CN113960170A - Method for determining motion response of tubular pile in saturated soil under action of earthquake P wave - Google Patents

Method for determining motion response of tubular pile in saturated soil under action of earthquake P wave Download PDF

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CN113960170A
CN113960170A CN202111239639.1A CN202111239639A CN113960170A CN 113960170 A CN113960170 A CN 113960170A CN 202111239639 A CN202111239639 A CN 202111239639A CN 113960170 A CN113960170 A CN 113960170A
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tubular pile
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张石平
徐站
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Changsha University of Science and Technology
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    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
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    • G01N29/4418—Processing the detected response signal, e.g. electronic circuits specially adapted therefor by comparison with a model, e.g. best-fit, regression analysis
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Abstract

The invention discloses a method for determining the motion response of a tubular pile in saturated soil under the action of earthquake P waves, which comprises the steps of firstly determining the amplitude and the excitation circle frequency of the movement of a bedrock under the action of the earthquake P waves; then determining the outer radius of the tubular pile, the inner radius of the tubular pile, the length of the tubular pile, the elastic modulus of the tubular pile and the Poisson ratio of the tubular pile, and calculating the shear modulus of the tubular pile, the cross-sectional area of the tubular pile and the polar inertia moment of the tubular pile; regarding soil around the tubular pile as a saturated three-dimensional porous elastic continuous body filled with fluid, and determining material parameters such as elastic modulus, Poisson ratio, density and the like of a soil framework and pore fluid of an external domain or an internal domain of the tubular pile; then, establishing a motion equation of the two-dimensional tubular pile considering radial and vertical deformation under steady-state vibration, and solving to obtain a characteristic value and a corresponding characteristic vector of the two-dimensional tubular pile; and finally, determining the motion response of the tubular pile in the saturated soil under the action of the earthquake P wave by adopting the parameters through an analytic method.

Description

Method for determining motion response of tubular pile in saturated soil under action of earthquake P wave
Technical Field
The invention belongs to the field of civil engineering, and relates to a method for determining motion response of a tubular pile in saturated soil under the action of earthquake P waves.
Background
Pile foundations, as an effective deep foundation, include different types such as solid concrete piles, tubular piles or special-shaped piles, and are widely applied to heavy projects such as roads, buildings and bridges. Due to the serious damage of earthquake to the engineering structure, the research on the dynamic interaction between the pile foundation and the surrounding soil body under the action of earthquake is always a hotspot and difficult problem in the field of geotechnical and earthquake engineering. Generally, researchers have studied more about the pile-soil dynamic interaction under the action of a horizontal earthquake, and study less about the pile-soil system dynamic response under the action of a vertical earthquake, and the influence of a vertical earthquake component is usually ignored in engineering design. However, it has been demonstrated that in the near field, i.e., within 20km from the epicenter, the influence of the vertical components (P-waves and SV-waves) is more pronounced, above 20km, Rayleigh waves become the main influencing factor. In Northbridge and Kobe earthquakes, structural damage caused by vertical seismic action has been reported. Therefore, pile-soil interaction under vertical seismic action is also worth considering. At present, the research on the vertical seismic response of a pile-soil system is mainly focused on a single-phase soil condition and is not suitable for coastal engineering (saturated soil). And regarding the soil body around the pile as a saturated porous elastic medium, some scholars also analyze the vertical seismic response of the pile in the saturated soil. However, in these studies, the peripile soil is considered to be a half-space medium of infinite thickness, and cannot reveal the resonance phenomenon of the bedrock thin soil layer which is common in practice, so that the solution of the half-space medium is inaccurate in the case of the bedrock thin soil layer. In addition, the above researches all consider the pile foundation as a one-dimensional structure, and do not consider the influence of pile radial deformation and pile bottom soil reaction force on the pile-soil interaction effect. Although one-dimensional structure theory has been widely used in many engineering practices due to its mathematical simplicity and practical value, its application to the problem of structure-continuum interaction will create some fundamental drawbacks, particularly for structures with relatively small major diameters. For example, the radial and axial displacements of the pile foundation are generally determined by the radial and axial forces generated by the surrounding soil mass medium. However, since the one-dimensional rod theory can only describe longitudinal deformation caused by axial loads, it is not possible to reveal a coordinated lateral displacement and contact force effect between the two interacting media. Besides causing non-physical risks such as inaccurate calculation, the approximate processing method can also have serious influence on the correlation analysis of problems such as pile-soil radial contact stress distribution or the response of a pile-soil system to the Poisson effect.
Disclosure of Invention
The embodiment of the invention aims to provide a method for determining motion response of a tubular pile in saturated soil under the action of an earthquake P wave, which aims to solve the problem that in the interaction analysis of the existing pile-soil system under the action of a vertical earthquake, the pile soil around the pile is regarded as a half-space medium with infinite thickness and is inconsistent with the actual common pile-soil resonance phenomenon of a thin soil layer of a lower bedrock, so that the solution of the half-space medium is inaccurate under the condition of the thin soil layer of the lower bedrock; and in the interaction analysis of the existing pile-soil system under the action of vertical earthquake, the pile foundation is regarded as a one-dimensional structure, so that the method is not suitable for the condition of interaction of the structure and the continuum.
The technical scheme adopted by the embodiment of the invention is as follows: a method for determining the motion response of a tubular pile in saturated soil under the action of seismic P waves comprises the following steps:
determining the amplitude w of the movement of the bedrock under the action of seismic P-wavesgAnd obtaining the excitation circle frequency omega according to the excitation frequency f, wherein omega is 2 pi f;
determining the outer radius r of the tubular pileOInner radius r of tubular pileILength L of tubular pile and density rho of tubular pilepElastic modulus E of tubular pilepPoisson ratio upsilon of tubular pilepAnd calculating the shear modulus mu of the pipe pilepCross section area A of tubular pilepPolar inertia moment J of tubular pilep；
Determining the elastic modulus E of the soil framework of the external or internal area of the pipe pileskPoisson ratio upsilon of soil body framework of tubular pile outer domain or tubular pile inner domainskCalculating the shear modulus G of the soil body framework of the external domain or the internal domain of the pipe pilesk(ii) a Determining hysteretic damping ratio eta of soil framework in external domain or internal domain of tubular pileskCalculating the complex Ramse constant mu of the soil framework of the external domain or the internal domain of the tubular pileskAnd λsk(ii) a Determining the volume fraction n of the soil body skeleton in the saturated soil of the external or internal area of the pipe pileskTrue density rho of soil framework of outer domain or inner domain of tubular pilesRkCalculating the volume density rho of the soil framework of the external area or the internal area of the pipe pilesk(ii) a Wherein, the upper subscript k and the subscript k of each parameter are I or O, and I representsThe former parameter is the parameter of the inner domain of the tubular pile, and O represents that the current parameter is the parameter of the outer domain of the tubular pile;
determining the true density rho of pore fluid in the external or internal area of the pipe pilefRkVolume fraction n of pore fluid in saturated soil in external or internal area of pipe pilefkCalculating the volume density rho of pore fluid in the external domain or the internal domain of the pipe pilefkLiquid-solid coupling coefficient s of external or internal tubular pile domainsvk；
Based on the Hamilton dynamics principle, a motion equation of the two-dimensional tubular pile considering radial and vertical deformation under steady-state vibration is established, the established motion equation is converted by adopting a variable separation method, then a coefficient determinant of the equation obtained by conversion is zero, and a characteristic value eta is obtained by solvingjJ is 1,2,3, 4; then eta is addedjSubstituting into the equation obtained by conversion to obtain the sum etajOne-to-one correspondence of feature vectors
Figure BDA0003318921540000021
Determining the radial displacement and the vertical displacement of the tubular pile according to the following formula:
Figure BDA0003318921540000022
the embodiment of the invention has the beneficial effects that: the method comprises the following steps of (1) researching the motion response of the tubular pile in saturated soil under the action of earthquake P waves by adopting an analytical method, regarding the tubular pile foundation as a two-dimensional hollow rod piece with vertical and radial deformation, and establishing a control equation of the tubular pile according to a Hamilton variation principle; and (3) regarding the soil bodies around the tubular pile and inside and outside the tubular pile as saturated three-dimensional porous elastic continuous bodies filled with fluid, and establishing a motion equation of the saturated soil bodies around the tubular pile and inside and outside the tubular pile by adopting a Boer porous medium model. And solving partial differential equations of the piles and the soil through a variable separation method. Aiming at the problem that the saturated soil and the tubular pile generate complex motion interaction and inertia interaction under the action of earthquake load, the embodiment of the invention takes the soil around the pile and the soil in the core of the pile as saturated porous media, separately considers the motion interaction and the inertia interaction based on the superposition principle to obtain respective soil response and superposes the soil response, then establishing the boundary and contact conditions of the tubular pile and the soil surrounding the pile and the soil in the core of the pile, applying the boundary and contact conditions of the tubular pile and the soil surrounding the pile and the soil in the core of the pile to obtain the motion response of the tubular pile in the saturated soil under the action of the earthquake P wave, and the reasonability and correctness of the solution of the embodiment of the invention are verified by defining a motion amplification coefficient reflecting the influence of the pile-soil system on the movement of the bedrock and a motion response coefficient reflecting the influence of the pile foundation on the movement of the free field of the soil body, and the method can be applied to the condition of structure-continuum interaction. The problem that in the interaction analysis of the existing pile-soil system under the action of a vertical earthquake, the pile soil around the pile is regarded as a half-space medium with infinite thickness and is inconsistent with the actual common pile-soil resonance phenomenon of the thin soil layer of the lower bedrock, so that the solution of the half-space medium is inaccurate under the condition of the thin soil layer of the lower bedrock is solved; and in the interaction analysis of the existing pile-soil system under the action of vertical earthquake, the pile foundation is regarded as a one-dimensional structure, so that the method is not suitable for the condition of interaction of the structure and the continuum.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a schematic diagram of an interaction model of saturated soil and a pipe pile system under vertical steady-state movement of bedrock.
Fig. 2 is a schematic diagram comparing results of a motion response solution of a simplified one-dimensional pile and a motion response solution of an existing one-dimensional pile according to an embodiment of the present invention, where (a) is a motion amplification factor comparison diagram of the motion response solution of the embodiment of the present invention and the existing one-dimensional pile, and (b) is a motion response factor comparison diagram of the motion response solution of the embodiment of the present invention and the existing one-dimensional pile.
Fig. 3 is a schematic diagram of an axis-symmetric ADINA finite element model of a pile-soil system established by an embodiment of the invention.
FIG. 4 is a graphical comparison of the motion response solution of an embodiment of the present invention with the results of calculation of an ADINA finite element model, wherein (a) is a graph comparing vertical pile displacement at different excitation frequencies to the solution of an embodiment of the present invention, and (b) is a graph comparing different pore fluid pressure distributions to the solution of an embodiment of the present invention.
Fig. 5 is a first schematic diagram of the distribution of the pile-soil model field variables along the pile body, wherein (a) is a graph of the real part of the radial displacement of the soil in the inner area of the pipe pile, (b) is a graph of the imaginary part of the radial displacement of the soil in the inner area of the pipe pile, (c) is a graph of the real part of the radial displacement of the soil in the outer area of the pipe pile, (d) is a graph of the imaginary part of the radial displacement of the soil in the outer area of the pipe pile, (e) is a graph of the real part of the vertical displacement of the soil in the outer area of the pipe pile, and (f) is a graph of the imaginary part of the vertical displacement of the soil in the outer area of the pipe pile.
Fig. 6 is a second schematic diagram of the distribution of the pile-soil model field variables along the pile body, wherein (a) is a graph of the real part of the pore fluid pressure in the outer domain of the pipe pile, (b) is a graph of the imaginary part of the pore fluid pressure in the outer domain of the pipe pile, (c) is a graph of the real part of the axial force of the pipe pile, (d) is a graph of the imaginary part of the axial force of the pipe pile, (e) is a graph of the real part of the shear force of the pipe pile, and (f) is a graph of the imaginary part of the shear force of the pipe pile.
Fig. 7 is a schematic diagram of motion amplification coefficients of a pile-soil system under different pile-soil conditions, wherein (a) is a graph of motion amplification coefficients of a solid pile with a pile-soil modulus ratio of 500 under different length-diameter ratios, (b) is a graph of motion amplification coefficients of a solid pile with a pile-soil modulus ratio of 1000 under different length-diameter ratios, (c) is a graph of motion amplification coefficients of a pile without an inner soil pipe with a pile-soil modulus ratio of 500 under different length-diameter ratios, (d) is a graph of motion amplification coefficients of a pile without an inner soil pipe with a pile-soil modulus ratio of 1000 under different length-diameter ratios, (e) is a graph of motion amplification coefficients of an inner soil pipe pile with a pile-soil modulus ratio of 500 under different length-diameter ratios, and (f) is a graph of motion amplification coefficients of an inner soil pipe pile with a pile-soil modulus ratio of 1000 under different length-diameter ratios.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
As shown in fig. 1, the embodiment of the invention considers the dynamic response problem that the elastic pipe pile is embedded in and completely bonded with the saturated porous elastic soil layer on the rigid bedrock under the action of the earthquake P wave. The action of the earthquake P wave is considered as the vertical time harmonic motion action of the bedrock under the soil body, and the action of the earthquake P wave passes through the bedrock by wg(t)＝wgeiωtForm of vertical motion representation (
Figure BDA0003318921540000041
wgIs the amplitude of the bedrock motion, ω ═ 2 π f, where f is the excitation frequency in Hz). r isOIs the outer radius of the tubular pile, rIThe inner radius of the tubular pile, L is the length of the tubular pile; t is trOShowing the radial acting force of the outer surface of the pipe pile, tzOShowing the soil body acting force of the outer surface of the tubular pile in the vertical direction; t is trIShowing the soil body acting force, t, of the inner surface of the pipe pile in the radial directionzIShowing the soil body acting force of the inner surface of the tubular pile in the vertical direction; u. ofpRepresenting the radial component of displacement, w, of the pipe pilepThe displacement component of the pipe pile in the vertical direction is shown,
Figure BDA0003318921540000042
and
Figure BDA0003318921540000043
is a unit vector.
In addition, the pipe pile and saturated soil system satisfy the following basic assumptions:
(1) the soil particles and the pore fluid are not compressible microscopically, and heat and mass exchange does not occur;
(2) ignoring the gravitational field of the pore fluid viscosity and liquid-solid aggregates;
(3) the liquid-solid interaction force is in direct proportion to the liquid-solid relative speed;
(4) the pores in the liquid-solid aggregate are uniformly distributed;
(5) the deformation of the pile-soil system is micro deformation so as to ensure the linear strain and displacement relation of the pile-soil system;
(6) in the vibration process, the pile foundation and the soil body are completely bonded, and relative sliding and separation behaviors do not exist.
1. Tubular pile control equation and solution thereof
Firstly, a kinetic variational principle is adopted to establish a motion equation of the two-dimensional pipe pile considering radial and vertical deformation under steady-state vibration. Under the loading conditions shown in fig. 1, the pile-soil system is axisymmetric about the z-axis. Under the condition of axial symmetry, pile-soil response is irrelevant to the coordinate theta, and the annular displacement is zero.
Generally, the deformation modes of an axially loaded pile with sufficient stiffness and slenderness ratio are primarily longitudinal translation and axial compression/tension. On this basis, it is feasible to take only the first order approximation of the pile axial displacement field, which is taken as a function of the ordinate z. However, under the Poisson effect, axial compression/tension of the stake typically produces a corresponding radial deformation response. In addition, due to the lateral confinement of the surrounding medium, the circumferential surface of the cylindrical pile foundation is subject to boundary side pressures caused by external loads, resulting in significant internal radial compression. Some scholars can reasonably reveal the deformation characteristics by assuming that the radial displacement of the pile foundation varies linearly along its cross section. In addition to enabling the above-mentioned physical correlation analysis, this kinematic assumption has been shown to model the radial shear phenomenon, which is very important in the wave propagation problem. Therefore, based on this assumption, the tube stake displacement field can be expressed as:
Figure BDA0003318921540000051
in the formula (1), up(r, z) represents the radial displacement of any point inside the tube pile, wp(r, z) represents the axial displacement of any point inside the tubular pile, and up(r, z) and wp(r, z) are functions of radial coordinate r and axial coordinate z; u. ofp(z) is the radial displacement component of the outer surface of the tubular pile, i.e. up(rO,z)；wp(z) is the vertical displacement component of the outer surface of the tubular pile, namely wp(rO,z)。
The axial force n (z) and the shear force q (z) of the cross section of the tube pile can be defined as:
Figure BDA0003318921540000052
in the formula (2), the reaction mixture is,
Figure BDA0003318921540000053
is the normal stress component on the cross section of the tubular pile,
Figure BDA0003318921540000054
the dS represents the integral of the area S, and is the shearing stress component on the cross section of the tubular pile; mu.spIs the shear modulus, mu, of the tube pilep＝Ep/2/(1+υp)，EpIs the elastic modulus of the tubular pile, upsilonpThe poisson ratio of the tubular pile is obtained; a. thepIs the cross-sectional area of the pipe pile,
Figure BDA0003318921540000055
Jpis the polar moment of inertia of the tubular pile,
Figure BDA0003318921540000056
the elastic energy P and the kinetic energy T of the tubular pile can be determined as follows:
Figure BDA0003318921540000057
Figure BDA0003318921540000058
in formula (3), σpIs the stress tensor of the pipe pile, epsilonpIs the strain tensor of the tubular pile, V is the volume of the tubular pile,
Figure BDA0003318921540000059
representing a positive stress acting on the r-plane and in the radial direction,
Figure BDA00033189215400000510
represents the positive strain acting on the r-plane and in the radial direction;
Figure BDA00033189215400000511
representing the shear stress acting on the r-plane and in the vertical direction,
Figure BDA00033189215400000512
represents the shear strain acting on the r-plane and in the vertical direction;
Figure BDA00033189215400000513
representing a positive stress acting in the theta plane and in the theta direction,
Figure BDA0003318921540000061
representing a positive strain acting in the theta plane and in the theta direction,
Figure BDA0003318921540000062
representing the shear strain acting in the z-plane and vertically,
Figure BDA0003318921540000063
representing a positive strain acting on the z-plane and in the vertical direction. In the formula (4), ρpIndicates the density, v, of the tube pilepRepresenting the velocity vector of the pipe pile, the point on the displacement sign representing the derivation of t, i.e.
Figure BDA0003318921540000064
The radial speed of the side surface of the pipe pile is shown,
Figure BDA0003318921540000065
is vpThe component in the radial direction is such that,
Figure BDA0003318921540000066
shaft for indicating side surface of tubular pileThe speed of the moving-direction is controlled,
Figure BDA0003318921540000067
is vpA component in the axial direction.
Under the action of the surface acting force and the end acting force of the surrounding soil body, the pile foundation externally acts as:
Figure BDA0003318921540000068
in the formula (5), tpRepresenting the external force vector u borne by the whole surface of the pipe pilepRepresenting the displacement vector of the whole surface of the tubular pile; q (z ═ 0) represents the shear force applied to the top of the tube pile, and u represents the shear force applied to the top of the tube pilep(z ═ 0) denotes the radial displacement of the top of the tube pile, Q (z ═ L) denotes the shear force to which the bottom of the tube pile is subjected, and u denotes the shear force to which the bottom of the tube pile is subjectedp(z ═ L) denotes the radial displacement of the bottom of the tube pile, N (z ═ 0) denotes the axial force to which the top of the tube pile is subjected, w denotes the axial force to which the top of the tube pile is subjectedp(z ═ 0) denotes the axial displacement of the top of the tube pile, N (z ═ L) denotes the axial force applied to the bottom of the tube pile, and w denotes the axial force applied to the bottom of the tube pilepAnd (z ═ L) represents the axial displacement of the bottom of the tube pile.
By using the virtual work principle, the virtual work of the derived formula (5) is:
Figure BDA0003318921540000069
combining equations (3) - (6) and the following Hamilton variational principle:
Figure BDA00033189215400000610
the control equation for the pipe pile can be derived as:
Figure BDA00033189215400000611
since the embodiment of the present invention considers the steady-state vibration, the further derivation formula (8) is:
Figure BDA0003318921540000071
for ease of analysis, e belowiωtAre omitted.
Then, solving the formula (9) by a variable separation method to obtain the characteristic value and the corresponding characteristic vector thereof, and obtaining a homogeneous solution of the response of the tubular pile.
Setting u according to the theory of variable separationp(z)＝Upeηz、wp(z)＝Wpeηz，eηzCan be understood as a shape function, wherein eta represents a parameter for controlling the shape function, the characteristic value of the equation after being substituted into the subsequent concrete equation is eta, and t is letrO＝trI＝0、tzO＝tzISubstitution of formula (9) with 0 yields:
Figure BDA0003318921540000072
if equation (10) has a non-zero solution, the determinant of coefficients of equation (10) is required to be zero, i.e.:
Figure BDA0003318921540000073
equation (11) is a quadratic equation for the variable η, containing 4 roots. So that the homogeneous solution of formula (9)
Figure BDA0003318921540000074
Comprises the following steps:
Figure BDA0003318921540000075
wherein e is a natural constant, UjAre four unknown constants, hjAnd kjIs a characteristic value etajComponent of corresponding feature vector, feature value ηjIs a root of formula (11), j is 1,2,3, 4. EtajContaining four values, i.e. eta1,η2,η3,η4Each value corresponds to a feature vector, which is a vector containing two elements, namely hjAnd kj，hjAnd kjRespectively correspond to U one by onepAnd Wp. Will etajSubstituting into formula (10) to obtain the feature vector
Figure BDA0003318921540000076
Eta in the embodiment of the inventionj、hjAnd kjAll the calculation results are obtained by mathematic calculation software such as MatLab, and the like, and the mathematic calculation software comprises the function of solving the matrix eigenvalue and the corresponding eigenvector, and the detailed solving process is not repeated here.
Equation (9) is a non-homogeneous equation that includes homogeneous and solution-specific solutions. The reaction force t of the soil acting on the surface of the tubular pile is obtained by solving the following equation of motion of the soilrO、trI、tzO、tzIThe final solution can be determined.
2. Equation of motion and boundary conditions of the earth
The soil bodies of the inner domain and the outer domain around the tubular pile are regarded as saturated three-dimensional porous elastic continuous bodies filled with fluid, the mechanical behavior of the saturated soil body around the tubular pile is described by adopting a Boer porous medium model, and the mechanical behavior is described by a soil body framework displacement vector u of the outer domain of the tubular pile or the inner domain of the tubular pileskPore fluid displacement vector u of either the external or internal tubular pile domainsfkPore fluid pressure p of the external or internal tubular pile domainsfkThe equation of motion expressed is:
Figure BDA0003318921540000081
Figure BDA0003318921540000082
Figure BDA0003318921540000083
wherein k is O or I, O represents the external domain of the tubular pile, and I represents the internal domain of the tubular pile; mu.sskAnd λskThe complex Ramse constant, mu, of the soil framework representing the external or internal domains of a tubular pilesk＝Gsk(1+2iηsk)，ηskThe hysteresis damping ratio G of the soil body skeleton of the outer domain or the inner domain of the tubular pilesk＝Esk/2/(1+υsk)，GskShear modulus of the soil framework of the outer or inner domain of a pipe pile, EskIs the elastic modulus of the soil body skeleton in the outer domain or the inner domain of the tubular pile, upsilonskIs the Poisson's ratio, lambda, of the soil skeleton in the external or internal area of the tubular pilesk＝2υskμsk/(1-2υsk)；ρskRepresenting the bulk density, rho, of the soil framework in the external or internal domain of the pipe pilesk＝ρsRknsk，ρsRkRepresenting the true density, n, of the soil framework in the outer or inner domain of the pipe pileskRepresenting the volume fraction of a soil body framework in saturated soil of the external domain or the internal domain of the pipe pile; rhofkVolume density, p, of pore fluid representing either the external or internal tubular pile domainsfk＝ρfRknfk，ρfRkRepresenting the true density, n, of the pore fluid in the outer or inner region of the tube pilefkRepresenting the volume fraction of pore fluid in saturated soil of the external pipe pile domain or the internal pipe pile domain; svkIs the liquid-solid coupling coefficient of the external domain or the internal domain of the pipe pile, which represents the interaction between the soil framework of the external domain or the internal domain of the pipe pile and the pore fluid, svk＝nfkρfkg/kfkG is the acceleration of gravity, kfkIs the Darcy permeability coefficient.
Figure BDA0003318921540000084
In order to be a gradient operator, the method comprises the following steps,
Figure BDA0003318921540000085
is a divergence operator. The displacement amount with a single point on the upper surface represents the velocity, and the displacement amount with a double point on the upper surface represents the acceleration. The formulas (13a) to (13b) are soil body frameworks of the external domain or the internal domain of the tubular pileEquation (13c) is the mass balance equation of the saturated soil in the external or internal tubular pile domain.
Under the property of axial symmetry, the expressions (13a) to (13c) can be extended to
Figure BDA0003318921540000086
Figure BDA0003318921540000087
Figure BDA0003318921540000088
Figure BDA0003318921540000089
Figure BDA00033189215400000810
In formulae (14a) to (14e), uskRadial displacement, w, of soil skeleton in the outer or inner region of the pipe pileskIs the vertical displacement of the soil skeleton of the external or internal area of the pipe pile, ufkRadial displacement of pore fluid, w, of the outer or inner region of the tube pilefkIs the vertical displacement of the pore fluid in the outer or inner area of the tube pile, eskIs the volume strain of the soil body skeleton in the external area or the internal area of the tubular pile,
Figure BDA0003318921540000091
is the laplacian operator, and is,
Figure BDA0003318921540000092
in fig. 1, the boundary and interface conditions of the pipe pile and the saturated soil are as follows:
at infinity, i.e., r → ∞, the response of the soil decays to zero, i.e.:
Figure BDA0003318921540000093
wherein the content of the first and second substances,
Figure BDA0003318921540000094
representing the effective stress component acting on the soil framework of the external domain of the tube pile in the radial plane and then pointing to the vertical direction,
Figure BDA0003318921540000095
and the effective stress component of the soil body framework acting on a z plane, namely a plane perpendicular to the z axis of the vertical coordinate axis and then pointing to the vertical tubular pile external domain is represented.
The top surface of the soil body is a free boundary, the normal stress is zero, and the soil body has permeability, namely:
Figure BDA0003318921540000096
wherein the content of the first and second substances,
Figure BDA0003318921540000097
and the effective stress component of the soil body framework acting on a z plane, namely a plane perpendicular to the z axis of the vertical coordinate axis and pointing to the inner area of the vertical pipe pile is represented.
The soil body bottom bonds with the rigidity bed rock, and the bed rock is waterproof, and vertical displacement is unanimous, promptly:
wsO(r,L,t)＝wsI(r,L,t)＝wg(t),wfO(r,L,t)＝wfI(r,L,t)＝wg(t)， (15c)
wherein, wsO(r, L, t) represents the vertical displacement of the soil framework of the external domain of the pipe pile at the bedrock location, wsI(r, L, t) represents the vertical displacement of the soil framework in the inner area of the pipe pile at the bedrock position; w is afO(r, L, t) represents the vertical displacement of pore fluid in the external area of the tubular pile at the bedrock location, wfI(r, L, t) represents pore fluid in the inner area of the pipe pile at the bedrock locationAnd (4) vertical displacement.
Since the pile foundation is impervious to water, the pore fluid and the radial displacement of the tube pile are identical, i.e.:
Figure BDA0003318921540000098
wherein u isfO(rOZ, t) represents the radial displacement of the pore fluid of the external area of the tube stake at the external surface of the tube stake, ufI(rIAnd z, t) represents the radial displacement of pore fluid in the inner area of the tube stake at the inner surface of the tube stake.
The pipe pile is in close contact with the saturated soil at the interface, so the displacement is continuous, namely:
Figure BDA0003318921540000099
wherein u issO(rOZ, t) denotes the radial displacement of the soil skeleton of the external area of the pipe pile at the external surface of the pipe pile, usI(rIAnd z, t) represents the radial displacement of the soil skeleton in the inner area of the pipe pile at the inner surface of the pipe pile.
The tubular pile stake top surface is smooth, and tubular pile stake top axial force equals vertical external load, promptly:
Q(z＝0,t)＝0,N(z＝0,t)＝p(t)， (15f)
in the formula (15f), p (t) is a vertical external load applied to the pile top, possibly an applied time-harmonic load or an action from a superstructure, and if the pile top has no external load action, p (t) is 0.
The tubular pile bottom is radial smooth, and the vertical direction and basement rock be in intimate contact, therefore its vertical displacement is unanimous with the motion of basement rock, and the shearing force is zero at the tubular pile bottom, promptly:
wp(z＝L,t)＝wg(t),Q(z＝L,t)＝0, (15g)
it should be noted that the pile base (e.g. driven pile in the foundation) considered here is only located on the bedrock and not embedded in the bedrock, and therefore the pile base may not be radially fixed.
3. General solution of pile surrounding soil body under vertical movement of bedrock
Under the action of vertical movement of bedrock, a pile-soil system consisting of soil bodies in the inner area of the tubular pile, the tubular pile and the soil body in the outer area of the tubular pile generates a complex interaction process. Firstly, when the bedrock starts to move, soil and a pile are driven to move synchronously, namely, the bedrock moves in a free field; however, due to the difference in stiffness between the pile and the soil, the pile moves further relative to the soil (pile vibration) which in turn produces pile-soil interaction and deformation of the soil mass, which may be referred to as motion interaction. When the movement of the lower bedrock reaches the pile top, the superstructure (if any) further acts on the pile foundation due to inertial movement (pile vibration), causing pile-soil interaction, called inertial interaction. For the linear pile-soil system considered in the embodiment of the invention, for convenient analysis, the soil response can be decomposed into two parts according to the pile-soil interaction process, namely the soil free field response state caused by bedrock movement and the soil response state caused by tubular pile vibration without bedrock movement. And finally, on the basis of a superposition principle, performing general solution superposition on the obtained soil response of the soil free field response state caused by bedrock movement and the soil response of the soil response state caused by the vibration of the tubular pile without bedrock movement to obtain a general solution of the dynamic response of the tubular pile and the saturated soil in the inner and outer regions under the action of the bedrock movement. It should be noted that the soil body free field response state caused by bedrock movement and the soil body response solution of the soil body response state caused by pipe pile vibration without bedrock movement both satisfy the control equation and the boundary condition of saturated soil.
3.1 vibration of tubular pile soil body response solution
Obtained from formulae (14c) to (14 d):
Figure BDA0003318921540000101
Figure BDA0003318921540000102
connection of equations (14a) to (14b)The vertical process is represented as
Figure BDA0003318921540000103
Then, the equations (16a) to (16b) are substituted to obtain:
Figure BDA0003318921540000111
in the formula (17), intermediate variables
Figure BDA0003318921540000112
Figure BDA0003318921540000113
Similarly, the simultaneous process of equations (16a) - (16b) is
Figure BDA0003318921540000114
Then substituting equation (14e) yields:
Figure BDA0003318921540000115
in the formula (18), intermediate variables
Figure BDA0003318921540000116
Substituting formula (18) for formula (17) to obtain:
Figure BDA0003318921540000117
in the formula (19), the intermediate variable
Figure BDA0003318921540000118
Let e by variable separationsk＝Rk(r)Zk(z) as for eskCarrying out variable separation, Rk(r) and Zk(z) is an unknown function, which is then generated by substituting it into equation (19):
Figure BDA0003318921540000119
in the formula (20), b1kAnd b2kAs an unknown constant, b1kAnd b2kAt later stage, the boundary condition and continuous condition of pile-soil system are solved, and
Figure BDA00033189215400001110
the solution of equation (20) is:
Figure BDA00033189215400001111
in the formula (21), A1k、A2k、B1kAnd B2kAre all unknown constants, are solved by boundary conditions and continuous conditions of the pile-soil system in the later period, I0() Representing zero-order modified Bessel functions of the first kind, K0() Representing a second class of zero-order modified bessel functions. Therefore:
esk＝[A1kK0(b1kr)+A2kI0(b1kr)][B1ksin(b2kz)+B2kcos(b2kz)]， (22)
from the formula (18):
Figure BDA00033189215400001112
using the variable separation method described above, a homogeneous solution of the non-homogeneous equation (23) with a homogeneous solution and a particular solution can be obtained as:
pfkh＝[A3kK0(b3kr)+A4kI0(b3kr)][B3ksin(b3kz)+B4kcos(b3kz)]， (24)
wherein p isfkhHomogeneous solution of pore water pressure of external or internal area of pipe pile。
In the formula (24), A3k、b3k、B3k、A4kAnd B4kAnd solving the undetermined coefficient in the later period by using the boundary condition and the continuous condition of the pile-soil system.
The special solution of equation (23) can be set as:
pfkt＝T1kesk＝T1k[A1kK0(b1kr)+A2kI0(b1kr)][B1ksin(b2kz)+B2kcos(b2kz)]， (25)
wherein p isfktIndicating the solution of pore water pressure in the external or internal tubular pile domains.
Substituting formula (25) for formula (23) to obtain intermediate variable T1kComprises the following steps:
Figure BDA0003318921540000121
Figure BDA0003318921540000122
then according to pfkSolving the formula (14) in the solving process, and obtaining the displacement and stress response of the saturated soil layer as follows:
Figure BDA0003318921540000123
Figure BDA0003318921540000124
Figure BDA0003318921540000125
Figure BDA0003318921540000126
Figure BDA0003318921540000131
Figure BDA0003318921540000132
Figure BDA0003318921540000133
in the formula (I), the compound is shown in the specification,
Figure BDA0003318921540000134
the effective stress component of the soil body framework acting on a radial plane, namely a plane perpendicular to a radial coordinate axis and pointing to a radial tubular pile inner domain or a radial tubular pile outer domain is represented;
Figure BDA0003318921540000135
the effective stress component of the soil body skeleton acting on a z plane, namely a plane perpendicular to a vertical coordinate axis and pointing to a vertical tubular pile inner domain or a vertical tubular pile outer domain is represented;
Figure BDA0003318921540000136
representing effective stress components of a soil body framework acting on a radial plane and then pointing to a vertical tubular pile inner domain or a vertical tubular pile outer domain;
Figure BDA0003318921540000137
representing the effective stress component of a soil framework acting on a z plane and then pointing to a radial inner tubular pile domain or an outer tubular pile domain; i is1() Is a first-order modified Bessel function of the first kind, K1() A modified Bessel function of a second type; b1k～b7k、A1k～A8k、B1k～B8kAll the parameters are undetermined constants, are obtained according to the boundary and continuity conditions of the pile-soil system, namely the expressions (15a) to (15g), specifically, the expressions (A1) to (A7) are substituted into the corresponding equations of the expressions (15a) to (15g),the derivation solution determines these unknown parameters.
Figure BDA0003318921540000138
Figure BDA0003318921540000139
Figure BDA0003318921540000141
In addition, the vertical displacement of the soil framework is substituted into the boundary condition equation (15c), and the obtained form such as cos (b) is deducednL) 0 (note: since the pile vibration response state is considered here, the boundary condition equations (15a) - (15f) do not include the movement of the bedrock, i.e. w is the time wheng(t) ═ 0). Then, a volume strain formula of a soil body framework of the external area and the internal area of the tubular pile is utilized
Figure BDA0003318921540000142
The boundary condition equations (15a) - (15b) and the response of the soil body in the pipe pile are bounded, and the expressions (a1) - (a7) can be rewritten as the following expressions:
for the soil body area outside the tubular pile:
Figure BDA0003318921540000143
Figure BDA0003318921540000144
Figure BDA0003318921540000145
Figure BDA0003318921540000146
Figure BDA0003318921540000147
Figure BDA0003318921540000148
for the soil body area inside the tubular pile:
Figure BDA0003318921540000149
Figure BDA00033189215400001410
Figure BDA00033189215400001411
Figure BDA00033189215400001412
Figure BDA0003318921540000151
Figure BDA0003318921540000152
wherein the content of the first and second substances,
Figure BDA0003318921540000153
the radial total stress component of saturated soil in the external area or the internal area of the pipe pile caused by the vibration of the pile foundation is shown,
Figure BDA0003318921540000154
Figure BDA0003318921540000155
the arrows above the variables indicate that these variables are the soil response caused by pile foundation vibration, i.e.
Figure BDA0003318921540000156
The radial displacement of the soil body skeleton of the external domain of the pipe pile caused by the vibration of the pile foundation is shown,
Figure BDA0003318921540000157
the vertical displacement of the soil body skeleton of the external domain of the pipe pile caused by the vibration of the pile foundation is shown,
Figure BDA0003318921540000158
the radial displacement of the pore fluid of the tubular pile external domain caused by the vibration of the pile foundation is shown,
Figure BDA0003318921540000159
showing the effective stress component of the soil body skeleton of the external domain of the pipe pile acting on the radial plane and then pointing to the vertical direction caused by the vibration of the pile foundation,
Figure BDA00033189215400001510
the effective stress component of the soil body framework which is caused by pile foundation vibration and acts on the z plane and then points to the radial external area of the pipe pile is shown,
Figure BDA00033189215400001511
the radial total stress component of saturated soil of the external area of the pipe pile caused by pile foundation vibration is shown,
Figure BDA00033189215400001512
the pore fluid pressure of the external area of the pipe pile caused by the vibration of the pile foundation is represented;
Figure BDA00033189215400001513
the radial displacement of the soil body framework in the inner area of the tubular pile caused by the vibration of the pile foundation,
Figure BDA00033189215400001514
caused by pile vibrationsThe vertical displacement of the soil body framework in the inner area of the tubular pile,
Figure BDA00033189215400001515
the radial displacement of the pore fluid in the pipe pile caused by pile foundation vibration is shown,
Figure BDA00033189215400001516
the effective stress component of the soil body framework in the inner area of the pipe pile, which acts on the radial plane and then points to the vertical direction, caused by the vibration of the pile foundation is shown,
Figure BDA00033189215400001517
the effective stress component of the soil body framework which is caused by the vibration of the pile foundation and acts on the z plane and then points to the inner area of the radial pipe pile is shown,
Figure BDA00033189215400001518
the radial total stress component of saturated soil in the pipe pile inner area caused by pile foundation vibration is shown,
Figure BDA00033189215400001519
the pressure of pore fluid in the pipe pile inner area caused by pile foundation vibration is shown.
Figure BDA00033189215400001520
C1n～C6nThe number of undetermined coefficients is 6, and the undetermined coefficients are determined by pile-soil boundaries and continuity conditions.
Figure BDA00033189215400001521
Figure BDA00033189215400001522
Figure BDA00033189215400001523
Figure BDA00033189215400001524
Figure BDA00033189215400001525
Figure BDA00033189215400001526
Figure BDA0003318921540000161
Figure BDA0003318921540000162
Figure BDA0003318921540000163
Figure BDA0003318921540000164
Figure BDA0003318921540000165
Figure BDA0003318921540000166
Figure BDA0003318921540000167
Figure BDA0003318921540000168
Figure BDA0003318921540000169
3.2 free field response solution of soil under bedrock movement
As shown in figure 1, under the action of vertical movement of bedrock, the free field of soil can be simulated into a plane strain state, namely
Figure BDA00033189215400001610
And
Figure BDA00033189215400001611
the equations of motion (13a) - (13c) for the saturated soil can be simplified as:
Figure BDA00033189215400001612
Figure BDA00033189215400001613
Figure BDA00033189215400001614
wherein, above each variable, the variable belongs to the response of the soil free field to the movement of the bedrock, i.e. the response
Figure BDA00033189215400001615
Is the vertical displacement of the soil body skeleton in the outer domain or the inner domain of the tubular pile caused by the movement of bedrock,
Figure BDA00033189215400001616
is the vertical displacement of pore fluid in the outer area or the inner area of the pipe pile caused by the movement of bedrock,
Figure BDA0003318921540000171
pore fluid pressure in the outer or inner tubular pile area caused by bedrock movement.
According to the vertical displacement of the soil framework in the inner area of the pipe pile caused by the vibration of the pile base in the boundary condition formula (15c) and the formula (29b)
Figure BDA0003318921540000172
The series expression of (2) can assume that the vertical displacement of the soil framework of the outer domain or the inner domain of the tubular pile under the movement of the bedrock is as follows:
Figure BDA0003318921540000173
in the formula (31), PknIs the undetermined coefficient.
By substituting formula (31) for formula (30 c):
Figure BDA0003318921540000174
combining formulae (30a), (30b), (31), (32) to yield:
Figure BDA0003318921540000175
in the formula (33), the reaction mixture,
Figure BDA0003318921540000176
trigonometric function sin (b)nz) and cos (b)nz) satisfies the following orthogonality:
Figure BDA0003318921540000177
substituting formula (34) for formula (33) to obtain:
Figure BDA0003318921540000178
from the formula (30 b):
Figure BDA0003318921540000179
the stress component can be:
Figure BDA00033189215400001710
Figure BDA00033189215400001711
Figure BDA0003318921540000181
wherein the content of the first and second substances,
Figure BDA0003318921540000182
representing effective stress components caused by bedrock movement, acting on a radial plane and then pointing to a radial external pipe pile domain or a soil framework of a pipe pile internal domain,
Figure BDA0003318921540000183
and the radial total stress component of the soil body framework of the outer domain or the inner domain of the pipe pile caused by the movement of the bedrock is represented.
Wherein the content of the first and second substances,
Figure BDA0003318921540000184
3.3 Final solution of saturated soil
Now, the solutions of the two states (the solution of the response of the soil body under the vibration of the pipe pile and the solution of the response of the free field of the soil body under the movement of the bedrock) are added, and the final displacement and stress components of the saturated soil are as follows:
for the external domain of the tubular pile: r is not less than rO
Figure BDA0003318921540000185
Figure BDA0003318921540000186
Figure BDA0003318921540000187
Figure BDA0003318921540000188
Figure BDA0003318921540000189
Figure BDA00033189215400001810
For the inner area of the tubular pile: r is less than or equal to rI
Figure BDA00033189215400001811
Figure BDA00033189215400001812
Figure BDA0003318921540000191
Figure BDA0003318921540000192
Figure BDA0003318921540000193
Figure BDA0003318921540000194
4. Solution for pile-soil interaction under vertical movement of bedrock
And finally, determining unknown parameters in the general solutions of the dynamic responses of the tubular pile and the saturated soil body in the inner and outer regions under the action of the movement of the bedrock by combining the boundary and continuity conditions of the tubular pile and the soil body, thereby obtaining the movement response solution of the tubular pile in the saturated soil under the action of the seismic P wave.
Considering the stress continuity of the pile-soil interface, determining the stress of the side wall of the tubular pile as follows:
Figure BDA0003318921540000195
wherein the content of the first and second substances,
Figure BDA0003318921540000196
shows the radial total stress component of the soil body skeleton at the outer wall of the pipe pile,
Figure BDA0003318921540000197
shows the radial total stress component of the soil framework at the inner wall of the pipe pile,
Figure BDA0003318921540000198
representing the effective stress component of the soil framework acting on the radial plane and then pointing to the outer wall of the vertical pipe pile,
Figure BDA0003318921540000199
the effective stress component acting on the soil body framework at the radial plane and then pointing to the inner wall of the vertical pipe pile is represented.
The formula (40) is substituted into the formula (9), and the radial displacement u of the outer surface of the tubular pile can be obtained by combining the formula (12)p(z) and vertical displacement wp(z) is:
Figure BDA00033189215400001910
wherein the content of the first and second substances,
Figure BDA00033189215400001911
P2n＝α1nC1n+α2nC2n-α3nC3n+α4nC4n+α5nC5n+α6nC6n+wgαgn，P3n＝β1nC1n+β2nC2n-β3nC3n+β4nC4n+β5nC5n+β6nC6n+wgβgnwherein, in the step (A),
Figure BDA0003318921540000201
Figure BDA0003318921540000202
Figure BDA0003318921540000203
Figure BDA0003318921540000204
Figure BDA0003318921540000205
Figure BDA0003318921540000206
Figure BDA0003318921540000207
Figure BDA0003318921540000208
Figure BDA0003318921540000209
Figure BDA00033189215400002010
Figure BDA00033189215400002011
Figure BDA00033189215400002012
Figure BDA00033189215400002013
Figure BDA0003318921540000211
Figure BDA0003318921540000212
according to equations (2) and (41), the axial force and shear force of the tube pile are:
Figure BDA0003318921540000213
Figure BDA0003318921540000214
in the formula (42a), the compound represented by the formula (A),
Figure BDA0003318921540000215
using boundary conditions and continuous condition formulas (15d) - (15g) of pile-soil system, and adopting sin (b)nz) and cos (b)nz) orthogonal equation (34) may determine 10 unknown constants C1n～C6nAnd U1～U4Respectively as follows:
Figure BDA0003318921540000216
Figure BDA0003318921540000217
Figure BDA0003318921540000218
Figure BDA0003318921540000219
Figure BDA00033189215400002110
Figure BDA00033189215400002111
Figure BDA00033189215400002112
Figure BDA0003318921540000221
Figure BDA0003318921540000222
Figure BDA0003318921540000223
wherein the content of the first and second substances,
Figure BDA0003318921540000224
Figure BDA0003318921540000225
Figure BDA0003318921540000226
Figure BDA0003318921540000227
Figure BDA0003318921540000228
Figure BDA0003318921540000229
Figure BDA00033189215400002210
Figure BDA00033189215400002211
Figure BDA00033189215400002212
Figure BDA00033189215400002213
Figure BDA00033189215400002214
Figure BDA0003318921540000231
Figure BDA0003318921540000232
Figure BDA0003318921540000233
Figure BDA0003318921540000234
Figure BDA0003318921540000235
Figure BDA0003318921540000236
Figure BDA0003318921540000237
Figure BDA0003318921540000238
Figure BDA0003318921540000239
Figure BDA00033189215400002310
Figure BDA00033189215400002311
Figure BDA00033189215400002312
Figure BDA00033189215400002313
Figure BDA00033189215400002314
Figure BDA00033189215400002315
Figure BDA00033189215400002316
Figure BDA0003318921540000241
Figure BDA0003318921540000242
Figure BDA0003318921540000243
Figure BDA0003318921540000244
Figure BDA0003318921540000245
Figure BDA0003318921540000246
Figure BDA0003318921540000247
mg3＝mg2(m1m5-m2m4)+mg1(m1m8-m2m7)，mg4＝mg2(m1m6-m3m4)+mg1(m1m9-m3m7)，
Figure BDA0003318921540000248
Figure BDA0003318921540000249
R＝(m1m8-m2m7)(m1m6-m3m4)-(m1m9-m3m7)(m1m5-m2m4)，
Figure BDA00033189215400002410
Figure BDA00033189215400002411
is a tubular pileThe magnitude of the external load at the top; at this point all unknown constants in the pile-soil solution have been determined.
Numerical results and discussion:
to represent the variation of seismic P-waves propagating from the bottom of the earth to the top, the motion amplification factor is defined as:
Figure BDA00033189215400002412
in order to reveal the influence of the tubular pile on seismic P wave propagation in a saturated soil free field, motion response factors are defined as follows:
Figure BDA00033189215400002413
the correctness of the solution of the motion response determining method of the embodiment of the invention is verified through the numerical calculation of the calculation example, and the seismic response characteristic of the pile-soil system is researched. Unless otherwise stated, the soil parameters in Table 1 were used for the calculation examples. In the following analysis, to facilitate comparison and disclosure of the resonance characteristics of the soil mass free field and the pile-soil system considered in the present invention, the excitation frequency of the abscissa is divided by the first-order natural frequency ω of the single-phase soil free field1Performing dimensionless analysis, VpIs single-phase soil longitudinal wave velocity, omega1＝πVp/2L、
Figure BDA00033189215400002414
TABLE 1 pile soil parameters
Figure BDA0003318921540000251
Protocol validation and comparison:
in order to verify the correctness of the solution of the embodiment of the invention, the poisson ratio and the inner radius of the pile foundation are set to be zero, the two-dimensional pipe pile solution is simplified into the one-dimensional solid pile solution in the saturated soil, and compared with the existing solution Zhang S P, Cui C Y, Yang GComparison of the results of the channel analysis of the interacting system structured resources, the pixel group and the structural under the vertical movement of the channels, the soil Dynamics and the earthquakes Engineering,2019,123: 425) 434 is shown in FIG. 2 (a), Zhang et al shows Zhang S P, Cui C Y, Yang G.coupled vision of the interacting system structured resources, the pixel group and the structural under the vertical movement of the channels, the soil Dynamics and the earthquakes Engineering,2019,123:425 and 434, and the Reduced shows AM obtained in the present embodiment. And the Poisson ratio, the pore fluid density and the pore volume fraction of the pile foundation are all set to be zero, the solution is simplified into a one-dimensional pipe pile solution in single-phase soil, and compared with the existing solutions such as Zheng C J, Kouretzis G, Luan L B, et al, kinetic response of pipe pile sub-subject to vertical promotion and section P-waves, acta Geotechnical, 2021,16: 895-channel 909, the result is shown in (B) of FIG. 2, and the results are shown in (B) of FIG. 2, Zheng et al, Zheng C J, Kouretzis G, Luan L B, et al, kinetic response of pipe pile sub-subject to vertical promotion and section P-waves, Ach, Geotechnical, 16: 895-channel I1, 2025: 16: 895-channelVReduced represents the I obtained in the examples of the present inventionV. As can be seen from FIGS. 2 (a) and (b), the solutions of the embodiments of the present invention have AM and I at different ratios of pile length to radiusVThe curve is better matched with the curve of the existing solution along with the change of the excitation frequency.
Next, an axisymmetric pile-soil finite element model corresponding to fig. 1 is established by using an ADINA numerical software, as shown in fig. 3, and vertical displacement of the pile and pore fluid pressure distribution at different excitation frequencies are compared with the solution of the embodiment of the present invention, and the comparison result is shown in fig. 4. In the finite element model, the meshing of the pipe pile and the saturated soil model is divided according to the length ratio (back/front) of the cell edge. Wherein, two grid parameters of 50 and 1 are used for the vertical side of the pile soil area, 4 and 1 are used for the horizontal side of the pile and inner soil area, and 50 and 20 are used for the horizontal side of the outer soil area; respectively simulating saturated soil and a tubular pile by adopting a 9-node linear porous elastic and pure elastic rectangular high-order unit; the upper surface of the model is set as a free boundary, and the lower surface of the model applies vertical time-harmonic displacement load with unit amplitude; the right boundary of the earth model is set to allow only vertical motion and the left boundary of the model (the axisymmetric boundary) is also set to allow only vertical motion. The above-described boundary setting is consistent with the boundary conditions shown in fig. 1. It should be noted that the width of the pile-soil model is 50m, which can eliminate the boundary effect caused by the boundary setting on the right side of the model and ensure that the steady-state amplitude of the calculation case can be obtained. As can be seen from (a) to (b) of fig. 4, the solutions of the embodiments of the present invention are well matched with the finite element calculation results, and the correctness of the solutions of the embodiments of the present invention is verified through the above comparison.
Fig. 5 to 6 show the distribution of the pile-soil model field variables along the pile body, and it can be seen from (a) to (f) of fig. 5 to 6 that the response of the pile changes nonlinearly along the pile body, and the response curve of the pile fluctuates with the increase of the excitation frequency. Because the movement of the bedrock directly acts on the pile bottom, and the pile bottom is smooth, the shearing effect near the pile bottom is most obvious. In addition, the radial displacement curves of the pile foundation, the soil framework and the pore fluid coincide, and the vertical displacement curves of the pile foundation and the soil framework overlap, which respectively correspond to the pile-soil boundary conditions in the formulas (15d) to (15 e). The vertical displacement of the pile bottom is equal to that of the bedrock, and corresponds to a pile-soil system boundary condition formula (15 g). The soil surface pore fluid pressure is zero, corresponding to the soil boundary condition equation (15 b). The pile tip axial force is zero, corresponding to the pile boundary condition equation (15 f). Note that the pile top does not take into account the external loading effect at this point. The shear forces at both ends of the pile foundation are zero, corresponding to the pile boundary condition formulas (15f) - (15 g). It can be seen that the numerical result meets the specified boundary condition and continuity condition, which also reflects the reasonableness of the calculation result of the embodiment of the present invention.
FIG. 7 is the effect of radial deformation of the pile foundation on the dynamic behavior of the pile-soil system, where EpExpressing the modulus of elasticity of the pipe pile, EsODenotes the modulus of elasticity of the soil body of the outer domain, Ep/EsOThe pile-soil modulus ratio is expressed, and the relative hardness degree of the pile foundation and the soil body is reflected. As can be seen from (a) to (f) of fig. 7, when the pile length and diameter are relatively small, the radial deformation of the pile has a significant influence on the dynamic characteristics of the pile soil. For solid piles and pipe piles without internal soilUnder the condition, the one-dimensional rod solution has a larger motion amplification coefficient peak value, which shows that in the condition, if the influence of radial deformation of a pile foundation is ignored, the amplification characteristic of the pile-soil system is overestimated, so that the calculation result is larger. For the tubular pile with the pile core soil, the influence of the radial deformation of the pile on the dynamic characteristics of the pile soil is relatively small due to the supporting effect of the soil body in the tubular pile. Along with the increase of the length-radius ratio of the pile, the difference value of the results of the two-dimensional pile and the one-dimensional pile is reduced. For the pile foundation with a larger long diameter, the radial deformation of the pile foundation can be ignored, and the conclusion that the existing pile foundation with the larger long diameter can be regarded as a one-dimensional rod is consistent.
The above description is only for the preferred embodiment of the present invention, and is not intended to limit the scope of the present invention. Any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention shall fall within the protection scope of the present invention.

Claims (7)

1. A method for determining motion response of a tubular pile in saturated soil under the action of seismic P waves is characterized by comprising the following steps:
determining the amplitude w of the movement of the bedrock under the action of seismic P-wavesgAnd obtaining the excitation circle frequency omega according to the excitation frequency f, wherein omega is 2 pi f;
determining the outer radius r of the tubular pileOInner radius r of tubular pileILength L of tubular pile and density rho of tubular pilepElastic modulus E of tubular pilepPoisson ratio upsilon of tubular pilepAnd calculating the shear modulus mu of the pipe pilepCross section area A of tubular pilepPolar inertia moment J of tubular pilep；
Determining the elastic modulus E of the soil framework of the external or internal area of the pipe pileskPoisson ratio upsilon of soil body framework of tubular pile outer domain or tubular pile inner domainskCalculating the shear modulus G of the soil body framework of the external domain or the internal domain of the pipe pilesk(ii) a Determining hysteretic damping ratio eta of soil framework in external domain or internal domain of tubular pileskCalculating the complex Ramse constant mu of the soil framework of the external domain or the internal domain of the tubular pileskAnd λsk(ii) a Determining saturation of external or internal tubular pile domainsAnd the volume fraction n of the soil skeleton in the soilskTrue density rho of soil framework of outer domain or inner domain of tubular pilesRkCalculating the volume density rho of the soil framework of the external area or the internal area of the pipe pilesk(ii) a The subscript k of each parameter is I or O, wherein I represents that the current parameter is the parameter of the inner domain of the tubular pile, and O represents that the current parameter is the parameter of the outer domain of the tubular pile;
determining the true density rho of pore fluid in the external or internal area of the pipe pilefRkVolume fraction n of pore fluid in saturated soil in external or internal area of pipe pilefkCalculating the volume density rho of pore fluid in the external domain or the internal domain of the pipe pilefkLiquid-solid coupling coefficient s of external or internal tubular pile domainsvk；
Based on the Hamilton dynamics principle, a motion equation of the two-dimensional tubular pile considering radial and vertical deformation under steady-state vibration is established, the established motion equation is converted by adopting a variable separation method, then a coefficient determinant of the equation obtained by conversion is zero, and a characteristic value eta is obtained by solvingjJ is 1,2,3, 4; then eta is addedjSubstituting into the equation obtained by conversion to obtain the sum etajOne-to-one correspondence of feature vectors
Figure FDA0003318921530000011
Determining the radial displacement and the vertical displacement of the outer surface of the tubular pile according to the following formula:
Figure FDA0003318921530000012
wherein u isp(z) is the radial displacement of the outer surface of the tubular pile, wp(z) is the vertical displacement of the outer surface of the tubular pile, z is an axial coordinate, and e is a natural constant; intermediate variables
Figure FDA0003318921530000021
Figure FDA0003318921530000022
Figure FDA0003318921530000023
Figure FDA0003318921530000024
The external load amplitude of the top of the tubular pile is shown; intermediate variables
Figure FDA0003318921530000025
Figure FDA0003318921530000026
R＝(m1m8-m2m7)(m1m6-m3m4)-(m1m9-m3m7)(m1m5-m2m4)；
Figure FDA0003318921530000027
Figure FDA0003318921530000028
Figure FDA0003318921530000029
Figure FDA00033189215300000210
Figure FDA00033189215300000211
n is an integer, n is 1,2,3, …, ∞;
Figure FDA00033189215300000212
Figure FDA00033189215300000213
Figure FDA00033189215300000214
Figure FDA00033189215300000215
Figure FDA00033189215300000216
Figure FDA0003318921530000031
Figure FDA0003318921530000032
Figure FDA0003318921530000033
Figure FDA0003318921530000034
Figure FDA0003318921530000035
Figure FDA0003318921530000036
Figure FDA0003318921530000037
Figure FDA0003318921530000038
Figure FDA0003318921530000039
Figure FDA00033189215300000310
Figure FDA00033189215300000311
Figure FDA00033189215300000312
Figure FDA00033189215300000313
Figure FDA00033189215300000314
Figure FDA0003318921530000041
Figure FDA0003318921530000042
Figure FDA0003318921530000043
Figure FDA0003318921530000044
Figure FDA0003318921530000045
Figure FDA0003318921530000046
Figure FDA0003318921530000047
Figure FDA0003318921530000048
Figure FDA0003318921530000049
Figure FDA00033189215300000410
Figure FDA00033189215300000411
Figure FDA00033189215300000412
Figure FDA00033189215300000413
Figure FDA0003318921530000051
Figure FDA0003318921530000052
Figure FDA0003318921530000053
Figure FDA0003318921530000054
Figure FDA0003318921530000055
Figure FDA0003318921530000056
Figure FDA0003318921530000057
Figure FDA0003318921530000058
Figure FDA0003318921530000059
Figure FDA00033189215300000510
Figure FDA00033189215300000511
Figure FDA00033189215300000512
mg3＝mg2(m1m5-m2m4)+mg1(m1m8-m2m7)，mg4＝mg2(m1m6-m3m4)+mg1(m1m9-m3m7)，
Figure FDA00033189215300000513
Figure FDA00033189215300000514
Figure FDA0003318921530000061
Figure FDA0003318921530000062
Figure FDA0003318921530000063
Figure FDA0003318921530000064
Figure FDA0003318921530000065
Figure FDA0003318921530000066
Figure FDA0003318921530000067
Figure FDA0003318921530000068
Figure FDA0003318921530000069
Figure FDA00033189215300000610
Figure FDA00033189215300000611
Figure FDA0003318921530000071
Figure FDA0003318921530000072
Figure FDA0003318921530000073
Figure FDA0003318921530000074
Figure FDA0003318921530000075
Figure FDA0003318921530000076
Figure FDA0003318921530000077
Figure FDA0003318921530000078
Figure FDA0003318921530000079
Figure FDA00033189215300000710
r is the radial coordinate, I0() Representing zero-order modified Bessel functions of the first kind, I1() Is a first-order modified Bessel function of the first kind, K0() For zero-order modification of Bessel function of the second kind, K1() A modified Bessel function of a second type;
Figure FDA00033189215300000711
P2n＝α1nC1n+α2nC2n-α3nC3n+α4nC4n+α5nC5n+α6nC6n+wgαgn，P3n＝β1nC1n+β2nC2n-β3nC3n+β4nC4n+β5nC5n+β6nC6n+wgβgn；
Figure FDA0003318921530000081
Figure FDA0003318921530000082
Figure FDA0003318921530000083
Figure FDA0003318921530000084
Figure FDA0003318921530000085
Figure FDA0003318921530000086
Figure FDA0003318921530000087
in the above formula, the first and second carbon atoms are,
Figure FDA0003318921530000088
Figure FDA0003318921530000089
all subscripts n ═ 1,2,3, …, infinity,
Figure FDA00033189215300000810
Figure FDA00033189215300000811
Figure FDA00033189215300000812
Figure FDA0003318921530000091
Figure FDA0003318921530000092
the upper subscript k and the subscript k of each parameter are I or O, wherein I represents that the current parameter is the parameter of the inner domain of the tubular pile, and O represents that the current parameter is the parameter of the outer domain of the tubular pile.
2. The method for determining the motion response of the tubular pile in the saturated soil under the action of the seismic P wave as claimed in claim 1, wherein the axial force and the shearing force of the tubular pile are calculated according to the following formula:
Figure FDA0003318921530000093
Figure FDA0003318921530000094
wherein, N (z) is the axial force of the pipe pile, and Q (z) is the shearing force of the pipe pile.
3. The method for determining the motion response of the tubular pile in the saturated soil under the action of the seismic P wave as claimed in claim 1, wherein the displacement and the stress component of the saturated soil in the external domain of the tubular pile are calculated according to the following formulas:
Figure FDA0003318921530000095
Figure FDA0003318921530000096
Figure FDA0003318921530000097
Figure FDA0003318921530000098
Figure FDA0003318921530000099
Figure FDA0003318921530000101
wherein u issORadial displacement of the soil skeleton of the external area of the pipe pile, wsOVertical displacement of the soil skeleton of the external area of the pipe pile, ufOIs the radial displacement of the pore fluid of the external area of the tube pile,
Figure FDA0003318921530000102
representing the effective stress component acting on the soil framework of the external domain of the tube pile in the radial plane and then pointing to the vertical direction,
Figure FDA0003318921530000103
represents the radial total stress component, p, of the saturated soil of the external area of the tubular pilefOThe pore fluid pressure of the external area of the pipe pile is expressed;
Figure FDA0003318921530000104
and all the upper and lower subscripts n are 1,2,3, … and infinity, the upper and lower subscripts k of each parameter are I or O, I represents that the current parameter is the parameter of the inner domain of the tube pile, and O represents that the current parameter is the parameter of the outer domain of the tube pile.
4. The method for determining the motion response of the tubular pile in the saturated soil under the action of the seismic P waves as claimed in claim 1, wherein the displacement and stress components of the saturated soil in the inner region of the tubular pile are calculated according to the following formula:
Figure FDA0003318921530000105
Figure FDA0003318921530000106
Figure FDA0003318921530000107
Figure FDA0003318921530000108
Figure FDA0003318921530000109
Figure FDA00033189215300001010
wherein u issIIs the radial displacement, w, of the soil skeleton in the inner region of the pipe pilesIIs the vertical displacement of the soil body skeleton in the inner area of the tubular pile, ufIIs the radial displacement of pore fluid in the inner area of the pipe pile,
Figure FDA0003318921530000111
representing the effective stress component of the soil framework acting on the radial plane and then pointing to the inner area of the vertical pipe pile,
Figure FDA0003318921530000112
represents the radial total stress component, p, of the saturated soil in the inner area of the pipe pilefIIndicating the pore fluid pressure in the interior of the tube stake.
5. The method for determining the motion response of the tubular pile in the saturated soil under the action of the seismic P waves as claimed in any one of claims 1 to 4, wherein the shear modulus mu of the tubular pilepCross section area A of tubular pilepPolar inertia moment J of tubular pilepThe following formula is adopted for calculation:
μp＝Ep/2/(1+υp)；
Figure FDA0003318921530000113
Figure FDA0003318921530000114
6. the method for determining the motion response of the tubular pile in the saturated soil under the action of the seismic P waves according to any one of claims 1 to 4, wherein the shear modulus G of the soil body framework of the outer domain or the inner domain of the tubular pileskThe following formula is adopted for calculation:
Gsk＝Esk/2/(1+υsk)；
the complex Ramse constant mu of the soil framework of the external domain or the internal domain of the tubular pileskAnd λskThe following formula is adopted for calculation:
μsk＝Gsk(1+2iηsk)；
λsk＝2υskμsk/(1-2υsk)；
the volume density rho of the soil body framework of the external pipe pile domain or the internal pipe pile domainskUsing the formula rhosk＝ρsRknskAnd (4) calculating.
7. The method for determining the motion response of the tubular pile in the saturated soil under the action of the seismic P waves according to any one of claims 1 to 4, wherein the volume density rho of pore fluid in the outer domain or the inner domain of the tubular pile isfkLiquid-solid coupling coefficient s of external or internal tubular pile domainsvkThe following formula is adopted for calculation:
ρfk＝ρfRknfk；
svk＝nfkρfkg/kfk；
wherein g is the acceleration of gravity, kfkIs the Darcy permeability coefficient.
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Cited By (1)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116306177A (en) * 2023-05-18 2023-06-23 石家庄铁道大学 Method and device for analyzing earthquake response of river-valley-crossing terrain bridge and terminal equipment

Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN116306177A (en) * 2023-05-18 2023-06-23 石家庄铁道大学 Method and device for analyzing earthquake response of river-valley-crossing terrain bridge and terminal equipment
CN116306177B (en) * 2023-05-18 2023-08-04 石家庄铁道大学 Method and device for analyzing earthquake response of river-valley-crossing terrain bridge and terminal equipment

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