CN107604957B - Complex heterogeneous soil-in-pipe pile longitudinal vibration analysis method based on viscous damping model - Google Patents

Complex heterogeneous soil-in-pipe pile longitudinal vibration analysis method based on viscous damping model Download PDF

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CN107604957B
CN107604957B CN201710955103.7A CN201710955103A CN107604957B CN 107604957 B CN107604957 B CN 107604957B CN 201710955103 A CN201710955103 A CN 201710955103A CN 107604957 B CN107604957 B CN 107604957B
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soil body
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赵密
崔春义
许成顺
杜修力
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Beijing University of Technology
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Abstract

The invention discloses a method for analyzing longitudinal vibration of a complicated heterogeneous soil-in-pipe pile based on a viscous damping model, and belongs to the field of civil engineering theory analysis. Firstly, a tubular pile-soil coupling vibration system is longitudinally divided into any number of intervals, soil around a pile of each longitudinal interval is radially divided into an internal disturbance area and an external area, the internal disturbance area is radially divided into any number of circle layers, each circle layer of soil is a homogeneous and isotropic linear viscoelastic body, the soil in the external area radially extends infinitely, and viscous damping is adopted for soil material damping; the displacement of the pile-soil interface and the two sides of the soil interface of each circle layer is continuous, the stress is balanced, and the vibration of the pile-soil system is small deformation; the pile body concrete is linear elastic, and the propagation of stress waves in the pile body meets the assumption of a flat section; and establishing longitudinal vibration equations of the pile periphery, the pile core soil body and the pile body, solving the three vibration equations, and obtaining a time domain speed response function of any exciting force acting on the pile top by utilizing the transmissibility of the impedance function of the pile body among longitudinal sections.

Description

Complex heterogeneous soil-in-pipe pile longitudinal vibration analysis method based on viscous damping model
Technical Field
The invention relates to the field of civil engineering theory analysis, in particular to a method for analyzing longitudinal vibration of a pipe pile in a radial heterogeneous and longitudinal stratified soil body based on a viscous damping model.
Background
The study on the coupling vibration characteristics of the pile foundation interaction system in the longitudinal stratified soil 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 always a cross hotspot problem in the fields of soil dynamics, geotechnical engineering and structure-foundation interaction.
The study on the pile-soil coupling vibration characteristics is a theoretical basis in the engineering technical fields of pile foundation earthquake resistance, earthquake-proof design, pile foundation power detection and the like, and is a hot point problem in geotechnical engineering and solid mechanics all the time.
As is known, in the process of pile foundation construction, due to the influence of soil squeezing, loosening and other disturbance factors, the soil body around the pile has a certain non-uniformity along the radial direction of the pile foundation, i.e. a radial non-uniform effect. In order to take such radial heterogeneous effect into consideration, many scholars at home and abroad have achieved a great deal of achievements. The achievements can be classified from different angles, and from the view of acting external load, the achievements can be divided into frequency domain response research under the harmonic load action and time domain and frequency domain response research under any load; from the view of material damping of soil, the damping method can be divided into hysteretic material damping and viscous material damping; from the viewpoint of the solving method, the method can be classified into an analytical method, a semi-analytical method and a numerical method.
The material damping of the soil body is energy dissipation caused by particle friction in the soil body, the internal friction is caused by defects of medium particle crystal structures, inelastic connection among medium particles and other thermoelastic processes, and is inevitable, and in order to consider the internal friction effect, the soil body linear constitutive equation considering the damping effect is adopted to study the influence of the material damping on the pile dynamic response.
Common linear damping constitutive equations established on the basis of observation and experiment can be divided into two types: a time domain constitutive equation and a frequency domain constitutive equation, wherein the time domain constitutive equation is directly established in a time domain from a macroscopic physical model linear viscoelastic body; the latter is established in the frequency domain by matching with classical frequency domain analysis methods.
The time domain constitutive model of the linear viscoelastic body can be composed of a linear spring and a linear damping element, the viscous stress of the linear damping element is in direct proportion to the strain rate, and various linear viscoelastic constitutive models can be composed of the two linear units and can reflect the stress-strain property of a real solid.
The linear hysteretic damping is mainly embodied in the hysteretic damping ratio of the frequency domain constitutive, the frequency domain constitutive can be understood as inverse Fourier transform of the time domain constitutive, the hysteretic damping ratio is generally assumed to be constant, namely, the change of the hysteretic damping ratio is not large or no obvious trend change is assumed to be in an elastic working region of a material. In addition, the frequency domain analysis of the steady-state vibration problem under harmonic and load can approximately reflect the material damping characteristics of the soil body. However, for the problems of anharmonic and vibration (transient vibration or random vibration), the hysteretic damping model is not suitable, particularly when the time-domain response of the pile under the transient excitation condition is researched, the soil damping force is related to the amplitude and the strain rate, and the adoption of the hysteretic damping model can cause contradiction conceptually, so that the so-called 'non-causality of the dynamic response' is generated, and the viscous damping model is suitable and is more reasonable physically.
In addition, most of the current researches are directed to solid piles, and for large-diameter tubular piles, due to the existence of pile core soil, the vibration characteristics of the solid piles and the solid piles are different. The vibration characteristics of the pipe pile in the radial homogeneous soil are solved by considering the soil around the pile and the soil in the pile core simultaneously, such as Dingming, etc. and Zhengchangjie, and the results are compared with the results of the solid pile, which shows that the pipe pile shows different dynamic characteristics from the solid pile under the action of vertical load.
Disclosure of Invention
The invention aims to overcome the technical defects in the prior art, and considers the construction disturbance of soil around a pile and the longitudinal layering characteristic of the soil, the soil adopts a viscous damping model, and a multi-circle layer plane strain model is transmitted based on complex stiffness, so that the longitudinal vibration characteristic of a radial heterogeneous and longitudinal layering viscous damping soil-in-pipe pile under the action of any exciting force is analyzed.
In order to achieve the purpose, the technical scheme of the invention is as follows:
a method for analyzing the longitudinal vibration of a tubular pile in a radial heterogeneous and longitudinal stratified soil body based on a viscous damping model comprises the following steps:
s1: the pile soil surrounding system of the tubular pile is divided into a pile soil surrounding body and a pile core soil body, the pile soil surrounding body and the pile core soil body are assumed to be a series of mutually independent thin layers, and the interaction among the thin layers is ignored;
s2: the pile-surrounding soil body is divided into any number of intervals along the longitudinal direction, the pile-surrounding soil body of each interval is divided into an internal disturbance area and an external area along the radial direction, the internal disturbance area is divided into any number of circle layers along the radial direction, the soil body in each circle layer is a homogeneous and isotropic linear viscoelastic body, the soil body in the external area extends in the radial direction infinitely, the damping of the soil body material adopts viscous damping, and the radial displacement of the soil body is ignored;
s3: the displacement of the soil interface around the pile and the two sides of the soil interface of each circle layer is continuous and the stress is balanced, and the vibration of the soil system around the pile is small deformation;
s4: the concrete of the tubular pile body is linear elastic, and the propagation of stress waves in the tubular pile body meets the assumption of a flat section;
s5: according to the basic theory of elastic dynamics, establishing a longitudinal vibration equation and boundary conditions of a pile surrounding soil body, a pile core soil body and a pile body of the pipe pile under a plane strain condition;
s6: and (3) solving the three vibration equations in the S5 by using Laplace (Laplace) transformation, and obtaining a time domain speed response function of any exciting force acting on the pile top by using the transmissibility of the impedance function of the pile body between the longitudinal sections so as to analyze the longitudinal vibration of the pipe pile.
The vibration equation of the soil body around the pile:
Figure BDA0001433788930000031
pile core soil body vibration equation:
Figure BDA0001433788930000032
the tubular pile body longitudinal vibration equation which accords with the assumption of the flat section is as follows:
Figure BDA0001433788930000033
wherein, the soil body around the pile is divided into m sections along the longitudinal direction, the pile length is H, the pile sections are numbered as 1, 2, …, i, … and m sections from the bottom to the top in sequence, and the thickness of each section is l1、l2、…、li、…、lmThe top buried depth of each layer section is h1、h2、…、hi…、hm. The inner diameter of the tubular pile, the outer diameter of the tubular pile, the sectional area of a tubular pile section, the density of the tubular pile and the elastic modulus of the tubular pile are respectively ri0、ri1、
Figure BDA0001433788930000034
And
Figure BDA0001433788930000035
the coefficient of stiffness of the viscoelastic support at the bottom of the pile isp、kp. In the longitudinal ith layer, the shear modulus, the viscous damping coefficient and the density of the pile core soil body are respectively Gi0、ηi0、ρi0. Meanwhile, the pile-surrounding soil body of the longitudinal ith layer is divided into an internal disturbance area and an external area along the radial direction, and the radial thickness of the internal disturbance area of the pile-surrounding soil body is biDividing the internal disturbance area into n circle layers along the radial direction, wherein the shear modulus, the viscous damping coefficient and the density of the soil body of the j-th circle layer are respectively Gij、ηij、ρijThe radius at the interface of the j-1 th circle layer and the j circle layer is rij. Radius at the interface of the inner zone and the outer zone is ri(n+1)The outer region is a radially semi-infinite uniform viscoelastic medium. The pile top of the pipe pile acts any exciting force p (t), and the shear stress generated by the pile core soil and the pile surrounding soil of the ith layer on the pile body is respectively
Figure BDA0001433788930000041
And
Figure BDA0001433788930000042
in the ith layer, the pile core soil body is set to be displaced into
Figure BDA0001433788930000043
The displacement of a certain point in the jth circle layer of the soil body around the pile is
Figure BDA0001433788930000044
The i-th layer has a pile body displacement of
Figure BDA0001433788930000045
r is radial displacement, t is time, z is longitudinal displacement, EpiIs the elastic modulus of the pile body of the ith layer section, ApiThe sectional area of the pile body of the ith layer is shown;
the boundary conditions in S5 include:
pile core soil body boundary conditions:
when r is 0, the displacement is finite:
Figure BDA0001433788930000046
the displacement and force continuous conditions of the pile core soil body and the pile are as follows:
Figure BDA0001433788930000047
Figure BDA0001433788930000048
ri0is the inner radius of the pile,
Figure BDA0001433788930000049
the shear stress of the pile core soil body to the pile body is generated,
Figure BDA00014337889300000410
the vertical shear stress of a pile core soil body on the inner wall of a tubular pile is positive clockwise;
boundary conditions of soil bodies around the pile:
when r ∞, the displacement is zero:
Figure BDA00014337889300000411
in the formula (I), the compound is shown in the specification,
Figure BDA00014337889300000412
representing the displacement of the outer region of the soil body of the ith layer.
The displacement and force continuous conditions of the soil body around the pile and the pile are as follows:
Figure BDA00014337889300000413
Figure BDA00014337889300000414
wherein r isi1Is the outer radius of the pile,
Figure BDA00014337889300000415
for the displacement of the soil body of the layer 1,
Figure BDA00014337889300000416
the shear stress of the soil body around the pile on the pile body is generated,
Figure BDA00014337889300000417
the vertical shear stress of the soil body around the pile on the outer wall of the tubular pile is positive clockwise;
pile body boundary conditions:
pile top acting force is p (t):
Figure BDA0001433788930000051
boundary conditions at pile ends:
Figure BDA0001433788930000052
Figure BDA0001433788930000053
the modulus of elasticity of the pile body is,
Figure BDA0001433788930000054
is the cross-sectional area of pile body, kp，pIs the viscoelastic support constant of the pile bottom.
S6 includes the following steps:
step 1: and performing Laplace (Laplace) conversion on the equation to obtain a soil layer shear stiffness recurrence formula of the multi-circle layer plane strain model of the ith longitudinal layer section based on viscous damping, wherein the soil layer shear stiffness recurrence formula comprises the following steps:
Figure BDA0001433788930000055
wherein
Figure BDA0001433788930000056
Wherein r isijIs the inner boundary of the ith layer interval and the jth circle layer soil, ri(j+1)Is the outer boundary of the ith layer and the jth circle of layer soil,
Figure BDA0001433788930000057
is the intrinsic parameters of the soil of the jth circle of the ith layer section, s is a complex variable,
Figure BDA0001433788930000058
the shear stiffness of the inner boundary of the jth circle layer soil of the ith layer section,
Figure BDA0001433788930000059
shear stiffness of the outer boundary of the ith interval and jth circle of layer soil, I0、I1Modified Bessel functions of the first kind, K, for zero and first order0、K1A second class of zero and first order modified Bessel (Bessel) functions;
step 2: and performing Laplace (Laplace) conversion on the equation sum to obtain a shear stiffness formula of the inner wall of the pipe pile of the ith layer subjected to the pile core soil body:
Figure BDA00014337889300000510
wherein the content of the first and second substances,
Figure BDA00014337889300000511
the intrinsic parameters of the pile core soil are obtained;
and step 3: and carrying out Laplace (Laplace) transformation on the equation sum, and obtaining an impedance function recurrence formula of the adjacent longitudinal layer section of the pile body according to the conditions of force balance and displacement continuity at the interface of the adjacent layer section of the pile body:
Figure BDA0001433788930000061
Figure BDA0001433788930000062
in the formula (I), the compound is shown in the specification,
Figure BDA0001433788930000063
is the pile body impedance function of the ith layer section and the ith-1 layer section,
Figure BDA0001433788930000064
αi、βito solve for the simplification parameters,/iIs the length of the pile body of the ith layer section hi、hi-1The lengths from the pile top to the pile body top of the ith layer and the ith-1 layer are respectively.
And 4, step 4: obtaining a pile top complex dynamic stiffness formula by utilizing the transmissibility of a pile body impedance function:
Figure BDA0001433788930000065
wherein
Figure BDA0001433788930000066
K′dComplex stiffness K for pile topdDimensionless parameter of, say K'd＝Kr+iKi，KrRepresenting the dynamic stiffness of the pile head, Kirepresenting dynamic damping of pile head, alpham、βmTo solve for the simplification parameters,/mThe length of the m section of pile is;
step 5: obtaining pile top velocity admittance function H according to formula (17)v：
Figure BDA0001433788930000067
Wherein the content of the first and second substances,
Figure BDA0001433788930000068
the density of the m-th section of pile body,
Figure BDA0001433788930000069
is the elastic wave velocity H of the m-th section of pile bodyvIs pile tip velocity admittance function HvDimensionless of(ii) a And omega is the longitudinal vibration circular frequency.
Step 6: the time domain response of the unit pulse excitation is obtained from (18) as:
Figure BDA00014337889300000610
wherein T ═ T/TcIs dimensionless time, theta is dimensionless frequency; IFT is the fast Fourier transform symbol;
and 7: obtaining a time domain speed response function of any exciting force p (t) acting on the pile top according to the convolution theorem
g(t)＝p(t)*h(t)＝IFT[P(iω)·H(iω)](20)
H (t) is time domain velocity response under the unit pulse excitation effect, and H (i omega) is a pile top velocity frequency response function.
And 8: the exciting force p (t) is half-sine pulse excitation
Figure BDA0001433788930000071
when T belongs to (0, T), T is pulse width, and the half-analytic solution of the pile top time domain velocity response is as follows:
Figure BDA0001433788930000072
wherein Q ismaxIs a half-sinusoidal pulse amplitude, Vv' is the dimensionless speed of the time domain response.
According to the technical scheme, the longitudinal vibration of the large-diameter tubular pile is analyzed by adopting the radial heterogeneous and longitudinal stratified viscous damping soil body model, the damping force of the viscous damping soil body model is related to the strain rate, the viscous damping soil body model can be suitable for the problems of non-resonance and excitation, particularly the problem of time domain vibration response of the pile body under the transient excitation condition, meanwhile, the radial heterogeneous performance considers the construction disturbance effect of the soil body around the pile, is closer to a real model, in addition, the influence of the pile core soil on the vibration characteristic of the tubular pile is considered, the calculation precision is higher, and theoretical guidance and reference effects can be provided for the power detection of the pile foundation.
Drawings
FIG. 1 is a flow chart of a method for analyzing longitudinal vibration of a pipe pile in a radially inhomogeneous and longitudinally stratified soil mass based on a viscous damping model.
Fig. 2 is a schematic diagram of the mechanical simplified model of the longitudinal coupling vibration of the pile-soil system.
Detailed Description
The following describes embodiments of the present invention in further detail with reference to the accompanying drawings.
Referring to fig. 1, fig. 1 is a flow chart of a method for analyzing longitudinal vibration of a pipe pile in a radially inhomogeneous and longitudinally stratified soil mass based on a viscous damping model according to the present invention. As shown in fig. 1, a method for analyzing longitudinal vibration of a pipe pile in a radially inhomogeneous and longitudinally stratified soil mass based on a viscous damping model includes the following steps:
s1: assuming that the soil around the pile and the soil in the pile core are a series of mutually independent thin layers, and neglecting the interaction between soil layers;
s2: the pile-surrounding soil body is longitudinally divided into any number of intervals, the pile-surrounding soil body of each interval is divided into an internal disturbance area and an external area along the radial direction, the internal disturbance area is divided into any number of circle layers along the radial direction, each circle of soil body is a homogeneous and isotropic linear viscoelastic body, the soil body of the external area extends in the radial direction infinitely, the soil body material damping adopts viscous damping, and the radial displacement of the soil body is ignored;
s3: the displacement of the pile-soil interface and the two sides of the soil interface of each circle layer is continuous, the stress is balanced, and the vibration of the pile-soil system is small deformation;
s4: the pile body concrete is linear elastic, and the propagation of stress waves in the pile body meets the assumption of a flat section;
the invention is based on a plane strain model, researches the longitudinal vibration characteristics of viscoelastic supporting pipe piles in soil of any longitudinal layer section and any radial ring layer, and a mechanical simplified model is shown in figure 2. Dividing the soil body around the pile into m sections along the longitudinal direction, numbering the pile length H tubular pile from the bottom of the pile body to the top in sequence as 1, 2, …, i, … and m sections, wherein the thickness of each section is l1、l2、…、li、…、lmOn top of each layer segmentBuried depths are respectively h1、h2、…、hi…、hm. The inner diameter, the outer diameter, the sectional area of a pile section, the density and the elastic modulus of the tubular pile are respectively ri0、ri1、Ai P、ρi PAnd Ei PThe coefficient of stiffness of the viscoelastic support at the bottom of the pile isp、kp. In the longitudinal ith layer, the shear modulus, the viscous damping coefficient and the density of the pile core soil body are respectively Gi0、ηi0、ρi0. Meanwhile, the pile-surrounding soil body of the longitudinal ith layer is divided into an internal disturbance area and an external area along the radial direction, and the radial thickness of the internal disturbance area of the pile-surrounding soil body is biDividing the internal disturbance area into n circle layers along the radial direction, wherein the shear modulus, the viscous damping coefficient and the density of the soil body of the j-th circle layer are respectively Gij、ηij、ρijThe radius at the interface of the j-1 th circle layer and the j circle layer is rij. In particular, the radius at the interface of the inner zone and the outer zone is ri(n+1)The outer region is a radially semi-infinite uniform viscoelastic medium. The pile top of the pipe pile acts any exciting force p (t), and the shear stress generated by the pile core soil and the pile surrounding soil of the ith layer on the pile body is respectively
Figure BDA0001433788930000081
And
Figure BDA0001433788930000082
s5: according to the basic theory of elastic dynamics, establishing a longitudinal vibration equation and boundary conditions of a pile surrounding soil body, a pile core soil body and a pile body under a plane strain condition;
s6: and (3) solving the three vibration equations in the S5 by using Laplace transformation, and obtaining a time domain speed response function of any exciting force acting on the pile top by using the transmissibility of the impedance function of the pile body between the longitudinal sections so as to analyze the longitudinal vibration of the pipe pile.
Specifically, the method comprises the following specific steps:
step 1: dividing the soil body around the pile into m sections along the longitudinal direction, and forming the pile length into H-shaped tubular piles from bottom to topThe numbers of the layers are 1, 2, …, i, … and m, the thickness of each layer is l1、l2、…、li、…、lmThe top buried depth of each layer section is h1、h2、…、hi…、hm. The inner diameter, the outer diameter, the sectional area of a pile section, the density and the elastic modulus of the tubular pile are respectively ri0、ri1、
Figure BDA0001433788930000091
And
Figure BDA0001433788930000092
the coefficient of stiffness of the viscoelastic support at the bottom of the pile isp、kp. In the longitudinal ith layer, the shear modulus, the viscous damping coefficient and the density of the pile core soil body are respectively Gi0、ηi0、ρi0. Meanwhile, the pile-surrounding soil body of the longitudinal ith layer is divided into an internal disturbance area and an external area along the radial direction, and the radial thickness of the internal disturbance area of the pile-surrounding soil body is biDividing the internal disturbance area into n circle layers along the radial direction, wherein the shear modulus, the viscous damping coefficient and the density of the soil body of the j-th circle layer are respectively Gij、ηij、ρijThe radius at the interface of the j-1 th circle layer and the j circle layer is rij. In particular, the radius at the interface of the inner zone and the outer zone is ri(n+1)The outer region is a radially semi-infinite uniform viscoelastic medium. The pile top of the pipe pile acts any exciting force p (t), and the shear stress generated by the pile core soil and the pile surrounding soil of the ith layer on the pile body is respectively
Figure BDA0001433788930000093
And
Figure BDA0001433788930000094
in the ith layer, the pile core soil body is set to be displaced into
Figure BDA0001433788930000095
The displacement of a certain point in the jth circle layer of the soil body around the pile is
Figure BDA0001433788930000096
The i-th layer has a pile body displacement of
Figure BDA0001433788930000097
r is radial displacement, t is time, z is longitudinal displacement, EpiIs the elastic modulus of the pile body of the ith layer section, ApiEstablishing longitudinal vibration equations and boundary conditions of the pile periphery, the pile core soil body and the pile body under the plane strain condition according to the basic theory of elastic dynamics for the sectional area of the pile body of the ith layer as follows:
the vibration equation of the soil body around the pile:
Figure BDA0001433788930000098
pile core soil body vibration equation:
Figure BDA0001433788930000099
the longitudinal vibration equation of the pile body according with the assumption of the flat section is as follows:
Figure BDA00014337889300000910
pile core soil boundary conditions:
when r is 0, the displacement is finite:
Figure BDA0001433788930000101
the displacement and force continuous conditions of the pile core soil and the pile are as follows:
Figure BDA0001433788930000102
Figure BDA0001433788930000103
ri0is the inner radius of the pile,
Figure BDA0001433788930000104
the shear stress of the pile core soil to the pile body is generated,
Figure BDA0001433788930000105
the vertical shear stress of the pile core soil on the inner wall of the tubular pile is positive clockwise;
boundary conditions of soil around the pile:
when r ∞, the displacement is zero:
Figure BDA0001433788930000106
in the formula (I), the compound is shown in the specification,
Figure BDA0001433788930000107
representing the displacement of the outer region of the soil body of the ith layer.
And (3) displacement and force continuous conditions of soil around the pile and the pile:
Figure BDA0001433788930000108
Figure BDA0001433788930000109
wherein r isi1Is the outer radius of the pile,
Figure BDA00014337889300001010
for the displacement of the soil body of the layer 1,
Figure BDA00014337889300001011
the shear stress of the soil around the pile on the pile body is generated,
Figure BDA00014337889300001012
the vertical shear stress of the soil around the pile on the outer wall of the tubular pile is positive clockwise;
pile body boundary conditions:
pile top acting force is p (t):
Figure BDA00014337889300001013
boundary conditions at pile ends:
Figure BDA00014337889300001014
wherein the content of the first and second substances,
Figure BDA00014337889300001015
the modulus of elasticity of the pile body is,
Figure BDA00014337889300001016
is the cross-sectional area of pile body, kp，pIs the viscoelastic support constant of the pile bottom.
Further, the S6 includes the following specific steps:
step 1: and performing Laplace transformation on the equation to obtain a soil layer shear stiffness recurrence formula of the longitudinal ith layer section multi-circle layer plane strain model based on viscous damping, wherein the soil layer shear stiffness recurrence formula is as follows:
Figure BDA0001433788930000111
wherein
Figure BDA0001433788930000112
Wherein r isijIs the inner boundary of the ith layer interval and the jth circle layer soil, ri(j+1)Is the outer boundary of the ith layer and the jth circle of layer soil,
Figure BDA0001433788930000113
is the intrinsic parameters of the soil of the jth circle of the ith layer section, s is a complex variable,
Figure BDA0001433788930000114
the shear stiffness of the inner boundary of the jth circle layer soil of the ith layer section,
Figure BDA0001433788930000115
is the ith layer sectionShear stiffness of the outer boundary of j circles of subsoil, I0、I1Modifying Bessel functions for the first class of zero and first orders, K0、K1A second class of zero and first modified Bessel functions;
step 2: and (3) carrying out Laplace transformation on the equation sum to obtain a shear stiffness formula of the inner wall of the pipe pile of the ith layer subjected to the pile core soil body:
Figure BDA0001433788930000116
wherein the content of the first and second substances,
Figure BDA0001433788930000117
the intrinsic parameters of the pile core soil are obtained;
and step 3: and carrying out Laplace transformation on the equation sum, and obtaining an impedance function recurrence formula of the adjacent longitudinal layer section of the pile body according to the conditions of force balance and displacement continuity at the interface of the adjacent layer section of the pile body:
Figure BDA0001433788930000118
Figure BDA0001433788930000119
in the formula (I), the compound is shown in the specification,
Figure BDA00014337889300001110
is the pile body impedance function of the ith layer section and the ith-1 layer section,
Figure BDA00014337889300001111
αi、βito solve for the simplification parameters,/iIs the length of the pile body of the ith layer section hi、hi-1The lengths from the pile top to the pile body top of the ith layer and the ith-1 layer are respectively.
And 4, step 4: obtaining a pile top complex dynamic stiffness formula by utilizing the transmissibility of a pile body impedance function:
Figure BDA0001433788930000121
wherein
Figure BDA0001433788930000122
K′dComplex stiffness K for pile topdDimensionless parameter of, say K'd＝Kr+iKi，KrRepresenting the dynamic stiffness of the pile head, Kirepresenting dynamic damping of pile head, alpham、βmTo solve for the simplification parameters,/mThe length of the m section of pile is;
and 5: obtaining a pile top velocity admittance function according to the formula (17):
Figure BDA0001433788930000123
wherein the content of the first and second substances,
Figure BDA0001433788930000124
the density of the m-th section of pile body,
Figure BDA0001433788930000125
is the elastic wave velocity H of the m-th section of pile bodyvIs pile tip velocity admittance function HvDimensionless of (a);
step 6: the time domain response of the unit pulse excitation is obtained from (18) as:
Figure BDA0001433788930000126
wherein T ═ T/TcIs dimensionless time, theta is dimensionless frequency; IFT is the fast Fourier transform symbol;
and 7: obtaining a time domain speed response function of any exciting force p (t) acting on the pile top according to the convolution theorem
g(t)＝p(t)*h(t)＝IFT[P(iω)·H(iω)](20)
H (t) is time domain velocity response under the unit pulse excitation effect, and H (i omega) is a pile top velocity frequency response function.
And 8: the exciting force p (t) is half-sine pulse excitation
Figure BDA0001433788930000127
when T belongs to (0, T) and T is the pulse width, the half-analytic solution of the pile top time domain velocity response is as follows:
Figure BDA0001433788930000128
wherein Q ismaxIs a half-sinusoidal pulse amplitude, Vv' is the dimensionless speed of the time domain response.
Further, based on the pile top speed admittance function and the pile top speed time domain response function, the vibration characteristic of the pile body and the integrity of the pile body can be evaluated.
In summary, the damping model adopted by the method for analyzing the longitudinal vibration of the tubular pile in the radially heterogeneous and longitudinally layered soil mass based on the viscous damping model is related to the strain rate, so that the method can be suitable for the non-harmonic and excitation problems, particularly the time domain vibration response problem of the pile body under the transient excitation condition, the construction disturbance effect of the soil mass around the pile is considered in the radial heterogeneous performance, the layering characteristic caused by natural deposition of the soil mass can be considered in the longitudinal layering, and theoretical guidance and reference effects can be provided for pile foundation power 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 (2)

1. A method for analyzing longitudinal vibration of a pipe pile in complex heterogeneous soil based on a viscous damping model is characterized by comprising the following steps: the method comprises the following steps:
s1: the pile soil surrounding system of the tubular pile is divided into a pile soil surrounding body and a pile core soil body, the pile soil surrounding body and the pile core soil body are assumed to be a series of mutually independent thin layers, and the interaction among the thin layers is ignored;
s2: the pile-surrounding soil body is divided into any number of intervals along the longitudinal direction, the pile-surrounding soil body of each interval is divided into an internal disturbance area and an external area along the radial direction, the internal disturbance area is divided into any number of circle layers along the radial direction, the soil body in each circle layer is a homogeneous and isotropic linear viscoelastic body, the soil body in the external area extends in the radial direction infinitely, the damping of the soil body material adopts viscous damping, and the radial displacement of the soil body is ignored;
s3: the displacement of the soil interface around the pile and the two sides of the soil interface of each circle layer is continuous and the stress is balanced, and the vibration of the soil system around the pile is small deformation;
s4: the concrete of the tubular pile body is linear elastic, and the propagation of stress waves in the tubular pile body meets the assumption of a flat section;
s5: according to the basic theory of elastic dynamics, establishing a longitudinal vibration equation and boundary conditions of a pile surrounding soil body, a pile core soil body and a pile body of the pipe pile under a plane strain condition;
s6: using Laplace (Laplace) transformation to solve a longitudinal vibration equation of a pile body around the pile, a pile core soil body and the pile body of the tubular pile in S5, and obtaining a time domain speed response function of any exciting force acting on the pile top by utilizing the transmissibility of a pile body impedance function between longitudinal sections so as to analyze the longitudinal vibration of the tubular pile;
the vibration equation of the soil body around the pile:
Figure FDA0002634813710000011
pile core soil body vibration equation:
Figure FDA0002634813710000012
the tubular pile body longitudinal vibration equation which accords with the assumption of the flat section is as follows:
Figure FDA0002634813710000013
wherein, the soil body around the pile is divided into m sections along the longitudinal direction, the pile length is H-shaped, and the pile is arranged from the bottom of the pile body to the bottomThe number of the layer sections is 1, 2, …, i, … and m in turn, and the thickness of each layer section is l1、l2、…、li、…、lmThe top buried depth of each layer section is h1、h2、…、hi…、hm(ii) a The inner diameter of the tubular pile, the outer diameter of the tubular pile, the sectional area of a tubular pile section, the density of the tubular pile and the elastic modulus of the tubular pile are respectively ri0、ri1、
Figure FDA0002634813710000021
And
Figure FDA0002634813710000022
the coefficient of stiffness of the viscoelastic support at the bottom of the pile isp、kp(ii) a In the longitudinal ith layer, the shear modulus, the viscous damping coefficient and the density of the pile core soil body are respectively Gi0、ηi0、ρi0(ii) a Meanwhile, the pile-surrounding soil body of the longitudinal ith layer is divided into an internal disturbance area and an external area along the radial direction, and the radial thickness of the internal disturbance area of the pile-surrounding soil body is biDividing the internal disturbance area into n circle layers along the radial direction, wherein the shear modulus, the viscous damping coefficient and the density of the soil body of the j-th circle layer are respectively Gij、ηij、ρijThe radius at the interface of the j-1 th circle layer and the j circle layer is rij(ii) a Radius at the interface of the inner zone and the outer zone is ri(n+1)The outer area is radial semi-infinite uniform visco-elastic medium; the pile top of the pipe pile acts any exciting force p (t), and the shear stress generated by the pile core soil and the pile surrounding soil of the ith layer on the pile body is respectively
Figure FDA0002634813710000023
And
Figure FDA0002634813710000024
in the ith layer, the pile core soil body is set to be displaced into
Figure FDA0002634813710000025
The displacement of a certain point in the jth circle layer of the soil body around the pile is
Figure FDA0002634813710000026
The i-th layer has a pile body displacement of
Figure FDA0002634813710000027
r is radial displacement, t is time, z is longitudinal displacement, EpiIs the elastic modulus of the pile body of the ith layer section, ApiThe sectional area of the pile body of the ith layer is shown;
the boundary conditions in S5 include:
pile core soil body boundary conditions:
when r is 0, the displacement is finite:
Figure FDA0002634813710000028
the displacement and force continuous conditions of the pile core soil body and the pile are as follows:
Figure FDA0002634813710000029
Figure FDA00026348137100000210
ri0is the inner radius of the pile,
Figure FDA00026348137100000211
the shear stress of the pile core soil body to the pile body is generated,
Figure FDA00026348137100000212
the vertical shear stress of a pile core soil body on the inner wall of a tubular pile is positive clockwise;
boundary conditions of soil bodies around the pile:
when r ∞, the displacement is zero:
Figure FDA00026348137100000213
in the formula (I), the compound is shown in the specification,
Figure FDA00026348137100000214
representing the displacement of the outer area of the soil body of the ith layer;
the displacement and force continuous conditions of the soil body around the pile and the pile are as follows:
Figure FDA00026348137100000215
Figure FDA0002634813710000031
wherein r isi1Is the outer radius of the pile,
Figure FDA0002634813710000032
for the displacement of the soil body of the layer 1,
Figure FDA0002634813710000033
the shear stress of the soil body around the pile on the pile body is generated,
Figure FDA0002634813710000034
the vertical shear stress of the soil body around the pile on the outer wall of the tubular pile is positive clockwise;
pile body boundary conditions:
pile top acting force is p (t):
Figure FDA0002634813710000035
boundary conditions at pile ends:
Figure FDA0002634813710000036
Figure FDA0002634813710000037
the modulus of elasticity of the pile body is,
Figure FDA0002634813710000038
is the cross-sectional area of pile body, kp，pIs the viscoelastic supporting constant of the pile bottom;
s6 includes the following steps:
step 1: performing Laplace transformation on the equations (1), (7), (8) and (9) to obtain a soil layer shear stiffness recurrence formula of the multi-circle layer plane strain model of the longitudinal ith layer section based on viscous damping, wherein the soil layer shear stiffness recurrence formula is as follows:
Figure FDA0002634813710000039
wherein
Figure FDA00026348137100000310
Wherein r isijIs the inner boundary of the ith layer interval and the jth circle layer soil, ri(j+1)Is the outer boundary of the ith layer and the jth circle of layer soil,
Figure FDA00026348137100000311
is the intrinsic parameters of the soil of the jth circle of the ith layer section, s is a complex variable,
Figure FDA00026348137100000312
the shear stiffness of the inner boundary of the jth circle layer soil of the ith layer section,
Figure FDA00026348137100000313
shear stiffness of the outer boundary of the ith interval and jth circle of layer soil, I0、I1Modified Bessel functions of the first kind, K, for zero and first order0、K1A second class of zero and first order modified Bessel (Bessel) functions;
step 2: and (3) performing Laplace (Laplace) conversion on the (2), (4), (5) and (6) to obtain a shear stiffness formula of the inner wall of the pipe pile of the ith layer subjected to the pile core soil body:
Figure FDA0002634813710000041
wherein the content of the first and second substances,
Figure FDA0002634813710000042
the intrinsic parameters of the pile core soil are obtained;
and step 3: performing Laplace transformation on the equations (3), (10) and (11), and obtaining an impedance function recurrence formula of the adjacent longitudinal layer sections of the pile body according to the conditions of force balance and displacement continuity at the interfaces of the adjacent layer sections of the pile body:
Figure FDA0002634813710000043
Figure FDA0002634813710000044
in the formula (I), the compound is shown in the specification,
Figure FDA0002634813710000045
is the pile body impedance function of the ith layer section and the ith-1 layer section,
Figure FDA0002634813710000046
αi、βito solve for the simplification parameters,/iIs the length of the pile body of the ith layer section hi、hi-1The lengths from the pile top to the pile body top of the ith layer and the ith-1 layer are respectively;
and 4, step 4: obtaining a pile top complex dynamic stiffness formula by utilizing the transmissibility of a pile body impedance function:
Figure FDA0002634813710000047
wherein
Figure FDA0002634813710000048
K′dComplex stiffness K for pile topdDimensionless parameter of, say K'd＝Kr+iKi，KrRepresenting the dynamic stiffness of the pile head, Kirepresenting dynamic damping of pile head, alpham、βmTo solve for the simplification parameters,/mThe length of the m section of pile is;
step 5: obtaining pile top velocity admittance function H according to formula (17)v：
Figure FDA0002634813710000049
Wherein the content of the first and second substances,
Figure FDA00026348137100000410
the density of the m-th section of pile body,
Figure FDA00026348137100000411
is the elastic wave velocity H of the m-th section of pile bodyvIs pile tip velocity admittance function HvDimensionless of (a); omega is the longitudinal vibration circular frequency;
step 6: the time domain response of the unit pulse excitation is obtained from (18) as:
Figure FDA0002634813710000051
wherein T ═ T/TcIs dimensionless time, theta is dimensionless frequency; IFT is the fast Fourier transform symbol;
and 7: obtaining a time domain speed response function of any exciting force p (t) acting on the pile top according to the convolution theorem
g(t)＝p(t)*h(t)＝IFT[P(iω)·H(iω)](20)
H (t) is time domain velocity response under the unit pulse excitation effect, and H (i omega) is a pile top velocity frequency response function;
and 8: the exciting force p (t) is half-sine pulse excitation
Figure FDA0002634813710000052
when T belongs to (0, T), T is pulse width, and the half-analytic solution of the pile top time domain velocity response is as follows:
Figure FDA0002634813710000053
wherein Q ismaxIs a half-sinusoidal pulse amplitude, Vv' is the dimensionless speed of the time domain response.
2. The complicated heterogeneous soil-in-pipe pile longitudinal vibration analysis method based on the viscous damping model as claimed in claim 1, wherein: based on the pile top speed admittance function and the pile top speed time domain response function, the vibration characteristic of the pile body and the integrity of the pile body can be evaluated.
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