CN110162908B - Horizontal vibration analysis method for radial heterogeneous saturated soil pipe pile - Google Patents

Horizontal vibration analysis method for radial heterogeneous saturated soil pipe pile Download PDF

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CN110162908B
CN110162908B CN201910459458.6A CN201910459458A CN110162908B CN 110162908 B CN110162908 B CN 110162908B CN 201910459458 A CN201910459458 A CN 201910459458A CN 110162908 B CN110162908 B CN 110162908B
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CN110162908A (en
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崔春义
梁志孟
王本龙
辛宇
孟坤
刘海龙
赵九野
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Dalian Maritime University
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Abstract

The invention discloses a horizontal vibration analysis method for a pipe pile in radial heterogeneous saturated soil, which comprises the steps of taking the coupling effect between solid and liquid phases into consideration by adopting a radial heterogeneous saturated soil model and a Biot porous medium theoretical model, establishing and solving a pile foundation horizontal vibration response model of the radial heterogeneous saturated soil model under a plane strain condition, and obtaining an impedance function of the pipe pile. The plane strain assumption adopted by the method can simply process the relatively complex actual engineering situation, and has clear concept and strong theoretical performance; meanwhile, the coupling effect between solid and liquid phases is considered in the Biot porous medium model, and the Biot porous medium model is closer to the actual working condition than a single-phase medium. The radial heterogeneous performance considers the construction disturbance effect of the soil body around the pile, and can provide theoretical guidance and reference for the research of more complex saturated soil-pile dynamic interaction problems.

Description

Horizontal vibration analysis method for radial heterogeneous saturated soil pipe-in-pipe pile
Technical Field
The invention relates to the field of civil engineering, in particular to a horizontal vibration analysis method for a radial heterogeneous saturated soil-in-pipe pile.
Background
The study of pile-soil coupling vibration characteristics is a theoretical basis in the engineering technical fields of pile foundation earthquake resistance, earthquake-proof design, pile foundation power detection and the like, and is a hot point problem of geotechnical engineering and solid mechanics all the time.
The research about the saturated soil-pile horizontal coupling vibration problem is developed based on a homogeneous saturated soil medium model, the model regards soil around the pile as homogeneous or longitudinal layering, and in the pile foundation construction process, due to the influences of soil squeezing, loosening and other factors, the properties and parameters of the soil can be changed to different degrees in different ranges around the pile, namely, the radial heterogeneous effect. At the moment, the adoption of a radial heterogeneous saturated soil model is more suitable, and the influence of a pile core soil body on the horizontal vibration of the pipe pile is considered. In addition, the Biot porous medium theoretical model considers the coupling effect between solid and liquid phases, and has more complexity and higher applicability compared with the traditional ideal (single-phase) medium.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a method for analyzing horizontal vibration of a tubular pile in radial heterogeneous saturated soil.
In order to realize the purpose, the technical scheme of the invention is as follows:
the horizontal vibration analysis method for the pipe pile in the radial heterogeneous saturated soil is characterized by comprising the following steps
S1: the following assumptions are introduced, and a vibration model of the radial heterogeneous saturated soil pipe-in-pipe pile under horizontal excitation under the plane strain condition is established:
(1) assuming that the tubular pile is a linear elastic homogeneous uniform-section circular Bernoulli-Euler beam model, neglecting the shear deformation of a pile body, and adopting a fixed support at the pile end;
(2) the soil body around the pile is divided into an inner area and an outer area, the inner area is divided into n circle layers, and the soil body of each circle layer and the pile core soil are homogeneous and isotropic two-phase saturated elastic media;
(3) the pile-soil system vibrates in small deformation, the pile-soil interface is in complete contact without separation and slippage, the pile-soil interface is impervious, the displacement of two sides of the soil interface of each ring layer is continuous, and the stress is balanced;
(4) when the pile foundation horizontally vibrates, the soil around the pile and the pile core soil have no vertical deformation;
s2: establishing a motion equation of saturated soil bodies of all circle layers, a motion equation of pile core soil and a pile body horizontal vibration basic equation under a plane strain condition based on a Biot two-phase medium fluctuation theory;
establishing pile-soil system boundary conditions according to the assumption in the step S1;
s3: using laplace transform to solve the motion equation of the saturated soil body of each circle layer and the motion equation of the pile core soil under the plane strain condition established in the step S2 to obtain the horizontal dynamic impedance of the tubular pile so as to analyze the horizontal vibration of the tubular pile in the radially inhomogeneous saturated soil.
Preferably, in the step S2, the equation of motion of the saturated soil body of each circle layer under the planar strain condition is
Figure BDA0002077602570000021
Figure BDA0002077602570000022
Figure BDA0002077602570000023
Figure BDA0002077602570000024
The equation of motion of the pile core soil is
Figure BDA0002077602570000025
Figure BDA0002077602570000026
Figure BDA0002077602570000027
Figure BDA0002077602570000028
The basic equation of horizontal vibration of the pile body is
Figure BDA0002077602570000029
In the above formulae, the symbols have the following meanings:
j is 1-n, the number sequence of the ring layers is 1, the ring layer adjacent to the tubular pile is numbered, and n is the total number of the ring layers;
r and theta are coordinates of a cylindrical coordinate system, wherein the zero point of a z axis of the cylindrical coordinate system is positioned at the center of the circle of the upper surface of the tubular pile, the positive direction of the z axis is vertically downward, the direction of the r axis is a horizontal direction, and the zero point is positioned at the center of the circle of the upper surface of the tubular pile;
Figure BDA0002077602570000031
are different operator symbols;
u rj is the radial displacement of the soil skeleton in the saturated soil body of the jth circle layer u θj Is the circumferential displacement, w, of the soil framework in the j-th circle of saturated soil body rj Is the radial displacement of the fluid in the saturated soil body of the j-th circle layer relative to the soil framework, w θj The circumferential displacement of the fluid in the j-th circle of saturated soil body relative to the soil framework is obtained;
u r0 is the radial displacement of the soil skeleton in the pile core soil, u θ0 Is the circumferential displacement, w, of the soil skeleton in the pile core soil r0 Is the radial displacement of the fluid in the pile core soil relative to the soil skeleton, w θ0 The circumferential displacement of the fluid in the pile core soil relative to the soil framework is obtained;
ρ j ＝(1-n j )ρ sj +n j ρ fj is the density of the saturated soil body of the jth circle layer, wherein rho fj The density and rho of the fluid in the saturated soil body of the jth circle layer sj The density of soil particles in the saturated soil body of the jth circle layer, n j The porosity of the saturated soil body of the jth circle layer;
ρ 0 ＝(1-n 0 )ρ s0 +n 0 ρ f0 is the soil mass density of the pile core soil, where p f0 Density, rho, of fluid in the pile core soil body s0 Is the density of soil particles in the pile core soil body, n 0 The porosity of the pile core soil body;
m j ＝ρ j /n j is the viscous coupling coefficient of the layer soil skeleton and pore fluid of the j-th circle, b j ＝ρ fj g/k dj Is the Darcy law permeability coefficient of the soil body of the jth circle layer, m 0 ＝ρ 0 /n 0 Is the viscous coupling coefficient of the soil skeleton and pore fluid in the pile core soil, b 0 ＝ρ f0 g/k d0 The permeability coefficient of the pile core soil body is Darcy's law, and g is the gravity acceleration;
λ j shear modulus of soil body, G, of saturated soil body of j circle layer j Is Lame constant, upsilon of a saturated soil body of a jth circle layer sj The poisson ratio of the saturated soil body of the jth circle layer is defined; k sj Is the volume compression modulus, K, of soil particles in the saturated soil body of the jth circle fj Is the volume compression modulus, K, of the fluid in the saturated soil body of the jth circle bj ＝λ j +(23)×G j The volume compression modulus of a soil framework in the j-th circle of saturated soil body; k dj ＝K sj [1+n j (K sj /K fj -1)]；
λ 0 Shear modulus of soil body, G, of pile core soil 0 Lame constant, upsilon, of pile core soil s0 The Poisson's ratio of the pile core soil is taken as the ratio; k s0 Is the volume compression modulus, K, of soil particles in the pile core soil f0 Is the bulk compression modulus, K, of the fluid in the core soil b0 ＝λ 0 +(23)×G 0 The volume compression modulus of a soil framework in the pile core soil is shown; k d0 ＝K s0 [1+n 0 (K s0 /K f0 -1)]；
α j ＝1-K bj /K sj ，
Figure BDA0002077602570000032
Respectively representing the compressibility of soil particles and fluid in the saturated soil body of the j-th circle of layer;
α 0 ＝1-K b0 /K s0 ，
Figure BDA0002077602570000033
respectively representing the compressibility of soil particles and fluid in the pile core soil;
u p horizontally shifting the pile body of the tubular pile;
E p is shear modulus, I, of the pipe pile p Is the inertia moment of the tubular pile, A p Is the cross-sectional area of the tubular pile, m p Is the mass per unit length of the tubular pile, N 1 The transverse acting force of the axial unit length of the soil around the pile on the pile body of the pipe pile during horizontal vibration is N 0 The transverse acting force of axial unit length pile core soil on a pile body of the pipe pile during horizontal vibration.
Preferably, in step S2, the boundary condition of the pile-soil system is a boundary condition of a pile top of the pipe pile
Figure BDA0002077602570000041
Figure BDA0002077602570000042
Boundary condition of pile bottom of tubular pile
u p | z＝H ＝0
Figure BDA0002077602570000043
Pile-soil full contact of the soil around the pile and waterproof conditions of the pile-soil interface
Figure BDA0002077602570000044
Continuous condition between loop layers
Figure BDA0002077602570000045
Limited displacement condition of pile core soil center
Figure BDA0002077602570000046
Pile-soil full contact of pile core soil and waterproof condition of pile-soil interface
Figure BDA0002077602570000047
In the above formulae, the symbols mean
j is 1-n, the number sequence of the ring layers is 1, the ring layer adjacent to the tubular pile is numbered, and n is the total number of the ring layers;
r and theta are coordinates of a cylindrical coordinate system, wherein the zero point of a z axis of the cylindrical coordinate system is positioned at the center of the circle of the upper surface of the tubular pile, the positive direction of the z axis is vertically downward, the direction of the r axis is a horizontal direction, and the zero point is positioned at the center of the circle of the upper surface of the tubular pile;
h is the length of the tubular pile, r 1 The radius of the tubular pile;
u p horizontally shifting the pile body of the tubular pile;
p (t) pile top excitation;
u rj is the radial displacement of the soil skeleton in the saturated soil body of the jth circle layer u θj Is the circumferential displacement, w, of the soil framework in the j-th circle of saturated soil body rj Is the radial displacement of the fluid in the saturated soil body of the j-th circle layer relative to the soil framework, w θj The circumferential displacement of the fluid in the saturated soil body of the jth circle layer relative to the soil framework is obtained; u. of r0 Is the radial displacement of the soil skeleton in the pile core soil, u θ0 Is the circumferential direction of a pile core soil-in-soil frameworkDisplacement, w r0 Is the radial displacement of the fluid in the pile core soil relative to the soil skeleton, w θ0 The circumferential displacement of the fluid in the pile core soil relative to the soil framework is obtained;
σ rj is the normal stress on the interface of the j circle saturated soil body and the j +1 circle saturated soil body θj The tangential shear stress on the interface of the j-th circle of saturated soil and the j + 1-th circle of saturated soil is obtained.
Preferably, in the step S3, solving the motion equation of the saturated soil body and the pile core soil of each circle layer and the basic equation of the horizontal vibration of the pile body under the plane strain condition includes the following steps
S31: respectively introducing potential functions to soil skeleton and fluid in the saturated soil body of the jth circle layer
Figure BDA0002077602570000051
ψ sj ，
Figure BDA0002077602570000052
ψ fj
Figure BDA0002077602570000053
S32: potential function
Figure BDA0002077602570000054
ψ sj ，
Figure BDA0002077602570000055
ψ fj Introducing a motion equation of a j-th circle saturated soil body under the condition of plane strain to perform Laplace transformation to obtain
Figure BDA0002077602570000056
Figure BDA0002077602570000057
Figure BDA0002077602570000058
ρ fj s 2 Ψ sj +m j s 2 Ψ fj +b j sΨ fj ＝0
S33: solving the potential function to obtain
Φ sj ＝[A j1 K 1 (β j1 r)+B j1 I 1 (β j1 r)+A j2 K 1 (β j2 r)+B j2 I 1 (β j2 r)]cosθ
Φ fj ＝[C j1 K 1 (β j1 r)+D j1 I 1 (β j1 r)+C j2 K 1 (β j2 r)+D j2 I 1 (β j2 r)]cosθ
Ψ sj ＝[A j3 K 1 (β j3 r)+B j3 I 1 (β j3 r)]sinθ
Ψ fj ＝[C j3 K 1 (β j3 r)+D j3 I 1 (β j3 r)]sinθ
S34: bringing back the potential function to obtain each displacement and stress expression containing undetermined coefficient
U rj ＝{A j1 [-β j1 K 0 (β j1 r)-K 1 (β j1 r)/r]+A j2 [-β j2 K 0 (β j2 r)-K 1 (β j2 r)/r]+A j3 K 1 (β j3 r)/r+B j1 [β j1 I 0 (β j1 r)-I 1 (β j1 r)/r]+B j2 [-β j2 I 0 (β j2 r)-I 1 (β j2 r)/r]+B j3 I 1 (β j3 r)/r}cosθU θj ＝{-A j1 K 1 (β j1 r)/r-A j2 K 1 (β j2 r)/r-A j3 [-β j3 K 0 (β j3 r)-K 1 (β j3 r)/r]-B j1 I 1 (β j1 r)-B j2 I 1 (β j2 r)/r-B j3 [β j3 I 0 (β j3 r)-I 1 (β j3 r)/r]}sinθ
W rj ＝{C j1 [-β j1 K 0 (β j1 r)-K 1 (β j1 r)/r]+C j2 [-β j2 K 0 (β j2 r)-K 1 (β j2 r)/r]+C j3 K 1 (β j3 r)/r+D j1 [β j1 I 0 (β j1 r)-I 1 (β j1 r)/r]+D j2 [-β j2 I 0 (β j2 r)-I 1 (β j2 r)/r]+D j3 I 1 (β j3 r)/r}cosθW θj ＝{-C j1 K 1 (β j1 r)/r-C j2 K 1 (β j2 r)/r-C j3 [-β j3 K 0 (β j3 r)-K 1 (β j3 r)/r]-D j1 I 1 (β j1 r)-D j2 I 1 (β j2 r)/r-D j3 [β j3 I 0 (β j3 r)-I 1 (β j3 r)/r]}sinθ
Figure BDA0002077602570000061
τ rθj ＝A j1 G j {-2[K 1 (β j1 r)]'/r+2K 1 (β j1 r)/r 2 }sinθ+A j2 G j {-2[K 1 (β j2 r)]'/r+2K 1 (β j2 r)/r 2 }sinθ+A j3 G j {[K 1 (β j3 r)]'/r-2K 1 (β j3 r)/r 2 -[K 1 (β j3 r)]”}sinθ+B j1 G j {-2[I 1 (β j1 r)]'/r+2I 1 (β j1 r)/r 2 }sinθ+B j2 G j {-2[I 1 (β j2 r)]'/r+2I 1 (β j2 r)/r 2 }sinθ+B j3 G j {[I 1 (β j3 r)]'/r-2I 1 (β j3 r)/r 2 -[I 1 (β j3 r)]”}sinθ
S35: substituting the expression of the previous step into the boundary condition, and solving to obtain the undetermined coefficient A j1 ，A j2 ，A j3 ，B j1 ，B j2 ，B j3 ，C j1 ，C j2 ，C j3 ，D j1 ，D j2 ，D j3 ；
S36: the transverse acting force of the soil around the pile on the pile body is
Figure BDA0002077602570000062
S37: respectively introducing potential functions to soil skeleton and fluid of pile core
Figure BDA0002077602570000063
ψ s0 ,
Figure BDA0002077602570000064
ψ f0
Figure BDA0002077602570000065
S38: potential function
Figure BDA0002077602570000066
ψ s0 ,
Figure BDA0002077602570000067
ψ f0 Introducing the motion equation of the pile core soil under the condition of plane strain to perform Laplace transformation to obtain
Figure BDA0002077602570000071
Figure BDA0002077602570000072
Figure BDA0002077602570000073
ρ f0 s 2 Ψ s0 +m 0 s 2 Ψ f0 +b 0 sΨ f0 ＝0
S39: solving the potential function to obtain
Φ s0 ＝[B 01 I 1 (β 01 r)+B 02 I 1 (β 02 r)]cosθ
Φ f0 ＝[B 03 I 1 (β 01 r)+B 04 I 1 (β 02 r)]cosθ
Ψ s0 ＝B 05 I 1 (β 03 r)sinθ
Ψ f0 ＝B 06 I 1 (β 03 r)sinθ
S310: bringing back the potential function to obtain each displacement expression containing undetermined coefficients
U r0 ＝{B 01 [β 01 I 0 (β 01 r)-I 1 (β 01 r)/r]+B 02 [β 02 I 0 (β 02 r)-I 1 (β 02 r)/r]+B 05 I 1 (β 03 r)/r}cosθ
W r0 ＝{B 01 γ 01 [β 01 I 0 (β 01 r)-I 1 (β 01 r)/r]+B 02 γ 02 [β 02 I 0 (β 02 r)-I 1 (β 02 r)/r]+B 05 γ 03 I 1 (β 03 r)/r}cosθ
U θ0 ＝{-B 01 I 1 (β 01 r)-B 02 I 1 (β 02 r)/r-B 05 [β 03 I 0 (β 03 r)-I 1 (β 03 r)/r]}sinθ
W θ0 ＝{-B 01 γ 01 I 1 (β 01 r)-B 02 γ 02 I 1 (β 02 r)/r-B 05 γ 03 [β 03 I 0 (β 03 r)-I 1 (β 03 r)/r]}sinθ
S311: substituting the expression of the last step into boundary conditions, and solving to obtain undetermined coefficients
B 01 ＝(X 3 X 5 +X 5 X 9 -X 2 X 6 -X 6 X 8 )/κ
B 02 ＝(X 1 X 6 +X 6 X 7 -X 3 X 4 -X 4 X 9 )/κ；
B 03 ＝(X 2 X 4 +X 4 X 8 -X 1 X 5 -X 5 X 7 )/κ
S312: the transverse acting force of the pile core soil on the pile body is
Figure BDA0002077602570000074
S313: laplace transformation is carried out on the basic equation of horizontal vibration of the pile body
Figure BDA0002077602570000075
And solving to obtain
U p (z)＝Y 1 cos(ηz)+Y 2 sin(ηz)+Y 3 cosh(ηz)+Y 4 sinh(ηz)
Θ p (z)＝η[-Y 1 sin(ηz)+Y 2 cos(ηz)+Y 3 sinh(ηz)+Y 4 cosh(ηz)]
M p (z)＝-E p I p η 2 [-Y 1 cos(ηz)-Y 2 sin(ηz)+Y 3 cosh(ηz)+Y 4 sinh(ηz)]
Q p (z)＝-E p I p η 3 [Y 1 sin(ηz)-Y 2 cos(ηz)+Y 3 sinh(ηz)+Y 4 cosh(ηz)]
S314: substituting the expression of the previous step into the boundary condition, and solving to obtain the undetermined coefficient
Figure BDA0002077602570000081
Figure BDA0002077602570000082
Figure BDA0002077602570000083
Figure BDA0002077602570000084
S315: calculating the horizontal dynamic impedance of the tubular pile
K QU ＝Q p (0)/U p (0)
Can also be expressed in dimensionless form as its real and imaginary parts
Figure BDA0002077602570000085
Figure BDA0002077602570000086
In the above expression, each symbol means
s is i omega, wherein s is Laplace transform, i is an imaginary number unit, and omega is the excitation load frequency;
j is 1-n, the number sequence of the ring layers is 1, the ring layer adjacent to the tubular pile is numbered, and n is the total number of the ring layers;
r and theta are coordinates of a cylindrical coordinate system, wherein the zero point of the z axis of the cylindrical coordinate system is positioned at the center of the circle of the upper surface of the tubular pile, the positive direction of the z axis is vertically downward, the direction of the r axis is the horizontal direction, and the zero point is positioned at the center of the circle of the upper surface of the tubular pile;
Figure BDA0002077602570000087
and psi sj Is the radial displacement u of the soil skeleton in the saturated soil body of the jth circle layer rj And the circumferential displacement u θj Is determined by the potential function of (a) a,
Figure BDA0002077602570000088
ψ fj the radial displacement w of the fluid in the saturated soil body of the j-th circle layer relative to the soil framework rj And circumferential displacement w θj A potential function of (d);
Figure BDA0002077602570000089
and psi s0 For radial displacement u of soil skeleton in pile core soil r0 And the circumferential displacement u θ0 Is determined by the potential function of (a) a,
Figure BDA00020776025700000810
ψ f0 for radial displacement w of fluid in pile core soil relative to soil skeleton r0 And circumferential displacement w θ0 A potential function of (a);
Φ sj is a potential function
Figure BDA00020776025700000811
Of the laplace transform, Ψ sj Is a potential function psi sj Laplace transform of phi fj Is a potential function
Figure BDA00020776025700000812
Of the laplace transform, Ψ fj Is a potential function psi fj (ii) a laplace transform of; phi (phi) of s0 Is a potential function
Figure BDA00020776025700000813
Of the laplace transform, Ψ s0 Is a potential function psi s0 Laplace transform of phi f0 Is a potential function
Figure BDA00020776025700000814
Of the laplace transform, Ψ f0 Is a potential function psi f0 (ii) a laplace transform of;
ρ j ＝(1-n j )ρ sj +n j ρ fj for the saturated soil density of the jth circle layer, where p fj The density and rho of the fluid in the saturated soil body of the jth circle layer sj The density of soil particles in the saturated soil body of the jth circle layer, n j The porosity of the saturated soil body of the jth circle layer;
ρ 0 ＝(1-n 0 )ρ s0 +n 0 ρ f0 is the soil mass density of the pile core soil, where p f0 Density, rho, of fluid in the pile core soil body s0 Is the density of soil particles in the pile core soil body, n 0 The porosity of the pile core soil body;
m j ＝ρ j /n j is the viscous coupling coefficient of the layer soil skeleton and pore fluid of the j-th circle, b j ＝ρ fj g/k dj Is the Darcy law permeability coefficient of the soil body of the jth circle layer, m 0 ＝ρ 0 /n 0 Is the viscous coupling coefficient of soil skeleton and pore fluid in the pile core soil, b 0 ＝ρ f0 g/k d0 The permeability coefficient of the pile core soil body is Darcy's law, and g is the gravity acceleration;
λ j shear modulus of soil body, G, of saturated soil body of j circle layer j Is Lame constant, upsilon of a saturated soil body of a jth circle layer sj The Poisson ratio of the saturated soil body of the jth circle layer is obtained; k sj Is the volume compression modulus, K, of soil particles in the saturated soil body of the jth circle fj Is the volume compression modulus, K, of the fluid in the saturated soil body of the jth circle bj ＝λ j +(2/3)×G j The volume compression modulus of the soil framework in the j-th circle of saturated soil body is obtained; k is dj ＝K sj [1+n j (K sj /K fj -1)]；
λ 0 Shear modulus of soil body, G, of pile core soil 0 Lame constant, upsilon, of pile core soil s0 The Poisson's ratio of the pile core soil is; k s0 For volume pressure of soil particles in pile core soilModulus of contraction, K f0 Is the bulk compression modulus, K, of the fluid in the core soil b0 ＝λ 0 +(2/3)×G 0 The volume compression modulus of the soil framework in the pile core soil is obtained; k d0 ＝K s0 [1+n 0 (K s0 /K f0 -1)]；
α j ＝1-K bj /K sj ，
Figure BDA0002077602570000091
Respectively representing the constants of compressibility of soil particles and fluid in the saturated soil body of the j-th circle of layer; alpha is alpha 0 ＝1-K b0 /K s0 ，
Figure BDA0002077602570000092
Respectively representing the compressibility of soil particles and fluid in the pile core soil;
Figure BDA0002077602570000093
is an operator symbol;
Figure BDA0002077602570000094
the first derivative of the expression in parentheses on r is shown;
Figure BDA0002077602570000095
the expression in parentheses takes the second derivative of r;
Figure BDA0002077602570000096
Figure BDA0002077602570000097
is a shorthand notation in the calculation process;
Figure BDA0002077602570000098
Figure BDA0002077602570000099
is also a shorthand notation in the calculation process;
Figure BDA00020776025700000910
Figure BDA00020776025700000911
is also a shorthand notation in the calculation process;
η 4 ＝(m p s 2 +f 1 +f 0 )/E p I p is also a shorthand notation in the calculation process;
Figure BDA0002077602570000101
the first order, first class and second class deformed Bessel functions are respectively;
A j1 ，A j2 ，A j3 ，B j1 ，B j2 ，B j3 ，C j1 ，C j2 ，C j3 ，D j1 ，D j2 ，D j3 is a symbol to be determined, and the following relationship exists
Figure BDA0002077602570000102
Y 1 ，Y 2 ，Y 3 ，Y 4 Is also a pending symbol;
X 1 ＝β 01 I 0 (β 01 r 0 )-I 1 (β 01 r 0 )/r 0 ，X 2 ＝β 02 I 0 (β 02 r 0 )-I 1 (β 02 r 0 )/r 0 ，X 3 ＝-I 1 (β 03 r 0 )/r 0 ，X 4 ＝γ 01 [β 01 I 0 (β 01 r 0 )-I 1 (β 01 r 0 )/r 0 ]，X 5 ＝γ 02 [β 02 I 0 (β 02 r 0 )-I 1 (β 02 r 0 )/r 0 ]，X 6 ＝γ 03 I 1 (β 03 r 0 )/r 0 ，X 7 ＝-I 1 (β 01 r 0 )/r 0 ，X 8 ＝-I 1 (β 02 r 0 )/r 0 ，X 9 ＝-[β 03 I 0 (β 03 r 0 )-I 1 (β 03 r 0 )/r 0 ]，κ＝X 1 X 5 X 9 +X 3 X 4 X 8 +X 2 X 6 X 7 -X 1 X 5 X 9 -X 1 X 6 X 8 -X 2 X 4 X 9 is also a shorthand notation in the calculation process;
Figure BDA0002077602570000103
Figure BDA0002077602570000104
a 5 ＝η[a 3 cos(ηH)-a 1 sin(ηH)]，a 6 ＝-η[a 2 sin(ηH)+a 4 cos(ηH)]and a 7 ＝η[sin(ηH)cosh(ηH)+cos(ηH)sinh(ηH)]And is also a shorthand notation in the calculation process.
Preferably, in the step S1, the method for determining the shear modulus of the soil body of the jth circle is to
Figure BDA0002077602570000105
Wherein G (r) is the shear modulus of the soil at the position where the distance between the jth circle of soil and the center of the pile-soil interface is r
Figure BDA0002077602570000106
Wherein f (r) is a function of the change in shear modulus of the earth
Figure BDA0002077602570000107
Wherein GR is a parameter for describing the disturbance degree of the construction of the soil body around the pile, and GR is a parameter for describing the disturbance degree of the construction of the soil body around the pile>1 is soil hardening GR<1 is softening of soil, GR is 1 is uniform soil, q is positive index, r is 1 The radius of the pipe pile is b, and the radius of the soil body in the radial heterogeneous region is b.
According to the technical scheme, the horizontal vibration of the pipe pile is analyzed by adopting the radial heterogeneous saturated soil model based on the plane strain model, the adopted plane strain hypothesis can simply process the relatively complex actual engineering situation, the concept is clear, and the theoretical performance is strong; meanwhile, the Biot porous medium model considers the coupling effect between solid and liquid phases and is closer to the actual working condition compared with a single-phase medium. The radial heterogeneous performance considers the construction disturbance effect of the soil body around the pile and the influence of the pile core soil body on the horizontal vibration of the pipe pile, and can provide theoretical guidance and reference for the research of the more complex saturated soil-pile dynamic interaction problem.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a model schematic of the present invention.
Detailed Description
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
In the following detailed description of the embodiments of the present invention, in order to clearly illustrate the structure of the present invention and to facilitate explanation, it should be understood that the structure shown in the drawings is not drawn to general scale and is partially enlarged, modified or simplified, so that the present invention is not limited thereto.
In the following detailed description of the present invention, reference is made to FIG. 1, which is a flow chart of the method of the present invention. As shown in the figure.
The horizontal vibration analysis method for the radial heterogeneous saturated soil-in-pipe pile is characterized by comprising the following steps
The following assumptions are introduced, and a vibration model of the radial heterogeneous saturated soil pipe-in-pipe pile under horizontal excitation under the plane strain condition is established:
(1) assuming that the tubular pile is a linear elastic homogeneous uniform-section circular Bernoulli-Euler beam model, neglecting the shear deformation of a pile body, and adopting a fixed support at the pile end;
(2) the soil around the pile is divided into an inner area and an outer area, the inner area is divided into n circle layers, and the soil of each circle layer and the pile core soil are homogeneous and isotropic two-phase saturated elastic media;
(3) the pile-soil system vibrates in small deformation, the pile-soil interface is in complete contact without separation and slippage, the pile-soil interface is impervious to water, and the two sides of the soil interface of each ring layer are continuously displaced and have balanced stress;
(4) when the pile foundation horizontally vibrates, the soil body around the pile and the pile core soil have no vertical deformation.
The method for determining the shear modulus of the soil body of the jth circle layer comprises the following steps
Figure BDA0002077602570000111
Wherein G (r) is the shear modulus of the soil at the position where the distance between the jth circle of soil and the center of the pile-soil interface is r
Figure BDA0002077602570000121
Wherein f (r) is a function of the change in shear modulus of the earth
Figure BDA0002077602570000122
Wherein GR is a parameter for describing the disturbance degree of the construction of the soil body around the pile, and GR is a parameter for describing the disturbance degree of the construction of the soil body around the pile>1 is hardening of soil, GR<1 is softening of soil, GR is 1 is uniform soil, q is positive index, r is 1 The radius of the pipe pile is b, and the radius of the soil body in the radial heterogeneous region is b.
S2: establishing a motion equation of saturated soil bodies of all circle layers, a motion equation of pile core soil and a pile body horizontal vibration basic equation under a plane strain condition based on a Biot two-phase medium fluctuation theory; and establishes pile-soil system boundary conditions based on the assumptions in step S1.
The equation of motion of the saturated soil body of each circle layer under the condition of plane strain is
Figure BDA0002077602570000123
Figure BDA0002077602570000124
Figure BDA0002077602570000125
Figure BDA0002077602570000126
The equation of motion of the pile core soil is
Figure BDA0002077602570000127
Figure BDA0002077602570000128
Figure BDA0002077602570000129
Figure BDA00020776025700001210
The basic equation of horizontal vibration of the pile body is
Figure BDA00020776025700001211
The boundary condition of the pile-soil system is the boundary condition of the pile top of the pipe pile
Figure BDA00020776025700001212
Figure BDA0002077602570000131
Boundary condition of pile bottom of tubular pile
u p | z＝H ＝0
Figure BDA0002077602570000132
Pile-soil full contact of the soil around the pile and waterproof conditions of the pile-soil interface
Figure BDA0002077602570000133
Continuous condition between loop layers
Figure BDA0002077602570000134
Limited displacement condition of pile core soil center
Figure BDA0002077602570000135
Pile-soil full contact of pile core soil and waterproof condition of pile-soil interface
Figure BDA0002077602570000136
S3: using laplace transform to solve the motion equation of the saturated soil body of each circle layer and the motion equation of the pile core soil under the plane strain condition established in the step S2 to obtain the horizontal dynamic impedance of the tubular pile so as to analyze the horizontal vibration of the tubular pile in the radially inhomogeneous saturated soil. Comprises the following specific steps
S31: respectively introducing potential functions to soil skeleton and fluid in the saturated soil body of the jth circle layer
Figure BDA0002077602570000137
ψ sj ，
Figure BDA0002077602570000138
ψ fj
Figure BDA0002077602570000139
S32: potential function
Figure BDA00020776025700001310
ψ sj ，
Figure BDA00020776025700001311
ψ fj Introducing the motion equation of the saturated soil body of the j-th circle layer under the plane strain condition, and performing Laplace transformation to obtain
Figure BDA00020776025700001312
Figure BDA00020776025700001313
Figure BDA00020776025700001314
ρ fj s 2 Ψ sj +m j s 2 Ψ fj +b j sΨ fj ＝0
S33: solving the potential function to obtain
Φ sj ＝[A j1 K 1 (β j1 r)+B j1 I 1 (β j1 r)+A j2 K 1 (β j2 r)+B j2 I 1 (β j2 r)]cosθ
Φ fj ＝[C j1 K 1 (β j1 r)+D j1 I 1 (β j1 r)+C j2 K 1 (β j2 r)+D j2 I 1 (β j2 r)]cosθ
Ψ sj ＝[A j3 K 1 (β j3 r)+B j3 I 1 (β j3 r)]sinθ
Ψ fj ＝[C j3 K 1 (β j3 r)+D j3 I 1 (β j3 r)]sinθ
S34: bringing back the potential function to obtain each displacement and stress expression containing undetermined coefficient
U rj ＝{A j1 [-β j1 K 0 (β j1 r)-K 1 (β j1 r)/r]+A j2 [-β j2 K 0 (β j2 r)-K 1 (β j2 r)/r]+A j3 K 1 (β j3 r)/r+B j1 [β j1 I 0 (β j1 r)-I 1 (β j1 r)/r]+B j2 [-β j2 I 0 (β j2 r)-I 1 (β j2 r)/r]+B j3 I 1 (β j3 r)/r}cosθ
U θj ＝{-A j1 K 1 (β j1 r)/r-A j2 K 1 (β j2 r)/r-A j3 [-β j3 K 0 (β j3 r)-K 1 (β j3 r)/r]-B j1 I 1 (β j1 r)-B j2 I 1 (β j2 r)/r-B j3 [β j3 I 0 (β j3 r)-I 1 (β j3 r)/r]}sinθ
W rj ＝{C j1 [-β j1 K 0 (β j1 r)-K 1 (β j1 r)/r]+C j2 [-β j2 K 0 (β j2 r)-K 1 (β j2 r)/r]+C j3 K 1 (β j3 r)/r+D j1 [β j1 I 0 (β j1 r)-I 1 (β j1 r)/r]+D j2 [-β j2 I 0 (β j2 r)-I 1 (β j2 r)/r]+D j3 I 1 (β j3 r)/r}cosθW θj ＝{-C j1 K 1 (β j1 r)/r-C j2 K 1 (β j2 r)/r-C j3 [-β j3 K 0 (β j3 r)-K 1 (β j3 r)/r]-D j1 I 1 (β j1 r)-D j2 I 1 (β j2 r)/r-D j3 [β j3 I 0 (β j3 r)-I 1 (β j3 r)/r]}sinθ
Figure BDA0002077602570000141
τ rθj ＝A j1 G j {-2[K 1 (β j1 r)]'/r+2K 1 (β j1 r)/r 2 }sinθ+A j2 G j {-2[K 1 (β j2 r)]'/r+2K 1 (β j2 r)/r 2 }sinθ+A j3 G j {[K 1 (β j3 r)]'/r-2K 1 (β j3 r)/r 2 -[K 1 (β j3 r)]”}sinθ+B j1 G j {-2[I 1 (β j1 r)]'/r+2I 1 (β j1 r)/r 2 }sinθ+B j2 G j {-2[I 1 (β j2 r)]'/r+2I 1 (β j2 r)/r 2 }sinθ+B j3 G j {[I 1 (β j3 r)]'/r-2I 1 (β j3 r)/r 2 -[I 1 (β j3 r)]”}sinθ
S35: substituting the expression of the last step into the boundary condition, and solving to obtain undetermined coefficient A j1 ，A j2 ，A j3 ，B j1 ，B j2 ，B j3 ，C j1 ，C j2 ，C j3 ，D j1 ，D j2 ，D j3 ；
S36: the transverse acting force of the soil around the pile on the pile body is
Figure BDA0002077602570000151
S37: respectively introducing potential functions to soil skeleton and fluid of pile core
Figure BDA0002077602570000152
ψ s0 ,
Figure BDA0002077602570000153
ψ f0
Figure BDA0002077602570000154
S38: potential function
Figure BDA0002077602570000155
ψ s0 ,
Figure BDA0002077602570000156
ψ f0 Introducing the motion equation of the pile core soil under the condition of plane strain to perform Laplace transformation to obtain
Figure BDA0002077602570000157
Figure BDA0002077602570000158
Figure BDA0002077602570000159
ρ f0 s 2 Ψ s0 +m 0 s 2 Ψ f0 +b 0 sΨ f0 ＝0
S39: solving the potential function to obtain
Φ s0 ＝[B 01 I 1 (β 01 r)+B 02 I 1 (β 02 r)]cosθ
Φ f0 ＝[B 03 I 1 (β 01 r)+B 04 I 1 (β 02 r)]cosθ
Ψ s0 ＝B 05 I 1 (β 03 r)sinθ
Ψ f0 ＝B 06 I 1 (β 03 r)sinθ
S310: bringing back the potential function to obtain each displacement and stress expression containing undetermined coefficient
U r0 ＝{B 01 [β 01 I 0 (β 01 r)-I 1 (β 01 r)/r]+B 02 [β 02 I 0 (β 02 r)-I 1 (β 02 r)/r]+B 05 I 1 (β 03 r)/r}cosθ
W r0 ＝{B 01 γ 01 [β 01 I 0 (β 01 r)-I 1 (β 01 r)/r]+B 02 γ 02 [β 02 I 0 (β 02 r)-I 1 (β 02 r)/r]+B 05 γ 03 I 1 (β 03 r)/r}cosθ
U θ0 ＝{-B 01 I 1 (β 01 r)-B 02 I 1 (β 02 r)/r-B 05 [β 03 I 0 (β 03 r)-I 1 (β 03 r)/r]}sinθ
W θ0 ＝{-B 01 γ 01 I 1 (β 01 r)-B 02 γ 02 I 1 (β 02 r)/r-B 05 γ 03 [β 03 I 0 (β 03 r)-I 1 (β 03 r)/r]}sinθ
S311: substituting the expression of the previous step into the boundary condition, and solving to obtain the undetermined coefficient
B 01 ＝(X 3 X 5 +X 5 X 9 -X 2 X 6 -X 6 X 8 )/κ
B 02 ＝(X 1 X 6 +X 6 X 7 -X 3 X 4 -X 4 X 9 )/κ；
B 03 ＝(X 2 X 4 +X 4 X 8 -X 1 X 5 -X 5 X 7 )/κ
S312: the transverse acting force of the pile core soil on the pile body is
Figure BDA0002077602570000161
S313: laplace transformation is carried out on the basic equation of horizontal vibration of the pile body
Figure BDA0002077602570000162
And solving to obtain
U p (z)＝Y 1 cos(ηz)+Y 2 sin(ηz)+Y 3 cosh(ηz)+Y 4 sinh(ηz)
Θ p (z)＝η[-Y 1 sin(ηz)+Y 2 cos(ηz)+Y 3 sinh(ηz)+Y 4 cosh(ηz)]
M p (z)＝-E p I p η 2 [-Y 1 cos(ηz)-Y 2 sin(ηz)+Y 3 cosh(ηz)+Y 4 sinh(ηz)]
Q p (z)＝-E p I p η 3 [Y 1 sin(ηz)-Y 2 cos(ηz)+Y 3 sinh(ηz)+Y 4 cosh(ηz)]
S314: substituting the expression of the previous step into the boundary condition, and solving to obtain the undetermined coefficient
Figure BDA0002077602570000163
Figure BDA0002077602570000164
Figure BDA0002077602570000165
Figure BDA0002077602570000166
S315: calculating the horizontal dynamic impedance of the tubular pile
K QU ＝Q p (0)/U p (0)
Can also be expressed in a dimensionless form of its real and imaginary parts
Figure BDA0002077602570000167
Figure BDA0002077602570000168
In the above formulae, the symbols have the following meanings:
j is 1-n and is the number sequence of the ring layers, the ring layer number adjacent to the tubular pile is 1, and n is the total number of the ring layers;
r and theta are coordinates of a cylindrical coordinate system, wherein the zero point of the z axis of the cylindrical coordinate system is positioned at the center of the circle of the upper surface of the tubular pile, the positive direction of the z axis is vertically downward, the direction of the r axis is the horizontal direction, and the zero point is positioned at the center of the circle of the upper surface of the tubular pile;
Figure BDA0002077602570000171
are different operator symbols;
u rj is the radial displacement of the soil skeleton in the saturated soil body of the jth circle layer u θj Is the circumferential displacement, w, of the soil framework in the j-th circle of saturated soil body rj Is the jth turnRadial displacement, w, of fluid in a layer saturated soil body relative to a soil skeleton θj The circumferential displacement of the fluid in the saturated soil body of the jth circle layer relative to the soil framework is obtained;
u r0 is the radial displacement of the soil skeleton in the pile core soil, u θ0 Is the circumferential displacement, w, of the soil skeleton in the pile core soil r0 Is the radial displacement of the fluid in the pile core soil relative to the soil skeleton, w θ0 The circumferential displacement of the fluid in the pile core soil relative to the soil framework is obtained; sigma rj Is the normal stress on the interface of the j circle saturated soil body and the j +1 circle saturated soil body θj The tangential shear stress on the interface of the j circle saturated soil body and the j +1 circle saturated soil body is measured;
ρ j ＝(1-n j )ρ sj +n j ρ fj is the density of the saturated soil body of the jth circle layer, wherein rho fj The density, rho, of the fluid in the saturated soil body of the jth circle layer sj The density of soil particles in the saturated soil body of the jth circle layer, n j The porosity of the saturated soil body of the jth circle layer;
ρ 0 ＝(1-n 0 )ρ s0 +n 0 ρ f0 is the body density of the pile core soil, where ρ f0 Density, rho, of fluid in the pile core soil body s0 Is the density of soil particles in the pile core soil body, n 0 The porosity of the pile core soil body;
m j ＝ρ j /n j is the viscous coupling coefficient of the layer soil skeleton and pore fluid of the j-th circle, b j ＝ρ fj g/k dj Is the Darcy law permeability coefficient of the soil body of the jth circle layer, m 0 ＝ρ 0 /n 0 Is the viscous coupling coefficient of soil skeleton and pore fluid in the pile core soil, b 0 ＝ρ f0 g/k d0 The permeability coefficient of the pile core soil body is Darcy's law, and g is the gravity acceleration;
λ j shear modulus of soil body, G, of saturated soil body of jth circle layer j Is Lame constant, upsilon of a saturated soil body of a jth circle layer sj The Poisson ratio of the saturated soil body of the jth circle layer is obtained; k is sj Is the volume compression modulus, K, of soil particles in the saturated soil body of the jth circle fj Is the volume of fluid in the saturated soil body of the jth circle layerModulus of compression, K bj ＝λ j +(23)×G j The volume compression modulus of the soil framework in the j-th circle of saturated soil body is obtained; k dj ＝K sj [1+n j (K sj /K fj -1)]；
λ 0 Shear modulus of soil body, G, of pile core soil 0 Lame constant, upsilon, of pile core soil s0 The Poisson's ratio of the pile core soil is; k is s0 Is the volume compression modulus, K, of soil particles in the pile core soil f0 Is the bulk compression modulus, K, of the fluid in the core soil b0 ＝λ 0 +(2/3)×G 0 The volume compression modulus of a soil framework in the pile core soil is shown; k is d0 ＝K s0 [1+n 0 (K s0 /K f0 -1)]；
α j ＝1-K bj /K sj ，
Figure BDA0002077602570000172
Respectively representing the compressibility of soil particles and fluid in the saturated soil body of the j-th circle of layer;
α 0 ＝1-K b0 /K s0 ，
Figure BDA0002077602570000173
respectively representing the compressibility of soil particles and fluid in the pile core soil;
u p horizontally shifting the pile body of the tubular pile; h is the length of the tubular pile, r 1 Is the radius of the tubular pile;
E p is shear modulus, I, of the pipe pile p Is the inertia moment of the tubular pile, A p Is the cross-sectional area of the tubular pile, m p Is the mass per unit length of the tubular pile, N 1 The transverse acting force of the soil around the axial unit length on the pile body of the pipe pile during horizontal vibration is adopted; n is a radical of 0 The transverse acting force of axial unit length of pile core soil on a pile body of the pipe pile is generated during horizontal vibration;
s-i ω, where s is laplace transform, i is an imaginary unit, and ω is the excitation load frequency;
Figure BDA0002077602570000181
and psi sj Is the radial displacement u of the soil framework in the saturated soil body of the jth circle layer rj And the circumferential displacement u θj Is determined by the potential function of (a) a,
Figure BDA0002077602570000182
ψ fj the radial displacement w of the fluid in the saturated soil body of the j-th circle layer relative to the soil framework rj And circumferential displacement w θj A potential function of (d);
Figure BDA0002077602570000183
and psi s0 For radial displacement u of soil skeleton in pile core soil r0 And the circumferential displacement u θ0 Is determined by the potential function of (a) a,
Figure BDA0002077602570000184
ψ f0 for radial displacement w of fluid in pile core soil relative to soil skeleton r0 And circumferential displacement w θ0 A potential function of (d);
Φ sj is a potential function
Figure BDA0002077602570000185
Of laplace transform, Ψ sj Is a potential function psi sj Laplace transform of phi fj Is a potential function
Figure BDA0002077602570000186
Of the laplace transform, Ψ fj Is a potential function psi fj (ii) a laplace transform of; phi s0 Is a potential function
Figure BDA0002077602570000187
Of the laplace transform, Ψ s0 Is a potential function psi s0 Laplace transform of phi f0 Is a potential function
Figure BDA0002077602570000188
Of laplace transform, Ψ f0 Is a potential function psi f0 (ii) a laplace transform of;
Figure BDA0002077602570000189
the first derivative of the expression in parentheses on r is shown;
Figure BDA00020776025700001810
the expression in parentheses takes the second derivative of r;
Figure BDA00020776025700001811
Figure BDA00020776025700001812
is a shorthand notation in the calculation process;
Figure BDA00020776025700001813
Figure BDA00020776025700001814
is also a shorthand notation in the calculation process;
Figure BDA00020776025700001815
Figure BDA00020776025700001816
is also a shorthand notation in the calculation process;
η 4 ＝(m p s 2 +f 1 +f 0 )/E p I p is also a shorthand notation in the calculation process;
Figure BDA00020776025700001817
the first order, first class and second class of deformed Bessel functions are respectively;
A j1 ，A j2 ，A j3 ，B j1 ，B j2 ，B j3 ，C j1 ，C j2 ，C j3 ，D j1 ，D j2 ，D j3 is a symbol to be determined, and the following relationship exists
Figure BDA00020776025700001818
Y 1 ，Y 2 ，Y 3 ，Y 4 Is also a pending symbol;
X 1 ＝β 01 I 0 (β 01 r 0 )-I 1 (β 01 r 0 )/r 0 ，X 2 ＝β 02 I 0 (β 02 r 0 )-I 1 (β 02 r 0 )/r 0 ，X 3 ＝-I 1 (β 03 r 0 )/r 0 ，X 4 ＝γ 01 [β 01 I 0 (β 01 r 0 )-I 1 (β 01 r 0 )/r 0 ]，X 5 ＝γ 02 [β 02 I 0 (β 02 r 0 )-I 1 (β 02 r 0 )/r 0 ]，X 6 ＝γ 03 I 1 (β 03 r 0 )/r 0 ，X 7 ＝-I 1 (β 01 r 0 )/r 0 ，X 8 ＝-I 1 (β 02 r 0 )/r 0 ，X 9 ＝-[β 03 I 0 (β 03 r 0 )-I 1 (β 03 r 0 )/r 0 ]，κ＝X 1 X 5 X 9 +X 3 X 4 X 8 +X 2 X 6 X 7 -X 1 X 5 X 9 -X 1 X 6 X 8 -X 2 X 4 X 9 is also a shorthand notation in the calculation process;
Figure BDA0002077602570000191
Figure BDA0002077602570000192
a 5 ＝η[a 3 cos(ηH)-a 1 sin(ηH)]，a 6 ＝-η[a 2 sin(ηH)+a 4 cos(ηH)]and a 7 ＝η[sin(ηH)cosh(ηH)+cos(ηH)sinh(ηH)]And is also a shorthand notation in the calculation process.
The above description is only for the preferred embodiment of the present invention, but the scope of the present invention is not limited thereto, and any person skilled in the art should be considered to be within the technical scope of the present invention, and the technical solutions and the inventive concepts thereof according to the present invention should be equivalent or changed within the scope of the present invention.

Claims (5)

1. The horizontal vibration analysis method for the pipe pile in the radial heterogeneous saturated soil is characterized by comprising the following steps
S1: the following assumptions are introduced, and a vibration model of the radial heterogeneous saturated soil pipe-in-pipe pile under horizontal excitation under the plane strain condition is established:
(1) assuming that the tubular pile is a linear elastic homogeneous uniform-section circular Bernoulli-Euler beam model, neglecting the shear deformation of a pile body, and adopting a fixed support at the pile end;
(2) the soil body around the pile is divided into an inner area and an outer area, the inner area is divided into n circle layers, and the soil body of each circle layer and the pile core soil are homogeneous and isotropic two-phase saturated elastic media;
(3) the pile-soil system vibrates in small deformation, the pile-soil interface is in complete contact without separation and slippage, the pile-soil interface is impervious to water, and the two sides of the soil interface of each ring layer are continuously displaced and have balanced stress;
(4) when the pile foundation horizontally vibrates, the soil body around the pile and the pile core soil have no vertical deformation;
s2: establishing a motion equation of saturated soil bodies of all ring layers, a motion equation of pile core soil and a pile body horizontal vibration basic equation under a plane strain condition based on a Biot two-phase medium fluctuation theory;
establishing pile-soil system boundary conditions according to the assumption in the step S1;
s3: and (4) solving the motion equation of the saturated soil body of each circle layer and the motion equation of the pile core soil under the plane strain condition established in the step S2 by using Laplace transformation to obtain the horizontal dynamic impedance of the tubular pile so as to analyze the horizontal vibration of the tubular pile in the radially inhomogeneous saturated soil.
2. The analysis method according to claim 1, wherein in step S2, the equation of motion of the saturated soil mass in each circle under the planar strain condition is
Figure FDA0002077602560000011
Figure FDA0002077602560000012
Figure FDA0002077602560000013
Figure FDA0002077602560000014
The equation of motion of the pile core soil is
Figure FDA0002077602560000021
Figure FDA0002077602560000022
Figure FDA0002077602560000023
Figure FDA0002077602560000024
The basic equation of horizontal vibration of the pile body is
Figure FDA0002077602560000025
In the above formulae, the symbols have the following meanings:
j is 1-n and is the number sequence of the ring layers, the ring layer number adjacent to the tubular pile is 1, and n is the total number of the ring layers;
r and theta are coordinates of a cylindrical coordinate system, wherein the zero point of the z axis of the cylindrical coordinate system is positioned at the center of the circle of the upper surface of the tubular pile, the positive direction of the z axis is vertically downward, the direction of the r axis is the horizontal direction, and the zero point is positioned at the center of the circle of the upper surface of the tubular pile;
Figure FDA0002077602560000026
different operator symbols;
u rj is the radial displacement of the soil skeleton in the saturated soil body of the jth circle layer u θj Is the circumferential displacement, w, of the soil framework in the j-th circle of saturated soil body rj Is the radial displacement of the fluid in the saturated soil body of the j-th circle layer relative to the soil framework, w θj The circumferential displacement of the fluid in the saturated soil body of the jth circle layer relative to the soil framework is obtained;
u r0 is the radial displacement of the soil skeleton in the pile core soil, u θ0 Is the circumferential displacement, w, of the soil skeleton in the pile core soil r0 Is the radial displacement of the fluid in the pile core soil relative to the soil skeleton, w θ0 The circumferential displacement of the fluid in the pile core soil relative to the soil framework is obtained;
ρ j ＝(1-n j )ρ sj +n j ρ fj is the density of the saturated soil body of the jth circle layer, wherein rho fj The density and rho of the fluid in the saturated soil body of the jth circle layer sj The density of soil particles in the saturated soil body of the jth circle layer, n j The porosity of the saturated soil body of the jth circle layer;
ρ 0 ＝(1-n 0 )ρ s0 +n 0 ρ f0 is the soil mass density of the pile core soil,where ρ is f0 Density, rho, of fluid in the pile core soil body s0 Is the density of soil particles in the pile core soil body, n 0 The porosity of the pile core soil body;
m j ＝ρ j /n j is the viscous coupling coefficient of the layer soil skeleton and pore fluid of the j-th circle, b j ＝ρ fj g/k dj Is the Darcy law permeability coefficient of the soil body of the jth circle layer, m 0 ＝ρ 0 /n 0 Is the viscous coupling coefficient of soil skeleton and pore fluid in the pile core soil, b 0 ＝ρ f0 g/k d0 The permeability coefficient of the pile core soil body is Darcy's law, and g is the gravity acceleration;
λ j shear modulus of soil body, G, of saturated soil body of jth circle layer j Is Lame constant, upsilon of a saturated soil body of a jth circle layer sj The Poisson ratio of the saturated soil body of the jth circle layer is obtained; k sj Is the volume compression modulus, K, of soil particles in the saturated soil body of the jth circle fj Is the volume compression modulus, K, of the fluid in the saturated soil body of the jth circle bj ＝λ j +(2/3)×G j The volume compression modulus of the soil framework in the j-th circle of saturated soil body is obtained; k is dj ＝K sj [1+n j (K sj /K fj -1)]；
λ 0 Shear modulus of soil body, G, of pile core soil 0 Lame constant, upsilon, of pile core soil s0 The Poisson's ratio of the pile core soil is taken as the ratio; k s0 Is the volume compression modulus, K, of soil particles in the pile core soil f0 Is the bulk compression modulus, K, of the fluid in the core soil b0 ＝λ 0 +(2/3)×G 0 The volume compression modulus of a soil framework in the pile core soil is shown; k d0 ＝K s0 [1+n 0 (K s0 /K f0 -1)]；
α j ＝1-K bj /K sj ，
Figure FDA0002077602560000036
Respectively representing the compressibility of soil particles and fluid in the saturated soil body of the j-th circle of layer;
α 0 ＝1-K b0 /K s0 ，
Figure FDA0002077602560000037
respectively representing the compressibility of soil particles and fluid in the pile core soil;
u p horizontally displacing the pile body of the pipe pile;
E p is shear modulus, I, of the pipe pile p Is the inertia moment of the tubular pile, A p Is the cross-sectional area of the tubular pile, m p Is the mass per unit length of the tubular pile, N 1 The transverse acting force of the soil around the axial unit length on the pile body of the pipe pile during horizontal vibration is N 0 The transverse acting force of axial unit length of pile core soil on a pile body of the pipe pile is generated during horizontal vibration.
3. The analysis method according to claim 2, wherein in step S2, the pile-soil system boundary condition is a pile top boundary condition of a pipe pile
Figure FDA0002077602560000031
Figure FDA0002077602560000032
Boundary condition of pile bottom of tubular pile
u p | z＝H ＝0
Figure FDA0002077602560000033
Pile-soil full contact of the soil around the pile and waterproof conditions of the pile-soil interface
Figure FDA0002077602560000034
Continuous condition between loop layers
Figure FDA0002077602560000035
Limited displacement condition of pile core soil center
Figure FDA0002077602560000041
Pile-soil full contact of pile core soil and waterproof condition of pile-soil interface
Figure FDA0002077602560000042
In the above formulae, the symbols mean
j is 1-n and is the number sequence of the ring layers, the ring layer number adjacent to the tubular pile is 1, and n is the total number of the ring layers;
r and theta are coordinates of a cylindrical coordinate system, wherein the zero point of the z axis of the cylindrical coordinate system is positioned at the center of the circle of the upper surface of the tubular pile, the positive direction of the z axis is vertically downward, the direction of the r axis is the horizontal direction, and the zero point is positioned at the center of the circle of the upper surface of the tubular pile;
h is the length of the tubular pile, r 1 Is the radius of the tubular pile;
u p horizontally displacing the pile body of the pipe pile;
p (t) exciting the pile top;
u rj is the radial displacement of the soil skeleton in the saturated soil body of the jth circle layer u θj Is the circumferential displacement, w, of the soil framework in the j-th circle of saturated soil body rj Is the radial displacement of the fluid in the saturated soil body of the j-th circle layer relative to the soil framework, w θj The circumferential displacement of the fluid in the j-th circle of saturated soil body relative to the soil framework is obtained; u. of r0 Is the radial displacement of the soil skeleton in the pile core soil, u θ0 Is the circumferential displacement, w, of the soil skeleton in the pile core soil r0 Is the radial displacement of the fluid in the pile core soil relative to the soil skeleton, w θ0 The circumferential displacement of the fluid in the pile core soil relative to the soil framework is obtained;
σ rj is the j circle layer saturated soil body and the j +Normal stress, tau, at the interface of 1 circle of saturated soil θj The tangential shear stress on the interface of the j circle saturated soil body and the j +1 circle saturated soil body is obtained.
4. The analysis method according to claim 3, wherein the step S3 of solving the motion equation of the saturated soil body and the pile core soil of each circle layer and the basic equation of the horizontal vibration of the pile body under the plane strain condition comprises the following steps
S31: respectively introducing potential functions to soil skeleton and fluid in the saturated soil body of the jth circle layer
Figure FDA0002077602560000049
ψ sj ，
Figure FDA0002077602560000047
ψ fj
Figure FDA0002077602560000043
S32: potential function
Figure FDA00020776025600000410
ψ sj ，
Figure FDA0002077602560000048
ψ fj Introducing a motion equation of a j-th circle saturated soil body under the condition of plane strain to perform Laplace transformation to obtain
Figure FDA0002077602560000044
Figure FDA0002077602560000045
Figure FDA0002077602560000046
ρ fj s 2 Ψ sj +m j s 2 Ψ fj +b j sΨ fj ＝0
S33: solving the potential function to obtain
Φ sj ＝[A j1 K 1 (β j1 r)+B j1 I 1 (β j1 r)+A j2 K 1 (β j2 r)+B j2 I 1 (β j2 r)]cosθ
Φ fj ＝[C j1 K 1 (β j1 r)+D j1 I 1 (β j1 r)+C j2 K 1 (β j2 r)+D j2 I 1 (β j2 r)]cosθ
Ψ sj ＝[A j3 K 1 (β j3 r)+B j3 I 1 (β j3 r)]sinθ
Ψ fj ＝[C j3 K 1 (β j3 r)+D j3 I 1 (β j3 r)]sinθ
S34: bringing back the potential function to obtain each displacement and stress expression containing undetermined coefficient
U rj ＝{A j1 [-β j1 K 0 (β j1 r)-K 1 (β j1 r)/r]+A j2 [-β j2 K 0 (β j2 r)-K 1 (β j2 r)/r]+A j3 K 1 (β j3 r)/r+B j1 [β j1 I 0 (β j1 r)-I 1 (β j1 r)/r]+B j2 [-β j2 I 0 (β j2 r)-I 1 (β j2 r)/r]+B j3 I 1 (β j3 r)/r}cosθ
U θj ＝{-A j1 K 1 (β j1 r)/r-A j2 K 1 (β j2 r)/r-A j3 [-β j3 K 0 (β j3 r)-K 1 (β j3 r)/r]-B j1 I 1 (β j1 r)-B j2 I 1 (β j2 r)/r-B j3 [β j3 I 0 (β j3 r)-I 1 (β j3 r)/r]}sinθ
W rj ＝{C j1 [-β j1 K 0 (β j1 r)-K 1 (β j1 r)/r]+C j2 [-β j2 K 0 (β j2 r)-K 1 (β j2 r)/r]+C j3 K 1 (β j3 r)/r+D j1 [β j1 I 0 (β j1 r)-I 1 (β j1 r)/r]+D j2 [-β j2 I 0 (β j2 r)-I 1 (β j2 r)/r]+D j3 I 1 (β j3 r)/r}cosθ
W θj ＝{-C j1 K 1 (β j1 r)/r-C j2 K 1 (β j2 r)/r-C j3 [-β j3 K 0 (β j3 r)-K 1 (β j3 r)/r]-D j1 I 1 (β j1 r)-D j2 I 1 (β j2 r)/r-D j3 [β j3 I 0 (β j3 r)-I 1 (β j3 r)/r]}sinθ
Figure FDA0002077602560000051
τ rθj ＝A j1 G j {-2[K 1 (β j1 r)]'/r+2K 1 (β j1 r)/r 2 }sinθ+A j2 G j {-2[K 1 (β j2 r)]'/r+2K 1 (β j2 r)/r 2 }sinθ+A j3 G j {[K 1 (β j3 r)]'/r-2K 1 (β j3 r)/r 2 -[K 1 (β j3 r)]”}sinθ+B j1 G j {-2[I 1 (β j1 r)]'/r+2I 1 (β j1 r)/r 2 }sinθ+B j2 G j {-2[I 1 (β j2 r)]'/r+2I 1 (β j2 r)/r 2 }sinθ+B j3 G j {[I 1 (β j3 r)]'/r-2I 1 (β j3 r)/r 2 -[I 1 (β j3 r)]”}sinθ
S35: substituting the expression of the previous step into the boundary condition, and solving to obtain the undetermined coefficient A j1 ，A j2 ，A j3 ，B j1 ，B j2 ，B j3 ，C j1 ，C j2 ，C j3 ，D j1 ，D j2 ，D j3 ；
S36: the transverse acting force of the soil around the pile on the pile body is
Figure FDA0002077602560000061
S37: respectively introducing potential functions to soil skeleton and fluid of pile core
Figure FDA0002077602560000066
ψ s0 ,
Figure FDA0002077602560000067
ψ f0
Figure FDA0002077602560000062
S38: potential function
Figure FDA0002077602560000068
ψ s0 ,
Figure FDA0002077602560000069
ψ f0 Introducing the motion equation of the pile core soil under the condition of plane strain to perform Laplace transformation to obtain
Figure FDA0002077602560000063
Figure FDA0002077602560000064
Figure FDA0002077602560000065
ρ f0 s 2 Ψ s0 +m 0 s 2 Ψ f0 +b 0 sΨ f0 ＝0
S39: solving the potential function to obtain
Φ s0 ＝[B 01 I 1 (β 01 r)+B 02 I 1 (β 02 r)]cosθ
Φ f0 ＝[B 03 I 1 (β 01 r)+B 04 I 1 (β 02 r)]cosθ
Ψ s0 ＝B 05 I 1 (β 03 r)sinθ
Ψ f0 ＝B 06 I 1 (β 03 r)sinθ
S310: bringing back the potential function to obtain each displacement expression containing undetermined coefficients
U r0 ＝{B 01 [β 01 I 0 (β 01 r)-I 1 (β 01 r)/r]+B 02 [β 02 I 0 (β 02 r)-I 1 (β 02 r)/r]+B 05 I 1 (β 03 r)/r}cosθ
W r0 ＝{B 01 γ 01 [β 01 I 0 (β 01 r)-I 1 (β 01 r)/r]+B 02 γ 02 [β 02 I 0 (β 02 r)-I 1 (β 02 r)/r]+B 05 γ 03 I 1 (β 03 r)/r}cosθ
U θ0 ＝{-B 01 I 1 (β 01 r)-B 02 I 1 (β 02 r)/r-B 05 [β 03 I 0 (β 03 r)-I 1 (β 03 r)/r]}sinθ
W θ0 ＝{-B 01 γ 01 I 1 (β 01 r)-B 02 γ 02 I 1 (β 02 r)/r-B 05 γ 03 [β 03 I 0 (β 03 r)-I 1 (β 03 r)/r]}sinθ
S311: substituting the expression of the previous step into the boundary condition, and solving to obtain the undetermined coefficient
B 01 ＝(X 3 X 5 +X 5 X 9 -X 2 X 6 -X 6 X 8 )/κ
B 02 ＝(X 1 X 6 +X 6 X 7 -X 3 X 4 -X 4 X 9 )/κ；
B 03 ＝(X 2 X 4 +X 4 X 8 -X 1 X 5 -X 5 X 7 )/κ
S312: the transverse acting force of the pile core soil on the pile body is
Figure FDA0002077602560000071
S313: laplace transformation is carried out on the basic equation of horizontal vibration of the pile body
Figure FDA0002077602560000072
And solving to obtain
U p (z)＝Y 1 cos(ηz)+Y 2 sin(ηz)+Y 3 cosh(ηz)+Y 4 sinh(ηz)
Θ p (z)＝η[-Y 1 sin(ηz)+Y 2 cos(ηz)+Y 3 sinh(ηz)+Y 4 cosh(ηz)]
M p (z)＝-E p I p η 2 [-Y 1 cos(ηz)-Y 2 sin(ηz)+Y 3 cosh(ηz)+Y 4 sinh(ηz)]
Q p (z)＝-E p I p η 3 [Y 1 sin(ηz)-Y 2 cos(ηz)+Y 3 sinh(ηz)+Y 4 cosh(ηz)]
S314: substituting the expression of the previous step into the boundary condition, and solving to obtain the undetermined coefficient
Figure FDA0002077602560000073
Figure FDA0002077602560000074
Figure FDA0002077602560000075
Figure FDA0002077602560000076
S315: calculating the horizontal dynamic impedance of the tubular pile
K QU ＝Q p (0)/U p (0)
Can also be expressed in dimensionless form as its real and imaginary parts
Figure FDA0002077602560000077
Figure FDA0002077602560000078
In the above expression, each symbol means
s is i omega, wherein s is Laplace transform, i is an imaginary number unit, and omega is the excitation load frequency;
j is 1-n and is the number sequence of the ring layers, the ring layer number adjacent to the tubular pile is 1, and n is the total number of the ring layers;
r and theta are coordinates of a cylindrical coordinate system, wherein the zero point of the z axis of the cylindrical coordinate system is positioned at the center of the circle of the upper surface of the tubular pile, the positive direction of the z axis is vertically downward, the direction of the r axis is the horizontal direction, and the zero point is positioned at the center of the circle of the upper surface of the tubular pile;
Figure FDA0002077602560000083
and psi sj Is the radial displacement u of the soil skeleton in the saturated soil body of the jth circle layer rj And the circumferential displacement u θj Is determined by the potential function of (a) a,
Figure FDA0002077602560000084
ψ fj the radial displacement w of the fluid in the saturated soil body of the j-th circle layer relative to the soil framework rj And circumferential displacement w θj A potential function of (d);
Figure FDA0002077602560000085
and psi s0 For radial displacement u of soil skeleton in pile core soil r0 And the circumferential displacement u θ0 Is determined by the potential function of (a) a,
Figure FDA00020776025600000810
ψ f0 for radial displacement w of fluid in pile core soil relative to soil skeleton r0 And circumferential displacement w θ0 A potential function of (a);
Φ sj is a potential function
Figure FDA00020776025600000813
Of laplace transform, Ψ sj Is a potential function psi sj Laplace transform of phi fj Is a potential function
Figure FDA0002077602560000086
Of the laplace transform, Ψ fj Is a potential function psi fj (ii) a laplace transform of; phi (phi) of s0 Is a potential function
Figure FDA0002077602560000089
Of the laplace transform, Ψ s0 Is a potential function psi s0 Laplace transform of phi f0 Is a potential function
Figure FDA00020776025600000814
Of laplace transform, Ψ f0 Is a potential function psi f0 Laplace transform of (d);
ρ j ＝(1-n j )ρ sj +n j ρ fj is the density of the saturated soil body of the jth circle layer, wherein rho fj The density and rho of the fluid in the saturated soil body of the jth circle layer sj The density of soil particles in the saturated soil body of the jth circle layer, n j The porosity of the saturated soil body of the jth circle layer;
ρ 0 ＝(1-n 0 )ρ s0 +n 0 ρ f0 is the soil mass density of the pile core soil, where p f0 Density, rho, of fluid in the pile core soil body s0 Is the density of soil particles in the pile core soil body, n 0 The porosity of the pile core soil body;
m j ＝ρ j /n j is the viscous coupling coefficient of the layer soil skeleton and pore fluid of the j-th circle, b j ＝ρ fj g/k dj Is the Darcy law permeability coefficient of the soil body of the jth circle layer, m 0 ＝ρ 0 /n 0 Is the viscous coupling coefficient of soil skeleton and pore fluid in the pile core soil, b 0 ＝ρ f0 g/k d0 The permeability coefficient of the pile core soil body is Darcy's law, and g is the gravity acceleration;
λ j shear modulus of soil body, G, of saturated soil body of jth circle layer j Is Lame constant, upsilon of a saturated soil body of a jth circle layer sj The Poisson ratio of the saturated soil body of the jth circle layer is obtained; k sj Is the volume compression modulus, K, of soil particles in the saturated soil body of the jth circle fj Is the volume compression modulus, K, of the fluid in the saturated soil body of the jth circle bj ＝λ j +(2/3)×G j Saturate for j-th circle layerThe volume compression modulus of a soil framework in a soil body; k is dj ＝K sj [1+n j (K sj /K fj -1)]；
λ 0 Shear modulus of soil body, G, of pile core soil 0 Lame constant, upsilon, of pile core soil s0 The Poisson's ratio of the pile core soil is taken as the ratio; k is s0 Is the volume compression modulus, K, of soil particles in the pile core soil f0 Is the bulk compression modulus, K, of the fluid in the core soil b0 ＝λ 0 +(2/3)×G 0 The volume compression modulus of a soil framework in the pile core soil is shown; k d0 ＝K s0 [1+n 0 (K s0 /K f0 -1)]；
α j ＝1-K bj /K sj ，
Figure FDA0002077602560000087
Respectively representing the compressibility of soil particles and fluid in the saturated soil body of the j-th circle of layer; alpha is alpha 0 ＝1-K b0 /K s0 ，
Figure FDA0002077602560000088
Respectively representing the compressibility of soil particles and fluid in the pile core soil;
Figure FDA0002077602560000081
is an operator symbol;
Figure FDA00020776025600000811
the first derivative of the expression in parentheses on r is shown;
Figure FDA00020776025600000812
the expression in parentheses takes the second derivative of r;
Figure FDA0002077602560000082
Figure FDA0002077602560000091
is a shorthand notation in the calculation process;
Figure FDA0002077602560000092
Figure FDA0002077602560000093
is also a shorthand notation in the calculation process;
Figure FDA0002077602560000094
Figure FDA0002077602560000095
is also a shorthand notation in the calculation process;
η 4 ＝(m p s 2 +f 1 +f 0 )/E p I p is also a shorthand notation in the calculation process;
Figure FDA00020776025600000910
the first order, first class and second class deformed Bessel functions are respectively;
A j1 ，A j2 ，A j3 ，B j1 ，B j2 ，B j3 ，C j1 ，C j2 ，C j3 ，D j1 ，D j2 ，D j3 is a symbol to be determined, and the following relationship exists
Figure FDA0002077602560000096
Y 1 ，Y 2 ，Y 3 ，Y 4 Is also a pending symbol;
X 1 ＝β 01 I 0 (β 01 r 0 )-I 1 (β 01 r 0 )/r 0 ，X 2 ＝β 02 I 0 (β 02 r 0 )-I 1 (β 02 r 0 )/r 0 ，X 3 ＝-I 1 (β 03 r 0 )/r 0 ，X 4 ＝γ 01 [β 01 I 0 (β 01 r 0 )-I 1 (β 01 r 0 )/r 0 ]，X 5 ＝γ 02 [β 02 I 0 (β 02 r 0 )-I 1 (β 02 r 0 )/r 0 ]，X 6 ＝γ 03 I 1 (β 03 r 0 )/r 0 ，X 7 ＝-I 1 (β 01 r 0 )/r 0 ，X 8 ＝-I 1 (β 02 r 0 )/r 0 ，X 9 ＝-[β 03 I 0 (β 03 r 0 )-I 1 (β 03 r 0 )/r 0 ]，κ＝X 1 X 5 X 9 +X 3 X 4 X 8 +X 2 X 6 X 7 -X 1 X 5 X 9 -X 1 X 6 X 8 -X 2 X 4 X 9 is also a shorthand notation in the calculation process;
Figure FDA0002077602560000097
Figure FDA0002077602560000098
a 5 ＝η[a 3 cos(ηH)-a 1 sin(ηH)]，a 6 ＝-η[a 2 sin(ηH)+a 4 cos(ηH)]and a 7 ＝η[sin(ηH)cosh(ηH)+cos(ηH)sinh(ηH)]And is also a shorthand notation in the calculation process.
5. The method of claim 1, wherein in step S1, the method for determining the shear modulus of the j-th circle of soil is
Figure FDA0002077602560000099
Wherein G (r) is the shear modulus of the soil at the position where the distance between the jth circle of soil and the center of the pile-soil interface is r
Figure FDA0002077602560000101
Wherein f (r) is a function of the change in shear modulus of the earth
Figure FDA0002077602560000102
Wherein GR is a parameter for describing the disturbance degree of the construction of the soil body around the pile, and GR is a parameter for describing the disturbance degree of the construction of the soil body around the pile>1 is soil hardening GR<1 is softening of soil, GR is 1 is uniform soil, q is positive index, r is 1 The radius of the pipe pile is b, and the radius of the soil body in the radial heterogeneous region is b.
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