coefficient.

8. The method for determining the vertical seismic response of friction type pipe piles in layered foundation considering soil plug effect according to claim 1, characterized in that: The fixed value in step S6 is 0.001.

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