CN108732242B - Floating bearing pile longitudinal vibration analysis method based on three-dimensional axisymmetric model of pile body - Google Patents

Floating bearing pile longitudinal vibration analysis method based on three-dimensional axisymmetric model of pile body Download PDF

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CN108732242B
CN108732242B CN201810551757.8A CN201810551757A CN108732242B CN 108732242 B CN108732242 B CN 108732242B CN 201810551757 A CN201810551757 A CN 201810551757A CN 108732242 B CN108732242 B CN 108732242B
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soil
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崔春义
孟坤
李玲玉
梁志孟
杨刚
张石平
王本龙
姚怡亦
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Dalian Maritime University
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Abstract

The invention provides a floating pile longitudinal vibration analysis method based on a three-dimensional axisymmetric model of a pile body. The method comprises the following steps: establishing a three-dimensional axial symmetry model of a pile body and a pile surrounding soil body, considering the longitudinal displacement and the change of strain of the pile body along the radial direction, establishing a longitudinal vibration equation of the pile surrounding soil body and the pile body under the three-dimensional axial symmetry condition, solving the vibration equation by a separation variable method, deducing frequency transcendental equations corresponding to different vibration modes of a pile soil system, determining undetermined functions which can simultaneously meet the viscoelastic supporting condition of a pile end and the inhomogeneous boundary condition of a pile top, obtaining the complex stiffness of the pile top, carrying out mean value processing on the analytic solution of the complex stiffness of the pile top along the radial direction, further obtaining a radial mean value analytic solution of a displacement frequency response function of the pile top and the complex stiffness of the pile top, and finishing the evaluation of the vibration characteristic of the pile body and the integrity of the pile body under the action of resonance and. The method is closer to a real model, has high calculation precision, and can provide theoretical guidance and reference for pile foundation power detection.

Description

Floating bearing pile longitudinal vibration analysis method based on three-dimensional axisymmetric model of pile body
Technical Field
The invention relates to the field of civil engineering, in particular to a floating pile longitudinal vibration analysis method based on a three-dimensional axisymmetric model of a pile body.
Background
The study on the pile-soil coupling longitudinal 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 in geotechnical engineering and solid mechanics all the time. How to establish a reasonable pile-soil coupled vibration model is the key of the research on the longitudinal vibration characteristics of a pile foundation, most of the existing researches are developed aiming at the improvement of a soil body model around the pile, and the researches particularly experience a Winkler model, a plane strain model considering the stress and strain continuity of the soil body along the radial direction of the periphery of the pile, a three-dimensional continuous medium model considering the displacement of the soil body and the change of the stress component along the depth, and a true three-dimensional continuous medium model considering the vertical displacement and the radial displacement of the soil body.
The pile body model is established on the basis of the classical Euler-Bernoulli theory, and the problem of processing the large-diameter pile by applying the one-dimensional stress wave theory causes larger deviation. Based on the consideration, Rayleigh provides a correction theory for the rod piece at first, and Love particularly deduces a Raleigh-Love rod motion equation which can consider the transverse inertia effect based on the energy.
However, although the Rayleigh-Love rod model can consider the inertia effect of the lateral motion of the pile body particle, the radial fluctuation effect of the pile body is not considered, that is, the radial change of the stress and displacement of the pile body caused by fluctuation cannot be considered, the Rayleigh-Love rod model is not strict in theory, and the influence of the three-dimensional fluctuation effect of the pile body is further considered by adopting a three-dimensional axisymmetric pile body model. However, for the existing research of considering the influence of the three-dimensional fluctuation effect of the pile body by adopting the three-dimensional axial symmetry model of the pile body, the existing research is developed around the end-bearing pile, and the existing solution based on the specific boundary condition of the pile end fixing is not suitable for the friction and floating bearing pile.
Disclosure of Invention
According to the technical problem, a floating pile longitudinal vibration analysis method based on a three-dimensional axisymmetric model of a pile body is provided. The method mainly treats the pile body and the soil around the pile as elastic and viscoelastic continuous media respectively, establishes a three-dimensional axisymmetric pile-soil coupled system longitudinal vibration analysis model considering the viscoelastic supporting condition of the pile end, and obtains a frequency transcendental equation corresponding to different vibration modes of the pile soil system through derivation, thereby analyzing the longitudinal vibration characteristics of the viscoelastic supporting pile under the action of resonance and excitation forces.
The technical means adopted by the invention are as follows:
a floating pile longitudinal vibration analysis method based on a three-dimensional axisymmetric model of a pile body is characterized by comprising the following steps:
s1, establishing a three-dimensional axisymmetric model of the pile body and the soil body around the pile, wherein the pile body is regarded as a homogeneous uniform-section elastic body, the bottom of the pile body is regarded as a viscoelastic support, the radial displacement of the pile body is ignored, the longitudinal displacement and the change of strain along the radial direction of the pile body are considered, the upper surface of the soil body around the pile is regarded as a free boundary, and the free boundary has no normal stress or shear stress;
s2, establishing a longitudinal vibration equation of the soil body around the pile and the pile body under the three-dimensional axial symmetry condition according to the basic theory of elastic dynamics, wherein the pile-soil coupling vibration system meets the conditions of linear elasticity and small deformation, and the displacement of the soil around the pile and the pile interface is continuous and the stress is balanced;
s3, solving a longitudinal vibration equation of the soil body around the pile and the pile body by a separation variable method, and deducing frequency transcendental equations corresponding to different vibration modes of the pile soil system;
s4, determining undetermined functions which can simultaneously meet the viscoelastic supporting conditions of the pile end and the heterogeneous boundary conditions of the pile top according to the pile-soil coupling conditions, and solving the undetermined functions to obtain the complex stiffness of the pile top;
and S5, carrying out mean value processing on the pile top complex stiffness analytic solution along the radial direction, further obtaining a pile top displacement frequency response function and a pile top complex stiffness radial mean value analytic solution, and finishing evaluation on the pile body vibration characteristic and the pile body integrity under the action of harmonic and excitation force.
Further, in step S2, the pile top is uniformly distributed with harmonic excitation force, the pile-soil system is vibrated with steady state, and the longitudinal displacements of the soil around the pile and the pile body respectively satisfy the following relations:
Figure BDA0001681444790000021
Figure BDA0001681444790000022
wherein the harmonic excitation force is
Figure BDA0001681444790000023
Representing the unit of an imaginary number, omega the circular frequency,
Figure BDA0001681444790000031
respectively representing the longitudinal vibration amplitude of the soil body around the pile and the pile body, i represents an imaginary number unit, omega represents the vibration exciting load frequency,
the longitudinal vibration equations of the soil body around the pile and the pile body are respectively as follows:
Figure BDA0001681444790000032
Figure BDA0001681444790000033
wherein λ isSExpressing the Lame constant, mu, of the soil body around the pileSExpressing complex shear modulus, ρSExpressing the density of the soil around the pile, lambdaPDenotes the lanmei constant of the pile body, GPDenotes the shear modulus, ρPRepresenting the pile body density, r is a radial coordinate, t is time, z is a longitudinal coordinate,
λS＝2vSGS/(1-2vS)，μS＝GS(1+2ξSi)，
wherein v isSDenotes the Poisson's ratio, GSExpressing the viscoelastic support constant of the bottom of the soil layer, ξSRepresenting hysteresis damping ratio, vPThe expression of the poisson's ratio,
the expressions (1) and (2) are simplified into the following expressions:
Figure BDA0001681444790000034
Figure BDA0001681444790000035
wherein,
Figure BDA0001681444790000036
the shear wave velocity of the soil body around the pile and the pile body are respectively.
Further, the soil body around the pile and the pile body meet the following boundary conditions:
the soil layer boundary conditions include:
the surface of the soil layer is free:
Figure BDA0001681444790000037
viscoelastic support of the bottom surface of the soil layer:
Figure BDA0001681444790000038
wherein: eSDenotes the modulus of elasticity, kS、SRepresents the viscoelastic support constant of the soil bottom, and when r → ∞ is reached, the displacement is zero:
Figure BDA0001681444790000039
the pile body boundary conditions include:
pile top action uniform distribution harmonic excitation force
Figure BDA0001681444790000041
Comprises the following steps:
Figure BDA0001681444790000042
wherein:
Figure BDA0001681444790000043
the harmonic excitation force amplitude of the pile top is represented,
and (3) viscoelastic support of the pile bottom:
Figure BDA0001681444790000044
wherein k isP、PDenotes the viscoelastic support constant of the pile bottom, EPThe modulus of elasticity is indicated.
Further, in the step S3, the process of solving the vibration equation of the soil around the pile includes:
order to
Figure BDA00016814447900000415
Equation (3) is thus decomposed into two ordinary differential equations:
Figure BDA0001681444790000045
Figure BDA0001681444790000046
wherein h isS，qSIs constant and satisfies the following relationship:
Figure BDA0001681444790000047
the solutions of equations (10) and (11) are obtained as follows:
ZS(z)＝CScos(hSz)+DSsin(hSz) (13)
RS(r)＝MSK0(qSr)+NSI0(qSr) (14)
wherein, I0(qSr)、K0(qSr) is a zero-order first, second class imaginary-vector Bessel function, CS、DS、MS、NSFor the pending integration constant determined by the boundary conditions,
the transcendental equation can be obtained by integrating the boundary condition equations (5) and (6):
Figure BDA0001681444790000048
wherein,
Figure BDA0001681444790000049
dimensionless parameter, K, representing the complex stiffness of the springs at the bottom of the soil layerS＝kS+iωSAnd H represents the pile length,
the transcendental equation passes MATObtaining infinite characteristic values through LAB programming solution and recording the infinite characteristic values as
Figure BDA00016814447900000410
n is 1,2,. infinity,. and will
Figure BDA00016814447900000411
Can be substituted by formula (12)
Figure BDA00016814447900000412
The above-mentioned
Figure BDA00016814447900000413
A non-dimensional parameter is represented by,
comprehensive soil layer boundary condition formulas (5), (6) and (7) can obtain longitudinal vibration displacement amplitude of soil around the pile
Figure BDA00016814447900000414
The expression of (a) is:
Figure BDA0001681444790000051
wherein,
Figure BDA0001681444790000052
is a series of undetermined functions.
Further, in the step S3, the process of solving the vibration equation of the soil around the pile includes:
s31, decomposing the solution problem for obtaining the longitudinal vibration displacement response amplitude of the pile body because the pile top is the heterogeneous boundary condition:
Figure BDA0001681444790000053
wherein,
Figure BDA0001681444790000054
satisfies the solution problem as listed in the formula (18),
Figure BDA0001681444790000055
in order to satisfy the undetermined function of the formula (4) and the boundary condition formulas (8) and (9) at the same time,
Figure BDA0001681444790000056
s32, solving the solution problem to obtain:
Figure BDA0001681444790000057
wherein,
Figure BDA0001681444790000058
n is 1,2, infinity to satisfy the transcendental equation
Figure BDA0001681444790000059
An infinite number of characteristic values of (a),
Figure BDA00016814447900000510
dimensionless parameter, K, representing the complex stiffness of the springs at the bottom of the pile bodyP＝kP+iωP，
Figure BDA00016814447900000511
And
Figure BDA00016814447900000512
satisfies the following relation:
Figure BDA00016814447900000513
will be provided with
Figure BDA00016814447900000514
Can be substituted by the formula (4):
Figure BDA00016814447900000515
wherein,
Figure BDA00016814447900000516
the general solution to solve formula (21) is:
Figure BDA00016814447900000517
wherein, aP、bPIn order to determine the coefficient to be determined,
substituting the formula (22) into the boundary conditions (8) and (9) can be solved:
Figure BDA0001681444790000061
Figure BDA0001681444790000062
substituting equations (19), (22), (23) and (24) for equation (17) can obtain the displacement response amplitude when the pile vibrates in longitudinal harmonic mode:
Figure BDA0001681444790000063
wherein,
Figure BDA0001681444790000064
is a series of undetermined functions.
Further, the S4 mainly includes the following steps:
s41, solving the undetermined coefficient according to the pile-soil coupling condition
Figure BDA0001681444790000065
The shear stresses of the soil around the pile and the pile body obtained from the formulas (16) and (25) are respectively:
Figure BDA0001681444790000066
Figure BDA0001681444790000067
s42, solving the longitudinal vibration displacement amplitude of the pile body:
assuming that the dimensionless supporting rigidity of the bottom of the soil around the pile is equal to the dimensionless supporting rigidity of the bottom of the pile body, i.e.
Figure BDA0001681444790000068
Then
Figure BDA0001681444790000069
The displacement continuity and stress balance conditions at the interface of the soil around the pile and the pile body can be obtained by combining the formulas (16), (25) and the formulas (26) and (27):
Figure BDA00016814447900000610
Figure BDA00016814447900000611
wherein r is0The radius is indicated as such and,
based on the orthogonality of the eigenfunction system, cos (h) is multiplied simultaneously at both ends of equation (28)nz-hnH) And is in [0, H ]]The integration above can give:
Figure BDA00016814447900000612
wherein,
Figure BDA00016814447900000613
the coupling type (29) and (30) can be solved as follows:
Figure BDA0001681444790000071
wherein,
Figure BDA0001681444790000072
solving the longitudinal vibration displacement amplitude of the pile body as follows:
Figure BDA0001681444790000073
s43, obtaining a pile top displacement frequency response function:
Figure BDA0001681444790000074
s44, calculating the complex stiffness of the pile top:
Figure BDA0001681444790000075
wherein Kr(r, omega) is the dynamic stiffness of the pile top, KiAnd (r, omega) is pile top dynamic damping.
Further, in step S5, the pile top complex stiffness analytic solution is subjected to mean processing along the radial direction, and then the pile top displacement frequency response function and the pile top complex stiffness radial mean analytic solution are obtained as follows:
Figure BDA0001681444790000076
Figure BDA0001681444790000077
wherein,
Figure BDA0001681444790000078
represents the radial mean value of the dynamic stiffness of the pile top,
Figure BDA0001681444790000079
represents the radial mean value of the dynamic damping of the pile top,
and evaluating the vibration characteristic of the pile body and the integrity of the pile body based on the pile top displacement frequency response function and the pile top complex stiffness radial mean value analytic solution.
The method comprises the steps of respectively treating a pile body and soil around the pile body as elastic and viscoelastic continuous media, considering a radial fluctuation effect based on a three-dimensional axial symmetry model of the pile body, establishing a three-dimensional axial symmetry pile-soil coupling system longitudinal vibration analysis model considering a viscoelastic supporting condition of a pile end, deriving frequency transcendental equations corresponding to different vibration modes of the system, determining undetermined functions capable of simultaneously meeting the viscoelastic supporting condition of the pile end and a non-homogeneous boundary condition of a pile top, further obtaining a basic solution of displacement of the pile body and the soil around the pile, deriving a dynamic impedance analytical solution of the pile top in a frequency domain by using a complete coupling condition of the pile soil on the basis, being suitable for a vibration response problem of the pile body in the frequency domain under a harmonic and excitation condition of a friction pile, being closer to a real model, having high calculation precision, and providing theoretical guidance and reference for pile foundation dynamic detection.
Based on the reasons, the invention can be widely popularized in the field of civil engineering.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings needed to be used in the description of the embodiments or the prior art will be briefly introduced below, and it is obvious that the drawings in the following description are some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to these drawings without creative efforts.
FIG. 1 is a flow chart of a floating pile longitudinal vibration analysis method based on a three-dimensional axisymmetric model of a pile body;
fig. 2 is a schematic diagram of the mechanical simplified model of the longitudinal coupling vibration of the pile-soil system.
Detailed Description
In order to make the objects, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are some, but not all, embodiments of the present invention. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
As shown in fig. 1, a floating pile longitudinal vibration analysis method based on a three-dimensional axisymmetric model of a pile body is characterized by comprising the following steps:
s1, as shown in figure 2, establishing a three-dimensional axisymmetric model of the pile body and the soil body around the pile, wherein the pile body is regarded as a homogeneous uniform-section elastic body, the bottom of the pile body is regarded as a viscoelastic support, the radial displacement of the pile body is ignored, the longitudinal displacement and the change of strain in the radial direction of the pile body are considered, the upper surface of the soil body around the pile is regarded as a free boundary, and the free boundary has no normal stress or shear stress;
s2, establishing a longitudinal vibration equation of the soil body around the pile and the pile body under the three-dimensional axial symmetry condition according to the basic theory of elastic dynamics, wherein the pile-soil coupling vibration system meets the conditions of linear elasticity and small deformation, and the displacement of the soil around the pile and the pile interface is continuous and the stress is balanced;
s3, solving a longitudinal vibration equation of the soil body around the pile and the pile body by a separation variable method, and deducing frequency transcendental equations corresponding to different vibration modes of the pile soil system;
s4, determining undetermined functions which can simultaneously meet the viscoelastic supporting conditions of the pile end and the heterogeneous boundary conditions of the pile top according to the pile-soil coupling conditions, and solving the undetermined functions to obtain the complex stiffness of the pile top;
and S5, carrying out mean value processing on the pile top complex stiffness analytic solution along the radial direction, further obtaining a pile top displacement frequency response function and a pile top complex stiffness radial mean value analytic solution, and finishing evaluation on the pile body vibration characteristic and the pile body integrity under the action of harmonic and excitation force.
In the step S2, the pile top is uniformly distributed with harmonic excitation force, the pile-soil system is vibrated in a steady state, and the longitudinal displacements of the soil around the pile and the pile body respectively satisfy the following relations:
Figure BDA0001681444790000091
Figure BDA0001681444790000092
wherein the harmonic excitation force is
Figure BDA0001681444790000093
Representing the unit of an imaginary number, omega the circular frequency,
Figure BDA0001681444790000094
respectively representing the longitudinal vibration amplitude of the soil body around the pile and the pile body, i represents an imaginary number unit, omega represents the vibration exciting load frequency,
the longitudinal vibration equations of the soil body around the pile and the pile body are respectively as follows:
Figure BDA0001681444790000095
Figure BDA0001681444790000096
wherein λ isSExpressing the Lame constant, mu, of the soil body around the pileSExpressing complex shear modulus, ρSExpressing the density of the soil around the pile, lambdaPDenotes the lanmei constant of the pile body, GPDenotes the shear modulus, ρPRepresenting the pile body density, r is a radial coordinate, t is time, z is a longitudinal coordinate,
λS＝2vSGS/(1-2vS)，μS＝GS(1+2ξSi)，
wherein v isSDenotes the Poisson's ratio, GSExpressing the viscoelastic support constant of the bottom of the soil layer, ξSRepresenting hysteresis damping ratio, vPThe expression of the poisson's ratio,
the expressions (1) and (2) are simplified into the following expressions:
Figure BDA0001681444790000101
Figure BDA0001681444790000102
wherein,
Figure BDA0001681444790000103
the shear wave velocity of the soil body around the pile and the pile body are respectively.
The soil body around the pile and the pile body meet the following boundary conditions:
the soil layer boundary conditions include:
the surface of the soil layer is free:
Figure BDA0001681444790000104
viscoelastic support of the bottom surface of the soil layer:
Figure BDA0001681444790000105
wherein: eSDenotes the modulus of elasticity, kS、SRepresents the viscoelastic support constant of the soil bottom, and when r → ∞ is reached, the displacement is zero:
Figure BDA0001681444790000106
the pile body boundary conditions include:
pile top action uniform distribution harmonic excitation force
Figure BDA0001681444790000107
Comprises the following steps:
Figure BDA0001681444790000108
wherein:
Figure BDA0001681444790000109
the harmonic excitation force amplitude of the pile top is represented,
and (3) viscoelastic support of the pile bottom:
Figure BDA00016814447900001010
wherein k isP、PDenotes the viscoelastic support constant of the pile bottom, EPThe modulus of elasticity is indicated.
Further, in the step S3, the process of solving the vibration equation of the soil around the pile includes:
order to
Figure BDA0001681444790000111
Equation (3) is thus decomposed into two ordinary differential equations:
Figure BDA0001681444790000112
Figure BDA0001681444790000113
wherein h isS，qSIs constant and satisfies the following relationship:
Figure BDA0001681444790000114
the solutions of equations (10) and (11) are obtained as follows:
ZS(z)＝CScos(hSz)+DSsin(hSz) (13)
RS(r)＝MSK0(qSr)+NSI0(qSr) (14)
wherein, I0(qSr)、K0(qSr) is a zero-order first, second class imaginary-vector Bessel function, CS、DS、MS、NSFor the pending integration constant determined by the boundary conditions,
the transcendental equation can be obtained by integrating the boundary condition equations (5) and (6):
Figure BDA0001681444790000115
wherein,
Figure BDA0001681444790000116
dimensionless parameter, K, representing the complex stiffness of the springs at the bottom of the soil layerS＝kS+iωSAnd H represents the pile length,
solving the transcendental equation by MATLAB programming to obtain infinite characteristic values, and recording the infinite characteristic values as
Figure BDA0001681444790000117
n is 1,2,. infinity,. and will
Figure BDA0001681444790000118
Can be substituted by formula (12)
Figure BDA0001681444790000119
The above-mentioned
Figure BDA00016814447900001110
A non-dimensional parameter is represented by,
comprehensive soil layer boundary condition formulas (5), (6) and (7) can obtain longitudinal vibration displacement amplitude of soil around the pile
Figure BDA00016814447900001111
The expression of (a) is:
Figure BDA00016814447900001112
wherein,
Figure BDA00016814447900001113
is a series of undetermined functions.
In the step S3, the process of solving the vibration equation of the soil around the pile is as follows:
s31, decomposing the solution problem for obtaining the longitudinal vibration displacement response amplitude of the pile body because the pile top is the heterogeneous boundary condition:
Figure BDA00016814447900001114
wherein,
Figure BDA0001681444790000121
satisfies the solution problem as listed in the formula (18),
Figure BDA0001681444790000122
in order to satisfy the undetermined function of the formula (4) and the boundary condition formulas (8) and (9) at the same time,
Figure BDA0001681444790000123
s32, solving the solution problem to obtain:
Figure BDA0001681444790000124
wherein,
Figure BDA0001681444790000125
n is 1,2, infinity to satisfy the transcendental equation
Figure BDA0001681444790000126
An infinite number of characteristic values of (a),
Figure BDA0001681444790000127
dimensionless parameter, K, representing the complex stiffness of the springs at the bottom of the pile bodyP＝kP+iωP，
Figure BDA0001681444790000128
And
Figure BDA0001681444790000129
satisfies the following relation:
Figure BDA00016814447900001210
will be provided with
Figure BDA00016814447900001211
Can be substituted by the formula (4):
Figure BDA00016814447900001212
wherein,
Figure BDA00016814447900001213
the general solution to solve formula (21) is:
Figure BDA00016814447900001214
wherein, aP、bPIn order to determine the coefficient to be determined,
substituting the formula (22) into the boundary conditions (8) and (9) can be solved:
Figure BDA00016814447900001215
Figure BDA00016814447900001216
substituting equations (19), (22), (23) and (24) for equation (17) can obtain the displacement response amplitude when the pile vibrates in longitudinal harmonic mode:
Figure BDA00016814447900001217
wherein,
Figure BDA0001681444790000131
is a series of undetermined functions.
The S4 mainly includes the following steps:
s41, solving the undetermined coefficient according to the pile-soil coupling condition
Figure BDA0001681444790000132
The shear stresses of the soil around the pile and the pile body obtained from the formulas (16) and (25) are respectively:
Figure BDA0001681444790000133
Figure BDA0001681444790000134
s42, solving the longitudinal vibration displacement amplitude of the pile body:
assuming that the dimensionless supporting rigidity of the bottom of the soil around the pile is equal to the dimensionless supporting rigidity of the bottom of the pile body, i.e.
Figure BDA0001681444790000135
Then
Figure BDA0001681444790000136
The displacement continuity and stress balance conditions at the interface of the soil around the pile and the pile body can be obtained by combining the formulas (16), (25) and the formulas (26) and (27):
Figure BDA0001681444790000137
Figure BDA0001681444790000138
wherein r is0The radius is indicated as such and,
based on the orthogonality of the eigenfunction system, cos (h) is multiplied simultaneously at both ends of equation (28)nz-hnH) And is in [0, H ]]The integration above can give:
Figure BDA0001681444790000139
wherein,
Figure BDA00016814447900001310
the coupling type (29) and (30) can be solved as follows:
Figure BDA00016814447900001311
wherein,
Figure BDA00016814447900001312
solving the longitudinal vibration displacement amplitude of the pile body as follows:
Figure BDA0001681444790000141
s43, obtaining a pile top displacement frequency response function:
Figure BDA0001681444790000142
s44, calculating the complex stiffness of the pile top:
Figure BDA0001681444790000143
wherein Kr(r, omega) is the dynamic stiffness of the pile top, KiAnd (r, omega) is pile top dynamic damping.
Further, in step S5, the pile top complex stiffness analytic solution is subjected to mean processing along the radial direction, and then the pile top displacement frequency response function and the pile top complex stiffness radial mean analytic solution are obtained as follows:
Figure BDA0001681444790000144
Figure BDA0001681444790000145
wherein,
Figure BDA0001681444790000146
represents the radial mean value of the dynamic stiffness of the pile top,
Figure BDA0001681444790000147
represents the radial mean value of the dynamic damping of the pile top,
and evaluating the vibration characteristic of the pile body and the integrity of the pile body based on the pile top displacement frequency response function and the pile top complex stiffness radial mean value analytic solution.
Finally, it should be noted that: the above embodiments are only used to illustrate the technical solution of the present invention, and not to limit the same; while the invention has been described in detail and with reference to the foregoing embodiments, it will be understood by those skilled in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some or all of the technical features may be equivalently replaced; and the modifications or the substitutions do not make the essence of the corresponding technical solutions depart from the scope of the technical solutions of the embodiments of the present invention.

Claims (1)

1. A floating pile longitudinal vibration analysis method based on a three-dimensional axisymmetric model of a pile body is characterized by comprising the following steps:
s1, establishing a three-dimensional axisymmetric model of the pile body and the soil body around the pile, wherein the pile body is regarded as a homogeneous uniform-section elastic body, the bottom of the pile body is regarded as a viscoelastic support, the radial displacement of the pile body is ignored, the longitudinal displacement and the change of strain along the radial direction are considered, the upper surface of the soil body around the pile is regarded as a free boundary, and the free boundary has no normal stress and shear stress;
s2, establishing a longitudinal vibration equation of the soil body around the pile and the pile body under the three-dimensional axial symmetry condition according to the basic theory of elastic dynamics, wherein the pile-soil coupling vibration system meets the conditions of linear elasticity and small deformation, and the displacement of the soil around the pile and the pile interface is continuous and the stress is balanced;
s3, solving a longitudinal vibration equation of the soil body around the pile and the pile body by a separation variable method, and deducing frequency transcendental equations corresponding to different vibration modes of the pile soil system;
s4, determining undetermined functions which can simultaneously meet the viscoelastic supporting condition of the bottom of the pile body and the inhomogeneous boundary condition of the pile top according to the pile-soil coupling condition, and solving the undetermined functions to obtain the complex stiffness of the pile top;
s5, carrying out mean value processing on the pile top complex stiffness analytic solution along the radial direction, further obtaining a pile top displacement frequency response function and a pile top complex stiffness radial mean value analytic solution, and finishing evaluation on pile body vibration characteristics and pile body integrity under the action of harmonic and excitation force
In the step S2, the pile top is uniformly distributed with harmonic excitation force, the pile-soil system is vibrated in a steady state, and the longitudinal displacements of the soil around the pile and the pile body respectively satisfy the following relations:
Figure FDA0002583776920000011
Figure FDA0002583776920000012
wherein the harmonic excitation force is
Figure FDA0002583776920000013
Figure FDA0002583776920000014
Representing the unit of an imaginary number, omega the circular frequency,
Figure FDA0002583776920000015
the method respectively expresses the longitudinal vibration amplitude of the soil body around the pile and the pile body, i expresses an imaginary number unit, and the longitudinal vibration equations of the soil body around the pile and the pile body are respectively as follows:
Figure FDA0002583776920000016
Figure FDA0002583776920000017
wherein λ isSExpressing the Lame constant, mu, of the soil body around the pileSExpressing complex shear modulus, ρSExpressing the density of the soil around the pile, lambdaPDenotes the lanmei constant of the pile body, GPDenotes the shear modulus, ρPRepresenting the pile body density, r is a radial coordinate, t is time, z is a longitudinal coordinate,
λS＝2vSGS/(1-2vS)，μS＝GS(1+2ξSi)，
wherein v isSDenotes the Poisson's ratio, GSExpressing the viscoelastic support constant of the bottom of the soil layer, ξSIndicating hysteresis resistanceNibbi, vPExpressing Poisson's ratio, and simplifying the expressions (1) and (2) as follows:
Figure FDA0002583776920000021
Figure FDA0002583776920000022
wherein,
Figure FDA0002583776920000023
respectively the shear wave velocity of the soil body around the pile and the pile body;
the soil body around the pile and the pile body meet the following boundary conditions:
the soil layer boundary conditions include:
the surface of the soil layer is free:
Figure FDA0002583776920000024
viscoelastic support of the bottom surface of the soil layer:
Figure FDA0002583776920000025
wherein: eSDenotes the modulus of elasticity, kS、SThe viscoelastic support constant of the bottom of the soil layer is shown,
when r → ∞ the displacement is zero:
Figure FDA0002583776920000026
the pile body boundary conditions include:
pile top action uniform distribution harmonic excitation force
Figure FDA0002583776920000027
Comprises the following steps:
Figure FDA0002583776920000028
wherein:
Figure FDA0002583776920000029
the harmonic excitation force amplitude of the pile top is represented,
viscoelastic support of the bottom of the pile body:
Figure FDA0002583776920000031
wherein k isP、PDenotes the viscoelastic support constant of the pile bottom, EPDenotes the modulus of elasticity;
in the step S3, the process of solving the vibration equation of the soil around the pile is as follows:
order to
Figure FDA0002583776920000032
Equation (3) is thus decomposed into two ordinary differential equations:
Figure FDA0002583776920000033
Figure FDA0002583776920000034
wherein h isS，qSIs constant and satisfies the following relationship:
Figure FDA0002583776920000035
the solutions of equations (10) and (11) are obtained as follows:
ZS(z)＝CScos(hSz)+DSsin(hSz) (13)
RS(r)＝MSK0(qSr)+NSI0(qSr) (14)
wherein, I0(qSr)、K0(qSr) is a zero-order first, second class imaginary-vector Bessel function, CS、DS、MS、NSFor undetermined integral constants determined by boundary conditions, the transcendental equation can be obtained by synthesizing the boundary condition equations (5) and (6):
Figure FDA0002583776920000036
wherein,
Figure FDA0002583776920000037
dimensionless parameter, K, representing the complex stiffness of the springs at the bottom of the soil layerS＝kS+iωSAnd H represents the pile length,
solving the transcendental equation by MATLAB programming to obtain infinite characteristic values, and recording the infinite characteristic values as
Figure FDA0002583776920000038
n is 1,2, …, ∞, and
Figure FDA0002583776920000039
can be substituted by formula (12)
Figure FDA00025837769200000310
The above-mentioned
Figure FDA00025837769200000311
A non-dimensional parameter is represented by,
comprehensive soil layer boundary condition formulas (5), (6) and (7) can obtain longitudinal vibration displacement amplitude of soil around the pile
Figure FDA00025837769200000312
The expression of (a) is:
Figure FDA00025837769200000313
wherein,
Figure FDA00025837769200000314
is a series of undetermined functions;
in step S3, the solution process of the vibration equation in the longitudinal direction of the pile body is as follows:
s31, decomposing the solution problem for obtaining the longitudinal vibration displacement response amplitude of the pile body because the pile top is the heterogeneous boundary condition:
Figure FDA0002583776920000041
wherein,
Figure FDA0002583776920000042
satisfies the solution problem as listed in the formula (18),
Figure FDA0002583776920000043
in order to satisfy the undetermined function of the formula (4) and the boundary condition formulas (8) and (9) at the same time,
Figure FDA0002583776920000044
s32, solving the solution problem to obtain:
Figure FDA0002583776920000045
wherein,
Figure FDA0002583776920000046
to satisfy transcendental equation
Figure FDA0002583776920000047
An infinite number of characteristic values of (a),
Figure FDA0002583776920000048
represents dimensionless parameters of the complex stiffness of the spring at the bottom of the pile body,KP＝kP+iωP，
Figure FDA0002583776920000049
and
Figure FDA00025837769200000410
satisfies the following relation:
Figure FDA00025837769200000411
will be provided with
Figure FDA00025837769200000412
Can be substituted by the formula (4):
Figure FDA00025837769200000413
wherein,
Figure FDA00025837769200000414
the general solution to solve formula (21) is:
Figure FDA00025837769200000415
wherein, aP、bPIn order to determine the coefficient to be determined,
substituting the formula (22) into the boundary conditions (8) and (9) can be solved:
Figure FDA00025837769200000416
Figure FDA00025837769200000417
substituting equations (19), (22), (23) and (24) for equation (17) can obtain the displacement response amplitude when the pile vibrates in longitudinal harmonic mode:
Figure FDA0002583776920000051
wherein,
Figure FDA0002583776920000052
is a series of undetermined functions;
the S4 mainly includes the following steps:
s41, solving the undetermined coefficient according to the pile-soil coupling condition
Figure FDA0002583776920000053
The shear stresses of the soil around the pile and the pile body obtained from the formulas (16) and (25) are respectively:
Figure FDA0002583776920000054
Figure FDA0002583776920000055
s42, solving the longitudinal vibration displacement amplitude of the pile body:
assuming that the dimensionless supporting rigidity of the bottom of the soil around the pile is equal to the dimensionless supporting rigidity of the bottom of the pile body, i.e.
Figure FDA0002583776920000056
Then
Figure FDA0002583776920000057
The displacement continuity and stress balance conditions at the interface of the soil around the pile and the pile body can be obtained by combining the formulas (16), (25) and the formulas (26) and (27):
Figure FDA0002583776920000058
Figure FDA0002583776920000059
wherein r is0The radius is indicated as such and,
based on the orthogonality of the eigenfunction system, cos (h) is multiplied simultaneously at both ends of equation (28)nz-hnH) And is in [0, H ]]The integration above can give:
Figure FDA00025837769200000510
wherein,
Figure FDA00025837769200000511
the coupling type (29) and (30) can be solved as follows:
Figure FDA00025837769200000512
wherein,
Figure FDA00025837769200000513
solving the longitudinal vibration displacement amplitude of the pile body as follows:
Figure FDA0002583776920000061
s43, obtaining a pile top displacement frequency response function:
Figure FDA0002583776920000062
s44, calculating the complex stiffness of the pile top:
Figure FDA0002583776920000063
wherein Kr(r, omega) is the dynamic stiffness of the pile top, Ki(r, ω) is pile top dynamic damping;
in the step S5, the pile top complex stiffness analytic solution is subjected to mean processing along the radial direction, and then the pile top displacement frequency response function and the pile top complex stiffness radial mean analytic solution are obtained as follows:
Figure FDA0002583776920000064
Figure FDA0002583776920000065
wherein,
Figure FDA0002583776920000066
represents the radial mean value of the dynamic stiffness of the pile top,
Figure FDA0002583776920000067
represents the radial mean value of the dynamic damping of the pile top,
and evaluating the vibration characteristic of the pile body and the integrity of the pile body based on the pile top displacement frequency response function and the pile top complex stiffness radial mean value analytic solution.
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