CN110093951B - Virtual soil pile model-based friction pile longitudinal vibration analysis method - Google Patents

Virtual soil pile model-based friction pile longitudinal vibration analysis method Download PDF

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CN110093951B
CN110093951B CN201910335274.9A CN201910335274A CN110093951B CN 110093951 B CN110093951 B CN 110093951B CN 201910335274 A CN201910335274 A CN 201910335274A CN 110093951 B CN110093951 B CN 110093951B
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soil
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CN110093951A (en
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崔春义
孟坤
梁志孟
辛宇
刘海龙
赵九野
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Dalian Maritime University
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    • E—FIXED CONSTRUCTIONS
    • E02—HYDRAULIC ENGINEERING; FOUNDATIONS; SOIL SHIFTING
    • E02D—FOUNDATIONS; EXCAVATIONS; EMBANKMENTS; UNDERGROUND OR UNDERWATER STRUCTURES
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Abstract

The invention discloses a friction pile longitudinal vibration analysis method based on a virtual soil pile model, which adopts a Kelvin model and a virtual soil pile model, assumes that an entity pile and a virtual soil pile are both homogeneous and circular visco-elastic bodies with equal cross sections, and the interface of the entity pile and the virtual soil pile has continuous displacement and balanced stress; the soil around the pile and the soil at the bottom of the pile are all isotropic linear viscoelastic bodies, and the soil material damping adopts viscous damping; the upper surface of the pile surrounding soil layer is a free boundary without normal stress and shear stress, and the bottom of the pile substrate soil layer is a rigid substrate; establishing a longitudinal vibration control equation of a pile bottom soil body and a pile periphery soil body under the condition of vertical axis symmetry according to a viscoelastic dynamics theory, establishing a longitudinal vibration control equation of a virtual soil pile and a solid pile according to an Euler-Bernoulli rod member theory, and solving the vibration control equation by using Laplace transformation to obtain a time domain speed response function of any exciting force acting on the pile top so as to analyze the longitudinal vibration of the friction pile.

Description

Virtual soil pile model-based friction pile longitudinal vibration analysis method
Technical Field
The invention relates to the field of civil engineering, in particular to a friction pile longitudinal vibration analysis method based on a virtual soil pile model.
Background
In terms of pile-pile soil interface interaction, various methods have been studied to simplify the pile-soil interface interaction. A series of discrete Winkler spring-damper models study the longitudinal vibration characteristics of a rigid foundation. Although this method is simple, the parameter values depend on experience. On the basis of improvement, a plane strain analysis model is provided for considering the stress strain continuity of the soil body along the circumferential direction of the pile, and the model has a certain theoretical basis but cannot consider the change of the soil around the pile along the depth. Based on the consideration, a three-dimensional continuous medium model of soil around the pile is developed, and the model can consider the change of soil displacement and stress components along the depth, ignore the radial displacement of the soil and study the longitudinal vibration characteristic of the pile. Then, the influence of vertical and radial displacement of the soil body is considered at the same time, and a three-dimensional continuous medium model of soil around the pile is improved. The research is based on the assumption that the pile soil is completely contacted, and the assumption can exaggerate the constraint effect of the soil body on the pile body. Therefore, a plurality of students adopt a pile end rigid support model and consider the longitudinal vibration characteristic of the pile under the condition of relative slippage of a pile-soil interface. However, the pile tip rigid support model is used, and the fluctuation effect of the pile bottom soil is ignored. In the past, numerous scholars simplify the interaction of the pile and the pile bottom soil, for example, the pile bottom soil is simplified into a spring, a damper and an elastic half-space model, but the two methods have respective limitations, and on the basis, the poplar and winter English propose a virtual soil pile model, strictly consider the fluctuation effect of the pile bottom soil, and can simulate the engineering conditions of pile end sediment, pile end soil compaction and the like. However, a method for considering the longitudinal vibration characteristics of the pile foundation caused by the relative slip of the pile-soil interface and the pile bottom soil fluctuation effect is still lack of disclosure.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provides a friction pile longitudinal vibration analysis method based on a virtual soil pile model.
In order to achieve the purpose, the technical scheme of the invention is as follows:
a friction pile longitudinal vibration analysis method based on a virtual soil pile model is characterized by comprising the following steps
S1: the following assumptions are introduced to establish a friction pile longitudinal vibration analysis model based on a virtual soil pile model: assuming that the solid pile and the virtual soil pile are homogeneous, round and uniform-section sticky elastomers, and the solid pile and the virtual soil pile have continuous displacement and balanced stress at the interface; assuming that the soil around the pile and the soil at the bottom of the pile are all isotropic linear viscoelastic bodies, and the soil material damping adopts viscous damping; assuming that the upper surface of the pile surrounding soil layer is a free boundary without normal stress and shear stress, and the bottom of the pile substrate soil layer is a rigid substrate;
s2: establishing a longitudinal vibration control equation of a pile bottom soil body and a pile periphery soil body under the condition of vertical shaft symmetry according to a viscoelastic dynamics theory;
establishing a virtual soil pile and solid pile longitudinal vibration control equation according to the Euler-Bernoulli rod piece theory;
pile-soil boundary conditions are established according to the assumption in step S1.
S3: and (4) solving the vibration equations of the soil body at the bottom of the pile and the soil body around the pile in the step S2 by using Laplace transformation, solving a control equation of longitudinal vibration of the virtual soil pile and the solid pile, and obtaining a time domain speed response function of any excitation force acting on the pile top so as to analyze the longitudinal vibration of the friction pile.
Preferably, the control equation of the longitudinal vibration of the soil at the bottom of the pile and the soil around the pile under the axial symmetry condition in the step S2 is
Figure GDA0003006221650000021
Wherein r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of the circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, t is a time coordinate,
Figure GDA0003006221650000022
the displacement is the longitudinal displacement of the soil body,
Figure GDA0003006221650000023
is the Lame constant of the soil body and has
Figure GDA0003006221650000024
Figure GDA0003006221650000025
Modulus of elasticity of the respective soil body
The poisson ratio, the viscous damping coefficient and the density correspond to the parameters of the soil at the bottom of the pile when j is 1, and correspond to the parameters of the soil around the pile when j is 2.
Preferably, the control equation of the longitudinal vibration of the soil pile in the step S2 is
Figure GDA0003006221650000026
The control equation of the longitudinal vibration of the solid pile is
Figure GDA0003006221650000027
In the formula, r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of a circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, and t is a time coordinate; u. ofSPAnd uPRespectively the longitudinal displacement of the virtual soil pile and the solid pile,
Figure GDA0003006221650000028
is the sectional area of pile body, r0Is the radius of the cross section of the pile,
Figure GDA0003006221650000029
respectively the modulus of elasticity, viscous damping coefficient and density of the soil-deficient pile, EP、ηP、ρPThe elastic modulus, viscous damping coefficient and density of the solid pile are respectively.
Preferably, in step S2, the pile-soil boundary conditions include pile bottom soil boundary conditions, pile soil boundary conditions, boundary conditions of solid piles and soil deficiency piles, and pile soil coupling conditions, which are respectively
Pile bottom soil boundary conditions:
Figure GDA0003006221650000031
Figure GDA0003006221650000032
boundary conditions of soil around the pile:
Figure GDA0003006221650000033
Figure GDA0003006221650000034
boundary conditions of the solid pile and the deficient soil pile are as follows:
Figure GDA0003006221650000035
uSP|z＝H＝0 (6b)
Figure GDA0003006221650000036
Figure GDA0003006221650000037
pile-soil coupling conditions:
Figure GDA0003006221650000038
Figure GDA0003006221650000039
Figure GDA00030062216500000310
Figure GDA00030062216500000311
in the formula, r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of a circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, and t is a time coordinate; hPThe soil layer around the pile is thickDegree HSPIs thick in the pile foundation layer, H is HP+HSPThe total thickness of the soil layer on the bedrock; q (t) any exciting force acts on the pile top;
Figure GDA00030062216500000312
is the sectional area of pile body, r0Is the pile section radius; k is a radical ofSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSThe damping coefficient of the distributed damper between the soil around the pile and the soil layer at the bottom of the pile is shown; k is a radical offIs the elasticity coefficient of Kelvin model at the boundary between solid pile and soil around the pile, cfThe coefficient of a Kelvin model damper at the boundary between the solid pile and the soil around the pile is obtained; f. ofSP(z, t) is unit side frictional resistance of pile bottom soil to the deficient soil pile,
Figure GDA00030062216500000313
the shear stress of the pile bottom soil at the interface of the pile bottom soil and the virtual soil pile is defined; f. ofP(z, t) is unit side friction resistance of soil around the pile to the solid pile,
Figure GDA0003006221650000041
is the shear stress of the soil around the pile at the interface of the soil around the pile-the solid pile,
Figure GDA0003006221650000042
the longitudinal relative slippage between the solid pile and the soil around the pile,
Figure GDA0003006221650000043
the relative slip speed between the solid pile and the soil around the pile is obtained; u. ofSPAnd uPRespectively the longitudinal displacement of the virtual soil pile and the solid pile,
Figure GDA0003006221650000044
respectively the modulus of elasticity, viscous damping coefficient and density of the pile bottom soil, EP、ηP、ρPRespectively the elastic modulus, viscous damping coefficient and density of the solid pile;
Figure GDA0003006221650000045
is the longitudinal displacement of the soil body of the pile bottom soil,
Figure GDA0003006221650000046
is the Lame constant of the soil body of the pile bottom soil and has
Figure GDA0003006221650000047
Figure GDA0003006221650000048
Figure GDA0003006221650000049
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
Figure GDA00030062216500000410
is the longitudinal displacement of the soil body of the soil around the pile,
Figure GDA00030062216500000411
is the Lame constant of the soil around the pile, and has
Figure GDA00030062216500000412
Figure GDA00030062216500000413
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the soil around the pile;
preferably, in the step S3, the solving includes the following steps
Step S31: let j be 1, perform laplace transform on the control equation of the pile bottom soil longitudinal vibration under the axisymmetric condition in the formula (1), and perform laplace transform on the boundary condition formulas (4a) and (4b), to obtain a longitudinal displacement function of the pile bottom soil as
Figure GDA00030062216500000414
And the shear stress of the pile bottom soil at the pile bottom soil-deficient soil pile interface is
Figure GDA00030062216500000415
Step S32: let j be 2, perform laplace transform on the control equation of the longitudinal vibration of the soil around the pile under the axisymmetric condition in the formula (1), perform laplace transform on the boundary condition formulas (5a) and (5b), and obtain a longitudinal displacement function of the soil around the pile as
Figure GDA00030062216500000416
And the shear stress of the soil around the pile at the soil-solid pile interface around the pile is
Figure GDA00030062216500000417
Step S33: laplace transformation is carried out on the control equation (2) of the longitudinal vibration of the virtual soil pile and the boundary condition (7a), and the longitudinal vibration displacement function of the virtual soil pile is obtained on the basis of the shearing stress (9a) of the pile bottom soil at the virtual soil pile interface, which is obtained in the step S31
Figure GDA00030062216500000418
Performing Laplace transformation on the control equation (3) of the longitudinal vibration of the solid pile and the boundary condition (7c) to obtain a longitudinal vibration displacement function of the solid pile
Figure GDA0003006221650000051
Step S34: performing Laplace transformation on the boundary condition formula (4b) to obtain a complex impedance function at the interface of the virtual soil pile and the solid pile
Figure GDA0003006221650000052
Carrying out Laplace transformation on the boundary condition formulas (6c and 6d) to obtain a displacement impedance function of the pile top of the solid pile
Figure GDA0003006221650000053
Step S35: obtaining the complex stiffness of the pile top of the solid pile as
Kd＝ZP＝Kr+iKi (12)
Step S36: obtaining a pile top displacement function of (11b) according to the displacement impedance function of the pile top of the solid pile
Figure GDA0003006221650000054
Step S37: according to the pile top displacement function (13), obtaining a pile top speed frequency response function
Figure GDA0003006221650000055
Step S38: obtaining the time domain response of the unit impulse excitation using a Fourier transform from the pile-top velocity frequency response function (14)
Figure GDA0003006221650000056
Step S39: according to the convolution theorem, the time domain response of the pile top speed under the action of any exciting force q (t) is obtained as
g(t)＝q(t)*h(t)＝IFT[Q(iω)·Hv(iω)] (16)
When the exciting force is half-sine pulse excitation
Figure GDA0003006221650000057
When T is pulse width, the time domain half-analysis of the pile top is solved into
Figure GDA0003006221650000061
In the above-mentioned steps, the step of,
z′＝z-HPthe vertical coordinate is a local longitudinal coordinate, the zero point of the vertical coordinate is the top of the soil body at the bottom of the pile, and the direction is positive downwards; i ω is a laplace transform constant, i is an imaginary number unit, and ω is an excitation load frequency; n is a subscript;
Figure GDA0003006221650000062
is the sectional area of pile body, r0Is the pile section radius; q (t) is an arbitrary excitation force;
Figure GDA0003006221650000063
for longitudinal displacement of the soil under the pile
Figure GDA0003006221650000064
(ii) a laplace transform of;
Figure GDA0003006221650000065
for longitudinal displacement of soil around pile
Figure GDA0003006221650000066
(ii) a laplace transform of; u shapeSP(z', s) is the displacement of the pile body of the deficient soil pile, uSP(z', t) laplace transform; u shapeP(z, s) solid pile shaft displacement uP(z, t) laplace transform; q (i ω) is the fourier transform of any excitation force Q (t);
K0(·)、K1(.) is a zero-order and first-order second-class imaginary vector Bessel function respectively;
IFT [ ] is to perform Fourier transform operation
Figure GDA0003006221650000068
One-dimensional compression wave velocity of the deficient soil pile;
Figure GDA0003006221650000069
is a solid pile oneThe wave velocity of the compression wave is measured;
EP、ηP、ρPrespectively the elastic modulus, viscous damping coefficient and density of the solid pile;
A1nis a constant determined by the coupling condition (7b) of the pile bottom soil and the deficient soil pile and the shear stress (9a) of the pile bottom soil at the interface of the pile bottom soil and the deficient soil pile; a. the2nIs a constant determined by the coupling conditions (7c, d) of the soil around the pile and the solid pile and the shear stress (9b) of the soil around the pile at the soil-solid pile interface;
β1n(n-1, 2, 3 …) is an equation of transcendence
Figure GDA00030062216500000610
Middle beta 11The solution of (a) is to be solved,
Figure GDA00030062216500000611
wherein k isSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSIs the damping coefficient of the distributed damper between the soil around the pile and the soil layer at the bottom of the pile,
Figure GDA00030062216500000612
is the Lame constant of the soil body of the pile bottom soil and has
Figure GDA00030062216500000613
Figure GDA00030062216500000614
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
β2n(n-1, 2, 3 …) is an equation of transcendence
Figure GDA00030062216500000615
Middle beta2The solution of (a) is to be solved,
Figure GDA00030062216500000616
wherein k isSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSIs between the soil around the pile and the soil layer under the pileThe damping coefficient of the distributed damper of (2),
Figure GDA00030062216500000617
is the Lame constant of the soil body of the pile bottom soil and has
Figure GDA00030062216500000618
Figure GDA00030062216500000619
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
MSP，NSPfor the coefficient to be determined, the following relationship is satisfied
Figure GDA0003006221650000071
MP，NPFor the coefficient to be determined, the following relationship is satisfied
Figure GDA0003006221650000072
The above steps also include the following symbol definitions
Figure GDA0003006221650000073
Figure GDA0003006221650000074
Figure GDA0003006221650000075
Figure GDA0003006221650000076
Figure GDA0003006221650000077
Figure GDA0003006221650000078
Figure GDA0003006221650000079
Figure GDA0003006221650000081
Figure GDA0003006221650000082
Figure GDA0003006221650000083
Figure GDA0003006221650000084
Figure GDA0003006221650000085
Figure GDA0003006221650000086
Figure GDA0003006221650000087
According to the technical scheme, the friction pile longitudinal vibration dynamic impedance algorithm system based on the virtual soil pile model adopts the Kelvin model and the virtual soil pile model, can simultaneously consider the relative slippage of a pile soil interface and the fluctuation effect of pile bottom soil, carries out rationalization analysis on factors influencing the pile foundation longitudinal vibration characteristic (such as the rigidity coefficient and the damping coefficient of the pile soil interface, the thickness of the pile bottom soil, the shear wave speed and the like), is suitable for the friction pile longitudinal vibration characteristic research, and can provide theoretical guidance and reference for pile foundation dynamic detection.
Drawings
FIG. 1 is a flow chart of the method of the present invention;
FIG. 2 is a model schematic of the present invention.
In the figure, 1 is soil around the pile, 2 is soil at the bottom of the pile, 3 is rigid foundation, 4 is solid pile, 5 is virtual soil pile, 6 is interface between soil around the pile and soil at the bottom of the pile, 7 is interface between soil around the pile and solid pile
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, the structure shown in the drawings is not drawn to a general scale and is partially enlarged, deformed and simplified, so that the present invention should not be construed as 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 first and second,
a friction pile longitudinal vibration analysis method based on a virtual soil pile model is characterized by comprising the following steps:
s1: the following assumption is introduced to facilitate the establishment of a friction pile longitudinal vibration analysis model based on the virtual soil pile model.
1) Assuming that the solid pile and the virtual soil pile are homogeneous, round and uniform-section sticky elastomers, and the solid pile and the virtual soil pile have continuous displacement and balanced stress at the interface;
2) assuming that the soil around the pile and the soil at the bottom of the pile are all isotropic linear viscoelastic bodies, and the soil material damping adopts viscous damping;
3) the upper surface of the pile soil layer is assumed to be a free boundary without normal stress and shear stress, and the bottom of the pile soil layer is a rigid substrate.
A friction pile longitudinal vibration analysis method based on a virtual soil pile model is characterized by comprising the following steps
S1: the following assumptions are introduced to establish a friction pile longitudinal vibration analysis model based on a virtual soil pile model: assuming that the solid pile and the virtual soil pile are homogeneous, round and uniform-section sticky elastomers, and the solid pile and the virtual soil pile have continuous displacement and balanced stress at the interface; assuming that the soil around the pile and the soil at the bottom of the pile are all isotropic linear viscoelastic bodies, and the soil material damping adopts viscous damping; assuming that the upper surface of the pile surrounding soil layer is a free boundary without normal stress and shear stress, and the bottom of the pile substrate soil layer is a rigid substrate;
s2: and establishing a longitudinal vibration control equation of the soil body at the bottom of the pile and the soil body around the pile under the condition of vertical axis symmetry according to a viscoelastic dynamics theory.
The control equation of the longitudinal vibration of the pile bottom soil and the pile surrounding soil under the condition of axial symmetry is
Figure GDA0003006221650000091
Wherein r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of the circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, t is a time coordinate,
Figure GDA0003006221650000092
the displacement is the longitudinal displacement of the soil body,
Figure GDA0003006221650000093
is the Lame constant of the soil body and has
Figure GDA0003006221650000094
Figure GDA0003006221650000095
The elastic modulus, the Poisson ratio, the viscous damping coefficient and the density of the soil body are respectively, the parameters correspond to the parameters of the soil at the bottom of the pile when j is 1, and the parameters correspond to the parameters of the soil around the pile when j is 2.
And establishing a control equation of the longitudinal vibration of the virtual soil pile and the solid pile according to the Euler-Bernoulli rod member theory. The control equation of the longitudinal vibration of the virtual soil pile is
Figure GDA0003006221650000096
The control equation of the longitudinal vibration of the solid pile is
Figure GDA0003006221650000101
In the formula, r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of a circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, and t is a time coordinate; u. ofSPAnd uPRespectively the longitudinal displacement of the virtual soil pile and the solid pile,
Figure GDA0003006221650000102
is the sectional area of pile body, r0Is the radius of the cross section of the pile,
Figure GDA0003006221650000103
respectively the modulus of elasticity, viscous damping coefficient and density of the soil-deficient pile, EP、ηP、ρPThe elastic modulus, viscous damping coefficient and density of the solid pile are respectively.
Pile-soil boundary conditions are established according to the assumption in step S1.
The pile-soil boundary conditions comprise pile bottom soil boundary conditions, pile soil boundary conditions, solid pile and deficient soil pile boundary conditions and pile soil coupling conditions which are respectively
Pile bottom soil boundary conditions:
Figure GDA0003006221650000104
Figure GDA0003006221650000105
boundary conditions of soil around the pile:
Figure GDA0003006221650000106
Figure GDA0003006221650000107
boundary conditions of the solid pile and the deficient soil pile are as follows:
Figure GDA0003006221650000108
uSP|z＝H＝0 (6b)
Figure GDA0003006221650000109
Figure GDA00030062216500001010
pile-soil coupling conditions:
Figure GDA00030062216500001011
Figure GDA00030062216500001012
Figure GDA00030062216500001013
Figure GDA00030062216500001014
in the formula, r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of a circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, and t is a time coordinate; hPIs the thickness of the soil layer around the pile, HSPIs thick in the pile foundation layer, H is HP+HSPThe total thickness of the soil layer on the bedrock; q (t) any exciting force acts on the pile top;
Figure GDA0003006221650000111
is the sectional area of pile body, r0Is the pile section radius; k is a radical ofSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSThe damping coefficient of the distributed damper between the soil around the pile and the soil layer at the bottom of the pile is shown; k is a radical offIs the elasticity coefficient of Kelvin model at the boundary between solid pile and soil around the pile, cfThe coefficient of a Kelvin model damper at the boundary between the solid pile and the soil around the pile is obtained; f. ofSP(z, t) is unit side frictional resistance of pile bottom soil to the deficient soil pile,
Figure GDA0003006221650000112
the shear stress of the pile bottom soil at the interface of the pile bottom soil and the virtual soil pile is defined; f. ofP(z, t) is unit side friction resistance of soil around the pile to the solid pile,
Figure GDA0003006221650000113
is the shear stress of the soil around the pile at the interface of the soil around the pile-the solid pile,
Figure GDA0003006221650000114
the longitudinal relative slippage between the solid pile and the soil around the pile,
Figure GDA0003006221650000115
the relative slip speed between the solid pile and the soil around the pile is obtained; u. ofSPAnd uPRespectively the longitudinal displacement of the virtual soil pile and the solid pile,
Figure GDA0003006221650000116
respectively the modulus of elasticity, viscous damping coefficient and density of the pile bottom soil, EP、ηP、ρPRespectively the elastic modulus, viscous damping coefficient and density of the solid pile;
Figure GDA0003006221650000117
is the longitudinal displacement of the soil body of the pile bottom soil,
Figure GDA0003006221650000118
is the Lame constant of the soil body of the pile bottom soil and has
Figure GDA0003006221650000119
Figure GDA00030062216500001110
Figure GDA00030062216500001111
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
Figure GDA00030062216500001112
is the longitudinal displacement of the soil body of the soil around the pile,
Figure GDA00030062216500001113
is the Lame constant of the soil around the pile, and has
Figure GDA00030062216500001114
Figure GDA00030062216500001115
The elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the soil around the pile are respectively.
S3: and (4) solving the vibration equations of the soil body at the bottom of the pile and the soil body around the pile in the step S2 by using Laplace transformation, solving a control equation of longitudinal vibration of the virtual soil pile and the solid pile, and obtaining a time domain speed response function of any excitation force acting on the pile top so as to analyze the longitudinal vibration of the friction pile.
The solution includes the following steps
Step S31: let j be 1, perform laplace transform on the control equation of the pile bottom soil longitudinal vibration under the axisymmetric condition in the formula (1), and perform laplace transform on the boundary condition formulas (4a) and (4b), to obtain a longitudinal displacement function of the pile bottom soil as
Figure GDA00030062216500001116
And the shear stress of the pile bottom soil at the pile bottom soil-deficient soil pile interface is
Figure GDA00030062216500001117
Step S32: let j be 2, perform laplace transform on the control equation of the longitudinal vibration of the soil around the pile under the axisymmetric condition in the formula (1), perform laplace transform on the boundary condition formulas (5a) and (5b), and obtain a longitudinal displacement function of the soil around the pile as
Figure GDA0003006221650000121
And the shear stress of the soil around the pile at the soil-solid pile interface around the pile is
Figure GDA0003006221650000122
Step S33: laplace transformation is carried out on the control equation (2) of the longitudinal vibration of the virtual soil pile and the boundary condition (7a), and the longitudinal vibration displacement function of the virtual soil pile is obtained on the basis of the shearing stress (9a) of the pile bottom soil at the virtual soil pile interface, which is obtained in the step S31
Figure GDA0003006221650000123
Performing Laplace transformation on the control equation (3) of the longitudinal vibration of the solid pile and the boundary condition (7c) to obtain a longitudinal vibration displacement function of the solid pile
Figure GDA0003006221650000124
Step S34: performing Laplace transformation on the boundary condition formula (4b) to obtain a complex impedance function at the interface of the virtual soil pile and the solid pile
Figure GDA0003006221650000125
Carrying out Laplace transformation on the boundary condition formulas (6c and 6d) to obtain a displacement impedance function of the pile top of the solid pile
Figure GDA0003006221650000126
Step S35: obtaining the complex stiffness of the pile top of the solid pile as
Kd＝ZP＝Kr+iKi (12)
Step S36: obtaining a pile top displacement function of (11b) according to the displacement impedance function of the pile top of the solid pile
Figure GDA0003006221650000127
Step S37: according to the pile top displacement function (13), obtaining a pile top speed frequency response function
Figure GDA0003006221650000128
Step S38: obtaining the time domain response of the unit impulse excitation using a Fourier transform from the pile-top velocity frequency response function (14)
Figure GDA0003006221650000131
Step S39: according to the convolution theorem, the time domain response of the pile top speed under the action of any exciting force q (t) is obtained as
g(t)＝q(t)*h(t)＝IFT[Q(iω)·Hv(iω)] (16)
When the exciting force is half-sine pulse excitation
Figure GDA0003006221650000132
When T is pulse width, the time domain half-analysis of the pile top is solved into
Figure GDA0003006221650000133
In the above-mentioned steps, the step of,
z′＝z-HPthe vertical coordinate is a local longitudinal coordinate, the zero point of the vertical coordinate is the top of the soil body at the bottom of the pile, and the direction is positive downwards; i ω is a laplace transform constant, i is an imaginary number unit, and ω is an excitation load frequency; n is a subscript;
Figure GDA0003006221650000134
is the sectional area of pile body, r0Is the pile section radius; q (t) is an arbitrary excitation force;
Figure GDA0003006221650000135
for longitudinal displacement of the soil under the pile
Figure GDA0003006221650000136
(ii) a laplace transform of;
Figure GDA0003006221650000137
for longitudinal displacement of soil around pile
Figure GDA0003006221650000138
(ii) a laplace transform of; u shapeSP(z', s) is the displacement of the pile body of the deficient soil pile, uSP(z', t) laplace transform; u shapeP(z, s) solid pile shaft displacement uP(z, t) laplace transform; q (i ω) is the fourier transform of any excitation force Q (t);
K0(·)、K1are each of zero orderAnd a first order, second class imaginary vector Bessel function;
IFT [. cndot ] is to perform Fourier transform operation;
Figure GDA00030062216500001310
one-dimensional compression wave velocity of the deficient soil pile;
Figure GDA00030062216500001311
the wave velocity of one-dimensional compression waves of the solid pile is obtained; eP、ηP、ρPRespectively the elastic modulus, viscous damping coefficient and density of the solid pile;
A1nis a constant determined by the coupling condition (7b) of the pile bottom soil and the deficient soil pile and the shear stress (9a) of the pile bottom soil at the interface of the pile bottom soil and the deficient soil pile; a. the2nIs a constant determined by the coupling conditions (7c, d) of the soil around the pile and the solid pile and the shear stress (9b) of the soil around the pile at the soil-solid pile interface;
β1n(n-1, 2, 3 …) is an equation of transcendence
Figure GDA00030062216500001312
Middle beta1The solution of (a) is to be solved,
Figure GDA00030062216500001313
wherein k isSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSIs the damping coefficient of the distributed damper between the soil around the pile and the soil layer at the bottom of the pile,
Figure GDA00030062216500001314
is the Lame constant of the soil body of the pile bottom soil and has
Figure GDA00030062216500001315
Figure GDA00030062216500001316
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
β2n(n1, 2, 3 …) is an equation of transcendence
Figure GDA0003006221650000141
Middle beta2The solution of (a) is to be solved,
Figure GDA0003006221650000142
wherein k isSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSIs the damping coefficient of the distributed damper between the soil around the pile and the soil layer at the bottom of the pile,
Figure GDA0003006221650000143
is the Lame constant of the soil body of the pile bottom soil and has
Figure GDA0003006221650000144
Figure GDA0003006221650000145
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
MSP，NSPfor the coefficient to be determined, the following relationship is satisfied
Figure GDA0003006221650000146
MP，NPFor the coefficient to be determined, the following relationship is satisfied
Figure GDA0003006221650000147
The above steps also include the following symbol definitions
Figure GDA0003006221650000151
Figure GDA0003006221650000152
Figure GDA0003006221650000153
Figure GDA0003006221650000154
Figure GDA0003006221650000155
Figure GDA0003006221650000156
Figure GDA0003006221650000157
Figure GDA0003006221650000158
Figure GDA0003006221650000159
Figure GDA00030062216500001510
Figure GDA00030062216500001511
Figure GDA00030062216500001512
Figure GDA00030062216500001513
Figure GDA00030062216500001514
In summary, the friction pile longitudinal vibration dynamic impedance algorithm system based on the virtual soil pile model can simultaneously consider the fluctuation effect of the soil body at the pile bottom and the relative slip of the pile soil interface by adopting the virtual soil pile model and the kelvin model, can be suitable for the longitudinal vibration problem when the pile soil interface of the friction pile slides relatively, and can provide theoretical guidance and reference for pile foundation dynamic detection.
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 (3)

1. A friction pile longitudinal vibration analysis method based on a virtual soil pile model is characterized by comprising the following steps
S1: the following assumptions are introduced to establish a friction pile longitudinal vibration analysis model based on a virtual soil pile model: assuming that the solid pile and the virtual soil pile are homogeneous, round and uniform-section sticky elastomers, and the solid pile and the virtual soil pile have continuous displacement and balanced stress at the interface; assuming that the soil around the pile and the soil at the bottom of the pile are all isotropic linear viscoelastic bodies, and the soil material damping adopts viscous damping; assuming that the upper surface of the pile surrounding soil layer is a free boundary without normal stress and shear stress, and the bottom of the pile substrate soil layer is a rigid substrate;
s2: establishing a longitudinal vibration control equation of a pile bottom soil body and a pile periphery soil body under the condition of vertical shaft symmetry according to a viscoelastic dynamics theory;
establishing a virtual soil pile and solid pile longitudinal vibration control equation according to the Euler-Bernoulli rod piece theory;
establishing pile-soil boundary conditions according to the assumption in step S1; the control equation of the longitudinal vibration of the soil at the bottom of the pile and the soil around the pile under the axial symmetry condition in the step S2 is
Figure FDA0003006221640000011
Wherein r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of the circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, t is a time coordinate,
Figure FDA0003006221640000012
the displacement is the longitudinal displacement of the soil body,
Figure FDA0003006221640000013
is the Lame constant of the soil body and has
Figure FDA0003006221640000014
Figure FDA0003006221640000015
Respectively representing the elastic modulus, the Poisson ratio, the viscous damping coefficient and the density of a soil body, wherein j is 1 and corresponds to a pile bottom soil parameter, and j is 2 and corresponds to a pile surrounding soil parameter; the control equation of the longitudinal vibration of the deficient soil pile in the step S2 is
Figure FDA0003006221640000016
The control equation of the longitudinal vibration of the solid pile is
Figure FDA0003006221640000017
In the formula, r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of a circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, and t is a time coordinate; u. ofSPAnd uPRespectively the longitudinal displacement of the virtual soil pile and the solid pile,
Figure FDA0003006221640000018
is the sectional area of pile body, r0Is the radius of the cross section of the pile,
Figure FDA0003006221640000019
respectively the modulus of elasticity, viscous damping coefficient and density of the soil-deficient pile, EP、ηP、ρPRespectively the elastic modulus, viscous damping coefficient and density of the solid pile; f. ofSP(z, t) is unit side frictional resistance of pile bottom soil to the deficient soil pile; f. ofP(z, t) is unit side frictional resistance of soil around the pile to the solid pile;
s3: and (4) solving the vibration equations of the soil body at the bottom of the pile and the soil body around the pile in the step S2 by using Laplace transformation, solving a control equation of longitudinal vibration of the virtual soil pile and the solid pile, and obtaining a time domain speed response function of any excitation force acting on the pile top so as to analyze the longitudinal vibration of the friction pile.
2. The analysis method according to claim 1, wherein in step S2, the pile-soil boundary conditions include pile subsoil boundary conditions, solid pile and soil deficiency pile boundary conditions, and pile-soil coupling conditions
Pile bottom soil boundary conditions:
Figure FDA0003006221640000021
Figure FDA0003006221640000022
boundary conditions of soil around the pile:
Figure FDA0003006221640000023
Figure FDA0003006221640000024
boundary conditions of the solid pile and the deficient soil pile are as follows:
Figure FDA0003006221640000025
uSP|z＝H＝0 (6b)
Figure FDA0003006221640000026
Figure FDA0003006221640000027
pile-soil coupling conditions:
Figure FDA0003006221640000028
Figure FDA0003006221640000029
Figure FDA00030062216400000210
Figure FDA00030062216400000211
in the formula, r is an axial coordinate, the zero point of the axial coordinate is positioned at the center of a circle of the section of the pile, z is a longitudinal coordinate, the zero point of the longitudinal coordinate is positioned on the free surface and is positive downwards, and t is a time coordinate; hPIs the thickness of the soil layer around the pile, HSPIs a pile bottomThickness of soil layer, H ═ HP+HSPThe total thickness of the soil layer on the bedrock; q (t) any exciting force acts on the pile top;
Figure FDA00030062216400000212
is the sectional area of pile body, r0Is the pile section radius; k is a radical ofSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSThe damping coefficient of the distributed damper between the soil around the pile and the soil layer at the bottom of the pile is shown; k is a radical offIs the elasticity coefficient of Kelvin model at the boundary between solid pile and soil around the pile, cfThe coefficient of a Kelvin model damper at the boundary between the solid pile and the soil around the pile is obtained; f. ofSP(z, t) is unit side frictional resistance of pile bottom soil to the deficient soil pile,
Figure FDA0003006221640000031
the shear stress of the pile bottom soil at the interface of the pile bottom soil and the virtual soil pile is defined; f. ofP(z, t) is unit side friction resistance of soil around the pile to the solid pile,
Figure FDA0003006221640000032
is the shear stress of the soil around the pile at the interface of the soil around the pile-the solid pile,
Figure FDA0003006221640000033
the longitudinal relative slippage between the solid pile and the soil around the pile,
Figure FDA0003006221640000034
the relative slip speed between the solid pile and the soil around the pile is obtained; u. ofSPAnd uPRespectively the longitudinal displacement of the virtual soil pile and the solid pile,
Figure FDA0003006221640000035
respectively the modulus of elasticity, viscous damping coefficient and density of the pile bottom soil, EP、ηP、ρPRespectively the elastic modulus, viscous damping coefficient and density of the solid pile;
Figure FDA0003006221640000036
is the longitudinal displacement of the soil body of the pile bottom soil,
Figure FDA0003006221640000037
Figure FDA0003006221640000038
is the Lame constant of the soil body of the pile bottom soil and has
Figure FDA0003006221640000039
Figure FDA00030062216400000310
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
Figure FDA00030062216400000311
is the longitudinal displacement of the soil body of the soil around the pile,
Figure FDA00030062216400000312
is the Lame constant of the soil around the pile, and has
Figure FDA00030062216400000313
Figure FDA00030062216400000314
The elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the soil around the pile are respectively.
3. The analysis method according to claim 2, wherein in the step S3, the solving includes the following steps
Step S31: let j be 1, perform laplace transform on the control equation of the pile bottom soil longitudinal vibration under the axisymmetric condition in the formula (1), and perform laplace transform on the boundary condition formulas (4a) and (4b), to obtain a longitudinal displacement function of the pile bottom soil as
Figure FDA00030062216400000316
And the shear stress of the pile bottom soil at the pile bottom soil-deficient soil pile interface is
Figure FDA00030062216400000317
Step S32: let j be 2, perform laplace transform on the control equation of the longitudinal vibration of the soil around the pile under the axisymmetric condition in the formula (1), perform laplace transform on the boundary condition formulas (5a) and (5b), and obtain a longitudinal displacement function of the soil around the pile as
Figure FDA00030062216400000318
And the shear stress of the soil around the pile at the soil-solid pile interface around the pile is
Figure FDA00030062216400000319
Step S33: laplace transformation is carried out on the control equation (2) of the longitudinal vibration of the virtual soil pile and the boundary condition (7a), and the longitudinal vibration displacement function of the virtual soil pile is obtained on the basis of the shearing stress (9a) of the pile bottom soil at the virtual soil pile interface, which is obtained in the step S31
Figure FDA0003006221640000041
Performing Laplace transformation on the control equation (3) of the longitudinal vibration of the solid pile and the boundary condition (7c) to obtain a longitudinal vibration displacement function of the solid pile
Figure FDA0003006221640000042
Step S34: performing Laplace transformation on the boundary condition formula (4b) to obtain a complex impedance function at the interface of the virtual soil pile and the solid pile
Figure FDA0003006221640000043
Carrying out Laplace transformation on the boundary condition formulas (6c and 6d) to obtain a displacement impedance function of the pile top of the solid pile
Figure FDA0003006221640000044
Step S35: obtaining the complex stiffness of the pile top of the solid pile as
Kd＝ZP＝Kr+iKi (12)
Step S36: obtaining a pile top displacement function of (11b) according to the displacement impedance function of the pile top of the solid pile
Figure FDA0003006221640000045
Step S37: according to the pile top displacement function (13), obtaining a pile top speed frequency response function
Figure FDA0003006221640000046
Step S38: obtaining the time domain response of the unit impulse excitation using a Fourier transform from the pile-top velocity frequency response function (14)
Figure FDA0003006221640000047
Step S39: according to the convolution theorem, the time domain response of the pile top speed under the action of any exciting force q (t) is obtained as
g(t)＝q(t)*h(t)＝IFT[Q(iω)·Hv(iω)] (16)
When the exciting force is half-sine pulse excitation
Figure FDA0003006221640000051
When T is pulse width, the time domain half-analysis of the pile top is solved into
Figure FDA0003006221640000052
In the above-mentioned steps, the step of,
z′＝z-HPthe vertical coordinate is a local longitudinal coordinate, the zero point of the vertical coordinate is the top of the soil body at the bottom of the pile, and the direction is positive downwards; i ω is a laplace transform constant, i is an imaginary number unit, and ω is an excitation load frequency; n is a subscript;
Figure FDA0003006221640000053
is the sectional area of pile body, r0Is the pile section radius; q (t) is an arbitrary excitation force;
Figure FDA0003006221640000054
for longitudinal displacement of the soil under the pile
Figure FDA0003006221640000055
(ii) a laplace transform of;
Figure FDA0003006221640000056
for longitudinal displacement of soil around pile
Figure FDA0003006221640000057
(ii) a laplace transform of; u shapeSP(z', s) is the displacement of the pile body of the deficient soil pile, uSP(z', t) laplace transform; u shapeP(z, s) solid pile shaft displacement uP(z, t) laplace transform; q (i ω) is the fourier transform of any excitation force Q (t);
K0(·)、K1(.) is a zero-order and first-order second-class imaginary vector Bessel function respectively;
IFT [. cndot ] is to perform Fourier transform operation;
Figure FDA0003006221640000058
one-dimensional compression wave velocity of the deficient soil pile;
Figure FDA0003006221640000059
the wave velocity of one-dimensional compression waves of the solid pile is obtained;
EP、ηP、ρPrespectively the elastic modulus, viscous damping coefficient and density of the solid pile;
A1nis a constant determined by the coupling condition (7b) of the pile bottom soil and the deficient soil pile and the shear stress (9a) of the pile bottom soil at the interface of the pile bottom soil and the deficient soil pile; a. the2nIs a constant determined by the coupling conditions (7c, d) of the soil around the pile and the solid pile and the shear stress (9b) of the soil around the pile at the soil-solid pile interface;
β1n(n-1, 2, 3 …) is an equation of transcendence
Figure FDA00030062216400000510
Middle beta1The solution of (a) is to be solved,
Figure FDA00030062216400000511
wherein k isSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSIs the damping coefficient of the distributed damper between the soil around the pile and the soil layer at the bottom of the pile,
Figure FDA00030062216400000512
is the Lame constant of the soil body of the pile bottom soil and has
Figure FDA00030062216400000513
Figure FDA00030062216400000514
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
β2n(n-1, 2, 3 …) is an equation of transcendence
Figure FDA00030062216400000515
Middle beta2The solution of (a) is to be solved,
Figure FDA00030062216400000516
wherein k isSIs the dynamic stiffness of the distributed spring between the soil around the pile and the soil layer at the bottom of the pile, cSIs the damping coefficient of the distributed damper between the soil around the pile and the soil layer at the bottom of the pile,
Figure FDA0003006221640000061
is the Lame constant of the soil body of the pile bottom soil and has
Figure FDA0003006221640000062
Figure FDA0003006221640000063
Respectively the elastic modulus, Poisson's ratio, viscous damping coefficient and density of the soil body of the pile bottom soil;
MSP，NSPfor the coefficient to be determined, the following relationship is satisfied
Figure FDA0003006221640000064
MP，NPFor the coefficient to be determined, the following relationship is satisfied
Figure FDA0003006221640000065
The above steps also include the following symbol definitions
Figure FDA0003006221640000066
Figure FDA0003006221640000067
Figure FDA0003006221640000068
Figure FDA0003006221640000069
Figure FDA00030062216400000610
Figure FDA00030062216400000611
Figure FDA00030062216400000612
Figure FDA0003006221640000071
Figure FDA0003006221640000072
Figure FDA0003006221640000073
Figure FDA0003006221640000074
Figure FDA0003006221640000075
Figure FDA0003006221640000076
Figure FDA0003006221640000077
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