CN111962572B - Horizontal-swing vibration impedance determination method for partially-embedded pile foundation - Google Patents

Horizontal-swing vibration impedance determination method for partially-embedded pile foundation Download PDF

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CN111962572B
CN111962572B CN202010787704.3A CN202010787704A CN111962572B CN 111962572 B CN111962572 B CN 111962572B CN 202010787704 A CN202010787704 A CN 202010787704A CN 111962572 B CN111962572 B CN 111962572B
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王珏
韩亦鸣
张玉全
叶会然
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Hohai University HHU
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Abstract

The invention discloses a method for determining horizontal-swing vibration impedance of a partially embedded pile foundation, which comprises the following steps: dispersing pile bodies and pile periphery laminar saturated soil along a z axis according to the horizontal layering condition of the pile periphery laminar saturated soil, and circularly dispersing the saturated soil in a weakening area along the radial direction of a pile foundation according to the weakening condition of each layer of pile periphery soil; and establishing a fluctuation equation of the saturated soil of each circle layer after dispersion according to a Biot theory, solving to obtain a matrix of a displacement field and a stress field of the saturated soil of each circle layer, calculating a unit horizontal dynamic reaction on a pile-soil contact surface, solving a pre-established free transverse vibration equation of the pile body of the non-embedded section and a forced transverse vibration equation of the pile body of the embedded section under the action of the horizontal dynamic reaction of the saturated soil to obtain the force, deformation continuous conditions and pile bottom boundary conditions of each pile section, and calculating the horizontal-swing vibration impedance of the partial embedded pile foundation. The method can improve the calculation precision of the horizontal-swing impedance of the partial embedded pile foundation in the pile foundation engineering.

Description

Horizontal-swing vibration impedance determination method for partially-embedded pile foundation
Technical Field
The invention belongs to the technical field of civil engineering, and relates to a method for determining horizontal-swinging vibration impedance of a partially embedded pile foundation, in particular to a method for determining the horizontal-swinging vibration impedance of the partially embedded pile foundation after calculating the circular weakening effect of layered saturated soil around a pile under the long-term circulation action of wind waves on the embedded pile foundation in coastal areas.
Background
With the development of bridge engineering and ocean engineering, the design of pile foundations is no longer limited to the completely embedded type, and as shown in fig. 1, the pile foundations partially embedded type is widely applied to structures such as deep-water bridges, wind power engineering, oil drilling platforms and the like. Under the excitation of external load, dynamic interaction exists between the saturated soil and the pile body, and the dynamic characteristic of the upper structure is further changed. It is therefore necessary to take into account the effect of soil-pile dynamic interaction on the dynamic characteristics of the superstructure when the superstructure is a power sensitive structure that is high-rise and has concentrated top mass. In the research of the dynamic interaction problem mainly in the horizontal direction such as earthquake, wind, sea wave and the like, the determination of the horizontal-swing vibration impedance for describing the relationship between the vibration displacement of the pile top of the partially embedded pile foundation and the external excitation load is an important link.
The horizontal-swing vibration impedance of the partially embedded pile foundation is a complex variable function matrix depending on the external load frequency, wherein the real part represents a stiffness coefficient, and the imaginary part represents a damping coefficient. In the prior art, the dynamic analysis of the pile foundation is mainly popularized from the theory of bearing the static load of the foundation and is corrected by adopting an empirical coefficient, and the theory is not strict enough. Therefore, scientific and technical workers carry out research to obtain certain results, and commonly used theories include a Timoshenko beam and a Biot theory, wherein the Timoshenko beam can be used for considering the shear deformation and the moment of inertia in the transverse vibration process of the pile body and accurately describing the relation between the vibration displacement of the pile body and the external load. The Biot theory is a wave propagation theory of a fluid saturated porous medium, and can accurately describe the displacement of a solid framework and a fluid in a soil body and the stress-strain relationship of the solid framework and the fluid.
At present, the horizontal-swing vibration impedance of a pile foundation mainly comprises two theoretical determination methods, one is a numerical method represented by a finite element model, but the elastic half-space foundation brings huge calculation amount to the problem of pile-soil interaction and is not easy to be used for the problem of actual engineering; the other is an analytic or semi-analytic method represented by an elastic foundation beam model, and the method is simple and easy to calculate and widely applied to pile-soil interaction problems. However, the problem of pile-soil interaction is simplified layer by layer, and the existing method cannot accurately consider the weakening effect of soil around the pile, the permeability characteristic of offshore saturated soil, the partial embedment effect of the pile and the shearing effect of the pile due to the long-term circulation of horizontal loads such as wind waves and the like when calculating the horizontal-swinging vibration impedance of the pile.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, provides a horizontal-swing vibration impedance determination method for a partially-embedded pile foundation, can reasonably reflect the mechanical characteristics of foundation soil and a vertical pile in a vibration process, and improves the calculation accuracy of the horizontal-swing impedance of the partially-embedded pile foundation in pile foundation engineering.
In order to achieve the purpose, the invention is realized by adopting the following technical scheme:
the invention provides a method for determining horizontal-swing vibration impedance of a partially embedded pile foundation, which comprises the following steps:
1. a method for determining horizontal-sway vibration impedance of a partially-embedded pile foundation is characterized by comprising the following steps:
step a: dispersing pile bodies and pile periphery laminar saturated soil along a z axis according to the horizontal layering condition of the pile periphery laminar saturated soil, and circularly dispersing the saturated soil in a weakening area along the radial direction of a pile foundation according to the weakening condition of each layer of pile periphery soil;
step b: establishing a wave equation of the saturated soil of each circle layer after dispersion according to a Biot theory, solving to obtain a matrix of a displacement field and a stress field of the saturated soil of each circle layer, and calculating unit horizontal dynamic reaction on a pile-soil contact surface based on continuous conditions of displacement and stress of each circle layer;
step c: based on the unit horizontal dynamic reaction force on the pile-soil contact surface, solving a pre-established free transverse vibration equation of the pile body of the non-embedded section and a forced transverse vibration equation of the pile body of the embedded section under the action of the horizontal dynamic reaction force of saturated soil to obtain the force, deformation continuous conditions and pile bottom boundary conditions of each pile section;
step d: and calculating the horizontal-swing vibration impedance of the partial embedded pile foundation by utilizing the force, the deformation continuous condition and the pile bottom boundary condition of each pile section.
Further, the radius of the part of the embedded pile foundation is defined as r0Length L, length L of pile body not buried in saturated soil1Partially embedded in a layered saturated viscoelastic half-space containing a weakened zone, the top of which is subjected to horizontal and rocking harmonic loads of
Figure BDA0002622617980000031
And
Figure BDA0002622617980000032
the layered saturated soil around the discrete pile body and the discrete pile body along the z axis is as follows: the pile soil system is divided into M layers from top to bottom, wherein the 1 st layer is the pile body length h of the un-embedded saturated soil1The layer height of the ith layer (i ═ 2, …, M) is the soil layer thickness hi；
The saturated soil in the ring-shaped discrete weakened area along the radial direction of the pile foundation is as follows: radially dispersing soil around each layer of pile into an inner ring weakened area with the width of delta r and an outer ring non-weakened area, radially dispersing the inner ring weakened area from inside to outside into N sub-ring layers, wherein the non-weakened area is marked with the number of N +1, and the outer radius r of the jth ring in each layerjExpression is rj＝r0+j△r/N；
Wherein Q and M are respectively horizontal and swing excitation amplitude, omega is excitation frequency and imaginary number
Figure BDA0002622617980000033
Furthermore, the N of the sub-circle layer is more than or equal to 30, and the saturated soil in the sub-circle layer is homogeneous.
Further, in the step b, a fluctuation equation of the viscoelastic saturated soil infinitesimal is established according to the Biot theory as follows:
Figure BDA0002622617980000034
Figure BDA0002622617980000035
Figure BDA0002622617980000036
wherein, Laplacian operator
Figure BDA0002622617980000041
Points on the variables represent the derivation operation over time, u and w are displacement vectors of the solid phase and the liquid phase of the saturated soil relative to the external excitation frequency omega, the soil skeleton, pfAlpha and M are Biot parameters related to the compression of saturated pore media; the Lame constant of saturated soil is respectively
Figure BDA0002622617980000042
μ ═ G (1+ i β) and λc＝λ+α2M, wherein vsG and beta are respectively the poisson ratio of saturated soil, the shear wave velocity and the viscosity coefficient; rho, rhosAnd ρfThe total density of soil, the density of solid phase and the density of liquid phase respectively satisfy rho ═ n rhof+ρs(1-n), wherein n is the porosity of the saturated soil, and the parameter m is rho for the coupling action of the solid phase and the liquid phase in the saturated soilfN and bc＝ρfg/kDIs represented by (a) wherein kDIs the Darcy permeability coefficient, g is the acceleration of gravity;
introducing a potential function, solving a fluctuation equation of the viscoelasticity saturated soil infinitesimal,
Figure BDA0002622617980000043
substituting the formula (2) into the formulas (1a), (1b) and (1c) to obtain the following two matrix equations
Figure BDA0002622617980000044
Figure BDA0002622617980000045
Wherein phi is1And phi2Displacement scalars of a solid phase and a liquid phase of saturated soil are respectively,
Figure BDA0002622617980000046
and
Figure BDA0002622617980000047
respectively displacement vectors of the solid phase and the liquid phase,
phi is obtained by operator decomposition theory and separation variable methodiAnd
Figure BDA0002622617980000048
substituting the solution into the formula (2) to obtain the solid phase radial displacement urSolid phase annular displacement uθRadial displacement w of the liquid phaserAnd liquid phase circumferential displacement wθGeneral solution of
Figure BDA0002622617980000049
Figure BDA00026226179800000410
Figure BDA00026226179800000411
Figure BDA0002622617980000051
Wherein A iss、Bs、Cs、Ds、Es、FsFor the undetermined coefficients of the general solution,
Figure BDA0002622617980000052
Figure BDA0002622617980000053
in the formula, K1() And I1() A first-order imaginary-quantity Bessel function of the first kind and a second kind, respectively, and the symbol operation in the formula]' denotes a first-order derivation operation on radial coordinates r;
substituting the solid phase and liquid phase displacement of the saturated soil in the formula (7) into the differential relation between the stress field and the displacement field of the saturated soil
Figure BDA0002622617980000054
And after sorting, obtaining a general solution of a j-th circle layer displacement field and a stress field, and writing the general solution into a matrix form as follows:
{Sj(r,θ)}＝[tj(r,θ)]{Xj} (8)
wherein the content of the first and second substances,
Figure BDA0002622617980000055
{Xj}＝{Asj Bsj Csj Dsj Esj Fsj}T。
further, let the determinant of the differential operator of equations (3) and (4) be zero, equation (3) is:
Figure BDA00026226179800000511
in the formula
Figure BDA0002622617980000056
Figure BDA0002622617980000057
Figure BDA0002622617980000058
Similarly, formula (4) is represented by
Figure BDA0002622617980000059
Wherein the content of the first and second substances,
Figure BDA00026226179800000510
further, the step b of calculating the unit horizontal dynamic reaction force on the pile-soil contact surface by using the displacement and stress continuous conditions of each circle layer comprises the following steps:
for the jth circle (j is 1,2, …, N) in the saturated soil in the weakening area, the general solution of the displacement field and the stress field of the saturated soil unit in the formula (8) is as follows:
{Sj(rj-1,θ)}＝[tj(rj-1,θ)]{Xj}，{Sjrj,θ)}＝[tj(rj,θ)]{Xj} (9)
for the j +1 th circle in the saturated soil of the weakening area, the following are included:
{Sj+1(rj,θ)}＝[tj+1(rj,θ)]{Xj+1}，{Sj+1(rj+1,θ)}＝[tj+1(rj+1,θ)]{Xj+1} (10)
the continuous displacement and stress conditions of the layers, namely the continuous displacement and stress conditions of the interface, are as follows: { Sj(rj,θ)}＝{Sj+1(rjθ), then obtained by the transfer matrix method:
{Xj}＝[tj(rj)]-1[tj+1(rj)]{Xj+1} (j＝1,2,…,N) (11)
the saturated soil in a non-weakening area (namely the N +1 th circle) needs to meet the requirement
Figure BDA0002622617980000061
Therefore, the undetermined coefficient A in the general solution of equation (8)s＝Cs＝EsWhen the displacement field and the stress field of the saturated soil in the non-weakening area in the formula (8) are 0, the general solution is as follows:
{SN+1(r,θ)}＝[tN+1(r,θ)]{XN+1} (12)
wherein, { XN+1}＝{Bs N+1Cs N+1Es N+1}T，
Condition ({ S) continuity by interface of weakened and non-weakened zonesN(rN,θ)}＝{SN+1(rNθ) }) and the transfer matrix method:
{XN}＝[tN(rN)]-1[tN+1(rN)]{XN+1} (13)
establishing the relation between the coefficients to be determined in the 1 st circle layer and the N +1 th circle layer by the recursion relation of the formula (11) and the formula (13) as follows:
Figure BDA0002622617980000062
wherein, Tj(rj)＝[tj(rj)]-1[tj+1(rj)]，
According to the boundary condition of soil around the pile, when the pile body section generates unit horizontal displacement and the contact surface of the pile body is impermeable, the undetermined coefficient in the 1 st circle of layer displacement field and stress field general solution is determined by the formula (14) as follows:
Figure BDA0002622617980000071
in the formula (I), the compound is shown in the specification,
Figure BDA0002622617980000072
is [ t ]1(r0)]The first 3 rows of elements in the matrix,
substituting undetermined coefficients in the general solution of the 1 st circle of layer into a formula (8) to obtain a stress field sigma in the 1 st circle of layerr(r, theta) and tauθ(r, theta) solution, r-r for the soil around the pile0The unit horizontal moving counter force on the pile-soil contact surface is obtained by solving the integral of the stress
Figure BDA0002622617980000073
7. The method for determining horizontal-sway vibration resistance of a partially embedded pile foundation of claim 1, wherein said steps c and d comprise the steps of:
according to the Timoshenko beam theory, the transverse vibration displacement ui (z, t) of the i-th section of pile body meets the following vibration control equation:
Figure BDA0002622617980000074
in the formula, Ap，ρp，Ep，GpAnd IpRespectively the sectional area and the density of the pile bodyElastic modulus, shear model and section moment of inertia, wherein
Figure BDA0002622617980000075
The unit horizontal dynamic reaction force on the contact surface of the i-th layer pile soil is calculated in step S2, and the unit horizontal dynamic reaction force is applied to the pile section not embedded in the soil body
Figure BDA0002622617980000076
When the pile foundation is under the action of simple harmonic load, the transverse vibration displacement of the pile body meets the requirement
Figure BDA0002622617980000077
Of form, the general solution of equation (17) is:
Figure BDA0002622617980000078
in the formula (I), the compound is shown in the specification,
Figure BDA0002622617980000081
Figure BDA0002622617980000082
Wp＝κGpAp，Jp＝ρpIpω2，
Figure BDA0002622617980000083
Ai、Bi、Ci、Diundetermined coefficients of the displacement general solution of the ith section of pile body;
the middle substituted pile body section corner of the formula (18)
Figure BDA0002622617980000084
Bending moment of cross section
Figure BDA0002622617980000085
And section shear force
Figure BDA0002622617980000086
The relationship among the deformation, the internal force and the undetermined coefficient of the i-th section of pile body unit is obtained after the differential relationship is arranged
Figure BDA0002622617980000087
And establishing the relationship between the displacement of the upper surface and the lower surface of the i-th layer of pile section and the axial force according to the formula and the differential relationship between the force and the deformation when the pile body vibrates transversely as follows:
Figure BDA0002622617980000088
according to the continuity condition of the adjacent pile body interface (U)i(hi),Θi(hi),Qi(hi),Mi(hi)}＝{Ui+1(0),Θi+1(0),Qi+1(0),Mi+1(0) And (3) establishing the relationship between the displacement and the force between the pile top and the pile bottom of the partially embedded pile by a vertical transfer matrix method:
Figure BDA0002622617980000089
wherein, the pile body transmits the matrix
Figure BDA0002622617980000091
Transferring pile body to matrix [ Tp]Decomposed into 4 sub-matrices of 2 x 2, i.e.
Figure BDA0002622617980000092
Then the formula (22) is decomposed into
Figure BDA0002622617980000093
Figure BDA0002622617980000094
For the end bearing pile with the foundation rock at the pile bottom, the pile bottom is a fixed end, namely U (L) is 0 and theta (L) is 0, and the relational expression of the jacking force and the displacement of the partially embedded pile foundation pile is obtained from the expression (23)
Figure BDA0002622617980000095
Therefore, the horizontal-swing vibration impedance of the pile foundation is
Figure BDA00026226179800000912
Figure BDA0002622617980000096
Is a 2 x 2 matrix, i.e.
Figure BDA0002622617980000097
Element(s)
Figure BDA0002622617980000098
The impedance of the horizontal vibration is represented,
Figure BDA0002622617980000099
the resistance to the vibration of the rocking motion is represented,
Figure BDA00026226179800000910
and
Figure BDA00026226179800000911
equality represents the horizontal-to-sway coupling impedance, the elements in the matrix are complex, the real part represents the stiffness coefficient and the imaginary part represents the damping coefficient.
Compared with the prior art, the invention has the following beneficial effects:
the invention provides a method for solving horizontal-swing vibration impedance of a part of embedded pile foundations in saturated soil containing a weakening area.
Drawings
FIG. 1 is a schematic diagram of lateral vibration of a partially submerged pile foundation in saturated soil having weakened zones;
FIG. 2 is a schematic diagram of radial discrete division of the i-th layer of saturated soil containing weakened areas;
FIG. 3 is a graph of horizontal translational stiffness of a partially embedded pile foundation as a function of external excitation frequency;
FIG. 4 is a graph of horizontal translational damping of a partially embedded pile foundation as a function of external excitation frequency;
FIG. 5 is a graph of partial buried pile foundation sway stiffness as a function of external excitation frequency;
FIG. 6 is a graph of partial buried pile foundation sway damping as a function of external excitation frequency;
FIG. 7 is a graph of horizontal-rocking dynamic stiffness of a partially embedded pile foundation as a function of external excitation frequency;
fig. 8 is a graph of the variation of horizontal-swing coupled vibration impedance of a partially buried pile foundation with the variation of external excitation frequency.
Detailed Description
The invention is further described below with reference to the accompanying drawings. The following examples are only for illustrating the technical solutions of the present invention more clearly, and the protection scope of the present invention is not limited thereby.
In this embodiment, the pile foundation is partially embedded in viscoelastic foundation soil composed of two layers of saturated soil. The parameters of the partially embedded pile foundation are as follows: the length L of the pile is 20m, wherein the length of the part which is not embedded into the soil body is L14m, pile body radius r00.5m, density ρpIs 2500kg/m3The elastic modulus E of pile body is 2.5X 1010Pa, pile body shear modulus G1.042X 1010Pa. The parameters of the saturated soil are as follows: poisson's ratio v of saturated soils0.3, viscoelasticity coefficient beta 0.01, and solid density rho of soil masss＝2700kg/m3Density of liquid phase of soil body rhof＝1000kg/m3Porosity n is 0.3 and permeability coefficient kD＝1×10-6M/s, two Biot parameters α ═ 0.99 and M ═ 4.5 × 109. The shear modulus of the upper saturated soil in a non-weakened region is G1＝1.8×108Pa, shear modulus of non-weakened region of lower saturated soil is G2＝5×108Pa. Affected by the long-term cyclic action of horizontal environment load, the soil around the pile has a weakening area with the same width as the radius of the pile body, namely, delta r is r0. In this embodiment, the ratio of the soil shear modulus of the outer side end to the inner side end of the weakened region is 0.25, and the soil shear modulus in the weakened region changes linearly.
The first embodiment is as follows:
considering that the pile foundation is partially embedded in a viscoelastic foundation consisting of upper and lower layers of saturated soil, the partially embedded pile foundation divides a pile soil system into 3 layers from top to bottom: the 1 st layer is a part which is not embedded with saturated soil and has the height of 4 m; the 2 nd layer is a part embedded into the upper saturated soil and has the height of 6 m; the 3 rd layer is a part embedded in the saturated soil of the lower layer and has the height of 10 m. In order to process radially inhomogeneous weakened area pile-surrounding soil, upper and lower layers of pile-surrounding soil are annularly dispersed into an inner ring weakened area and an outer ring non-weakened area with the width of 0.5m, and the inner ring weakened area is further radially dispersed into a plurality of sub-ring layers, in the present embodiment, N is 30 as an example, j is 1,2, …, 30 from inside to outside, and j is 31 as a non-weakened area. Therefore, the outer radius r of the jth circle of each layer of saturated soiljExpression is rj＝0.5+0.5j/30。
Example two:
and establishing a saturated soil wave equation of each circle layer described by a Biot theory, solving the equation, and pushing the unit horizontal dynamic reaction on the pile soil contact surface by utilizing the continuous conditions of displacement and stress of each circle layer on the basis of the equation. According to the formula (16) in the second embodiment,
Figure BDA0002622617980000111
in the formula, σr(r0Theta) and tauθ(r0And theta) are respectively the positive stress and the shear stress on the soil contact surface around each layer of pile. Substituting the material parameters of the soil mass in the 1 st layer and the 2 nd layer into the formulas (1) to (15) in the invention content, respectively calculating the stress of the soil around each layer of pile, and further integrating to obtain the unit horizontal dynamic reaction force on the soil contact surface of the 1 st layer and the 2 nd layer of pile respectively
Figure BDA0002622617980000112
And
Figure BDA0002622617980000113
the dynamic reaction is complex and its real part is used separately
Figure BDA0002622617980000114
And
Figure BDA0002622617980000115
expressed, imaginary parts are respectively
Figure BDA0002622617980000116
And
Figure BDA0002622617980000117
and (4) showing. The unit horizontal dynamic reaction force on the contact surface of the pile soil of the 1 st layer and the 2 nd layer under different external load excitation frequencies omega can be calculated according to the parameters of the foundation soil and the vertical pile in the embodiment as shown in the table 1. In Table 1, the frequency ω of the external load is dimensionless0＝ωr0/VpWherein
Figure BDA0002622617980000118
The shear wave velocity of the pile foundation.
TABLE 1 unit horizontal dynamic reaction force on each layer of pile-soil contact surface under different excitation frequencies
Figure BDA0002622617980000119
Figure BDA0002622617980000121
Example three:
after the unit horizontal dynamic reaction force on the pile-soil contact surface of the layer 1 and the layer 2 is obtained by calculation in the second embodiment, the transverse vibration control equation of the un-embedded section and the embedded two sections of pile bodies can be respectively established according to the formula (17) in the invention content. According to the derivation of the formula (18) to the formula (26) in the summary of the invention, the horizontal-sway vibration resistance of the partially buried pile foundation can be finally obtained as follows:
Figure BDA0002622617980000122
in the formula, [ Tp ]11]And [ Tp12]Are all transfer matrices [ Tp]The submatrix in (1) can be calculated by using the formula (18) to the formula (22) in the summary of the invention. Horizontal-rocking vibration resistance
Figure BDA0002622617980000123
Is a 2 x 2 matrix, i.e.
Figure BDA0002622617980000124
Element(s)
Figure BDA0002622617980000125
The impedance of the horizontal vibration is represented,
Figure BDA0002622617980000126
the resistance to the vibration of the rocking motion is represented,
Figure BDA0002622617980000127
and
Figure BDA0002622617980000128
equality represents the horizontal-swing coupling impedance. The impedance elements are complex numbers, the real part of the impedance elements represents dynamic stiffness, and the imaginary part represents dynamic damping. Fig. 3-8 show the horizontal impedance, the sway impedance and the horizontal-sway coupling impedance of the partially buried pile foundation according to the embodiment of the invention.
The invention aims to solve the problem of dynamic interaction between weakened saturated soil and a part of embedded pile foundation in coastal areas, and provides a horizontal-swinging vibration impedance determination method which is high in efficiency, convenient to apply and capable of reasonably reflecting mechanical characteristics of foundation soil and a vertical pile in a vibration process for engineering designers and scientific researchers. Therefore, the problems of weakening effect, soil body internal seepage characteristic, pile body part embedding effect, pile body shearing effect and the like caused by long-term action of loads such as wind waves and the like on the layered soil saturated soil cannot be considered in the conventional horizontal-swing impedance determination method, and the calculation accuracy of the horizontal-swing impedance of the part embedded pile foundation in the pile foundation engineering in the coastal region is improved.
The above description is only a preferred embodiment of the present invention, and it should be noted that, for those skilled in the art, several modifications and variations can be made without departing from the technical principle of the present invention, and these modifications and variations should also be regarded as the protection scope of the present invention.

Claims (7)

1. A method for determining horizontal-sway vibration impedance of a partially-embedded pile foundation is characterized by comprising the following steps:
step a: dispersing pile bodies and pile periphery laminar saturated soil along a z axis according to the horizontal layering condition of the pile periphery laminar saturated soil, and circularly dispersing the saturated soil in a weakening area along the radial direction of a pile foundation according to the weakening condition of each layer of pile periphery soil;
step b: establishing a wave equation of the saturated soil of each circle layer after dispersion according to a Biot theory, solving to obtain a matrix of a displacement field and a stress field of the saturated soil of each circle layer, and calculating unit horizontal dynamic reaction on a pile-soil contact surface based on continuous conditions of displacement and stress of each circle layer;
step c: based on the unit horizontal dynamic reaction force on the pile-soil contact surface, solving a pre-established free transverse vibration equation of the pile body of the non-embedded section and a forced transverse vibration equation of the pile body of the embedded section under the action of the horizontal dynamic reaction force of saturated soil to obtain the force, deformation continuous conditions and pile bottom boundary conditions of each pile section;
step d: and calculating the horizontal-swing vibration impedance of the partial embedded pile foundation by utilizing the force, the deformation continuous condition and the pile bottom boundary condition of each pile section.
2. The method of claim 1, wherein the radius of the base of the partial embedded pile is defined as r0Length L, length L of pile body not buried in saturated soil1Partially embedded in a layered saturated viscoelastic half-space containing a weakened domain, and the top part of the layered saturated viscoelastic half-space is subjected to horizontal and swinging simple harmonic loads QeiωtAnd Meiωt；
The layered saturated soil around the discrete pile body and the discrete pile body along the z axis is as follows: the pile soil system is divided into M layers from top to bottom, wherein the 1 st layer is the pile body length h of the un-embedded saturated soil1The layer height of the ith layer (i ═ 2, …, M) is the soil layer thickness hi；
The saturated soil in the ring-shaped discrete weakened area along the radial direction of the pile foundation is as follows: radially dispersing soil around each layer of pile into an inner ring weakened area with the width of delta r and an outer ring non-weakened area, radially dispersing the inner ring weakened area from inside to outside into N sub-ring layers, wherein the non-weakened area is marked with the number of N +1, and the outer radius r of the jth ring in each layerjExpression is rj＝r0+j△r/N；
Wherein Q and M are respectively horizontal and swing excitation amplitude, omega is excitation frequency and imaginary number
Figure FDA0003147606320000021
3. The method for determining horizontal-rocking vibration resistance of a partially embedded pile foundation as claimed in claim 2, wherein the sub-layer N is greater than or equal to 30.
4. The method for determining horizontal-sway vibration impedance of a partially-embedded pile foundation of claim 1, wherein in step b, the wave equation of viscoelastic saturated soil infinitesimal is established according to Biot theory as follows:
Figure FDA0003147606320000022
Figure FDA0003147606320000023
Figure FDA0003147606320000024
wherein, Laplacian operator
Figure FDA0003147606320000025
The points on the variables represent the derivation operations over time, u and w are the displacement vectors of the solid and liquid phases of the saturated soil relative to the external excitation frequency omega, pfAlpha and M are Biot parameters related to the compression of saturated pore media; the Lame constant of saturated soil is respectively
Figure FDA0003147606320000026
μ ═ G (1+ i β) and λc＝λ+α2M, wherein vsG and beta are respectively the poisson ratio of saturated soil, the shear wave velocity and the viscosity coefficient; rho, rhosAnd ρfThe total density of soil, the density of solid phase and the density of liquid phase respectively satisfy rho ═ n rhof+ρs(1-n), wherein n is the porosity of the saturated soil, and the parameter m is rho for the coupling action of the solid phase and the liquid phase in the saturated soilfN and bc＝ρfg/kDIs represented by (a) wherein kDIs the Darcy permeability coefficient, g is the acceleration of gravity;
introducing a potential function, solving a fluctuation equation of the viscoelasticity saturated soil infinitesimal,
Figure FDA0003147606320000027
substituting the formula (2) into the formulas (1a), (1b) and (1c) to obtain the following two matrix equations
Figure FDA0003147606320000028
Figure FDA0003147606320000031
Wherein phi is1And phi2Displacement scalars of a solid phase and a liquid phase of saturated soil are respectively,
Figure FDA0003147606320000039
and
Figure FDA00031476063200000310
respectively displacement vectors of the solid phase and the liquid phase,
phi is obtained by operator decomposition theory and separation variable methodiAnd
Figure FDA00031476063200000311
substituting the solution into the formula (2) to obtain the solid phase radial displacement urSolid phase annular displacement uθRadial displacement w of the liquid phaserAnd liquid phase circumferential displacement wθGeneral solution of
Figure FDA0003147606320000032
Figure FDA0003147606320000033
Figure FDA0003147606320000034
Figure FDA0003147606320000035
Wherein A iss、Bs、Cs、Ds、Es、FsFor the undetermined coefficients of the general solution,
Figure FDA0003147606320000036
Figure FDA0003147606320000037
in the formula, K1() And I1() A first-order imaginary-quantity Bessel function of the first kind and a second kind, respectively, and the symbol operation in the formula]' denotes a first-order derivation operation on radial coordinates r;
substituting the solid phase and liquid phase displacement of the saturated soil in the formula (7) into the differential relation between the stress field and the displacement field of the saturated soil
Figure FDA0003147606320000038
And after sorting, obtaining a general solution of a j-th circle layer displacement field and a stress field, and writing the general solution into a matrix form as follows:
{Sj(r,θ)}＝[tj(r,θ)]{Xj} (8)
wherein, { Sj(r,θ)}＝{urj(r,θ) uθj(r,θ) wrj(r,θ) wθj(r,θ) σrj(r,θ) τθj(r,θ)}T，
{Xj}＝{Asj Bsj Csj Dsj Esj Fsj}T。
5. The method for determining horizontal-sway vibration impedance of a partially embedded pile foundation of claim 4, wherein the differential operator determinant of equations (3) and (4) is zero, and equation (3) is:
Figure FDA0003147606320000041
in the formula
Figure FDA0003147606320000042
Figure FDA0003147606320000043
Figure FDA0003147606320000044
Similarly, formula (4) is represented by
Figure FDA0003147606320000045
Wherein the content of the first and second substances,
Figure FDA0003147606320000046
6. the method of claim 4, wherein the step b of calculating the unit horizontal dynamic reaction force on the pile-soil contact surface by using the circle layer displacement and stress continuity conditions comprises:
for the jth circle (j is 1,2, …, N) in the saturated soil in the weakening area, the general solution of the displacement field and the stress field of the saturated soil unit in the formula (8) is as follows:
{Sj(rj-1,θ)}＝[tj(rj-1,θ)]{Xj}，{Sjrj,θ)}＝[tj(rj,θ)]{Xj} (9)
for the j +1 th circle in the saturated soil of the weakening area, the following are included:
{Sj+1(rj,θ)}＝[tj+1(rj,θ)]{Xj+1}，{Sj+1(rj+1,θ)}＝[tj+1(rj+1,θ)]{Xj+1} (10)
the continuous displacement and stress conditions of the layers, namely the continuous displacement and stress conditions of the interface, are as follows:
{Sj(rj,θ)}＝{Sj+1(rjθ), then obtained by the transfer matrix method:
{Xj}＝[tj(rj)]-1[tj+1(rj)]{Xj+1} (j＝1,2,…,N) (11)
the saturated soil in a non-weakening area (namely the N +1 th circle) needs to meet the requirement
Figure FDA0003147606320000047
Therefore, the undetermined coefficient A in the general solution of equation (8)s＝Cs＝EsWhen the displacement field and the stress field of the saturated soil in the non-weakening area in the formula (8) are 0, the general solution is as follows:
{SN+1(r,θ)}＝[tN+1(r,θ)]{XN+1} (12)
wherein, { XN+1}＝{BsN+1 CsN+1 EsN+1}T，
Condition ({ S) continuity by interface of weakened and non-weakened zonesN(rN,θ)}＝{SN+1(rNθ) }) and the transfer matrix method:
{XN}＝[tN(rN)]-1[tN+1(rN)]{XN+1} (13)
establishing the relation between the coefficients to be determined in the 1 st circle layer and the N +1 th circle layer by the recursion relation of the formula (11) and the formula (13) as follows:
Figure FDA0003147606320000051
wherein, Tj(rj)＝[tj(rj)]-1[tj+1(rj)]，
According to the boundary condition of soil around the pile, when the pile body section generates unit horizontal displacement and the contact surface of the pile body is impermeable, the undetermined coefficient in the 1 st circle of layer displacement field and stress field general solution is determined by the formula (14) as follows:
Figure FDA0003147606320000052
in the formula (I), the compound is shown in the specification,
Figure FDA0003147606320000054
is [ t ]1(r0)]The first 3 rows of elements in the matrix,
substituting undetermined coefficients in the general solution of the 1 st circle of layer into a formula (8) to obtain a stress field sigma in the 1 st circle of layerr(r, theta) and tauθ(r, theta) solution, r-r for the soil around the pile0The unit horizontal moving counter force on the pile-soil contact surface is obtained by solving the integral of the stress
Figure FDA0003147606320000053
7. The method for determining horizontal-sway vibration resistance of a partially embedded pile foundation of claim 1, wherein said steps c and d comprise the steps of:
according to the Timoshenko beam theory, the i-th section of pile body transversely vibrates and displaces ui(z, t) satisfies the following vibration control equation:
Figure FDA0003147606320000061
in the formula, Ap，ρp，Ep，GpAnd IpRespectively the sectional area, density, elastic modulus, shear model and section moment of inertia of the pile body, wherein
Figure FDA0003147606320000062
The unit level on the i-th layer pile-soil contact surface is calculated in step S2Dynamic reaction force, unit horizontal dynamic reaction force for pile section not embedded in soil body
Figure FDA0003147606320000063
When the pile foundation is under the action of simple harmonic load, the transverse vibration displacement of the pile body meets ui(z,t)＝Ui(z)eiωtOf form, the general solution of equation (17) is:
Figure FDA0003147606320000064
in the formula (I), the compound is shown in the specification,
Figure FDA0003147606320000065
Wp＝κGpAp，Jp＝ρpIpω2，
Figure FDA0003147606320000066
Ai、Bi、Ci、Diundetermined coefficients of the displacement general solution of the ith section of pile body;
the middle substituted pile body section corner of the formula (18)
Figure FDA0003147606320000067
Bending moment of cross section
Figure FDA0003147606320000068
And section shear force
Figure FDA0003147606320000069
The relationship among the deformation, the internal force and the undetermined coefficient of the i-th section of pile body unit is obtained after the differential relationship is arranged
Figure FDA0003147606320000071
And establishing the relationship between the displacement of the upper surface and the lower surface of the i-th layer of pile section and the axial force according to the formula and the differential relationship between the force and the deformation when the pile body vibrates transversely as follows:
Figure FDA0003147606320000072
according to the continuity condition of the adjacent pile body interface (U)i(hi),Θi(hi),Qi(hi),Mi(hi)}＝{Ui+1(0),Θi+1(0),Qi+1(0),Mi+1(0) And (3) establishing the relationship between the displacement and the force between the pile top and the pile bottom of the partially embedded pile by a vertical transfer matrix method:
Figure FDA0003147606320000073
wherein, the pile body transmits the matrix
Figure FDA0003147606320000074
Transferring pile body to matrix [ Tp]Decomposed into 4 sub-matrices of 2 x 2, i.e.
Figure FDA0003147606320000075
Then the formula (22) is decomposed into
Figure FDA0003147606320000076
Figure FDA0003147606320000077
For the end bearing pile with the foundation rock at the pile bottom, the pile bottom is a fixed end, namely U (L) is 0 and theta (L) is 0, and the relational expression of the jacking force and the displacement of the partially embedded pile foundation pile is obtained from the expression (23)
Figure FDA0003147606320000081
Therefore, the horizontal-swing vibration impedance of the pile foundation is
Figure FDA0003147606320000082
Figure FDA0003147606320000083
Is a 2 x 2 matrix, i.e.
Figure FDA0003147606320000084
Element(s)
Figure FDA0003147606320000085
The impedance of the horizontal vibration is represented,
Figure FDA0003147606320000086
the resistance to the vibration of the rocking motion is represented,
Figure FDA0003147606320000087
and
Figure FDA0003147606320000088
equality represents the horizontal-to-sway coupling impedance, the elements in the matrix are complex, the real part represents the stiffness coefficient and the imaginary part represents the damping coefficient.
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Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR100653433B1 (en) * 2005-12-21 2006-12-04 연세대학교 산학협력단 Finite difference method in beam analysis
CN109214123A (en) * 2018-10-18 2019-01-15 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 model
CN109344526A (en) * 2018-10-18 2019-02-15 大连海事大学 A kind of saturation laminated soil pile foundation extensional vibration research method based on loosened soil stake model
CN110598262A (en) * 2019-08-16 2019-12-20 河海大学 Calculation method of vertical impedance of vertical pile
JP2020013494A (en) * 2018-07-20 2020-01-23 ジャパンホームシールド株式会社 Foundation work design system, and optimization method of pile arrangement
CN111177933A (en) * 2019-12-31 2020-05-19 大连海事大学 Longitudinal vibration analysis system and method for large-diameter pipe pile in axial-symmetry radial heterogeneous soil

Patent Citations (6)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
KR100653433B1 (en) * 2005-12-21 2006-12-04 연세대학교 산학협력단 Finite difference method in beam analysis
JP2020013494A (en) * 2018-07-20 2020-01-23 ジャパンホームシールド株式会社 Foundation work design system, and optimization method of pile arrangement
CN109214123A (en) * 2018-10-18 2019-01-15 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 model
CN109344526A (en) * 2018-10-18 2019-02-15 大连海事大学 A kind of saturation laminated soil pile foundation extensional vibration research method based on loosened soil stake model
CN110598262A (en) * 2019-08-16 2019-12-20 河海大学 Calculation method of vertical impedance of vertical pile
CN111177933A (en) * 2019-12-31 2020-05-19 大连海事大学 Longitudinal vibration analysis system and method for large-diameter pipe pile in axial-symmetry radial heterogeneous soil

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