CN110093951A - A kind of friction pile Longitudinal vibration analysis method based on loosened soil stake model - Google Patents
A kind of friction pile Longitudinal vibration analysis method based on loosened soil stake model Download PDFInfo
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Abstract
The invention discloses a kind of friction pile Longitudinal vibration analysis methods based on loosened soil stake model, using Kelvin model and loosened soil stake model, it is assumed that entity stake and loosened soil stake are homogeneous, round cross-section viscoelastic body, and entity stake and loosened soil stake interface are displaced continuous, stress equilibrium；Soil around pile and stake subsoil are isotropism linear viscoelasticity body, and soil body material damping uses viscous damping；Pile soil horizon upper surface is free boundary, no direct stress and shear stress, and stake substratum bottom is rigid basement；Stake subsoil body and pile peripheral earth extensional vibration governing equation under axial-symmetric condition are established, according to Euler-Bernoulli rod piece theory according to viscoelastic Earth model theory, establish loosened soil stake and entity stake extensional vibration governing equation, it is converted using Laplace, solve above-mentioned vibration control equation, the time domain speed responsive function that any exciting force acts on stake top is obtained, to analyze friction longitudinal vibration o f pile.
Description
Technical field
The present invention relates to civil engineering fields, more particularly, to a kind of friction pile extensional vibration based on loosened soil stake model point
Analysis method.
Background technique
In terms of stake-soil around pile interfacial interaction, existing research simplifies stake Soil Interface using a variety of methods and interacts.
Series of discrete Winkler spring-dampers model, studies rigid foundation longitudinal vibration characteristics.Though such method
It is so easy, but parameter value relies on experience.It improves on this basis, proposes plane strain analysis model and consider that resistance to shear of soil is answered
Become along stake week radially continuous property, which has certain theoretical basis, but soil around pile can not be considered along the variation of depth.It is based on
Considerations above has developed a kind of soil around pile three-dimensional continuum Model, which can be considered land movement, the components of stress along deep
Degree variation, ignores soil body radial displacement, studies stake longitudinal vibration characteristics.Hereafter occur considering simultaneously again the soil body it is vertical with
Radial displacement influences, and improves soil around pile three-dimensional continuum Model.The above research is based on stake soil and completely attaches to it is assumed that such
Assuming that the soil body can be exaggerated to the effect of contraction of pile body.Therefore there are many scholars using stake end rigid support model, consider stake pedosphere
Longitudinal vibration o f pile characteristic in the case of the Relative sliding of face.However stake end rigid support model is used, ignore the fluctuation effect of a subsoil
It answers.Also there are numerous scholars to simplify stake-stake subsoil interaction in the past, such as by stake subsoil be simplified to spring and damper,
Semi-infinite elastic foundation model, but both methods has respective limitation, and Yang Dongying proposes loosened soil stake model on this basis, sternly
Lattice consider the fluctuation effect of a subsoil, and can simulate sediment of the hole bottom, pile-end soil by the project situations such as compacted.But it examines simultaneously
Stake Soil Interface Relative sliding and stake subsoil fluctuation effect are considered to the method for pile foundation longitudinal vibration characteristics, still lack disclosure.
Summary of the invention
It is an object of the invention to overcome drawbacks described above of the existing technology, a kind of rubbing based on loosened soil stake model is provided
Stake Longitudinal vibration analysis method is wiped, stake Soil Interface Relative sliding is simulated using Kelvin model, stake is considered using loosened soil stake model
The fluctuation effect of subsoil.
To achieve the above object, technical scheme is as follows:
A kind of friction pile Longitudinal vibration analysis method based on loosened soil stake model, which is characterized in that include the following steps
S1: introduce following it is assumed that establishing the friction pile Longitudinal vibration analysis model based on loosened soil stake model: it is assumed that entity stake
It is homogeneous, round cross-section viscoelastic body with loosened soil stake, and entity stake and loosened soil stake interface are displaced continuous, stress equilibrium；
It is assumed that soil around pile and stake subsoil are isotropism linear viscoelasticity body, soil body material damping uses viscous damping；It is assumed that stake is all
Soil layer upper surface is free boundary, no direct stress and shear stress, and stake substratum bottom is rigid basement；
S2: stake subsoil body and pile peripheral earth extensional vibration under axial-symmetric condition are established according to viscoelastic Earth model theory and controlled
Equation；
It is theoretical according to Euler-Bernoulli rod piece, establish loosened soil stake and entity stake extensional vibration governing equation；
According in step S1 it is assumed that establishing Pile Soil boundary condition.
S3: being converted using Laplace, stake subsoil body described in solution procedure S2 and pile peripheral earth vibration equation, and is asked
Loosened soil stake and entity stake extensional vibration governing equation are solved, the time domain speed responsive function that any exciting force acts on stake top is obtained,
To analyze friction longitudinal vibration o f pile.
Preferably, stake subsoil and soil around pile extensional vibration governing equation are under the conditions of the step S2 formed symmetrical
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, longitudinal coordinate zero point
Positioned at Free Surface, it is positive downwards, t is time coordinate,For soil body length travel,For soil body Lame constant, and
Have The respectively elasticity modulus of the soil body, Poisson's ratio, viscosity
Damped coefficient and density, when j=1, correspond to stake subsoil parameter, and when j=2 corresponds to soil around pile parameter.
Preferably, loosened soil stake extensional vibration governing equation is in the step S2
Entity stake extensional vibration governing equation is
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, longitudinal coordinate zero point
It positioned at Free Surface, is positive downwards, t is time coordinate；uSPAnd uPThe respectively length travel of loosened soil stake and entity stake,For pile body sectional area, r0For stake section radius,The respectively elasticity modulus of loosened soil stake, viscosity resistance
Buddhist nun's coefficient and density, EP、ηP、ρPThe respectively elasticity modulus, viscous damping coefficient and density of entity stake.
Preferably, in the step S2, Pile Soil boundary condition includes stake subsoil boundary condition, soil around pile boundary condition, reality
Body stake and loosened soil stake boundary condition, stake soil coupling condition, respectively
Stake subsoil boundary condition:
Soil around pile boundary condition:
Entity stake and loosened soil stake boundary condition:
uSP|Z=H=0 (6b)
The native coupling condition of stake:
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, longitudinal coordinate zero point
It positioned at Free Surface, is positive downwards, t is time coordinate；HPFor pile soil horizon thickness, HSPFor stake subsoil thickness, H=HP+HSPFor
Soil layer overall thickness on basement rock；Q (t) is that stake top acts on any exciting force；For pile body sectional area, r0For stake section half
Diameter；kSFor the distributed spring dynamic stiffness of soil around pile and stake subsoil interlayer, cSIt is damped for soil around pile and the distributed of stake subsoil interlayer
The damped coefficient of device；kfFor the Kelvin model coefficient of elasticity of entity stake and soil around pile interface, cfFor entity stake and soil around pile circle
Kelvin model damper coefficient at face；fSP(z, t) is unit side friction of the stake subsoil to loosened soil stake,For stake subsoil
Shear stress in stake subsoil-loosened soil stake interface；fP(z, t) is unit side friction of the soil around pile to entity stake,For stake
All native shear stress in soil around pile-entity stake interface,It is longitudinally opposed between entity stake and soil around pile
Sliding,The Relative sliding speed between entity stake and soil around pile；uSPAnd uPThe respectively length travel of loosened soil stake and entity stake,The respectively elasticity modulus of stake subsoil, viscous damping coefficient and density, EP、ηP、ρPRespectively entity stake
Elasticity modulus, viscous damping coefficient and density；For the soil body length travel of stake subsoil,For the soil body of stake subsoil
Lame constant, and have The respectively bullet of the soil body of stake subsoil
Property modulus, Poisson's ratio, viscous damping coefficient and density；For the soil body length travel of soil around pile,For soil around pile
Soil body Lame constant, and have The respectively soil body of soil around pile
Elasticity modulus, Poisson's ratio, viscous damping coefficient and density；
Preferably, in the step S3, solution includes the following steps
Step S31: enabling j=1, draw to stake subsoil extensional vibration governing equation under the axial-symmetric condition in formula (1) general
Lars transformation, and Laplace transform is carried out to boundary conditional (4a) and (4b), the cross displacement function for obtaining a subsoil is
And stake subsoil is in stake subsoil-loosened soil stake interface shear stress
Step S32: enabling j=2, draw to soil around pile extensional vibration governing equation under the axial-symmetric condition in formula (1) general
Lars transformation carries out Laplace transform to boundary conditional (5a) and (5b), and the cross displacement function for obtaining soil around pile is
And soil around pile is in soil around pile-entity stake interface shear stress
Step S33: carrying out Laplace transform to loosened soil stake extensional vibration governing equation (2) and boundary condition (7a), and
Based on the stake subsoil obtained in step S31 in loosened soil stake interface shear stress (9a), loosened soil longitudinal vibration o f pile displacement letter is obtained
Number
Laplace transformation is carried out to entity stake extensional vibration governing equation (3) and boundary condition (7c), obtains entity stake
Extensional vibration displacement function
Step S34: Laplace transform is carried out to boundary conditional (4b), obtains the multiple resistance of loosened soil stake Yu entity stake interface
Anti- function
Laplace transform is carried out to boundary conditional (6c, 6d), obtains the displacement impedance function of entity stake top
Step S35: obtaining entity stake top Complex modes according to the displacement impedance function (11b) of entity stake top is
Kd=ZP=Kr+iKi (12)
Step S36: obtaining displacement at pile top function according to the displacement impedance function (11b) of entity stake top is
Step S37: according to displacement at pile top function (13), stake top speed in frequency receptance function is obtained
Step S38: unit pulse excitation is obtained using Fourier transformation according to stake top speed in frequency receptance function (14)
Time domain response
Step S39: it according to convolution theorem, obtains under any exciting force q (t) effect, stake top speed time domain response is
G (t)=q (t) * h (t)=IFT [Q (i ω) Hv(iω)] (16)
When exciting force is half-sine pulse excitationWhen T is pulse width, when stake top
Domain semi-analytical solution is
In above-mentioned steps,
Z '=z-HPFor local longitudinal coordinate, zero point is at the top of stake subsoil body, and direction is positive downwards；S=i ω is that drawing is general
Lars transformation constant, i are imaginary unit, and ω is exciting Loading frequency；N is subscript；For pile body sectional area, r0For stake
Section radius；Q (t) is any exciting force；
For stake subsoil length travelLaplace transform；For soil around pile longitudinal direction
DisplacementLaplace transform；USP(z ', s) is the displacement of loosened soil stake body, uSPThe Laplace transform of (z ', t)；UP
(z, s) entity stake body displacement components uPThe Laplace transform of (z, t)；Q (i ω) is the Fourier transformation of any exciting force q (t)；
K0(·)、K1() is respectively zeroth order and first rank the second class void argument Bessel function；
To carry out Fourier transform operation；
For the one-dimensional compressional wave velocity of wave of loosened soil stake；For the one-dimensional compressional wave wave of entity stake
Speed；
EP、ηP、ρPThe respectively elasticity modulus, viscous damping coefficient and density of entity stake；
A1nFor by stake subsoil and loosened soil stake coupling condition (7b) and stake subsoil in stake subsoil-loosened soil stake interface shear stress
Constant determined by (9a)；A2nFor by soil around pile and entity stake coupling condition (7c, d) and soil around pile in soil around pile-entity stake
Constant determined by interface shear stress (9b)；
β1n(n=1,2,3 ...) is transcendental equationMiddle β1Solution,Wherein kSFor stake week
The distributed spring dynamic stiffness of soil and stake subsoil interlayer, cSFor the damping system of soil around pile and the distributed vibration damper of stake subsoil interlayer
Number,For the soil body Lame constant of stake subsoil, and have
The respectively elasticity modulus of the soil body of stake subsoil, Poisson's ratio, viscous damping coefficient and density；
β2n(n=1,2,3 ...) is transcendental equationMiddle β2Solution,Wherein kSFor stake
The distributed spring dynamic stiffness of Zhou Tuyu subsoil interlayers, cSFor the damping of soil around pile and the distributed vibration damper of stake subsoil interlayer
Coefficient,For the soil body Lame constant of stake subsoil, and have
The respectively elasticity modulus of the soil body of stake subsoil, Poisson's ratio, viscous damping coefficient and density；
MSP, NSPFor undetermined coefficient, meet following relationship
MP, NPFor undetermined coefficient, meet following relationship
It further include following symbol definition in above-mentioned steps
It can be seen from the above technical proposal that the present invention is based on the calculations of the friction pile extensional vibration dynamic impedance of loosened soil stake model
Method system can consider the fluctuation effect of a Soil Interface Relative sliding and stake subsoil using Kelvin model and loosened soil stake model simultaneously
It answers, the factor for influencing pile foundation longitudinal vibration characteristics (such as the stiffness coefficient of stake Soil Interface, damped coefficient and stake subsoil thickness and is cut
Cut velocity of wave etc.) rationalization analysis is carried out, it is suitable for the research of friction pile longitudinal vibration characteristics, theory can be provided for dynamic pile detection
Guidance and reference role.
Detailed description of the invention
Fig. 1 is the method for the present invention flow chart；
Fig. 2 is model schematic of the invention.
In figure, 1 is soil around pile, and 2 be a subsoil, and 3 be rigid foundation, and 4 be entity stake, and 5 be loosened soil stake, 6 be soil around pile and
The interface of stake subsoil, 7 be the interface of soil around pile and entity stake
Specific embodiment
With reference to the accompanying drawing, specific embodiments of the present invention will be described in further detail.
It should be noted that in following specific embodiments, when describing embodiments of the invention in detail, in order to clear
Ground indicates structure of the invention in order to illustrate, spy does not draw to the structure in attached drawing according to general proportion, and has carried out part
Amplification, deformation and simplified processing, therefore, should be avoided in this, as limitation of the invention to understand.
In specific embodiment of the invention below, referring to Fig. 1, Fig. 1 is flow chart of the method for the present invention.As schemed
Show,
A kind of friction pile Longitudinal vibration analysis method based on loosened soil stake model, which comprises the following steps:
S1: it introduces following it is assumed that convenient for establishing the friction pile Longitudinal vibration analysis model based on loosened soil stake model.
1) assume that entity stake and loosened soil stake are homogeneous, round cross-section viscoelastic body, and entity stake and loosened soil stake interface
Place is displaced continuous, stress equilibrium；
2) assume soil around pile and stake subsoil is isotropism linear viscoelasticity body, soil body material damping is using viscosity resistance
Buddhist nun；
3) assume that pile soil horizon upper surface is free boundary, no direct stress and shear stress, stake substratum bottom are rigid base
Bottom.
A kind of friction pile Longitudinal vibration analysis method based on loosened soil stake model, which is characterized in that include the following steps
S1: introduce following it is assumed that establishing the friction pile Longitudinal vibration analysis model based on loosened soil stake model: it is assumed that entity stake
It is homogeneous, round cross-section viscoelastic body with loosened soil stake, and entity stake and loosened soil stake interface are displaced continuous, stress equilibrium；
It is assumed that soil around pile and stake subsoil are isotropism linear viscoelasticity body, soil body material damping uses viscous damping；It is assumed that stake is all
Soil layer upper surface is free boundary, no direct stress and shear stress, and stake substratum bottom is rigid basement；
S2: stake subsoil body and pile peripheral earth extensional vibration under axial-symmetric condition are established according to viscoelastic Earth model theory and controlled
Equation.
Stake subsoil and soil around pile extensional vibration governing equation are under axial-symmetric condition
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, longitudinal coordinate zero point
Positioned at Free Surface, it is positive downwards, t is time coordinate,For soil body length travel,For soil body Lame constant, and
Have The respectively elasticity modulus of the soil body, Poisson's ratio, viscosity
Damped coefficient and density, when j=1, correspond to stake subsoil parameter, and when j=2 corresponds to soil around pile parameter.
It is theoretical according to Euler-Bernoulli rod piece, establish loosened soil stake and entity stake extensional vibration governing equation.Loosened soil stake
Extensional vibration governing equation is
Entity stake extensional vibration governing equation is
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, longitudinal coordinate zero point
It positioned at Free Surface, is positive downwards, t is time coordinate；uSPAnd uPThe respectively length travel of loosened soil stake and entity stake,For pile body sectional area, r0For stake section radius,The respectively elasticity modulus of loosened soil stake, viscosity resistance
Buddhist nun's coefficient and density, EP、ηP、ρPThe respectively elasticity modulus, viscous damping coefficient and density of entity stake.
According in step S1 it is assumed that establishing Pile Soil boundary condition.
Pile Soil boundary condition includes stake subsoil boundary condition, soil around pile boundary condition, entity stake and loosened soil stake perimeter strip
Part, stake soil coupling condition, respectively
Stake subsoil boundary condition:
Soil around pile boundary condition:
Entity stake and loosened soil stake boundary condition:
uSP|Z=H=0 (6b)
The native coupling condition of stake:
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, longitudinal coordinate zero point
It positioned at Free Surface, is positive downwards, t is time coordinate；HPFor pile soil horizon thickness, HSPFor stake subsoil thickness, H=HP+HSPFor
Soil layer overall thickness on basement rock；Q (t) is that stake top acts on any exciting force；For pile body sectional area, r0For stake section half
Diameter；kSFor the distributed spring dynamic stiffness of soil around pile and stake subsoil interlayer, cSIt is damped for soil around pile and the distributed of stake subsoil interlayer
The damped coefficient of device；kfFor the Kelvin model coefficient of elasticity of entity stake and soil around pile interface, cfFor entity stake and soil around pile circle
Kelvin model damper coefficient at face；fSP(z, t) is unit side friction of the stake subsoil to loosened soil stake,For stake subsoil
Shear stress in stake subsoil-loosened soil stake interface；fP(z, t) is unit side friction of the soil around pile to entity stake,For stake
All native shear stress in soil around pile-entity stake interface,It is longitudinally opposed between entity stake and soil around pile
Sliding,The Relative sliding speed between entity stake and soil around pile；uSPAnd uPThe respectively length travel of loosened soil stake and entity stake,The respectively elasticity modulus of stake subsoil, viscous damping coefficient and density, EP、ηP、ρPThe respectively bullet of entity stake
Property modulus, viscous damping coefficient and density；For the soil body length travel of stake subsoil,For the soil body Lame of stake subsoil
Constant, and have The respectively springform of the soil body of stake subsoil
Amount, Poisson's ratio, viscous damping coefficient and density；For the soil body length travel of soil around pile,For the soil body of soil around pile
Lame constant, and have The respectively bullet of the soil body of soil around pile
Property modulus, Poisson's ratio, viscous damping coefficient and density.
S3: being converted using Laplace, stake subsoil body described in solution procedure S2 and pile peripheral earth vibration equation, and is asked
Loosened soil stake and entity stake extensional vibration governing equation are solved, the time domain speed responsive function that any exciting force acts on stake top is obtained,
To analyze friction longitudinal vibration o f pile.
Solution includes the following steps
Step S31: enabling j=1, draw to stake subsoil extensional vibration governing equation under the axial-symmetric condition in formula (1) general
Lars transformation, and Laplace transform is carried out to boundary conditional (4a) and (4b), the cross displacement function for obtaining a subsoil is
And stake subsoil is in stake subsoil-loosened soil stake interface shear stress
Step S32: enabling j=2, draw to soil around pile extensional vibration governing equation under the axial-symmetric condition in formula (1) general
Lars transformation carries out Laplace transform to boundary conditional (5a) and (5b), and the cross displacement function for obtaining soil around pile is
And soil around pile is in soil around pile-entity stake interface shear stress
Step S33: carrying out Laplace transform to loosened soil stake extensional vibration governing equation (2) and boundary condition (7a), and
Based on the stake subsoil obtained in step S31 in loosened soil stake interface shear stress (9a), loosened soil longitudinal vibration o f pile displacement letter is obtained
Number
Laplace transformation is carried out to entity stake extensional vibration governing equation (3) and boundary condition (7c), obtains entity stake
Extensional vibration displacement function
Step S34: Laplace transform is carried out to boundary conditional (4b), obtains the multiple resistance of loosened soil stake Yu entity stake interface
Anti- function
Laplace transform is carried out to boundary conditional (6c, 6d), obtains the displacement impedance function of entity stake top
Step S35: obtaining entity stake top Complex modes according to the displacement impedance function (11b) of entity stake top is
Kd=ZP=Kr+iKi (12)
Step S36: obtaining displacement at pile top function according to the displacement impedance function (11b) of entity stake top is
Step S37: according to displacement at pile top function (13), stake top speed in frequency receptance function is obtained
Step S38: unit pulse excitation is obtained using Fourier transformation according to stake top speed in frequency receptance function (14)
Time domain response
Step S39: it according to convolution theorem, obtains under any exciting force q (t) effect, stake top speed time domain response is
G (t)=q (t) * h (t)=IFT [Q (i ω) Hv(iω)] (16)
When exciting force is half-sine pulse excitationWhen T is pulse width, when stake top
Domain semi-analytical solution is
In above-mentioned steps,
Z '=z-HPFor local longitudinal coordinate, zero point is at the top of stake subsoil body, and direction is positive downwards；S=i ω is that drawing is general
Lars transformation constant, i are imaginary unit, and ω is exciting Loading frequency；N is subscript；For pile body sectional area, r0For stake
Section radius；Q (t) is any exciting force；
For stake subsoil length travelLaplace transform；For soil around pile longitudinal direction
DisplacementLaplace transform；USP(z ', s) is the displacement of loosened soil stake body, uSPThe Laplace transform of (z ', t)；UP
(z, s) entity stake body displacement components uPThe Laplace transform of (z, t)；Q (i ω) is the Fourier transformation of any exciting force q (t)；
K0(·)、K1() is respectively zeroth order and first rank the second class void argument Bessel function；
To carry out Fourier transform operation；
For the one-dimensional compressional wave velocity of wave of loosened soil stake；For the one-dimensional compressional wave wave of entity stake
Speed；EP、ηP、ρPThe respectively elasticity modulus, viscous damping coefficient and density of entity stake；
A1nFor by stake subsoil and loosened soil stake coupling condition (7b) and stake subsoil in stake subsoil-loosened soil stake interface shear stress
Constant determined by (9a)；A2nFor by soil around pile and entity stake coupling condition (7c, d) and soil around pile in soil around pile-entity stake
Constant determined by interface shear stress (9b)；
β1n(n=1,2,3 ...) is transcendental equationMiddle β1Solution,Wherein kSFor stake week
The distributed spring dynamic stiffness of soil and stake subsoil interlayer, cSFor the damping system of soil around pile and the distributed vibration damper of stake subsoil interlayer
Number,For the soil body Lame constant of stake subsoil, and have
The respectively elasticity modulus of the soil body of stake subsoil, Poisson's ratio, viscous damping coefficient and density；
β2n(n=1,2,3 ...) is transcendental equationMiddle β2Solution,Wherein kSFor
The distributed spring dynamic stiffness of soil around pile and stake subsoil interlayer, cSFor the resistance of soil around pile and the distributed vibration damper of stake subsoil interlayer
Buddhist nun's coefficient,For the soil body Lame constant of stake subsoil, and have The respectively elasticity modulus of the soil body of stake subsoil, Poisson's ratio, viscous damping coefficient and density；
MSP, NSPFor undetermined coefficient, meet following relationship
MP, NPFor undetermined coefficient, meet following relationship
It further include following symbol definition in above-mentioned steps
In conclusion the present invention is based on the friction pile extensional vibration dynamic impedance algorithmic systems of loosened soil stake model, using void
Native stake model and Kelvin model can consider a Relative sliding for the fluctuation effect of subsoil body and stake Soil Interface simultaneously, can be applicable in and rub
Longitudinal Vibration of Integrated when wiping stake stake Soil Interface Relative sliding, can provide theoretical direction and reference role for dynamic pile detection.
The foregoing is only a preferred embodiment of the present invention, but scope of protection of the present invention is not limited thereto,
Anyone skilled in the art in the technical scope disclosed by the present invention, according to the technique and scheme of the present invention and its
Inventive concept is subject to equivalent substitution or change, should be covered by the protection scope of the present invention.
Claims (5)
1. a kind of friction pile Longitudinal vibration analysis method based on loosened soil stake model, which is characterized in that include the following steps
S1: introduce following it is assumed that establishing the friction pile Longitudinal vibration analysis model based on loosened soil stake model: it is assumed that entity stake and void
Native stake is homogeneous, round cross-section viscoelastic body, and entity stake and loosened soil stake interface are displaced continuous, stress equilibrium；It is assumed that
Soil around pile and stake subsoil are isotropism linear viscoelasticity body, and soil body material damping uses viscous damping；It is assumed that pile soil horizon
Upper surface is free boundary, no direct stress and shear stress, and stake substratum bottom is rigid basement；
S2: stake subsoil body and pile peripheral earth extensional vibration controlling party under axial-symmetric condition are established according to viscoelastic Earth model theory
Journey；
It is theoretical according to Euler-Bernoulli rod piece, establish loosened soil stake and entity stake extensional vibration governing equation；
According in step S1 it is assumed that establishing Pile Soil boundary condition；
S3: being converted using Laplace, stake subsoil body described in solution procedure S2 and pile peripheral earth vibration equation, and solves void
Native stake and entity stake extensional vibration governing equation obtain the time domain speed responsive function that any exciting force acts on stake top, with right
Friction longitudinal vibration o f pile is analyzed.
2. analysis method according to claim 1, which is characterized in that under the conditions of the step S2 formed symmetrical stake subsoil and
Soil around pile extensional vibration governing equation is
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, and longitudinal coordinate zero point is located at
Free Surface is positive downwards, and t is time coordinate,For soil body length travel,For soil body Lame constant, and have The respectively elasticity modulus of the soil body, Poisson's ratio, viscosity resistance
Buddhist nun's coefficient and density, when j=1, correspond to stake subsoil parameter, and when j=2 corresponds to soil around pile parameter.
3. analysis method according to claim 1, which is characterized in that loosened soil stake extensional vibration controlling party in the step S2
Cheng Wei
Entity stake extensional vibration governing equation is
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, and longitudinal coordinate zero point is located at
Free Surface is positive downwards, and t is time coordinate；uSPAnd uPThe respectively length travel of loosened soil stake and entity stake,For
Pile body sectional area, r0For stake section radius,The respectively elasticity modulus, viscous damping coefficient of loosened soil stake and close
Degree, EP、ηP、ρPThe respectively elasticity modulus, viscous damping coefficient and density of entity stake.
4. analysis method according to claim 1, which is characterized in that in the step S2, Pile Soil boundary condition includes stake
Subsoil boundary condition, soil around pile boundary condition, entity stake and loosened soil stake boundary condition, stake soil coupling condition, respectively
Stake subsoil boundary condition:
Soil around pile boundary condition:
Entity stake and loosened soil stake boundary condition:
uSP|Z=H=0 (6b)
The native coupling condition of stake:
In formula, r is axial coordinate, and axial coordinate zero point is located at the stake section center of circle, and z is longitudinal coordinate, and longitudinal coordinate zero point is located at
Free Surface is positive downwards, and t is time coordinate；HPFor pile soil horizon thickness, HSPFor stake subsoil thickness, H=HP+HSPFor basement rock
Upper soil layer overall thickness；Q (t) is that stake top acts on any exciting force；For pile body sectional area, r0For stake section radius；kSFor
The distributed spring dynamic stiffness of soil around pile and stake subsoil interlayer, cSFor the resistance of soil around pile and the distributed vibration damper of stake subsoil interlayer
Buddhist nun's coefficient；kfFor the Kelvin model coefficient of elasticity of entity stake and soil around pile interface, cfFor entity stake and soil around pile interface
Kelvin model damper coefficient；fSP(z, t) is unit side friction of the stake subsoil to loosened soil stake,It is stake subsoil at stake bottom
The shear stress of soil-loosened soil stake interface；fP(z, t) is unit side friction of the soil around pile to entity stake,Exist for soil around pile
The shear stress of soil around pile-entity stake interface,Longitudinally opposed sliding between entity stake and soil around pile,The Relative sliding speed between entity stake and soil around pile；uSPAnd uPThe respectively length travel of loosened soil stake and entity stake,The respectively elasticity modulus of stake subsoil, viscous damping coefficient and density, EP、ηP、ρPRespectively entity stake
Elasticity modulus, viscous damping coefficient and density；For the soil body length travel of stake subsoil,For the soil body of stake subsoil
Lame constant, and have The respectively elasticity of the soil body of stake subsoil
Modulus, Poisson's ratio, viscous damping coefficient and density；For the soil body length travel of soil around pile,For the soil of soil around pile
Body Lame constant, and have The respectively soil body of soil around pile
Elasticity modulus, Poisson's ratio, viscous damping coefficient and density.
5. analysis method according to claim 1, which is characterized in that in the step S3, solution includes the following steps
Step S31: enabling j=1, carries out Laplce to stake subsoil extensional vibration governing equation under the axial-symmetric condition in formula (1)
Transformation, and Laplace transform is carried out to boundary conditional (4a) and (4b), the cross displacement function for obtaining a subsoil is
And stake subsoil is in stake subsoil-loosened soil stake interface shear stress
Step S32: enabling j=2, carries out Laplce to soil around pile extensional vibration governing equation under the axial-symmetric condition in formula (1)
Transformation carries out Laplace transform to boundary conditional (5a) and (5b), and the cross displacement function for obtaining soil around pile is
And soil around pile is in soil around pile-entity stake interface shear stress
Step S33: Laplace transform is carried out to loosened soil stake extensional vibration governing equation (2) and boundary condition (7a), and is based on
The stake subsoil obtained in step S31 obtains loosened soil longitudinal vibration o f pile displacement function in loosened soil stake interface shear stress (9a)
Laplace transformation is carried out to entity stake extensional vibration governing equation (3) and boundary condition (7c), obtains the longitudinal direction of entity stake
Vibration displacement function
Step S34: Laplace transform is carried out to boundary conditional (4b), obtains the complex impedance letter of loosened soil stake Yu entity stake interface
Number
Laplace transform is carried out to boundary conditional (6c, 6d), obtains the displacement impedance function of entity stake top
Step S35: obtaining entity stake top Complex modes according to the displacement impedance function (11b) of entity stake top is
Kd=ZP=Kr+iKi (12)
Step S36: obtaining displacement at pile top function according to the displacement impedance function (11b) of entity stake top is
Step S37: according to displacement at pile top function (13), stake top speed in frequency receptance function is obtained
Step S38: according to stake top speed in frequency receptance function (14), using Fourier transformation, obtain unit pulse excitation when
Domain response
Step S39: it according to convolution theorem, obtains under any exciting force q (t) effect, stake top speed time domain response is
G (t)=q (t) * h (t)=IFT [Q (i ω) Hv(iω)] (16)
When exciting force is half-sine pulse excitationWhen t ∈ (0, T), T are pulse width, stake top time domain half is solved
Analysing solution is
In above-mentioned steps,
Z '=z-HPFor local longitudinal coordinate, zero point is at the top of stake subsoil body, and direction is positive downwards；S=i ω is Laplce
Transformation constant, i are imaginary unit, and ω is exciting Loading frequency；N is subscript；For pile body sectional area, r0For stake section
Radius；Q (t) is any exciting force；
For stake subsoil length travelLaplace transform；For soil around pile length travelLaplace transform；USP(z ', s) is the displacement of loosened soil stake body, uSPThe Laplace transform of (z ', t)；UP(z,
S) entity stake body displacement components uPThe Laplace transform of (z, t)；Q (i ω) is the Fourier transformation of any exciting force q (t)；
K0(·)、K1() is respectively zeroth order and first rank the second class void argument Bessel function；
To carry out Fourier transform operation；
For the one-dimensional compressional wave velocity of wave of loosened soil stake；For the one-dimensional compressional wave velocity of wave of entity stake；
EP、ηP、ρPThe respectively elasticity modulus, viscous damping coefficient and density of entity stake；
A1nFor by stake subsoil and loosened soil stake coupling condition (7b) and stake subsoil in stake subsoil-loosened soil stake interface shear stress (9a)
Identified constant；A2nFor by soil around pile and entity stake coupling condition (7c, d) and soil around pile at soil around pile-entity stake interface
Locate constant determined by shear stress (9b)；
β1n(n=1,2,3 ...) is transcendental equationMiddle β1Solution,Wherein kSFor stake week
The distributed spring dynamic stiffness of soil and stake subsoil interlayer, cSFor the damping system of soil around pile and the distributed vibration damper of stake subsoil interlayer
Number,For the soil body Lame constant of stake subsoil, and have
The respectively elasticity modulus of the soil body of stake subsoil, Poisson's ratio, viscous damping coefficient and density；
β2n(n=1,2,3 ...) is transcendental equationMiddle β2Solution,Wherein kSFor
The distributed spring dynamic stiffness of soil around pile and stake subsoil interlayer, cSFor the resistance of soil around pile and the distributed vibration damper of stake subsoil interlayer
Buddhist nun's coefficient,For the soil body Lame constant of stake subsoil, and have The respectively elasticity modulus of the soil body of stake subsoil, Poisson's ratio, viscous damping coefficient and density；
MSP, NSPFor undetermined coefficient, meet following relationship
MP, NPFor undetermined coefficient, meet following relationship
It further include following symbol definition in above-mentioned steps
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