CN107506564A - Consider stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil - Google Patents
Consider stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil Download PDFInfo
- Publication number
- CN107506564A CN107506564A CN201710954506.XA CN201710954506A CN107506564A CN 107506564 A CN107506564 A CN 107506564A CN 201710954506 A CN201710954506 A CN 201710954506A CN 107506564 A CN107506564 A CN 107506564A
- Authority
- CN
- China
- Prior art keywords
- mrow
- msubsup
- msub
- msup
- mfrac
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Pending
Links
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Computer Hardware Design (AREA)
- Evolutionary Computation (AREA)
- Geometry (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Nitrogen And Oxygen Or Sulfur-Condensed Heterocyclic Ring Systems (AREA)
Abstract
The invention discloses stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil is considered, it is related to civil engineering theory analysis techniques field.Pile peripheral earth considers vertical fluctuation effect using Three-dimensional Axisymmetric model；Pile peripheral earth is divided into interior zone and perimeter, interior zone divides any ring layer, and each ring layer soil body is respectively homogeneous, isotropism linear viscoelasticity body, the radially unlimited extension of the perimeter soil body, soil body material damping uses viscous damping, ignores soil body radial displacement；Stake Soil Interface and each ring layer Soil Interface both sides displacement is continuous, stress equilibrium, and stake soil system vibration is small deformation；Pile concrete is linear elasticity, communication satisfaction plane cross-section assumption of the stress wave in pile body；According to elastodynamics basic theories, pile peripheral earth and pile body equation of longitudinal under the conditions of Three-dimensional Axisymmetric are established；Using Laplace conversion and the separation of variable, two vibration equations are solved, obtain the time domain speed responsive function that any exciting force acts on stake top.
Description
Technical field
The present invention relates to civil engineering theory analysis techniques field, is considered more particularly, to one kind based on viscous damping model
Vertical fluctuation effect radial direction non-homogeneous soil pile foundation Longitudinal vibration analysis method.
Background technology
Pile Soil coupled vibrations characteristic research is the field of engineering technology such as Anti-seismic Pile Foundation, aseismatic design and dynamic pile detection
Theoretical foundation, be also all the time Geotechnical Engineering and Solid Mechanics hot issue.
It is well known that during pile foundation construction, due to soil compaction, relaxation and the influence of other disturbance factors so that stake
Radially there is certain inhomogeneities in Zhou Tuti, i.e. the heterogeneous effect of radial direction along pile foundation.To consider such a heterogeneous effect of radial direction, state
Inside and outside many scholars achieve a large amount of achievements.These achievements can be classified from different perspectives, can from the point of view of the external load of effect
It is divided into time domain under frequency domain response research and the Arbitrary Load under harmonious load action, frequency domain response research；Hindered from the material of the soil body
From the point of view of Buddhist nun, hysteresis material damping and cohesive material damping can be divided into；From the point of view of method for solving, analytic method, semi analytical method can be divided into
And numerical method.
The material damping of the soil body is that this interior friction is by medium as the energy dissipation caused by inside soil body particle friction
It is inevitable caused by non-resilient connection and other thermoelasticity processes between the defect of grain crystalline structure, media particle
, in order to consider this intrinsic friction, using the soil body Linear Constitutive for considering damping effect, carry out research material damping pair
The influence of dynamic response of pile is very important.
The conventional linear damping constitutive equation established in observation and experiment basis can be divided into two classes：Time domain constitution equation and
Frequency domain constitutive equation, the former directly establishes from the linear viscoelastic body of Macroscopic physical model in time domain；The latter then by with warp
The frequency-domain analysis method of allusion quotation matches to be established in frequency domain.
The time domain constitution model of linear viscoelasticity body, it can be made up of Hookean spring and linear damping element, linear damping
The viscous stress of element is directly proportional to strain rate, and various linear viscoelasticity constitutive models are may be constructed by both linear units,
The stress-strain property of true solid can be reflected.
Linear hysteretic damping is mainly reflected in the hysteretic damping in frequency domain this structure, and this structure of frequency domain can be understood as time domain sheet
The inverse Fourier transform of structure, hysteretic damping usually assumes that to be in elastic working region for constant, i.e. hypothesis material, hysteresis resistance
Buddhist nun than change it is little, or without obvious taxis change.In addition, the frequency-domain analysis to the steady-state vibration problems under harmonious load,
The material damping characteristic of the soil body can approx be reflected.However, to non-harmonic vibration (transient oscillation or random vibration) problem, it is stagnant
It is unsuitable to return damper model, particularly in the time domain response of stake under the conditions of studying transient excitation, native damping force and amplitude
About also relevant with strain rate, contradiction can conceptually be caused using Hysteretic Type Damping model, so as to produce it is so-called " dynamic response
Non-causality ", and now viscous damping model is then relatively adapted to, it is physically also more reasonable.
The content of the invention
It is an object of the invention to overcome drawbacks described above existing for prior art, pile peripheral earth construction disturbance, the soil body are considered
Using viscous damping model, more ring layer Three-dimensional Axisymmetric models are transmitted based on Complex modes, any exciting force are acted on lower radially non-
Homogeneous viscous damping soil pile foundation longitudinal vibration characteristics carries out analytic theory research.
To achieve the above object, technical scheme is as follows：
One kind considers vertical fluctuation effect radial direction non-homogeneous soil pile foundation Longitudinal vibration analysis based on viscous damping model
Method, comprise the following steps：
S1：Pile peripheral earth is using Three-dimensional Axisymmetric model and considers vertical fluctuation effect；
S2：The native system vibration of stake includes pile peripheral earth and pile body, and pile peripheral earth is divided into interior zone and perimeter, internal
Any ring layer of region division, the soil body of each ring layer are respectively homogeneous, isotropism linear viscoelasticity body, perimeter soil body footpath
To unlimited extension, soil body material damping uses viscous damping, ignores the radial displacement of the soil body；
S3：Stake Soil Interface and each ring layer Soil Interface both sides displacement is continuous, stress equilibrium, and stake soil system vibration is small change
Shape；
S4：Pile concrete is linear elasticity, communication satisfaction plane cross-section assumption of the stress wave in pile body；
S5：According to elastodynamics basic theories, the pile peripheral earth and pile body established under the conditions of Three-dimensional Axisymmetric longitudinally shake
Dynamic equation and boundary condition；
S6：Using Laplace (Laplce) conversion and the separation of variable, two vibration equations in S5 are solved, are obtained
Any exciting force acts on the time domain speed responsive function of stake top, is analyzed with the extensional vibration to pile foundation.
Pile peripheral earth and pile body equation of longitudinal in S5 are respectively：
Pile peripheral earth vibration equation：
The pile body equation of longitudinal for meeting plane cross-section assumption is：
Stake is long, radius, pile body density, modulus of elasticity and stake bottom viscoelasticity supporting constant are respectively H, r1、ρP、EPAnd kP、δP,
Stake top acts on any exciting force p (t).Pile peripheral earth is radially divided into internal disturbance region and perimeter, pile peripheral earth
Internal disturbance region radial thickness is b, and internal disturbance region is radially divided into m ring layer, and jth ring layer soil body Lame is normal
Number, modulus of shearing, viscous damping coefficient, modulus of elasticity, density and soil layer bottom viscoelasticity supporting constant are respectivelyWithSoil around pile is that frictional resistance is f to the sidewall shear stress of pile bodyS(r1,
z,t).The interface radius of -1 ring layer of jth and jth ring layer is rj.If the jth ring layer land movement of stake week isStake
Body displacement is uP(z, t), r are radial displacement, and t is the time, and z is length travel, EPFor pile body modulus of elasticity, APFor pile body section
Product；
Boundary condition in S5 includes：
Soil layer boundaries condition：
As r → ∞, displacement zero：
In formula,Represent perimeter land movement.
Pile body boundary condition：
Stake top active force is p (t)：
Boundary condition at stake end：
Pile body and pile peripheral earth are stake soil coupling compatibility conditions
Laplace (Laplce) conversion is carried out to equation (3), formula (4), formula (5), obtained based on the more of viscous damping
The soil layer shearing rigidity formula of ring layer model is：
Abbreviation is carried out to formula (9) and can be calculated constantWithRatioAs j=m
Work as j=m-1 ..., when 2,1
Wherein,Shearing rigidity between soil layer,For a series of undetermined coefficients of equation solution, rjJth
The inner boundary of ring layer soil, rj+1For jth ring layer soil external boundary, For the intrinsic parameter of jth ring layer soil, s is complex variable, I0、
I1For the zero and first order first kind amendment Bessel (Bezier) function, K0、K1Zero and first order the second class amendment Bessel (shellfishes
Sai Er) function；
Step 2：Laplace (Laplce) conversion is carried out to equation, and combines boundary condition formula (6) and formula (7) and stake
Native coupling condition (8) obtains displacement at pile top impedance function：
In formula, Tc=H/VP,θ=ω Tc, It is dimensionless group,γn、γ′n、γ″nIt is related for stake soil coupling
Coefficient, ω are extensional vibration circular frequency, VPFor stake elastic wave velocity,For stake bottom dimensionless branch
Hold rigidity and damped coefficient.
Step 3：Stake top velocity admittance function is obtained according to formula (13)：
Wherein, Hv' it is stake top velocity admittance function HvNondimensionalization；
Step 4：According to obtain unit pulse excitation time domain response be：
T '=t/T in formulacFor nondimensional time, θ is dimensionless frequency；IFT is inverse fast Fourier transform symbol；
Step 5：The time domain speed responsive function that any exciting force p (t) acts on stake top is obtained according to convolution theorem
G (t)=p (t) * h (t)=IFT [P (i ω) H (i ω)] (16)
Wherein, h (t) is the lower time domain speed responsive of unit pulse excitation effect, and H (i ω) is that stake top speed in frequency responds letter
Number.
Exciting force p (t) in step 5 encourages for half-sine pulseDuring t ∈ (0, T), T is arteries and veins
Width is rushed, the semi analytic answer of stake top time domain speed responsive is：
Wherein, QmaxFor half-sine pulse amplitude, Vv' it is time domain response nondimensional velocity.
It can be seen from the above technical proposal that the present invention is by using the radially heterogeneous viscous damping Three-dimensional Axisymmetric soil body
Extensional vibration of the model to pile foundation is analyzed, and the damping force of viscous damping soil model is related to strain rate, can be suitably used for
Non- harmonious exciting problem, particularly under the conditions of transient excitation when pile body time domain vibratory response problem, meanwhile, radial direction anisotropism
Pile peripheral earth construction disturbance effect can be considered, closer to real model, additionally, it is contemplated that the vertical fluctuation effect of the soil body, makes calculating smart
Du Genggao, theoretical direction and reference role are provided for dynamic pile detection.
Brief description of the drawings
Fig. 1 is the present invention based on the vertical fluctuation effect radial direction non-homogeneous soil pile foundation longitudinal direction of viscous damping model consideration
The flow chart of vibration analysis method.
Fig. 2 is the schematic diagram of the stake soil series system coupled longitudinal vibration mechanics simplified model of the present invention；
Embodiment
Below in conjunction with the accompanying drawings, the embodiment of the present invention is described in further detail.
Referring to Fig. 1, Fig. 1, which is the present invention, based on viscous damping model considers vertical fluctuation effect radial direction non-homogeneous soil
The flow chart of pile foundation Longitudinal vibration analysis method.As shown in figure 1, a kind of consider vertical fluctuation effect based on viscous damping model
Radial direction non-homogeneous soil pile foundation Longitudinal vibration analysis method, comprises the following steps：
S1：Pile peripheral earth considers vertical fluctuation effect using Three-dimensional Axisymmetric model；
S2：Pile peripheral earth is divided into interior zone and perimeter, and interior zone divides any ring layer, and each ring layer soil body is each
Hindered from for homogeneous, isotropism linear viscoelasticity body, the radially unlimited extension of the perimeter soil body, soil body material damping using stickiness
Buddhist nun, ignore soil body radial displacement；
S3：Stake Soil Interface and each ring layer Soil Interface both sides displacement is continuous, stress equilibrium, and stake soil system vibration is small change
Shape；
S4：Pile concrete is linear elasticity, communication satisfaction plane cross-section assumption of the stress wave in pile body；
The present invention is based on Three-dimensional Axisymmetric model, and the longitudinal vibration characteristics of pile foundation is supported to the viscoelasticity in any ring layer soil
Studied, mechanics simplified model is as shown in Figure 2.Pile peripheral earth is radially divided into internal disturbance region and perimeter,
Pile peripheral earth internal disturbance region radial thickness is b, and internal disturbance region is radially divided into m ring layer, the jth ring layer soil body
Lame constants, modulus of shearing, viscous damping coefficient, modulus of elasticity, density and soil layer bottom viscoelasticity supporting constant are respectivelyWithSoil around pile is f to the sidewall shear stress (frictional resistance) of pile bodyS(r1,z,
t).The interface radius of -1 ring layer of jth and jth ring layer is rj。
S5：According to elastodynamics basic theories, the pile peripheral earth and pile body established under the conditions of Three-dimensional Axisymmetric longitudinally shake
Dynamic equation and boundary condition；
S6：Using Laplace (Laplce) conversion and the separation of variable, two vibration sides described in solution procedure 5
Journey, the time domain speed responsive function that any exciting force acts on stake top is obtained, is analyzed with the extensional vibration to pile foundation.
Specifically, including in detail below step：
Step 1：Pile peripheral earth is radially divided into internal disturbance region and perimeter, pile peripheral earth internal disturbance area
Domain radial thickness is b, and internal disturbance region radially divided into m ring layer, jth ring layer soil body Lame constants, modulus of shearing,
Viscous damping coefficient, modulus of elasticity, density and soil layer bottom viscoelasticity supporting constant are respectively
WithSoil around pile is f to the sidewall shear stress (frictional resistance) of pile bodyS(r1,z,t).- 1 ring layer of jth and jth ring layer
Interface radius be rj.If the jth ring layer land movement of stake week isPile body displacement is uP(z, t), according to elasticity
Dynamics basic theories, the pile peripheral earth established under the conditions of Three-dimensional Axisymmetric and pile body equation of longitudinal and boundary condition difference
It is as follows：
Pile peripheral earth vibration equation：
The pile body equation of longitudinal for meeting plane cross-section assumption is：
Stake is long, radius, pile body density, modulus of elasticity and stake bottom viscoelasticity supporting constant are respectively H, r1、ρP、EPAnd kP、δP,
Stake top acts on any exciting force p (t).R is radial displacement, and t is the time, and z is length travel, EPFor pile body modulus of elasticity, APFor stake
Body sectional area；
Boundary condition in S5 includes：
Soil layer boundaries condition：
As r → ∞, displacement zero：
In formula,Represent perimeter land movement.
Pile body boundary condition：
Stake top active force is p (t)：
Boundary condition at stake end：
Stake, soil coupling compatibility conditions
Step 2：Laplace (Laplce) conversion is carried out to formula (3), formula (4), formula (5), obtained based on viscous damping
The soil layer shearing rigidity formula of multi-turn layer model is：
Abbreviation is carried out to formula (9) and can be calculated constantWithRatioAs j=m
Work as j=m-1 ..., when 2,1
Wherein,Shearing rigidity between soil layer,For a series of undetermined coefficients of equation solution, rjJth
The inner boundary of ring layer soil, rj+1For jth ring layer soil external boundary, For the intrinsic parameter of jth ring layer soil, s is complex variable, I0、
I1For the zero and first order first kind amendment Bessel (Bezier) function, K0、K1Zero and first order the second class amendment Bessel (shellfishes
Sai Er) function；
Step 3：Laplace (Laplce) conversion is carried out to equation, and combines boundary condition formula (6) and formula (7) and stake
Native coupling condition formula (8) obtains displacement at pile top impedance function：
In formula, Tc=H/VP,θ=ω Tc, It is dimensionless group,γn、γ′n、γ″nIt is related for stake soil coupling
Coefficient, ω are extensional vibration circular frequency, VPFor stake elastic wave velocity,For stake bottom dimensionless branch
Hold rigidity and damped coefficient.
Step 4：Stake top velocity admittance function is obtained according to (13) formula：
Wherein, Hv' it is stake top velocity admittance function HvNondimensionalization；
Step 5：According to obtain unit pulse excitation time domain response be：
T '=t/T in formulacFor nondimensional time, θ is dimensionless frequency；IFT is inverse fast Fourier transform symbol；
Step 6：The time domain speed responsive function that any exciting force p (t) acts on stake top is obtained according to convolution theorem
G (t)=p (t) * h (t)=IFT [P (i ω) H (i ω)] (16)
Wherein, h (t) is the lower time domain speed responsive of unit pulse excitation effect, and H (i ω) is that stake top speed in frequency responds letter
Number.
Exciting force p (t) in step 6 encourages for half-sine pulseDuring t ∈ (0, T), T is arteries and veins
When rushing width, the semi analytic answer of stake top time domain speed responsive is：
Wherein, QmaxFor half-sine pulse amplitude, Vv' it is time domain response nondimensional velocity.
, can be complete to pile body vibration characteristics and pile body based on stake top velocity admittance function and stake top speed time domain response function
Whole property is evaluated.
In summary, it is of the invention that vertical fluctuation effect radial direction non-homogeneous soil pile foundation is considered based on viscous damping model
Longitudinal vibration analysis method, the damping force and strain rate phase that its damper model used provides for stake soil coupled vibrations system
Close, can be suitably used for non-harmonious exciting problem, particularly under the conditions of transient excitation when, pile body time domain vibratory response problem, and radially
Anisotropism can consider pile peripheral earth construction disturbance effect, can provide theoretical direction and reference role for dynamic pile detection.
The foregoing is only a preferred embodiment of the present invention, but protection scope of the present invention be not limited thereto,
Any one skilled in the art the invention discloses technical scope in, technique according to the invention scheme and its
Inventive concept is subject to equivalent substitution or change, should all be included within the scope of the present invention.
Claims (2)
1. consider stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil, it is characterised in that：Including following step
Suddenly,
S1：Pile peripheral earth is using Three-dimensional Axisymmetric model and considers vertical fluctuation effect；
S2：The native system vibration of stake includes pile peripheral earth and pile body, and pile peripheral earth is divided into interior zone and perimeter, interior zone
Any ring layer is divided, the soil body of each ring layer is respectively homogeneous, isotropism linear viscoelasticity body, perimeter soil body radial direction nothing
Limit extension, soil body material damping use viscous damping, ignore the radial displacement of the soil body；
S3：Stake Soil Interface and each ring layer Soil Interface both sides displacement is continuous, stress equilibrium, and stake soil system vibration is small deformation；
S4：Pile concrete is linear elasticity, communication satisfaction plane cross-section assumption of the stress wave in pile body；
S5：The pile peripheral earth established under the conditions of Three-dimensional Axisymmetric according to elastodynamics basic theories and pile body extensional vibration side
Journey and boundary condition；
S6：Using Laplace (Laplce) conversion and the separation of variable, two vibration equations in S5 are solved, are obtained any
Exciting force acts on the time domain speed responsive function of stake top, is analyzed with the extensional vibration to pile foundation；
Pile peripheral earth and pile body equation of longitudinal in S5 are respectively：
Pile peripheral earth vibration equation：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msubsup>
<mi>&lambda;</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>+</mo>
<mn>2</mn>
<msubsup>
<mi>G</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>)</mo>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msubsup>
<mi>u</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
<mo>+</mo>
<msubsup>
<mi>G</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>(</mo>
<mrow>
<mfrac>
<mn>1</mn>
<mi>r</mi>
</mfrac>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>r</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
<msubsup>
<mi>u</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>t</mi>
</mrow>
</mfrac>
<mo>&lsqb;</mo>
<mrow>
<mo>(</mo>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mi>r</mi>
</mfrac>
<mfrac>
<mo>&part;</mo>
<mrow>
<mo>&part;</mo>
<mi>r</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>r</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>)</mo>
</mrow>
<msubsup>
<mi>u</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>=</mo>
<msubsup>
<mi>&rho;</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mfrac>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<mrow>
<mo>&part;</mo>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<msubsup>
<mi>u</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
</mrow>
The pile body equation of longitudinal for meeting plane cross-section assumption is：
<mrow>
<msup>
<mi>E</mi>
<mi>P</mi>
</msup>
<msup>
<mi>A</mi>
<mi>P</mi>
</msup>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msup>
<mi>u</mi>
<mi>P</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<msup>
<mi>z</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>-</mo>
<msup>
<mi>m</mi>
<mi>P</mi>
</msup>
<mfrac>
<mrow>
<msup>
<mo>&part;</mo>
<mn>2</mn>
</msup>
<msup>
<mi>u</mi>
<mi>P</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<msup>
<mi>t</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mo>-</mo>
<mn>2</mn>
<msub>
<mi>&pi;r</mi>
<mn>1</mn>
</msub>
<msup>
<mi>f</mi>
<mi>S</mi>
</msup>
<mrow>
<mo>(</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>2</mn>
<mo>)</mo>
</mrow>
</mrow>
Stake is long, radius, pile body density, modulus of elasticity and stake bottom viscoelasticity supporting constant are respectively H, r1、ρP、EPAnd kP、δP,
Stake top acts on any exciting force p (t)；Pile peripheral earth is radially divided into internal disturbance region and perimeter, soil around pile
Internal portion disturbance region radial thickness is b, and internal disturbance region is radially divided into m ring layer, jth ring layer soil body Lame
Constant, modulus of shearing, viscous damping coefficient, modulus of elasticity, density and soil layer bottom viscoelasticity supporting constant are respectivelyWithSoil around pile is that frictional resistance is f to the sidewall shear stress of pile bodyS(r1,z,
t)；The interface radius of -1 ring layer of jth and jth ring layer is rj；If the jth ring layer land movement of stake week isPile body
Displacement is uP(z, t), r are radial displacement, and t is the time, and z is length travel, EPFor pile body modulus of elasticity, APFor pile body sectional area；
Boundary condition in S5 includes：
Soil layer boundaries condition：
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>u</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>z</mi>
</mrow>
</mfrac>
<msub>
<mo>|</mo>
<mrow>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>3</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>u</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>z</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<mrow>
<msubsup>
<mi>k</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<msub>
<mi>u</mi>
<mi>j</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<msubsup>
<mi>E</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
</mfrac>
<mo>+</mo>
<mfrac>
<msubsup>
<mi>&delta;</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<msubsup>
<mi>E</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
</mfrac>
<mfrac>
<mrow>
<mo>&part;</mo>
<msubsup>
<mi>u</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>t</mi>
</mrow>
</mfrac>
<msub>
<mo>|</mo>
<mrow>
<mi>z</mi>
<mo>=</mo>
<mi>H</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>4</mn>
<mo>)</mo>
</mrow>
</mrow>
As r → ∞, displacement zero：
<mrow>
<munder>
<mi>lim</mi>
<mrow>
<mi>r</mi>
<mo>&RightArrow;</mo>
<mi>&infin;</mi>
</mrow>
</munder>
<msubsup>
<mi>u</mi>
<mrow>
<mi>m</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>S</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>5</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula,Represent perimeter land movement；
Pile body boundary condition：
Stake top active force is p (t)：
<mrow>
<msup>
<mi>E</mi>
<mi>P</mi>
</msup>
<msup>
<mi>A</mi>
<mi>P</mi>
</msup>
<mfrac>
<mrow>
<mo>&part;</mo>
<msup>
<mi>u</mi>
<mi>P</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>z</mi>
</mrow>
</mfrac>
<msub>
<mo>|</mo>
<mrow>
<mi>z</mi>
<mo>=</mo>
<mn>0</mn>
</mrow>
</msub>
<mo>=</mo>
<mo>-</mo>
<mi>p</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>6</mn>
<mo>)</mo>
</mrow>
</mrow>
Boundary condition at stake end：
<mrow>
<mfrac>
<mrow>
<mo>&part;</mo>
<msup>
<mi>u</mi>
<mi>P</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>z</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<msup>
<mi>&delta;</mi>
<mi>P</mi>
</msup>
<mrow>
<msup>
<mi>E</mi>
<mi>P</mi>
</msup>
<msup>
<mi>A</mi>
<mi>P</mi>
</msup>
</mrow>
</mfrac>
<mfrac>
<mrow>
<mo>&part;</mo>
<msup>
<mi>u</mi>
<mi>P</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<mo>&part;</mo>
<mi>t</mi>
</mrow>
</mfrac>
<mo>+</mo>
<mfrac>
<msup>
<mi>k</mi>
<mi>P</mi>
</msup>
<mrow>
<msup>
<mi>E</mi>
<mi>P</mi>
</msup>
<msup>
<mi>A</mi>
<mi>P</mi>
</msup>
</mrow>
</mfrac>
<msup>
<mi>u</mi>
<mi>P</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<msub>
<mo>|</mo>
<mrow>
<mi>z</mi>
<mo>=</mo>
<mi>H</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>7</mn>
<mo>)</mo>
</mrow>
</mrow>
Pile body and pile peripheral earth are stake soil coupling compatibility conditions
<mrow>
<msubsup>
<mi>u</mi>
<mn>1</mn>
<mi>S</mi>
</msubsup>
<mrow>
<mo>(</mo>
<mi>r</mi>
<mo>,</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<msub>
<mo>|</mo>
<mrow>
<mi>r</mi>
<mo>=</mo>
<msub>
<mi>r</mi>
<mn>1</mn>
</msub>
</mrow>
</msub>
<mo>=</mo>
<msup>
<mi>u</mi>
<mi>P</mi>
</msup>
<mrow>
<mo>(</mo>
<mi>z</mi>
<mo>,</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>8</mn>
<mo>)</mo>
</mrow>
</mrow>
Laplace (Laplce) conversion is carried out to equation (3), formula (4), formula (5), obtains more ring layers based on viscous damping
The soil layer shearing rigidity formula of model is：
<mrow>
<msubsup>
<mi>&tau;</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>=</mo>
<mfenced open = "{" close = "">
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msubsup>
<mi>G</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mi>s</mi>
<mo>)</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mo>&lsqb;</mo>
<msubsup>
<mi>A</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mi>r</mi>
<mo>)</mo>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mo>(</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mi>z</mi>
<mo>)</mo>
<mo>&rsqb;</mo>
<mo>,</mo>
<mi>j</mi>
<mo>=</mo>
<mi>m</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msubsup>
<mi>G</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mi>s</mi>
<mo>)</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mo>{</mo>
<mo>&lsqb;</mo>
<mo>-</mo>
<msubsup>
<mi>B</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mi>r</mi>
<mo>)</mo>
<mo>+</mo>
<msubsup>
<mi>C</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mi>r</mi>
<mo>)</mo>
<mo>&rsqb;</mo>
<mi>cos</mi>
<mo>(</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mi>z</mi>
<mo>)</mo>
<mo>}</mo>
<mo>,</mo>
<mi>j</mi>
<mo>=</mo>
<mi>m</mi>
<mo>,</mo>
<mo>...</mo>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>1</mn>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>9</mn>
<mo>)</mo>
</mrow>
</mrow>
Abbreviation is carried out to formula (9) and can be calculated constantWithRatioAs j=m
Work as j=m-1 ..., when 2,1
<mrow>
<msubsup>
<mi>p</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mo>=</mo>
<mfrac>
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msubsup>
<mi>G</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mi>s</mi>
<mo>)</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
<mo>&lsqb;</mo>
<msubsup>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>I</mi>
<mn>0</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
<mo>+</mo>
<msub>
<mi>K</mi>
<mn>0</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>S</mi>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mrow>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>S</mi>
</msubsup>
<mi>s</mi>
<mo>)</mo>
</mrow>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>K</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&lsqb;</mo>
<msubsup>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
</mtable>
<mtable>
<mtr>
<mtd>
<mrow>
<mo>(</mo>
<msubsup>
<mi>G</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mi>j</mi>
<mi>S</mi>
</msubsup>
<mi>s</mi>
<mo>)</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
<mo>&lsqb;</mo>
<msubsup>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>I</mi>
<mn>0</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
<mo>+</mo>
<msub>
<mi>K</mi>
<mn>0</mn>
</msub>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msubsup>
<mi>G</mi>
<mrow>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>S</mi>
</msubsup>
<mo>+</mo>
<msubsup>
<mi>c</mi>
<mrow>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
<mi>S</mi>
</msubsup>
<mi>s</mi>
<mo>)</mo>
</mrow>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>I</mi>
<mn>0</mn>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mi>j</mi>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&lsqb;</mo>
<msubsup>
<mi>P</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mi>q</mi>
<mrow>
<mo>(</mo>
<mi>j</mi>
<mo>+</mo>
<mn>1</mn>
<mo>)</mo>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>r</mi>
<mi>j</mi>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>11</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein,Shearing rigidity between soil layer,For a series of undetermined coefficients of equation solution, rjJth ring layer
The inner boundary of soil, rj+1For jth ring layer soil external boundary, For the intrinsic parameter of jth ring layer soil, s is complex variable, I0、I1For
The zero and first order first kind amendment Bessel (Bezier) function, K0、K1Zero and first order the second class amendment Bessel (Bezier)
Function；
Step 2：Laplace (Laplce) conversion is carried out to equation, and combines boundary condition formula (6) and formula (7) and stake soil coupling
Conjunction condition (8) obtains displacement at pile top impedance function：
<mrow>
<mi>Z</mi>
<mrow>
<mo>(</mo>
<mi>s</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mrow>
<msup>
<mi>E</mi>
<mi>P</mi>
</msup>
<msup>
<mi>A</mi>
<mi>P</mi>
</msup>
<mi>&theta;</mi>
</mrow>
<mi>H</mi>
</mfrac>
<mo>&lsqb;</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mo>(</mo>
<msubsup>
<mi>D</mi>
<mn>1</mn>
<mi>P</mi>
</msubsup>
<mo>/</mo>
<msubsup>
<mi>D</mi>
<mn>2</mn>
<mi>P</mi>
</msubsup>
<mo>)</mo>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mo>&prime;</mo>
</msubsup>
<mo>)</mo>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mrow>
<mo>&prime;</mo>
<mo>&prime;</mo>
</mrow>
</msubsup>
</mrow>
</mfrac>
<mo>&rsqb;</mo>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>12</mn>
<mo>)</mo>
</mrow>
</mrow>
<mrow>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mo>&prime;</mo>
</msubsup>
<mo>=</mo>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mo>&lsqb;</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>&omega;</mi>
<mo>/</mo>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
<mo>-</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
</mrow>
</mfrac>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<mo>(</mo>
<mrow>
<mi>&omega;</mi>
<mo>/</mo>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
<mo>-</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
</mrow>
<mo>)</mo>
<mi>H</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>&omega;</mi>
<mo>/</mo>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
<mo>+</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
</mrow>
</mfrac>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<mo>(</mo>
<mrow>
<mi>&omega;</mi>
<mo>/</mo>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
<mo>+</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
</mrow>
<mo>)</mo>
<mi>H</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mrow>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mrow>
<mo>&prime;</mo>
<mo>&prime;</mo>
</mrow>
</msubsup>
<mo>=</mo>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mo>&lsqb;</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>&omega;</mi>
<mo>/</mo>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
<mo>+</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mi>cos</mi>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>&omega;</mi>
<mo>/</mo>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
<mo>+</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
</mrow>
<mo>)</mo>
</mrow>
<mi>H</mi>
</mrow>
<mo>)</mo>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>+</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mi>&omega;</mi>
<mo>/</mo>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
<mo>-</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mi>cos</mi>
<mo>(</mo>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>&omega;</mi>
<mo>/</mo>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
<mo>-</mo>
<msubsup>
<mi>h</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
</mrow>
<mo>)</mo>
</mrow>
<mi>H</mi>
</mrow>
<mo>)</mo>
<mo>-</mo>
<mn>1</mn>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mrow>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<msubsup>
<mi>iG</mi>
<mi>c</mi>
<mo>&prime;</mo>
</msubsup>
<mi>&theta;</mi>
<mo>)</mo>
<msubsup>
<mover>
<mi>q</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mover>
<mi>&rho;</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<msubsup>
<mover>
<mi>v</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
<mn>2</mn>
</msubsup>
</mrow>
<mrow>
<msub>
<mover>
<mi>r</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msup>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>h</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<mi>&theta;</mi>
<mn>2</mn>
</msup>
<mo>)</mo>
</mrow>
<msubsup>
<mi>&phi;</mi>
<mi>n</mi>
<mi>S</mi>
</msubsup>
<msubsup>
<mi>L</mi>
<mi>n</mi>
<mi>S</mi>
</msubsup>
</mrow>
</mfrac>
<mo>&lsqb;</mo>
<msub>
<mi>K</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>q</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mover>
<mi>r</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>-</mo>
<msubsup>
<mi>p</mi>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mi>I</mi>
<mn>1</mn>
</msub>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>q</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<msub>
<mover>
<mi>r</mi>
<mo>&OverBar;</mo>
</mover>
<mn>1</mn>
</msub>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
<mrow>
<mfrac>
<msubsup>
<mi>D</mi>
<mn>1</mn>
<mi>P</mi>
</msubsup>
<msubsup>
<mi>D</mi>
<mn>2</mn>
<mi>P</mi>
</msubsup>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>&theta;</mi>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<mi>&theta;</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mrow>
<mo>&prime;</mo>
<mo>&prime;</mo>
</mrow>
</msubsup>
<msubsup>
<mover>
<mi>h</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mi>s</mi>
<mi>i</mi>
<mi>n</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>h</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>+</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>sA</mi>
<mi>b</mi>
</msub>
<mo>+</mo>
<mi>R</mi>
<mo>)</mo>
</mrow>
<mo>&lsqb;</mo>
<mi>s</mi>
<mi>i</mi>
<mi>n</mi>
<mrow>
<mo>(</mo>
<mi>&theta;</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mrow>
<mo>&prime;</mo>
<mo>&prime;</mo>
</mrow>
</msubsup>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>h</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
<mrow>
<mi>&theta;</mi>
<mi>sin</mi>
<mrow>
<mo>(</mo>
<mi>&theta;</mi>
<mo>)</mo>
</mrow>
<mo>+</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<mtable>
<mtr>
<mtd>
<mrow>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mo>&prime;</mo>
</msubsup>
<msubsup>
<mover>
<mi>h</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mi>s</mi>
<mi>i</mi>
<mi>n</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>h</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mo>)</mo>
</mrow>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mrow>
<mo>(</mo>
<msub>
<mi>sA</mi>
<mi>b</mi>
</msub>
<mo>+</mo>
<mi>R</mi>
<mo>)</mo>
</mrow>
<mo>&lsqb;</mo>
<mi>cos</mi>
<mrow>
<mo>(</mo>
<mi>&theta;</mi>
<mo>)</mo>
</mrow>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msubsup>
<mi>&gamma;</mi>
<mi>n</mi>
<mo>&prime;</mo>
</msubsup>
<mi>c</mi>
<mi>o</mi>
<mi>s</mi>
<mrow>
<mo>(</mo>
<msubsup>
<mover>
<mi>h</mi>
<mo>&OverBar;</mo>
</mover>
<mrow>
<mn>1</mn>
<mi>n</mi>
</mrow>
<mi>S</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mfrac>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>13</mn>
<mo>)</mo>
</mrow>
</mrow>
In formula, Tc=H/VP,θ=ω Tc, It is dimensionless group,γn、γn′、γnIt is " related for stake soil coupling
Coefficient, ω are extensional vibration circular frequency, VPFor stake elastic wave velocity,For stake bottom dimensionless branch
Hold rigidity and damped coefficient；
Step 3：Stake top velocity admittance function is obtained according to formula (13)：
<mrow>
<mtable>
<mtr>
<mtd>
<mrow>
<msub>
<mi>H</mi>
<mi>v</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mi>&omega;</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<mi>i</mi>
<mi>&omega;</mi>
</mrow>
<mrow>
<mi>Z</mi>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mi>&omega;</mi>
<mo>)</mo>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mrow>
<mi>H</mi>
<mi>i</mi>
<mi>&omega;</mi>
</mrow>
<mrow>
<msup>
<mi>E</mi>
<mi>P</mi>
</msup>
<msup>
<mi>A</mi>
<mi>P</mi>
</msup>
<mi>&theta;</mi>
</mrow>
</mfrac>
<mo>&lsqb;</mo>
<mrow>
<mo>(</mo>
<msubsup>
<mi>D</mi>
<mn>1</mn>
<mi>P</mi>
</msubsup>
<mo>/</mo>
<msubsup>
<mi>D</mi>
<mn>2</mn>
<mi>P</mi>
</msubsup>
<mo>)</mo>
</mrow>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msup>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mo>)</mo>
</mrow>
<mo>-</mo>
<munderover>
<mo>&Sigma;</mo>
<mrow>
<mi>n</mi>
<mo>=</mo>
<mn>1</mn>
</mrow>
<mi>&infin;</mi>
</munderover>
<msup>
<msub>
<mi>&gamma;</mi>
<mi>n</mi>
</msub>
<mrow>
<mo>&prime;</mo>
<mo>&prime;</mo>
</mrow>
</msup>
<mo>&rsqb;</mo>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>=</mo>
<mo>-</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msup>
<mi>&rho;</mi>
<mi>P</mi>
</msup>
<msup>
<mi>A</mi>
<mi>P</mi>
</msup>
<msup>
<mi>V</mi>
<mi>P</mi>
</msup>
</mrow>
</mfrac>
<msubsup>
<mi>H</mi>
<mi>v</mi>
<mo>&prime;</mo>
</msubsup>
</mrow>
</mtd>
</mtr>
</mtable>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>14</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, Hv' it is stake top velocity admittance function HvNondimensionalization；
Step 4：According to obtain unit pulse excitation time domain response be：
<mrow>
<mi>h</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<mi>I</mi>
<mi>F</mi>
<mi>T</mi>
<mo>&lsqb;</mo>
<msub>
<mi>H</mi>
<mi>v</mi>
</msub>
<mrow>
<mo>(</mo>
<mi>i</mi>
<mi>&omega;</mi>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mn>2</mn>
<mi>&pi;</mi>
</mrow>
</mfrac>
<munderover>
<mo>&Integral;</mo>
<mrow>
<mo>-</mo>
<mi>&infin;</mi>
</mrow>
<mi>&infin;</mi>
</munderover>
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>&rho;</mi>
<mi>p</mi>
</msub>
<msub>
<mi>A</mi>
<mi>p</mi>
</msub>
<msub>
<mi>V</mi>
<mi>P</mi>
</msub>
</mrow>
</mfrac>
<msup>
<msub>
<mi>H</mi>
<mi>v</mi>
</msub>
<mo>&prime;</mo>
</msup>
<msup>
<mi>e</mi>
<mrow>
<msup>
<mi>i&theta;t</mi>
<mo>&prime;</mo>
</msup>
</mrow>
</msup>
<mi>d</mi>
<mi>&theta;</mi>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>15</mn>
<mo>)</mo>
</mrow>
</mrow>
T '=t/T in formulacFor nondimensional time, θ is dimensionless frequency；IFT is inverse fast Fourier transform symbol；
Step 5：The time domain speed responsive function that any exciting force p (t) acts on stake top is obtained according to convolution theorem
G (t)=p (t) * h (t)=IFT [P (i ω) H (i ω)] (16)
Wherein, h (t) is the lower time domain speed responsive of unit pulse excitation effect, and H (i ω) is stake top speed in frequency receptance function；
Exciting force p (t) in step 5 encourages for half-sine pulseDuring t ∈ (0, T), T is that pulse is wide
Degree, the semi analytic answer of stake top time domain speed responsive are：
<mrow>
<mi>g</mi>
<mrow>
<mo>(</mo>
<mi>t</mi>
<mo>)</mo>
</mrow>
<mo>=</mo>
<msub>
<mi>Q</mi>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
<mi>I</mi>
<mi>F</mi>
<mi>T</mi>
<mo>&lsqb;</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msub>
<mi>&rho;</mi>
<mi>p</mi>
</msub>
<msub>
<mi>A</mi>
<mi>p</mi>
</msub>
<msub>
<mi>V</mi>
<mi>P</mi>
</msub>
</mrow>
</mfrac>
<msup>
<msub>
<mi>H</mi>
<mi>v</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mfrac>
<mrow>
<mi>&pi;</mi>
<mi>T</mi>
</mrow>
<mrow>
<msup>
<mi>&pi;</mi>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<mi>T</mi>
<mn>2</mn>
</msup>
<msup>
<mi>&omega;</mi>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
<mrow>
<mo>(</mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow>
<mo>-</mo>
<mi>i</mi>
<mi>&omega;</mi>
<mi>T</mi>
</mrow>
</msup>
<mo>)</mo>
</mrow>
<mo>&rsqb;</mo>
<mo>=</mo>
<mfrac>
<msub>
<mi>Q</mi>
<mrow>
<mi>m</mi>
<mi>a</mi>
<mi>x</mi>
</mrow>
</msub>
<mrow>
<msub>
<mi>&rho;</mi>
<mi>p</mi>
</msub>
<msub>
<mi>A</mi>
<mi>p</mi>
</msub>
<msub>
<mi>V</mi>
<mi>P</mi>
</msub>
</mrow>
</mfrac>
<msup>
<msub>
<mi>V</mi>
<mi>v</mi>
</msub>
<mo>&prime;</mo>
</msup>
<mo>-</mo>
<mo>-</mo>
<mo>-</mo>
<mrow>
<mo>(</mo>
<mn>17</mn>
<mo>)</mo>
</mrow>
</mrow>
Wherein, QmaxFor half-sine pulse amplitude, Vv' it is time domain response nondimensional velocity.
2. according to claim 1 consider stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil, its
It is characterised by：, can be to pile body vibration characteristics and pile body based on stake top velocity admittance function and stake top speed time domain response function
Integrality is evaluated.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201710954506.XA CN107506564A (en) | 2017-10-13 | 2017-10-13 | Consider stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201710954506.XA CN107506564A (en) | 2017-10-13 | 2017-10-13 | Consider stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CN107506564A true CN107506564A (en) | 2017-12-22 |
Family
ID=60701657
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201710954506.XA Pending CN107506564A (en) | 2017-10-13 | 2017-10-13 | Consider stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN107506564A (en) |
Cited By (11)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108416130A (en) * | 2018-02-27 | 2018-08-17 | 大连海事大学 | Large diameter pile Longitudinal vibration analysis method in axial symmetry radial direction heterogeneous soil |
| CN108732242A (en) * | 2018-05-31 | 2018-11-02 | 大连海事大学 | Floating based on pile body Three-dimensional Axisymmetric model holds a Longitudinal vibration analysis method |
| CN108875157A (en) * | 2018-05-30 | 2018-11-23 | 北京工业大学 | A saturated soil-clump of piles-superstructure system dynamic response analysis method |
| CN109372035A (en) * | 2018-10-24 | 2019-02-22 | 大连海事大学 | It is a kind of that a Longitudinal vibration analysis method is held based on saturation the floating of loosened soil stake |
| CN110320221A (en) * | 2019-07-24 | 2019-10-11 | 水利部交通运输部国家能源局南京水利科学研究院 | A kind of steel shell and inhomogeneous structure body interface, which come to nothing, quantitatively determines method |
| CN110359469A (en) * | 2019-07-29 | 2019-10-22 | 中铁二院工程集团有限责任公司 | A kind of damping anti-slide pile structure and design method |
| CN110598262A (en) * | 2019-08-16 | 2019-12-20 | 河海大学 | Calculation method of vertical impedance of vertical pile |
| CN111122086A (en) * | 2019-12-31 | 2020-05-08 | 大连海事大学 | Method and system for analyzing torsional vibration of tubular pile in axisymmetric bidirectional heterogeneous viscous damping soil |
| CN112302061A (en) * | 2020-09-29 | 2021-02-02 | 安徽省（水利部淮河水利委员会）水利科学研究院（安徽省水利工程质量检测中心站） | Intelligent rapid interpretation method for integrity detection signal of low-strain foundation pile |
| CN114358091A (en) * | 2022-03-03 | 2022-04-15 | 中山大学 | Pile damage identification method, equipment and medium based on convolutional neural network |
| CN116090257A (en) * | 2023-03-07 | 2023-05-09 | 中大智能科技股份有限公司 | Theoretical model of road roller-soil-based material system |
-
2017
- 2017-10-13 CN CN201710954506.XA patent/CN107506564A/en active Pending
Non-Patent Citations (2)
| Title |
|---|
| 吴文兵: "《基于虚土桩法的桩土纵向耦合振动理论及应用研究》", 《中国博士学位论文全文数据库 工程科技II辑》 * |
| 胡昌斌: "《考虑土竖向波动效应的桩土纵向耦合振动理论》", 《中国优秀博硕士学位论文全文数据库（博士） 工程科技II辑》 * |
Cited By (16)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN108416130B (en) * | 2018-02-27 | 2020-06-16 | 大连海事大学 | Longitudinal vibration analysis method for large-diameter pile in axisymmetric radial heterogeneous soil |
| CN108416130A (en) * | 2018-02-27 | 2018-08-17 | 大连海事大学 | Large diameter pile Longitudinal vibration analysis method in axial symmetry radial direction heterogeneous soil |
| CN108875157B (en) * | 2018-05-30 | 2022-04-22 | 北京工业大学 | Dynamic response analysis method of saturated soil-pile group-upper structure system |
| CN108875157A (en) * | 2018-05-30 | 2018-11-23 | 北京工业大学 | A saturated soil-clump of piles-superstructure system dynamic response analysis method |
| CN108732242A (en) * | 2018-05-31 | 2018-11-02 | 大连海事大学 | Floating based on pile body Three-dimensional Axisymmetric model holds a Longitudinal vibration analysis method |
| CN108732242B (en) * | 2018-05-31 | 2020-09-01 | 大连海事大学 | Floating bearing pile longitudinal vibration analysis method based on three-dimensional axisymmetric model of pile body |
| CN109372035A (en) * | 2018-10-24 | 2019-02-22 | 大连海事大学 | It is a kind of that a Longitudinal vibration analysis method is held based on saturation the floating of loosened soil stake |
| CN110320221A (en) * | 2019-07-24 | 2019-10-11 | 水利部交通运输部国家能源局南京水利科学研究院 | A kind of steel shell and inhomogeneous structure body interface, which come to nothing, quantitatively determines method |
| CN110359469A (en) * | 2019-07-29 | 2019-10-22 | 中铁二院工程集团有限责任公司 | A kind of damping anti-slide pile structure and design method |
| CN110359469B (en) * | 2019-07-29 | 2023-10-13 | 中铁二院工程集团有限责任公司 | Damping slide-resistant pile structure and design method |
| CN110598262A (en) * | 2019-08-16 | 2019-12-20 | 河海大学 | Calculation method of vertical impedance of vertical pile |
| CN111122086A (en) * | 2019-12-31 | 2020-05-08 | 大连海事大学 | Method and system for analyzing torsional vibration of tubular pile in axisymmetric bidirectional heterogeneous viscous damping soil |
| CN112302061A (en) * | 2020-09-29 | 2021-02-02 | 安徽省（水利部淮河水利委员会）水利科学研究院（安徽省水利工程质量检测中心站） | Intelligent rapid interpretation method for integrity detection signal of low-strain foundation pile |
| CN114358091A (en) * | 2022-03-03 | 2022-04-15 | 中山大学 | Pile damage identification method, equipment and medium based on convolutional neural network |
| CN114358091B (en) * | 2022-03-03 | 2022-06-10 | 中山大学 | Pile damage identification method, equipment and medium based on convolutional neural network |
| CN116090257A (en) * | 2023-03-07 | 2023-05-09 | 中大智能科技股份有限公司 | Theoretical model of road roller-soil-based material system |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| CN107506564A (en) | Consider stake Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil | |
| CN107620329A (en) | Consider pile tube Longitudinal vibration analysis method in vertical fluctuation effect radial direction heterogeneous soil | |
| CN107330223A (en) | Pile tube Longitudinal vibration analysis method based on radially heterogeneous viscous damping soil model | |
| CN108446460A (en) | A kind of radial direction is heterogeneous, the layered soil body pile foundation Longitudinal vibration analysis method in longitudinal direction | |
| CN108416130B (en) | Longitudinal vibration analysis method for large-diameter pile in axisymmetric radial heterogeneous soil | |
| CN107604957A (en) | Based on pile tube Longitudinal vibration analysis method in viscous damping model complexity heterogeneous soil | |
| Nasr | Experimental and theoretical studies of laterally loaded finned piles in sand | |
| Veletsos et al. | Dynamic modeling and response of soil-wall systems | |
| Wang et al. | Comparative study on buffeting performance of Sutong Bridge based on design and measured spectrum | |
| CN108732242A (en) | Floating based on pile body Three-dimensional Axisymmetric model holds a Longitudinal vibration analysis method | |
| CN109344526B (en) | Virtual soil pile model-based longitudinal vibration research method for pile foundation in saturated layered soil | |
| Schanz et al. | Dynamic analysis of a one-dimensional poroviscoelastic column | |
| Wang et al. | Horizontal impedance of pile groups considering shear behavior of multilayered soils | |
| Wang et al. | Analytical model of vertical vibrations in piles for different tip boundary conditions: parametric study and applicationsx | |
| Khalil et al. | Dynamic behavior of pile foundations under vertical and lateral vibrations | |
| CN109359390B (en) | Axial-symmetry bidirectional heterogeneous viscous damping soil pile foundation torsional vibration analysis method | |
| CN110598262A (en) | Calculation method of vertical impedance of vertical pile | |
| Messioud et al. | Influence of the pile toe condition on the dynamic response of a group of pile foundations | |
| Qian et al. | Dynamic shakedown limits for flexible pavement with cross-anisotropic materials | |
| CN110147630A (en) | A kind of tubular pole Longitudinal vibration analysis method in transverse inertia domino effect radial direction heterogeneous soil | |
| CN110222400B (en) | Method for analyzing longitudinal vibration of bidirectional heterogeneous soil-in-pipe pile under transverse inertia effect | |
| Kjolsing et al. | Damping of a fluid-conveying pipe surrounded by a viscous annulus fluid | |
| Tripathy et al. | Investigation of dynamic behaviour for turbo generator frame foundation through experimental and computational approach | |
| Nguyen et al. | Reciprocal absorbing boundary condition for the time-domain numerical analysis of wave motion in unbounded layered media | |
| CN110093951B (en) | Virtual soil pile model-based friction pile longitudinal vibration analysis method |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| RJ01 | Rejection of invention patent application after publication |
Application publication date: 20171222 |
|
| RJ01 | Rejection of invention patent application after publication |