CN108732242A - Floating based on pile body Three-dimensional Axisymmetric model holds a Longitudinal vibration analysis method - Google Patents
Floating based on pile body Three-dimensional Axisymmetric model holds a Longitudinal vibration analysis method Download PDFInfo
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Abstract
The present invention provides a kind of floating based on pile body Three-dimensional Axisymmetric model and holds a Longitudinal vibration analysis method.The method of the present invention, including：Establish the Three-dimensional Axisymmetric model of pile body and pile peripheral earth, consider the variation of pile body length travel and strain radially, establish the equation of longitudinal of pile peripheral earth and pile body under the conditions of Three-dimensional Axisymmetric, the vibration equation is solved by the separation of variable, and it is derived from the frequency transcendental equation that a soil series is united corresponding to the different vibration vibration shapes, determination can meet a unJeiermined function for end viscoelasticity support conditions and stake top inhomogeneous boundary condition simultaneously, obtain the Complex modes of stake top, stake top Complex modes analytical solution radially carries out average value processing, and then obtain displacement at pile top frequency response function and stake top Complex modes radial direction mean value analytical solution, complete the evaluation to pile body vibration characteristics and pile body integrity under harmonious exciting force effect.For the present invention closer to real model, computational accuracy is high, can provide theoretical direction and reference role for dynamic pile detection.
Description
Technical field
The present invention relates to civil engineering field more particularly to a kind of extensional vibration is held based on the floating of pile body Three-dimensional Axisymmetric model
Analysis method.
Background technology
It is the engineering technology such as Anti-seismic Pile Foundation, aseismatic design and dynamic pile detection that Pile Soil, which couples longitudinal vibration characteristics research,
The theoretical foundation in field is also the hot issue of geotechnical engineering and Solid Mechanics all the time.How rational Pile Soil coupling is established
It is the key that pile foundation longitudinal vibration characteristics research to close model of vibration, and existing research is most for pile peripheral earth model refinement exhibition
It opens, specific experience Winkler models, considers that resistance to shear of soil strain along the plane strain model of stake week radially continuous property, considers
Land movement, the components of stress and consider that the soil body is vertical and radial displacement simultaneously along the three-dimensional continuum Model of change in depth
Very three-dimensional continuum Model.
And pile body model is established based on classical Euler-Bernoulli theories, and this is applied one-dimensional to answer
Reeb theoretical treatment large diameter pile problem can then cause relatively large deviation.Consider that Rayleigh proposes one kind earliest based on this point
To the correcting theory of rod piece, Love is then set out based on energy and has specifically derived the Raleigh-Love that can consider transverse inertia domino effect
The bar equation of motion.
Though however, Rayleigh-Love rod models it is contemplated that pile body particle transverse movement effect of inertia, do not consider
Pile body radial undulation effect can not consider that fluctuation causes the radial variations of pile body stress and displacement, theoretically also not stringent enough,
Three-dimensional Axisymmetric pile body model should be used further to consider the influence of pile body three-dimensional fluctuation effect.And for using pile body three-dimensional shaft
Symmetry model considers the existing research of pile body three-dimensional fluctuation effects, surrounds end-bearing pile and is unfolded, and is fixed based on stake end special
Determine the existing answer of boundary condition for rubbing, floating hold stake and be not suitable for.
Invention content
According to technical problem set forth above, and a kind of floating based on pile body Three-dimensional Axisymmetric model is provided holds stake and longitudinally shake
Dynamic analysis method.Pile body and soil around pile are mainly respectively seen as elasticity and viscoelasticity continuous media by the present invention, are established and are considered stake end
The Three-dimensional Axisymmetric Pile Soil coupled system Longitudinal vibration analysis model of viscoelasticity support conditions is united by being derived from a soil series
Frequency transcendental equation corresponding to the difference vibration vibration shape, to special to viscoelasticity supporting pile extensional vibration under harmonious exciting force effect
Property is analyzed.
The technological means that the present invention uses is as follows：
A kind of floating based on pile body Three-dimensional Axisymmetric model holds a Longitudinal vibration analysis method, which is characterized in that including with
Lower step：
S1, the Three-dimensional Axisymmetric model for establishing pile body and pile peripheral earth, wherein pile body is considered as homogeneous cross-section elasticity
Body, pile body bottom are considered as viscoelasticity bearing, ignore pile body radial displacement, consider the change of pile body length travel and strain radially
Change, pile peripheral earth upper surface is considered as free boundary, no direct stress, shear stress；
S2, according to elastodynamics basic theories, the longitudinal direction of pile peripheral earth and pile body shakes under the conditions of establishing Three-dimensional Axisymmetric
Dynamic equation, wherein Pile Soil Coupled Vibration System meets linear elasticity and small deformation condition, soil around pile with stake interfacial displacement continuously, answer
Dynamic balance；
S3, the vibration equation that the pile peripheral earth and pile body longitudinal direction are solved by the separation of variable, and it is derived from stake soil
System difference vibrates the frequency transcendental equation corresponding to the vibration shape；
S4, end viscoelasticity support conditions and a stake top nonhomogeneous boundary can be met according to the determination of Pile Soil coupling condition simultaneously
The unJeiermined function of condition solves the unJeiermined function and obtains the Complex modes of stake top；
S5, stake top Complex modes analytical solution radially carry out average value processing, and then obtain displacement at pile top frequency response function
With stake top Complex modes radial direction mean value analytical solution, complete to pile body vibration characteristics under harmonious exciting force effect and pile body integrity
Evaluation.
Further, in the step S2, uniformly distributed harmonious exciting force is acted on to stake top, Pile soil System shakes as stable state therewith
Dynamic, pile peripheral earth and pile body length travel meet following relationship respectively：
Wherein, the harmonious exciting force isIndicate that imaginary unit, ω indicate circular frequency,
Indicate that pile peripheral earth and pile body extensional vibration amplitude, i indicate that imaginary unit, ω indicate exciting lotus respectively
Carrier frequency rate,
Pile peripheral earth and pile body equation of longitudinal are respectively：
Wherein, λSIndicate pile peripheral earth Lame constants, μSIndicate complex value shear modulus, ρSIndicate pile peripheral earth density, λPTable
Show pile body Lame constants, GPIndicate modulus of shearing, ρPIndicating pile body density, r is radial coordinate, and t is the time, and z is longitudinal coordinate,
λS=2vSGS/(1-2vS), μS=GS(1+2ξSI),
Wherein, vSIndicate Poisson's ratio, GSIndicate that soil layer bottom viscoelasticity supports constant, ξSIndicate hysteretic damping, vPIt indicates
Poisson's ratio,
It is following expression by formula (1), (2) abbreviation：
Wherein,The respectively shear wave velocity of pile peripheral earth and pile body.
Further, pile peripheral earth and pile body meet such as downstream condition：
Soil layer boundaries condition includes：
Soil layer surface is free：
The viscoelasticity bearing of soil layer bottom surface：
Wherein：ESIndicate elasticity modulus, kS、δSIndicate that soil layer bottom viscoelasticity supports constant, as r → ∞, displacement is
Zero：
Pile body boundary condition includes：
The uniformly distributed harmonious exciting force of stake top effectFor：
Wherein：Indicate the harmonious amplitude of exciting force of stake top,
The viscoelasticity bearing of stake bottom：
Wherein, kP、δPIndicate that stake bottom viscoelasticity supports constant, EPIndicate elasticity modulus.
Further, as follows to pile peripheral earth vibration equation solution procedure in the step S3：
It enablesTo which formula (3) is decomposed into two ODEs：
Wherein, hS, qSFor constant, and meet following relationship：
Show that formula (10), the solution of (11) are respectively：
ZS(z)=CScos(hSz)+DSsin(hSz) (13)
RS(r)=MSK0(qSr)+NSI0(qSr) (14)
Wherein, I0(qSr)、K0(qSR) it is the zeroth order first kind, the second class Bessel function of imaginary argument, CS、DS、MS、NSIt serves as reasons
The integral constant undetermined that boundary condition determines,
Comprehensive boundary condition formula (5), (6) can obtain transcendental equation：
Wherein,Indicate the dimensionless group of soil layer bottom spring Complex modes, KS=kS+iωδS, H tables
Show that stake is long,
The transcendental equation obtains infinite multiple characteristic values by MATLAB Programs, is denoted asN=1,
2, ∞, and willSubstitution formula (12) can obtainIt is describedIndicate dimensionless group,
Comprehensive soil layer boundaries conditional (5), (6) and (7) can obtain soil around pile extensional vibration displacement amplitudeExpression formula be：
Wherein,For a series of unJeiermined functions.
Further, as follows to pile peripheral earth vibration equation solution procedure in the step S3：
S31, due to stake top be inhomogeneous boundary condition will determine solution to obtain pile body extensional vibration dynamic respond amplitude and ask
Topic is decomposed：
Wherein,Meet the Definite problem as listed by formula (18),To meet formula (4) and boundary condition formula simultaneously
(8), the unJeiermined function of (9),
S32, the Definite problem is solved and can be obtained：
Wherein,N=1,2, ∞ is to meet transcendental equationIt is infinite multiple
Characteristic value,Indicate the dimensionless group of pile body bottom spring Complex modes, KP=kP+iωδP,WithIt is full
The following relational expression of foot：
It willSubstitution formula (4) can obtain：
Wherein,
The general solution for solving formula (21) is：
Wherein, aP、bPFor undetermined coefficient,
Formula (22) is substituted into boundary condition formula (8), (9) can solve：
Formula (19), formula (22), formula (23) and formula (24) are substituted into displacement when formula (17) can obtain the longitudinal direction harmonic vibration of stake work to ring
Answer amplitude：
Wherein,For a series of unJeiermined functions.
Further, the S4 mainly includes the following steps：
S41, the undetermined coefficient is solved according to stake soil coupling conditionSoil around pile and stake can be obtained by formula (16) and formula (25)
The shear stress of body is respectively：
S42, pile body extensional vibration displacement amplitude is solved：
Assuming that soil around pile bottom dimensionless support stiffness is equal with pile body bottom dimensionless support stiffness, i.e.,
ThenContinuous, the stress equilibrium condition by soil around pile and the displacement of pile body interface, and convolution (16), (25) and formula
(26), (27) can obtain：
Wherein, r0Indicate radius,
According to the orthogonality of intrinsic function system, cos (h are multiplied by simultaneously at formula (28) both endsnz-hnH), and it is enterprising in [0, H]
Row integral can obtain：
Wherein,
Simultaneous formula (29) and (30) can solve：
Wherein,
Acquiring pile body extensional vibration displacement amplitude is：
S43, displacement at pile top frequency response function is acquired to obtain：
S44, the Complex modes for acquiring stake top：
Wherein Kr(r, ω) is stake top dynamic stiffness, Ki(r, ω) is stake top dynamic damping.
Further, in the step S5, the stake top Complex modes analytical solution radially carries out average value processing, in turn
Show that displacement at pile top frequency response function and stake top Complex modes radial direction mean value analytical solution are respectively：
Wherein,Stake top dynamic stiffness radial direction mean value is represented,Stake top dynamic damping radial direction mean value is represented,
Based on displacement at pile top frequency response function and stake top Complex modes radial direction mean value analytical solution, to pile body vibration characteristics and
Pile body integrity is evaluated.
Pile body and soil around pile are respectively seen as elasticity and viscoelasticity continuous media by the present invention, are based on pile body Three-dimensional Axisymmetric mould
Type considers its radial undulation effect, establishes and considers that the Three-dimensional Axisymmetric Pile Soil coupled system of stake end viscoelasticity support conditions is longitudinal
Vibration analysis model by the frequency transcendental equation being derived from corresponding to the system difference vibration shape, and is determined simultaneously to meet
The unJeiermined function of stake end viscoelasticity support conditions and stake top inhomogeneous boundary condition, and then acquire pile body and soil around pile displacement is basic
Solution is derived from stake top dynamic impedance analytical solution in frequency domain using stake soil coupling condition completely on this basis, can be suitably used for
Pile body frequency domain vibratory response problem under the conditions of the harmonious exciting of friction pile, closer to real model, computational accuracy is high, can be pile foundation
Dynamic testing provides theoretical direction and reference role.
The present invention can be widely popularized in civil engineering field based on the above reasons.
Description of the drawings
In order to more clearly explain the embodiment of the invention or the technical proposal in the existing technology, to embodiment or will show below
There is attached drawing needed in technology description to do simply to introduce, it should be apparent that, the accompanying drawings in the following description is this hair
Some bright embodiments for those of ordinary skill in the art without having to pay creative labor, can be with
Obtain other attached drawings according to these attached drawings.
Fig. 1 is that the present invention is based on the floating of pile body Three-dimensional Axisymmetric model to hold a Longitudinal vibration analysis method flow diagram；
Fig. 2 is the schematic diagram of the stake soil series system coupled longitudinal vibration mechanics simplified model of the present invention.
Specific implementation mode
In order to make the object, technical scheme and advantages of the embodiment of the invention clearer, below in conjunction with the embodiment of the present invention
In attached drawing, technical scheme in the embodiment of the invention is clearly and completely described, it is clear that described embodiment is
A part of the embodiment of the present invention, instead of all the embodiments.Based on the embodiments of the present invention, those of ordinary skill in the art
The every other embodiment obtained without making creative work, shall fall within the protection scope of the present invention.
As shown in Figure 1, a kind of floating based on pile body Three-dimensional Axisymmetric model holds a Longitudinal vibration analysis method, feature exists
In including the following steps：
S1, as shown in Fig. 2, establishing the Three-dimensional Axisymmetric model of pile body and pile peripheral earth, wherein pile body is considered as homogeneous etc.
Cross-section elasticities body, pile body bottom are considered as viscoelasticity bearing, ignore pile body radial displacement, consider pile body length travel and strain along diameter
To variation, pile peripheral earth upper surface is considered as free boundary, no direct stress, shear stress；
S2, according to elastodynamics basic theories, the longitudinal direction of pile peripheral earth and pile body shakes under the conditions of establishing Three-dimensional Axisymmetric
Dynamic equation, wherein Pile Soil Coupled Vibration System meets linear elasticity and small deformation condition, soil around pile with stake interfacial displacement continuously, answer
Dynamic balance；
S3, the vibration equation that the pile peripheral earth and pile body longitudinal direction are solved by the separation of variable, and it is derived from stake soil
System difference vibrates the frequency transcendental equation corresponding to the vibration shape；
S4, end viscoelasticity support conditions and a stake top nonhomogeneous boundary can be met according to the determination of Pile Soil coupling condition simultaneously
The unJeiermined function of condition solves the unJeiermined function and obtains the Complex modes of stake top；
S5, stake top Complex modes analytical solution radially carry out average value processing, and then obtain displacement at pile top frequency response function
With stake top Complex modes radial direction mean value analytical solution, complete to pile body vibration characteristics under harmonious exciting force effect and pile body integrity
Evaluation.
In the step S2, uniformly distributed harmonious exciting force is acted on to stake top, Pile soil System makees steady-state vibration, soil around pile therewith
Body and pile body length travel meet following relationship respectively：
Wherein, the harmonious exciting force isIndicate that imaginary unit, ω indicate circular frequency,
Indicate that pile peripheral earth and pile body extensional vibration amplitude, i indicate that imaginary unit, ω indicate exciting lotus respectively
Carrier frequency rate,
Pile peripheral earth and pile body equation of longitudinal are respectively：
Wherein, λSIndicate pile peripheral earth Lame constants, μSIndicate complex value shear modulus, ρSIndicate pile peripheral earth density, λPTable
Show pile body Lame constants, GPIndicate modulus of shearing, ρPIndicating pile body density, r is radial coordinate, and t is the time, and z is longitudinal coordinate,
λS=2vSGS/(1-2vS), μS=GS(1+2ξSI),
Wherein, vSIndicate Poisson's ratio, GSIndicate that soil layer bottom viscoelasticity supports constant, ξSIndicate hysteretic damping, vPIt indicates
Poisson's ratio,
It is following expression by formula (1), (2) abbreviation：
Wherein,The respectively shear wave velocity of pile peripheral earth and pile body.
Pile peripheral earth and pile body meet such as downstream condition：
Soil layer boundaries condition includes：
Soil layer surface is free：
The viscoelasticity bearing of soil layer bottom surface：
Wherein：ESIndicate elasticity modulus, kS、δSIndicate that soil layer bottom viscoelasticity supports constant, as r → ∞, displacement is
Zero：
Pile body boundary condition includes：
The uniformly distributed harmonious exciting force of stake top effectFor：
Wherein：Indicate the harmonious amplitude of exciting force of stake top,
The viscoelasticity bearing of stake bottom：
Wherein, kP、δPIndicate that stake bottom viscoelasticity supports constant, EPIndicate elasticity modulus.
Further, as follows to pile peripheral earth vibration equation solution procedure in the step S3：
It enablesTo which formula (3) is decomposed into two ODEs：
Wherein, hS, qSFor constant, and meet following relationship：
Show that formula (10), the solution of (11) are respectively：
ZS(z)=CScos(hSz)+DSsin(hSz) (13)
RS(r)=MSK0(qSr)+NSI0(qSr) (14)
Wherein, I0(qSr)、K0(qSR) it is the zeroth order first kind, the second class Bessel function of imaginary argument, CS、DS、MS、NSIt serves as reasons
The integral constant undetermined that boundary condition determines,
Comprehensive boundary condition formula (5), (6) can obtain transcendental equation：
Wherein,Indicate the dimensionless group of soil layer bottom spring Complex modes, KS=kS+iωδS, H expressions
Stake is long,
The transcendental equation obtains infinite multiple characteristic values by MATLAB Programs, is denoted asN=1,
2, ∞, and willSubstitution formula (12) can obtainIt is describedIndicate dimensionless group,
Comprehensive soil layer boundaries conditional (5), (6) and (7) can obtain soil around pile extensional vibration displacement amplitudeExpression formula be：
Wherein,For a series of unJeiermined functions.
It is as follows to pile peripheral earth vibration equation solution procedure in the step S3：
S31, due to stake top be inhomogeneous boundary condition will determine solution to obtain pile body extensional vibration dynamic respond amplitude and ask
Topic is decomposed：
Wherein,Meet the Definite problem as listed by formula (18),To meet formula (4) and boundary condition formula simultaneously
(8), the unJeiermined function of (9),
S32, the Definite problem is solved and can be obtained：
Wherein,N=1,2, ∞ is to meet transcendental equationIt is infinite multiple
Characteristic value,Indicate the dimensionless group of pile body bottom spring Complex modes, KP=kP+iωδP,WithMeet
Following relational expression：
It willSubstitution formula (4) can obtain：
Wherein,
The general solution for solving formula (21) is：
Wherein, aP、bPFor undetermined coefficient,
Formula (22) is substituted into boundary condition formula (8), (9) can solve：
Formula (19), formula (22), formula (23) and formula (24) are substituted into displacement when formula (17) can obtain the longitudinal direction harmonic vibration of stake work to ring
Answer amplitude：
Wherein,For a series of unJeiermined functions.
The S4 mainly includes the following steps：
S41, the undetermined coefficient is solved according to stake soil coupling conditionSoil around pile and stake can be obtained by formula (16) and formula (25)
The shear stress of body is respectively：
S42, pile body extensional vibration displacement amplitude is solved：
Assuming that soil around pile bottom dimensionless support stiffness is equal with pile body bottom dimensionless support stiffness, i.e.,
ThenContinuous, the stress equilibrium condition by soil around pile and the displacement of pile body interface, and convolution (16), (25) and formula
(26), (27) can obtain：
Wherein, r0Indicate radius,
According to the orthogonality of intrinsic function system, cos (h are multiplied by simultaneously at formula (28) both endsnz-hnH), and it is enterprising in [0, H]
Row integral can obtain：
Wherein,
Simultaneous formula (29) and (30) can solve：
Wherein,
Acquiring pile body extensional vibration displacement amplitude is：
S43, displacement at pile top frequency response function is acquired to obtain：
S44, the Complex modes for acquiring stake top：
Wherein Kr(r, ω) is stake top dynamic stiffness, Ki(r, ω) is stake top dynamic damping.
Further, in the step S5, the stake top Complex modes analytical solution radially carries out average value processing, in turn
Show that displacement at pile top frequency response function and stake top Complex modes radial direction mean value analytical solution are respectively：
Wherein,Stake top dynamic stiffness radial direction mean value is represented,Stake top dynamic damping radial direction mean value is represented,
Based on displacement at pile top frequency response function and stake top Complex modes radial direction mean value analytical solution, to pile body vibration characteristics and
Pile body integrity is evaluated.
Finally it should be noted that：The above embodiments are only used to illustrate the technical solution of the present invention., rather than its limitations；To the greatest extent
Present invention has been described in detail with reference to the aforementioned embodiments for pipe, it will be understood by those of ordinary skill in the art that：Its according to
So can with technical scheme described in the above embodiments is modified, either to which part or all technical features into
Row equivalent replacement；And these modifications or replacements, various embodiments of the present invention technology that it does not separate the essence of the corresponding technical solution
The range of scheme.
Claims (7)
1. a kind of floating based on pile body Three-dimensional Axisymmetric model holds a Longitudinal vibration analysis method, which is characterized in that including following
Step：
S1, the Three-dimensional Axisymmetric model for establishing pile body and pile peripheral earth, wherein pile body is considered as homogeneous cross-section elastomer, stake
Body bottom is considered as viscoelasticity bearing, ignores pile body radial displacement, the variation of pile body length travel and strain radially is considered, by stake
The upper surfaces Zhou Tuti are considered as free boundary, no direct stress and shear stress；
S2, according to elastodynamics basic theories, establish the extensional vibration side of pile peripheral earth and pile body under the conditions of Three-dimensional Axisymmetric
Journey, wherein Pile Soil Coupled Vibration System meets linear elasticity and small deformation condition, and soil around pile and stake interfacial displacement are continuous, stress is flat
Weighing apparatus；
S3, the vibration equation that the pile peripheral earth and pile body longitudinal direction are solved by the separation of variable, and it is derived from a soil series system
Frequency transcendental equation corresponding to the difference vibration vibration shape；
S4, end viscoelasticity support conditions and a stake top inhomogeneous boundary condition can be met according to the determination of Pile Soil coupling condition simultaneously
UnJeiermined function, solve the unJeiermined function and obtain the Complex modes of stake top；
S5, stake top Complex modes analytical solution radially carry out average value processing, and then obtain displacement at pile top frequency response function and stake
Complex modes radial direction mean value analytical solution is pushed up, pile body vibration characteristics under harmonious exciting force effect and pile body integrity are commented in completion
Valence.
2. according to the method described in claim 1, it is characterized in that, in the step S2, uniformly distributed harmonious exciting is acted on to stake top
Power, Pile soil System make steady-state vibration therewith, and pile peripheral earth and pile body length travel meet following relationship respectively：
Wherein, the harmonious exciting force isIndicate that imaginary unit, ω indicate circular frequency,
Indicate that pile peripheral earth and pile body extensional vibration amplitude, i indicate that imaginary unit, ω indicate exciting load frequency respectively
Rate,
Pile peripheral earth and pile body equation of longitudinal are respectively：
Wherein, λSIndicate pile peripheral earth Lame constants, μSIndicate complex value shear modulus, ρSIndicate pile peripheral earth density, λPIndicate stake
Body Lame constants, GPIndicate modulus of shearing, ρPIndicating pile body density, r is radial coordinate, and t is the time, and z is longitudinal coordinate,
λS=2vSGS/(1-2vS), μS=GS(1+2ξS i),
Wherein, vSIndicate Poisson's ratio, GSIndicate that soil layer bottom viscoelasticity supports constant, ξSIndicate hysteretic damping, vPIndicate Poisson
Than being following expression by formula (1), (2) abbreviation：
Wherein,The respectively shear wave velocity of pile peripheral earth and pile body.
3. according to the method described in claim 2, it is characterized in that, pile peripheral earth and pile body meet such as downstream condition：
Soil layer boundaries condition includes：
Soil layer surface is free：
The viscoelasticity bearing of soil layer bottom surface：
Wherein：ESIndicate elasticity modulus, kS、δSIndicate that soil layer bottom viscoelasticity supports constant,
As r → ∞, displacement zero：
Pile body boundary condition includes：
The uniformly distributed harmonious exciting force of stake top effectFor：
Wherein：Indicate the harmonious amplitude of exciting force of stake top,
The viscoelasticity bearing of stake bottom：
Wherein, kP、δPIndicate that stake bottom viscoelasticity supports constant, EPIndicate elasticity modulus.
4. according to the method described in claim 2, it is characterized in that, in the step S3, pile peripheral earth vibration equation is solved
Process is as follows：
It enablesTo which formula (3) is decomposed into two ODEs：
Wherein, hS, qSFor constant, and meet following relationship：
Show that formula (10), the solution of (11) are respectively：
ZS(z)=CScos(hSz)+DSsin(hSz) (13)
RS(r)=MSK0(qSr)+NSI0(qSr) (14)
Wherein, I0(qSr)、K0(qSR) it is the zeroth order first kind, the second class Bessel function of imaginary argument, CS、DS、MS、NSFor by boundary
The integral constant undetermined of conditional decision, comprehensive boundary condition formula (5), (6) can obtain transcendental equation：
Wherein,Indicate the dimensionless group of soil layer bottom spring Complex modes, KS=kS+iωδS, H expression stakes
It is long,
The transcendental equation obtains infinite multiple characteristic values by MATLAB Programs, is denoted as And it will
Substitution formula (12) can obtainIt is describedIndicate dimensionless group,
Comprehensive soil layer boundaries conditional (5), (6) and (7) can obtain soil around pile extensional vibration displacement amplitudeExpression formula be：
Wherein,For a series of unJeiermined functions.
5. according to the method described in claim 4, it is characterized in that, in the step S3, pile peripheral earth vibration equation is solved
Process is as follows：
S31, due to stake top be inhomogeneous boundary condition, to obtain pile body extensional vibration dynamic respond amplitude, by Definite problem into
Row decomposes：
Wherein,Meet the Definite problem as listed by formula (18),For meet simultaneously formula (4) and boundary condition formula (8),
(9) unJeiermined function,
S32, the Definite problem is solved and can be obtained：
Wherein,To meet transcendental equationInfinite multiple characteristic values,Indicate the dimensionless group of pile body bottom spring Complex modes, KP=kP+iωδP,WithMeet such as ShiShimonoseki
It is formula：
It willSubstitution formula (4) can obtain：
Wherein,
The general solution for solving formula (21) is：
Wherein, aP、bPFor undetermined coefficient,
Formula (22) is substituted into boundary condition formula (8), (9) can solve：
Formula (19), formula (22), formula (23) and formula (24) are substituted into dynamic respond width when formula (17) can obtain the longitudinal direction harmonic vibration of stake work
Value：
Wherein,For a series of unJeiermined functions.
6. according to the method described in claim 5, it is characterized in that, the S4 mainly includes the following steps：
S41, the undetermined coefficient is solved according to stake soil coupling conditionSoil around pile and pile body can be obtained by formula (16) and formula (25)
Shear stress is respectively：
S42, pile body extensional vibration displacement amplitude is solved：
Assuming that soil around pile bottom dimensionless support stiffness is equal with pile body bottom dimensionless support stiffness, i.e.,ThenContinuous, the stress equilibrium condition by soil around pile and the displacement of pile body interface, and convolution (16), (25) and formula
(26), (27) can obtain：
Wherein, r0Indicate radius,
According to the orthogonality of intrinsic function system, cos (h are multiplied by simultaneously at formula (28) both endsnz-hnH it), and on [0, H] is accumulated
Dividing can obtain：
Wherein,
Simultaneous formula (29) and (30) can solve：
Wherein,
Acquiring pile body extensional vibration displacement amplitude is：
S43, displacement at pile top frequency response function is acquired to obtain：
S44, the Complex modes for acquiring stake top：
Wherein Kr(r, ω) is stake top dynamic stiffness, Ki(r, ω) is stake top dynamic damping.
7. according to the method described in claim 6, it is characterized in that, in the step S5, the stake top Complex modes analytic solutions
It answers and radially carries out average value processing, and then obtain displacement at pile top frequency response function and stake top Complex modes radial direction mean value analytical solution
Respectively：
Wherein,Stake top dynamic stiffness radial direction mean value is represented,Stake top dynamic damping radial direction mean value is represented,
Based on displacement at pile top frequency response function and stake top Complex modes radial direction mean value analytical solution, to pile body vibration characteristics and pile body
Integrality is evaluated.
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| CN201810551757.8A CN108732242B (en) | 2018-05-31 | 2018-05-31 | Floating bearing pile longitudinal vibration analysis method based on three-dimensional axisymmetric model of pile body |
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| CN111310264A (en) * | 2020-02-07 | 2020-06-19 | 大连海事大学 | Single-pile horizontal dynamic response analysis method in layered soil based on Passternak foundation model |
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| CN114358091A (en) * | 2022-03-03 | 2022-04-15 | 中山大学 | Pile damage identification method, equipment and medium based on convolutional neural network |
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