CN112199905B - Method for determining axisymmetric dynamic response of two-dimensional socketed pile in saturated soil - Google Patents

Method for determining axisymmetric dynamic response of two-dimensional socketed pile in saturated soil Download PDF

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CN112199905B
CN112199905B CN202011057523.1A CN202011057523A CN112199905B CN 112199905 B CN112199905 B CN 112199905B CN 202011057523 A CN202011057523 A CN 202011057523A CN 112199905 B CN112199905 B CN 112199905B
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张石平
张军辉
林晨
徐站
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Changsha University of Science and Technology
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Abstract

The invention discloses a method for determining axisymmetric dynamic response of a two-dimensional socketed pile in saturated soil, which comprises the steps of firstly establishing and solving a motion equation of a two-dimensional elastic pile considering radial deformation based on a Hamilton dynamics principle to obtain a homogeneous solution of the displacement of the pile containing unknown constants; then, establishing and solving a motion equation of the soil based on a Boer porous medium theory to obtain a general solution of displacement and stress of the soil containing unknown constants; after the boundary condition and the continuity condition of the pile-soil system are uniformly determined, the general solution of the pile foundation displacement containing the unknown constant is obtained based on the boundary condition of the soil body, the general solution of the displacement and the stress of the soil containing the unknown constant and the homogeneous solution of the displacement of the pile containing the unknown constant; and determining all unknown constants in the pile foundation displacement general solution according to the boundary condition of the pile foundation and the continuity condition between pile soils to obtain the solution of the pile foundation displacement, and further obtaining the frequency domain dynamic impedance and the time domain speed response of the pile top, wherein the obtained result is more reasonable and accurate.

Description

Method for determining axisymmetric dynamic response of two-dimensional socketed pile in saturated soil
Technical Field
The invention belongs to the technical field of pile-soil interaction research, and relates to a method for determining axial symmetry dynamic response of a two-dimensional rock-socketed pile in saturated soil by considering radial deformation influence.
Background
Under the action of vertical harmonic load, the dynamic interaction of soil and the rock-socketed pile plays an important role in geotechnical engineering, earthquake engineering and structural engineering. The method is characterized in that a foundation is simulated as an elastic continuum, piles are simulated as beams, a basic solution is obtained by solving a motion equation of an elastic medium, pile-soil interaction problems are solved according to boundary conditions and continuity conditions between pile soil, the problems are generally expressed by an impedance function or a speed response of a pile-soil system, and the method can be used for dynamic design of structures and the foundation or integrity detection of pile foundations and the like.
Since Tajimi pioneered research on dynamic response of single piles in elastic soil layers in 1969, researchers put forward that classical processing methods such as a plane strain method, a radial simplification method, a potential function method, a Green function method and the like carry out extensive research on dynamic interaction of the piles and a pure elastic medium. By assuming soil as a porous medium containing saturated fluid, researchers have conducted a great deal of research on the problem of axisymmetric interaction between a pile and saturated soil under axial load by using the method described above. However, in these methods, the plane strain method uses a plane strain model to simulate the soil around the pile, and assumes that the soil around the pile is in a plane strain state, i.e., the soil around the pile does not deform along the radial direction, and the deformation generated along the vertical direction does not change with the depth, so that the method is greatly simplified. The radial simplification method neglects the radial deformation of the soil body, only considers the vertical displacement of the soil body, and the potential function method considers the soil body as a three-dimensional continuous medium with limited thickness containing the radial deformation and the vertical deformation and solves the equation by using the potential function. The three methods are strong in analysis and convenient for engineering practice, but because the methods are originally proposed aiming at the assumption of monophasic soil, some methods such as a plane strain method and a radial simplification method ignore the radial deformation influence of the soil. Therefore, it is necessary to compare the feasibility and the range of the application in the saturated soil. The Green function method generally treats a soil body as a three-dimensional elastic semi-infinite semi-space medium, and solves a related control equation of the pile and the soil by adopting an integral transformation method, which is a numerical method with large calculation amount and is mainly used for the analysis of the floating pile.
On the other hand, in the above-described study, the socketed pile was assumed to be a one-dimensional structure. Thus, the problem of the effect of radial deformation and bottom reaction forces of the pile on the mechanical interaction of the rest of the surrounding medium has not been solved. However, the authors Pak and Gobert (1993), Masoumi et al (2007), and Masoumi and digrande (2008) have demonstrated in practical terms that we need to take into account the effects of the above factors by conducting careful studies on the pile subjected to axial loading and fully embedded in the elastic medium. Although basic structural theories are widely used in engineering practice because of their mathematical simplicity and practical value, their use in this type of structure-continuum interaction problem suffers from some fundamental drawbacks, particularly when the aspect ratio of the structure is not large enough. For example, the axial and radial displacements of the embedded parts are generally dependent on the tangential and lateral boundary forces exerted on them by the surrounding medium, however, the basic rod theory can only describe axial deformation due to longitudinal loading, in fact, due to the poisson effect, there are also radial deformations and compression of the soil on the pile, which neglects radial deformation, so the basic rod theory essentially suppresses appropriate lateral displacement and radial surface force effects between the pile and the soil, but the basic rod theory regards the pile foundation as a one-dimensional structure with only axial deformation, neglects deformation of the pile foundation in the radial direction, and cannot reasonably describe the actual deformation of the pile foundation for short piles with larger diameters, thereby resulting in inaccurate results. Since such pile foundations undergo significant radial deformation in addition to axial deformation, the basic rod theory cannot in this case accurately describe the pile-soil interaction. In addition to the above-mentioned risk of non-compliance with physical laws, this approximation method also poses serious limitations on correlation analysis of some important issues, such as the influence of radial stress distribution and poisson effect on the system response.
Disclosure of Invention
The embodiment of the invention aims to provide a method for determining the axisymmetric dynamic response of a socketed pile considering the influence of radial deformation in saturated soil, so as to solve the problem that the axisymmetric interaction of the pile and the saturated soil under the action of axial load is inaccurate due to the fact that the radial deformation of the pile and the soil is not considered in the conventional method for researching the axisymmetric interaction of the pile and the saturated soil under the action of axial load.
The technical scheme adopted by the embodiment of the invention is that the method for determining the axisymmetric dynamic response of the two-dimensional socketed pile in the saturated soil is carried out according to the following steps:
s1, establishing a cylindrical coordinate system, analyzing the stress condition of the two-dimensional socketed pile foundation in saturated soil, determining a calculation formula of axial force and shearing force along the pile body, establishing a motion equation of the two-dimensional elastic pile considering radial deformation based on the Hamilton dynamics principle, then solving the motion equation of the two-dimensional elastic pile considering radial deformation, and obtaining a homogeneous solution of the displacement of the pile containing unknown constants;
s2, establishing a soil motion equation based on the Boer porous medium theory, and solving the soil motion equation to obtain a general solution of the displacement and the stress of the soil containing unknown constants;
step S3, uniformly determining boundary conditions and continuity conditions of a pile-soil system, and then obtaining a general solution of pile foundation displacement containing unknown constants based on the boundary conditions of a soil body, the general solution of the displacement and stress of soil containing unknown constants and the homogeneous solution of the displacement of piles containing unknown constants;
and S4, determining all unknown constants in the pile foundation displacement general solution according to the boundary condition of the pile foundation and the continuity condition between pile soils to obtain the definite solution of the pile foundation displacement, and determining the definite solution of the axial force and the shearing force of the pile and the frequency domain dynamic impedance and the time domain speed response of the pile top based on the definite solution of the pile foundation displacement.
The method has the advantages that the method for determining the dynamic response of the saturated soil and the rock-socketed pile considering the radial deformation of the pile soil under the action of the vertical load is provided, the soil is assumed to be a three-dimensional porous continuous medium, the mechanical behavior of the soil is described by using a Boer porous medium theory, the pile is regarded as a two-dimensional rod with radial and vertical deformation, and the equation of motion of the pile is obtained by using a Hamilton variation principle; under the condition of not introducing a potential function, firstly processing a motion equation of soil by taking the volume strain of a soil framework and the pore fluid pressure as intermediate variables, and then further solving the motion equation of the pile and the soil by separating the variables; and combining boundary conditions and continuity conditions of a pile-soil system to obtain displacement and stress solutions of soil and the pile, and further obtaining frequency domain dynamic impedance and time domain dynamic response of the pile top. Compared with the existing method, the embodiment of the invention considers the radial and vertical deformation of the saturated soil body and the pile foundation simultaneously, can more accurately describe the actual deformation state of the pile soil and truly reflect the actual working state of the pile soil, so that the obtained result is more reasonable and accurate, the calculation and application are convenient, and the problem of inaccurate axial symmetry interaction of the pile and the saturated soil under the axial load action caused by the fact that the axial symmetry interaction of the pile and the saturated soil is not considered in the existing axial load action axial symmetry interaction research method is effectively solved. And from the results obtained the radial deformation of the pile soil has a significant influence on the dynamic response of the pile soil system.
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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 mechanical model schematic diagram of a two-dimensional pile in saturated soil under the action of a vertical load.
FIG. 2 is a schematic diagram of a mechanical model of a two-dimensional pile in saturated soil under the action of axisymmetric loads.
Fig. 3 is a schematic diagram of an axisymmetric finite element model of the pile-soil system built in ADINA.
FIG. 4 is a comparison graph of axisymmetric dynamic response of a socketed pile in saturated soil obtained by solving the problems of the embodiment of the invention and a finite element numerical method.
FIG. 5 is a comparison graph of dynamic impedance of the solution of the embodiment of the present invention and the simplified model method, the planar strain method.
Fig. 6 is a diagram comparing the pile top velocity response of the embodiment of the present invention with the solutions of the simplified model method and the plane strain method.
FIG. 7 is a static stiffness comparison graph of a solution of an embodiment of the present invention and a simplified model method, a planar strain method.
FIG. 8 is a graph comparing the dynamic impedance of solutions of two-dimensional piles and one-dimensional piles of an embodiment of the present invention.
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.
The embodiment of the invention provides a method for determining axial symmetry dynamic response of a rock-socketed pile considering radial deformation influence in saturated soil, which comprises the following specific steps:
1. based on the Hamilton dynamics principle, an equation of motion of the two-dimensional elastic pile considering radial deformation and vertical deformation is established:
as shown in FIG. 1, one has an elastic modulus EpPoisson ratio is upsilonpDensity of rhopRadius of r0The elastic pile with the length of L is completely embedded into a soil layer which is uniform, saturated by fluid, porous and elastic and is positioned above bedrock, a cylindrical coordinate system is established by taking the center of the pile top as the center of a circle, and the center of the pile top, namely the position where r is 0 and z is 0, is subjected to vertical harmonic excitation force P (t) is P0eiωtThe function of (A) is to perform,
Figure BDA0002711244000000041
P0amplitude of external load p (t), eiωtFor the time factor, the circular frequency ω of the exciting force is 2 pi f, f represents the frequency of the exciting force, r is a radial coordinate in the cylindrical coordinate system, and z is a vertical coordinate in the cylindrical coordinate system.
Furthermore, the basic assumptions of this pile-soil system are as follows: (1) the soil particles and pore fluid are microscopically incompressible, there is no mass and heat exchange between the soil and the liquid; (2) neglecting viscosity and gravity of the pore fluid; (3) the resistance between the soil framework and pore fluid is in direct proportion to the relative speed of liquid and solid; (4) the pores in the soil are uniformly distributed; (5) the vibration of the pile soil system is small deformation vibration; (6) during the vibration process, the pile and the soil are always closely connected at the interface.
An equation of motion for the pile is first established that considers radial and vertical deformation. Under such loading, the pile-soil system is axisymmetric about the z-axis, as shown in fig. 1. For convenience, willThe pile is separated from the pile-soil system, the stress condition of the pile is obtained as shown in figure 2, and the pile is subjected to the radial surface force t on the pile siderPile side axial force tzPile top shearing force Q (0) and pile top axial force N (0), pile side radial surface force trAnd pile side axial face force tzIs the force from the earth to which the pile is subjected. Due to the axially symmetric nature, all variables are independent of the circumferential coordinate θ and angular displacement, i.e., displacement occurring in the θ direction, is ignored.
For an embedded pile with a slenderness ratio and enough rigidity, under the action of axial load, mechanical behaviors such as axial compression and displacement are mainly generated. In view of this, it is reasonable to express the longitudinal displacement of the pile as a function related only to z. However, during axial compression of the pile, a corresponding radial displacement field is clearly generated due to the poisson effect. In addition, because of the lateral constraint of the surrounding soil body, the surface force is generated on the peripheral surface of the pile, and the pile can be subjected to a larger internal radial compression effect. As a preliminary attempt to take the above factors into account without introducing unnecessary complexity, a first non-trivial approximation of the radial variation of the axial displacement of the pile over its cross-section is used. In addition to the mechanical dependencies described above, these kinematic assumptions have also proven to model the radial shear phenomena, which are important in the wave propagation problem. Therefore, firstly:
Figure BDA0002711244000000051
wherein u isp(r, z) represents the radial displacement of any point of the pile foundation, wp(r, z) represents axial displacement of pile foundation at any point, and up(r, z) and wp(r, z) are functions of radial coordinate r and axial coordinate z; u. up(z) denotes radial displacement of the pile flanks, wp(z) represents axial displacement of the pile sides; r is0Representing the pile foundation radius. It can be seen from formula (1) that when the pile foundation is deformed by a vertical external load, the cross section of the pile foundation is kept as a plane, so that the vertical displacements on the cross section are all equal; while the radial deformation varies linearly along the cross-section, i.e. pile foundation centreWhen r is 0, the radial displacement on the axis is maximum, while the radial displacement of other points on the cross section keeps linear variation relation with the radial coordinate r. The inside vertical displacement and the radial displacement that contain of pile foundation to this vertical displacement and radial displacement all are the function of axial coordinate z and radial coordinate r, and formula (1) is just modeling according to the atress characteristics of pile foundation under the vertical load effect, handles the inside displacement field of pile foundation and turns into the displacement that only needs to obtain pile foundation outside surface, can confirm the displacement of pile foundation arbitrary point.
The axial force N (z) and the shearing force Q (z) along the pile body are as follows:
Figure BDA0002711244000000052
wherein,
Figure BDA0002711244000000053
is the axial stress on the cross section of the pile foundation,
Figure BDA0002711244000000054
shear stress on the cross section of the pile foundation; mu.spLame constant, mu, of pile foundationp＝Ep/2/(1+υp) Representing the Lame constant of the pile foundation; a. thepIs the cross-sectional area of the pile foundation,
Figure BDA0002711244000000055
j represents the polar moment of inertia of the pile foundation,
Figure BDA0002711244000000056
the elastic potential energy of the pile foundation is as follows:
Figure BDA0002711244000000057
the kinetic energy of the pile is:
Figure BDA0002711244000000061
wherein σpRepresenting the stress tensor, epsilon, of the pilepThe strain tensor of the pile is represented,
Figure BDA0002711244000000062
and
Figure BDA0002711244000000063
is a specific component; v. ofpIs the velocity vector of the pile and the point on the displacement sign represents the derivation of t, i.e.
Figure BDA0002711244000000064
The radial velocity of the pile side is indicated,
Figure BDA0002711244000000065
representing axial velocity of the pile side; rhopAnd the pile body density is shown. V is the volume of the pile,
Figure BDA0002711244000000066
indicating a positive stress acting on the r-plane and in the r-direction,
Figure BDA0002711244000000067
represents the positive strain acting on the r-plane and in the radial direction;
Figure BDA0002711244000000068
represents the shear stress acting on the r-plane and in the z-direction,
Figure BDA0002711244000000069
representing the shear strain acting in the r-plane and in the z-direction,
Figure BDA00027112440000000610
representing a positive stress acting in the theta plane and in the theta direction,
Figure BDA00027112440000000611
representing a positive strain acting in the theta plane and in the theta direction,
Figure BDA00027112440000000612
representing a positive strain acting in the z-plane and in the z-direction.
Because the pile is subjected to external forces such as surface force of soil around the pile, pile top load, pile bottom counter force and the like, the non-conservative force does work as follows:
Figure BDA00027112440000000613
wherein, tpRepresenting the external force vector, u, experienced by the entire pile surfacepA displacement vector representing the entire pile surface; u. ofp(0) Indicating radial displacement of pile head on pile-foundation side surface, up(L) represents the radial displacement of the pile bottom on the pile foundation side surface; q (l) represents pile bottom shear force.
The virtual work of the external load can be expressed as (Morse and Feshbach, 1953):
Figure BDA00027112440000000614
wherein, delta is a variation symbol, delta up(z) represents upVariation of (z), δ wp(z) represents wp(z) variation.
According to the Hamilton kinetic principle, the method comprises the following steps:
Figure BDA00027112440000000615
wherein T represents the kinetic energy of the pile, P represents the elastic potential energy of the pile, and T1、t2Two different time points are arbitrarily taken during the pile movement process.
According to the Hamilton dynamics principle, the motion equation of the pile is obtained as follows:
Figure BDA0002711244000000071
since the current consideration is the factor e over timeiωtThe changing steady state vibration, equation (8) can be further written as:
Figure BDA0002711244000000072
wherein f isr(z) represents the total load in the radial direction due to the pile foundation inertia and the soil reaction force, fz(z) represents the total load generated by pile foundation inertia force and soil reaction force in the axial direction;
Figure BDA0002711244000000073
trthe pile side radial surface force is obtained;
Figure BDA0002711244000000074
tzis the pile flank axial face force. For simple analysis, the time factor e is uniformly omitted lateri ωt。
Suppose up(z)＝Upeηz，wp(z)＝Wpeηz，UpAmplitude, W, of a distribution form function of radial displacement of the pile side along an axial coordinate zpThe amplitude of the distribution form function along the axial coordinate z is the axial displacement of the pile side surface, and eta represents the characteristic value of the distribution form function. The expression converts the solving equivalence of the displacement of the pile foundation into solving eta and Up、Wp。
Let tr＝0、t z0, and substituting them into formula (9):
Figure BDA0002711244000000075
if the equation (10)) has a non-trivial solution, its coefficient matrix determinant value is required to be zero, i.e.:
Figure BDA0002711244000000076
a homogeneous solution of the displacement of the pile can thus be obtained:
Figure BDA0002711244000000081
wherein Dj(1. ltoreq. j. ltoreq.4) represents 4 unknown constants, solution eta of equation (11)j(1. ltoreq. j. ltoreq.4) is a characteristic value of the coefficient matrix of formula (10), hj(1. ltoreq. j. ltoreq.4) and kj(1. ltoreq. j. ltoreq.4) is the corresponding characteristic value etajElement of the feature vector of (1), ηjContaining 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 BDA0002711244000000082
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.
2. The method comprises the following steps of establishing a motion equation of soil (soil around a pile) based on a Boer porous medium theory, solving the motion equation of the soil to obtain a general solution of displacement and stress of the soil containing unknown parameters, and specifically realizing the following steps:
1) based on the Boer porous medium theory, an equation of motion of soil under an axisymmetric condition is established:
the Boer porous medium theory can describe the mechanical behavior of the saturated soil layer, so that the power control equation of the saturated soil layer can use the displacement vector u of the soil frameworksPore fluid displacement vector ufAnd pore fluid pressure pfExpressed as:
Figure BDA0002711244000000083
Figure BDA0002711244000000084
Figure BDA0002711244000000085
wherein λ issAnd musLame constant, which represents the earth skeleton, is well known in the art, lambdas＝2υsμs/(1-2υs)，μs＝Gs，GsIs the shear modulus, upsilon, of the soil skeletonsIs the poisson's ratio of the soil framework; rhofDenotes the bulk density, ρ, of the pore fluidf＝ρfRnf；ρsRepresenting the bulk density, ρ, of the soil skeletons＝ρsRnsWhere ρ issRRepresenting the true density, ρ, of the earth skeletonfRRepresenting the true density of the pore fluid, nsRepresenting the volume fraction of the soil skeleton, nfRepresents the volume fraction of pore fluid; svIs the liquid-solid coupling coefficient, sv＝nfρfg/kfDenotes the interaction of the soil skeleton and pore fluid, wherein kfThe permeability coefficient of soil Darcy (Darcy) and g is the gravity acceleration;
Figure BDA0002711244000000086
the expression of the gradient operator is used to indicate,
Figure BDA0002711244000000087
representing a divergence operator; one point on the displacement vector symbols represents the first derivative of these symbols with respect to time t, and two points on the displacement vector symbols represent the second derivative of these symbols with respect to time t; formula (13a) represents the momentum balance of the solid phase, i.e., the soil framework, figure (13b) represents the momentum balance of the liquid phase, i.e., the pore fluid, and formula (13c) represents the mass balance of the liquid-solid mixture, i.e., the soil framework-pore fluid system.
On-axis symmetrical (pile-soil system is symmetrical about z axis) stripUnder the condition, the field variable is independent of the angular coordinate theta and the angular displacement vs、vfZero, so equation (13) can be further written as the following components:
Figure BDA0002711244000000091
Figure BDA0002711244000000092
Figure BDA0002711244000000093
Figure BDA0002711244000000094
Figure BDA0002711244000000095
wherein e issIs the volume strain of the soil framework,
Figure BDA0002711244000000096
usis the radial displacement (component) of the soil skeleton, wsIs the vertical displacement (component) of the soil skeleton; u. offIs the radial displacement (component) of the pore fluid, wfIs the vertical displacement (component) of the pore fluid;
Figure BDA0002711244000000097
is the laplacian operator, and is,
Figure BDA0002711244000000098
2) processing a motion equation of soil by taking the volume strain of a soil framework and the pore fluid pressure as intermediate variables, solving the motion equation of the soil body by adopting a variable separation method, and obtaining a general solution of the displacement and the stress of the soil containing unknown parameters:
from the expressions (14c) to (14d), it is possible to obtain:
Figure BDA0002711244000000099
Figure BDA00027112440000000910
the simultaneous process between the expressions (14a) to (14b) is abbreviated as
Figure BDA00027112440000000911
Substituting the formulas (15a) to (15b) into the formula (16) to obtain the volume strain and pore fluid pressure of the soil skeleton:
Figure BDA00027112440000000912
wherein the intermediate variable
Figure BDA0002711244000000101
Similarly, the simultaneous processes between the expressions (15a) to (15b) are abbreviated as
Figure BDA0002711244000000102
Substituting the formula (14e) into the formula (17), wherein the volume strain and the pore fluid pressure of the soil framework are obtained:
Figure BDA0002711244000000103
wherein the intermediate variable
Figure BDA0002711244000000104
Substituting formula (17) for formula (16) by:
Figure BDA0002711244000000105
wherein the intermediate variable alpha2＝a2-a1a3。
By a separation variable method, let esR (r) Z (z), i.e. to esPerforming variable separation, wherein R (r) and Z (z) are unknown functions, and are determined by the following derivation and substituted into equation (18), to obtain:
Figure BDA0002711244000000106
wherein, b1、b2Is an unknown constant and satisfies
Figure BDA0002711244000000107
The solution of equation (19) is:
Figure BDA0002711244000000108
wherein A is1、A2、B1And B2Is an unknown constant, I0(. represents a zero-order transformed Bessel function of the first kind, K0(. cndot.) denotes a zero-order, second-class, deformed Bessel function.
Then:
es＝[A1K0(b1r)+A2I0(b1r)][B1sin(b2z)+B2cos(b2z)]； (21)
it can be derived from equation (17):
Figure BDA0002711244000000109
the formula (22) is a heterogeneous equation, the general solution of the heterogeneous equation comprises general solution of the homogeneous equation and special solution of the heterogeneous equation, and the homogeneous solution of the formula (22) is obtained by using a separation variable method:
pfh＝[A3K0(b3r)+A4I0(b3r)][B3sin(b3z)+B4cos(b3z)]； (23)
wherein, b3、A3、A4、B3And B4Are unknown constants.
Let the special solution of equation (22) be:
pft＝T1es＝T1[A1K0(b1r)+A2I0(b1r)][B1sin(b2z)+B2cos(b2z)]； (24)
since p is required to ensure that equation (22) holdsfSpecial solution form of (c) and (e)sAccordingly, the formula (24) is defined. Formula (24) represents pfSpecial solution of and esAre of uniform form but of different size, intermediate parameter T1This difference in size is quantified.
Substituting equation (24) into equation (22) yields:
Figure BDA0002711244000000111
therefore, there are:
Figure BDA0002711244000000112
volume strain e of soil skeletonsAnd pore fluid pressure pfHaving found that the volume strain e in equation (14)sAnd pore fluid pressure pfMoving to the right of the equation, this pattern (14) is a shifted heterogeneous equation, and it has been described in the foregoing how to solve the general solution of the heterogeneous equation (first solving the homogeneous solution by the discrete variable method, and then adding the special solution) to similarly solve for pfBy solving equation (14), a general solution of all field variables in the equation of motion in saturated soil can be obtained (the volume strain e of the soil framework obtained in the foregoing is usedsAnd pore fluid pressure pfThe radial displacement u related to the soil framework is obtained by substituting the formula (15a) into the formula (14a)sThen adopting similar solution to solve the volume strain e of the soil frameworksAnd pore fluid pressure pfThe non-homogeneous differential equation can be solved to obtain usThe solution of (1); subjecting the obtained volume strain e of the soil frameworksAnd pore fluid pressure pfThe solution of (2) and the substitution of the formula (15b) into the formula (14b) are arranged to obtain the vertical displacement w related to the soil frameworksThen adopting similar solution to solve the volume strain e of the soil frameworksAnd pore fluid pressure pfThe non-homogeneous differential equation can be solved to obtain wsThe solution of (1); the resulting pore fluid pressure pfAnd radial displacement u of the soil skeletonsVertical displacement w of earth skeletonsCan be substituted into the formula (15a-15b) respectively to obtain the radial u of the pore fluidfAnd the vertical displacement w of the pore fluidf) And solving to obtain the following steps:
(1) and (3) soil displacement component:
Figure BDA0002711244000000121
Figure BDA0002711244000000122
Figure BDA0002711244000000123
Figure BDA0002711244000000124
(2) component of stress in the soil
Figure BDA0002711244000000125
Figure BDA0002711244000000126
Figure BDA0002711244000000127
Wherein,
Figure BDA0002711244000000128
the effective stress component of the soil framework acting on the r plane, namely the plane perpendicular to the r axis of the radial coordinate axis and pointing to the r direction is represented;
Figure BDA0002711244000000131
the effective stress component of the soil framework acting on a z plane, namely a plane vertical to the z axis of the vertical coordinate axis and pointing to the z direction is represented;
Figure BDA0002711244000000132
representing the effective stress component of the soil framework acting on the r plane and then pointing to the z direction;
Figure BDA0002711244000000133
the effective stress component of the soil framework acting on the z plane and then pointing to the r direction is represented. In the embodiment of the invention, a stress-strain relation model of linear elasticity is adopted to describe the mechanical behavior of the soil framework, the stress-strain relation model of linear elasticity is well known in the field of mechanics, and a specific expression can be written as sigmas＝2μsEs+λsesI, where σsRepresenting stress tensor of soil framework and containing components under cylindrical coordinate system
Figure BDA0002711244000000134
Such as described above
Figure BDA0002711244000000135
I.e. the components when i ═ r and j ═ r, the other components are the same; i is a unity diagonal matrix, well known in the mathematical art, EsIs tensor of strain of soil skeleton, and
Figure BDA0002711244000000136
wherein u issRepresents the displacement vector of the soil framework, and u is under the condition of axial symmetrysWith only two displacement components, i.e. u, both radial and verticals＝(us,ws) Where the symbol T in the upper right corner represents the transpose of the matrix. On the basis of obtaining displacement solutions of the soil framework and the pore fluid, the displacement solutions are only required to be substituted into the stress-strain relation model of the linear elasticity to obtain the displacement solution
Figure BDA0002711244000000137
b1～b7,A1～A8And B1～B8Is a constant number, b1～b7,A1～A8And B1～B8All are obtained according to boundary conditions and continuity conditions of piles and soil. I is1(. represents a first order deformed Bessel function of the first kind, K1(. cndot.) respectively represents a first-order transformed Bessel function of the second kind.
Figure BDA0002711244000000138
Figure BDA0002711244000000139
B can be determined from the boundary and continuity conditions (28a) to (28g)5、b7To determine b therefrom4、b6(ii) a Intermediate variables
Figure BDA00027112440000001310
Figure BDA00027112440000001311
3. Determining boundary conditions and continuity conditions of a pile-soil system, and obtaining a general solution of pile foundation displacement containing unknown constants based on the boundary conditions of soil bodies, general solutions of soil displacement and stress and a homogeneous solution of pile displacement containing unknown constants.
As shown in fig. 1, the boundary and continuity conditions of the pile-soil system are as follows:
all field variables decay to zero at r → ∞, i.e.:
Figure BDA0002711244000000141
saturated soil layer surface (r is more than or equal to r)0) The normal stress is zero and the surface is permeable, i.e.:
Figure BDA0002711244000000142
the displacement of the saturated soil layer on the rigid bedrock is zero, namely:
ws(r,z＝L,t)＝0，wf(r,z＝L,t)＝0； (27c)
the pile is water impermeable, so the radial displacement of the pore fluid at the pile foundation side surface is equal to the radial displacement of the pile, i.e.:
uf(r0,z,t)＝up(z,t)； (27d)
uf(r0z, t) on the pile-side surface, i.e. r ═ r0Radial displacement of pore fluid.
The pile and the soil are completely bonded at the contact surface, and the displacement is continuous, namely:
us(r0,z,t)＝up(z,t)，ws(r0,z,t)＝wp(z,t)； (27e)
the position of the pile-soil contact surface is r ═ r0The piles and soil are completely bonded at the contact surface, i.e. deformed together without being separated, so that their displacements are equal and continuous, i.e. the radial displacement and vertical displacement of the soil framework at this point are equal to the radial displacement and vertical displacement of the pile foundation side surface, us(r0Z, t) is the radial displacement of the soil skeleton, ws(r0Z, t) is the vertical displacement of the soil skeleton, up(z, t) is the radial displacement of the pile side surface, wp(z, t) is the vertical displacement of the pile side surface.
The load p (t) acts on the pile top, and the surface of the pile top is smooth, namely:
N(z＝0,t)＝p(t)，Q(z＝0,t)＝0； (27f)
n (z is 0, t) represents an axial force at the pile top z is 0, and Q (z is 0, t) represents a shear force at the pile top z is 0.
The displacement of the socketed pile at the bedrock should be zero, i.e.:
up(z＝L,t)＝0，wp(z＝L,t)＝0； (27g)。
when the embodiment of the invention considers the steady load action, each variable can be expressed as f (r, z, t) f (r, z) eiωtSuch that the time factor eiωtBy a common factor, it can be eliminated, so time no longer appears in the formula; if the load is a transient load, the time t cannot be eliminated, so there will be t in the above equation.
Unknown constants in the general solution of the equation of motion of the pile and the soil are determined from the boundary conditions and continuity conditions of the pile-soil system, i.e., equations (27a) to (27 g). Firstly, substituting the vertical displacement equation (formula (A2)) of the soil framework into the boundary condition, namely substituting the formula (A2) into the formula (27c), so as to obtain the characteristic equation cos (b)nL) is 0, then the volume strain equation is used
Figure BDA0002711244000000151
And boundary conditions (27a) - (27b), the solution for which the field variable can be inferred is:
Figure BDA0002711244000000152
Figure BDA0002711244000000153
Figure BDA0002711244000000154
Figure BDA0002711244000000155
Figure BDA0002711244000000156
Figure BDA0002711244000000157
Figure BDA0002711244000000158
wherein,
Figure BDA0002711244000000159
then, the pile-side radial surface force t is obtained by using equations (28e) to (28g) which are general solutions of the soil stressrAnd pile side axial face force tzAnd according to the pile-side radial surface force trPile side axial force tzAnd obtaining the general solution of the displacement of the pile foundation.
The stress at the pile-soil interface is continuous, so that pile side surface forces can be obtained:
Figure BDA00027112440000001510
wherein,
Figure BDA00027112440000001511
for the total radial stress component acting on the pile foundation side surface,
Figure BDA00027112440000001512
which is obtained according to the principle of effective stress.
By substituting equation (29) for equation (9), in conjunction with equations (12), (28e) to (28g), one can deduce that the displacement of the pile is:
Figure BDA0002711244000000161
wherein, C1n～C3nIs an unknown constant, β, determined by boundary conditions and continuity conditions1n～β6nAre all intermediate variables, and
Figure BDA0002711244000000162
Figure BDA0002711244000000163
Figure BDA0002711244000000164
Figure BDA0002711244000000165
Figure BDA0002711244000000166
Figure BDA0002711244000000167
△3n＝c2nc3n+c1nc4nto do so
Figure BDA0002711244000000168
Figure BDA0002711244000000169
4. Determining all unknown constants in the pile foundation displacement general solution according to the boundary condition of the pile foundation and the continuity condition between piles and soils to obtain the definite solution of the pile foundation displacement, and specifically solving the unknown constant C according to the boundary condition of the pile foundation and the continuity condition between piles and soils1n、C2nAnd C3nAnd solving to obtain an unknown constant C1n、C2nAnd C3nThen the solution is driven into (30) to obtain the definite solution of the displacement of the pile foundation and the definite solution of the displacement and the stress of the soil, and the obtained C is solved1n、C2nAnd C3nThe method specifically comprises the following steps:
Figure BDA0002711244000000171
Figure BDA0002711244000000172
Figure BDA0002711244000000173
wherein,
Figure BDA0002711244000000174
P0the amplitude of the external load p (t),
Figure BDA0002711244000000175
Figure BDA0002711244000000176
D1～D4and C1n～C3nThe 7 unknown constants are represented by the boundary condition and continuity condition equations (27d) to (27g), and the function sin (b)nz) and cos (b)nz) orthogonality property
Figure BDA0002711244000000177
And (4) performing extrapolation. Intermediate variables of coefficients in pile foundation displacement solution obtained from pile foundation boundary condition equations (27f) and (27g)
Figure BDA0002711244000000178
Comprises the following steps:
Figure BDA0002711244000000179
Figure BDA00027112440000001710
Figure BDA00027112440000001711
wherein, X1＝h1η1+h1Mq1-k1Mq2，X2＝h2η2+h2Mq3-k2Mq4，X3＝h3η3+h3Mq5-k3Mq6，X4＝h4η4+h4Mq7-k4Mq8，
Figure BDA00027112440000001712
Figure BDA00027112440000001713
Figure BDA00027112440000001714
Figure BDA0002711244000000181
Figure BDA0002711244000000182
Figure BDA0002711244000000183
Figure BDA0002711244000000184
Figure BDA0002711244000000185
Figure BDA0002711244000000186
Intermediate variable Delta of coefficient in pile foundation displacement solution obtained from pile-soil continuity conditional expressions (27d) and (27e)1n＝β1n(γ5nγ9n+γ6nγ8n)-β2n(γ6nγ7n-γ4nγ9n)-β3n(γ4nγ8n+γ5nγ7n)，△2n＝β1n(γ3nγ5n+γ2nγ6n)+β2n(γ3nγ4n-γ1nγ6n)-β3n(γ2nγ4n+γ1nγ5n) Wherein γ is1n＝β1n-T2b1nK1(b1nr0)，γ2n＝β2n-T3bnK1(bnr0)，γ3n＝β3n-K1(b4nr0)，γ4n＝(ρfω2T2α2+nfa3)b1nK1(b1nr0)，γ5n＝α2(nf-ρfω2T3)bnK1(bnr0)，γ6n＝α2ρfω2K1(b4nr0)，γ7n＝T4bnK0(b1nr0)+β4n，γ8n＝T5bnK0(bnr0)+β5n，
Figure BDA0002711244000000187
Rn＝(γ2nγ4n+γ1nγ5n)(γ6nγ7n-γ4nγ9n)+(γ3nγ4n-γ1nγ6n)(γ4nγ8n+γ5nγ7n)；
Figure BDA0002711244000000188
Figure BDA0002711244000000189
Figure BDA00027112440000001810
5. And determining the solution of the axial force and the shearing force along the pile body and the frequency domain dynamic impedance and the time domain speed response of the pile top based on the solution of the pile foundation displacement.
The axial force and the shear force along the pile body can be obtained according to the formula (2) and the formula (30):
Figure BDA0002711244000000191
Figure BDA0002711244000000192
wherein the intermediate variable
Figure BDA0002711244000000193
Figure BDA0002711244000000194
In pile foundation engineering, the dynamic impedance and speed response of the pile top play a crucial role. For example, dynamic impedance is often used to assess the load and energy carrying capacity of pile soil systems, while velocity response can be used to calculate stress wave velocity, examine test data, and detect pile body defects in non-destructive inspection of piles.
Frequency domain dynamic impedance K of piled(ω) is defined as:
Figure BDA0002711244000000195
wherein, KR(omega) represents the true dynamic stiffness and ability of the pile-soil system to resist axial strain, KR(ω)＝real(Kd(ω))；CIRepresenting the vibration radiation damping of the pile-soil system and the damping generated by the relative motion of the pore fluid and the soil framework, reflecting the energy dissipation of the pile-soil system, CI＝imag(Kd(ω))。
When the low dynamic survey that meets an emergency of pile foundation, can simplify pile bolck excitation (hammering load, for transient load) into half sinusoidal pulse load, promptly:
Figure BDA0002711244000000196
wherein T is the pulse width, QmaxIs the pulse amplitude.
The pile top speed frequency domain response under the action of unit strength load
Figure BDA0002711244000000197
Comprises the following steps:
Figure BDA0002711244000000198
therefore, by convolving the pile top load with the unit pile top velocity time domain response, the pile top velocity time domain semi-analytic solution can be expressed as follows:
Figure BDA0002711244000000201
6. numerical results and discussion
The reasonability of the solution of the embodiment of the invention is verified by adopting the pile-soil system and the load parameters shown in the table 1 and through numerical calculation, and the dynamic characteristics of the pile-soil system are researched. It is worth noting that the loading parameter for transient response analysis corresponds to an excitation force generated by a polyethylene hammer in pile foundation integrity detection.
TABLE 1 pile-soil System and load parameters
Figure BDA0002711244000000202
(1) Numerical verification
In order to verify the correctness of the method proposed by the embodiment of the present invention, an axisymmetric finite element model of the pile-soil system in fig. 1 was established by using the ADINA software, and the model is shown in fig. 3. Fig. 4 shows the comparison result of the pile body displacement and pile top velocity response of the method proposed by the embodiment of the invention and the finite element method. In the finite element model, soil is assumed to be a porous elastic material, a soil body is simulated by using a 9-node rectangular unit, the left side of the model is set as an axisymmetric boundary, the surface of the soil layer is set as a free boundary, the bottom of the model is a watertight fixed boundary, and the right side of the model is set as a pore-pressure-free fixed boundary, so that an infinite boundary condition is simulated, thereby keeping the same with the boundary condition specified in fig. 1. It is noted that in this example, a model width of 50m can already obtain the steady-state response amplitude, and the right-side boundary effect is better eliminated. As can be seen from the comparison of FIG. 4, the dynamic response solution of the embodiment of the invention can be well matched with the finite element simulation result, thereby verifying the rationality of the method.
(2) Comparison of solutions
FIG. 5 shows a comparison of the dynamic impedance obtained by the method of the embodiment of the present invention with a simplified model solution, including a radial simplified solution obtained by ignoring the radial displacement of the soil mass, and a plane strain solution obtained by applying a plane strain assumption to the soil mass. It can be seen from the figure that when the excitation frequency f is greater than 5Hz, the dynamic impedance of the embodiment of the invention has good coincidence with the dynamic impedance obtained by the plane strain model; when f is less than or equal to 5Hz, the solution of the embodiment of the invention has larger dynamic stiffness than that of the solution obtained by the plane strain model, and the solution of the embodiment of the invention can reflect the behavior of cut-off frequency, namely, the pile-soil system has no damping. The dynamic impedance of the radial simplified solution is obviously different from the other two methods; in monophasic soil, the dynamic impedance of the embodiment of the invention is basically the same as that of the radial simplified model, and the comparison result of the dynamic impedance of the plane strain model and the embodiment of the invention is similar to that of saturated soil. These conclusions can be further confirmed by comparing the pile tip velocity response and the static impedance of the different methods, as shown in fig. 6 and 7. Furthermore, it can be seen from fig. 7 that as the pile length increases, the static stiffness of the pile eventually tends to a steady value, which indicates that the influence of the pile length on the static resistance is limited. With the increase of the length-diameter ratio of the pile and the modulus ratio of the pile to the soil, the difference between the solution of the embodiment of the invention and the solution of the plane strain model is gradually reduced, and when the rigidity of the pile is increased, the soil body around the pile is obviously more consistent with the plane strain assumption.
The radial deformation influence of the soil body is ignored by both the radial simplification method and the plane strain method, while the radial deformation and the vertical deformation of the pile soil are considered in the embodiment of the invention, as can be seen from fig. 5-6, the solution of the radial simplification method is closer to that of the embodiment of the invention under the condition of single-phase soil, but the radial simplification method generates larger difference under the condition of saturated soil, and cannot be adopted. Although the plane strain method is better compared with the results of the embodiment of the invention for the conditions of single-phase soil and saturated soil at the high-frequency stage, the plane strain method cannot reflect the physical phenomenon of the cutoff frequency of the pile soil system, has defects, and is obviously different from the embodiment of the invention at the low-frequency stage, which means that the plane strain method cannot obtain sufficiently accurate results at the low-frequency stage.
As shown in fig. 8, comparing the dynamic impedance of the two-dimensional pile obtained by the embodiment of the present invention with the dynamic impedance of the one-dimensional pile obtained by Liu, h.l., Zheng, c.j., Ding, x.m., Qin, h.y.,2014, Vertical dynamic response of a pipe pile in a structured soil layer, composite, geotech.,61,57-66, it can be seen that the radial deformation of the pile has a significant effect on the impedance function of the pile. The static stiffness of a two-dimensional pile is less than that of a one-dimensional pile because the lateral boundaries of the one-dimensional pile are fixed. However, the peak dynamic impedance of a two-dimensional pile is larger than that of a one-dimensional pile, which means that assuming the pile as a one-dimensional rod actually underestimates the maximum value of the dynamic impedance of the pile-soil system.
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 (9)

1. A method for determining axisymmetric dynamic response of a two-dimensional socketed pile in saturated soil is characterized by comprising the following steps:
s1, establishing a cylindrical coordinate system, analyzing the stress condition of the two-dimensional socketed pile foundation in saturated soil, determining a calculation formula of axial force and shearing force along the pile body, establishing a motion equation of the two-dimensional elastic pile considering radial deformation based on the Hamilton dynamics principle, then solving the motion equation of the two-dimensional elastic pile considering radial deformation, and obtaining a homogeneous solution of the displacement of the pile containing unknown constants;
s2, establishing a soil motion equation based on the Boer porous medium theory, and solving the soil motion equation to obtain a general solution of the displacement and the stress of the soil containing unknown constants;
step S3, uniformly determining boundary conditions and continuity conditions of a pile-soil system, and then obtaining a general solution of pile foundation displacement containing unknown constants based on the boundary conditions of soil bodies, general solutions of displacement and stress of soil containing unknown constants and homogeneous solutions of displacement of piles containing unknown constants;
s4, determining all unknown constants in the pile foundation displacement general solution according to boundary conditions of the pile foundation and continuity conditions among pile soils to obtain a definite solution of the pile foundation displacement, and determining a definite solution of the axial force and the shearing force of the pile and frequency domain dynamic impedance and time domain speed response of the pile top based on the definite solution of the pile foundation displacement;
the specific implementation process of step S2 is as follows:
step S21, establishing a motion equation of soil under an axisymmetric condition based on the Boer porous medium theory:
Figure FDA0003643450540000011
Figure FDA0003643450540000012
Figure FDA0003643450540000013
Figure FDA0003643450540000014
Figure FDA0003643450540000015
wherein p isfIs pore fluid pressure, λsAnd musLame constant, lambda, representing the earth skeletons＝2υsμs/(1-2υs)，μs＝Gs，GsIs the shear modulus, upsilon, of the soil skeletonsIs the poisson's ratio of the soil framework; rhofDenotes the bulk density, ρ, of the pore fluidf＝ρfRnf，ρfRRepresenting the true density of the pore fluid, nfRepresents the volume fraction of pore fluid; rhosRepresenting the bulk density, ρ, of the soil skeletons＝ρsRns，ρsRRepresenting the true density of the soil skeleton, nsRepresenting the volume fraction of the soil skeleton; svIs the liquid-solid coupling coefficient, sv＝nfρfg/kfDenotes the interaction of the soil skeleton and pore fluid, wherein kfThe soil body Darcy permeability coefficient is adopted, and g is the gravity acceleration; e.g. of a cylindersIs the volume strain of the soil framework,
Figure FDA0003643450540000021
usis the radial displacement, w, of the soil skeletonsIs the vertical displacement of the soil skeleton; u. ufIs the radial displacement of the pore fluid, wfIs the vertical displacement of the pore fluid; v2Is a function of the laplacian of arithmetic,
Figure FDA0003643450540000022
step S22, solving the volume strain and pore fluid pressure of the soil framework by a variable separation method according to the motion equation of the soil;
step S23, processing a motion equation of soil by taking the volume strain of a soil framework and the pore fluid pressure as intermediate variables, and solving the motion equation of a soil body by sequentially adopting a variable separation method to obtain a general solution of the displacement and the stress of the soil containing unknown parameters;
the implementation process of step S22 is as follows:
from equations (14c) to (14d) of the equation of motion of the soil under the axisymmetric condition, it can be obtained:
Figure FDA0003643450540000023
Figure FDA0003643450540000024
the simultaneous process between the expressions (14a) to (14b) is abbreviated
Figure FDA0003643450540000025
Substituting the formulas (15a) to (15b) into the formula (16) to obtain the volume strain and pore fluid pressure of the soil skeleton:
▽2es-a1▽2pf+a2es＝0； (16)
wherein the intermediate variable
Figure FDA0003643450540000026
Similarly, the simultaneous processes between the expressions (15a) to (15b) are abbreviated as
Figure FDA0003643450540000027
Substituting the formula (14e) into the formula (17), wherein the volume strain and the pore fluid pressure of the soil framework are obtained:
▽2pf-a3es＝0； (17)
wherein the intermediate variable
Figure FDA0003643450540000028
Substituting formula (17) for formula (16) by:
▽2es+α2es＝0； (18)
wherein the intermediate variable alpha2＝a2-a1a3；
By a separation variable method, let esR (r) Z (z), i.e. to esPerforming variable separation, where R (r) and Z (z) are unknown functions, and determining them by the following derivation and substituting them into equation (18) to obtain:
Figure FDA0003643450540000031
wherein, b1、b2Is an unknown constant and satisfies
Figure FDA0003643450540000032
The solution of equation (19) is:
Figure FDA0003643450540000033
wherein A is1、A2、B1And B2Is an unknown constant, I0(. cndot.) denotes a zero-order Bessel function of the first kind, K0(. cndot.) represents a zeroth order transformed Bessel function of the second type;
then:
es＝[A1K0(b1r)+A2I0(b1r)][B1sin(b2z)+B2cos(b2z)]； (21)
it can be derived from equation (17):
Figure FDA0003643450540000034
the formula (22) is an inhomogeneous equation, the general solution of which comprises general solution of the homogeneous equation and special solution of the inhomogeneous equation, and the homogeneous solution of the formula (22) is obtained by using a separation variable method:
pfh＝[A3K0(b3r)+A4I0(b3r)][B3sin(b3z)+B4cos(b3z)]； (23)
wherein, b3、A3、A4、B3And B4Is an unknown constant;
p is required to ensure that the formula (22) is satisfiedfSpecial solution form of and esIn agreement, therefore let the special solution of equation (22) be:
pft＝T1es＝T1[A1K0(b1r)+A2I0(b1r)][B1sin(b2z)+B2cos(b2z)]； (24)
substituting equation (24) into equation (22) yields an intermediate variable T1：
Figure FDA0003643450540000035
Therefore, there are:
Figure FDA0003643450540000041
2. the method for determining the axisymmetric dynamic response of the two-dimensional socketed pile in the saturated soil as claimed in claim 1, wherein the specific implementation process in the step S1 is as follows:
step S11, setting one elastic modulus as EpPoisson ratio is upsilonpDensity of rhopRadius of r0The elastic pile with the length of L is completely embedded into a uniform, fluid-saturated, porous and elastic soil layer which is located above the bedrock, and a cylindrical coordinate system is established by taking the center of the pile top as the center of a circle;
step S12, in the center of pile top, namely radial directionThe coordinate r is 0 and the vertical coordinate z is 0, the vertical harmonic excitation force P (t) is loaded at P0eiωtSeparating the pile from the pile-soil system, and analyzing the stress condition of the pile, wherein
Figure FDA0003643450540000042
P0The magnitude of the external load p (t); e.g. of the typeiωtThe time factor is, omega is the circular frequency of the exciting force, omega is 2 pi f, and f represents the frequency of the exciting force;
step S13, according to the stress condition of the pile, obtaining a calculation formula of axial force and shearing force of the pile, a calculation formula of elastic potential energy and kinetic energy of the pile and virtual work of external load, and based on the kinetic energy and the elastic potential energy of the pile and the virtual work of the external load, according to the Hamilton dynamics principle, obtaining a motion equation of the two-dimensional elastic pile considering radial deformation as follows:
Figure FDA0003643450540000043
wherein u isp(z) denotes radial displacement of the pile flanks, wp(z) represents axial displacement of the pile sides; mu.spLame constant, mu, of pile foundationp＝Ep/2/(1+υp)；ApIs the cross-sectional area of the pile foundation,
Figure FDA0003643450540000044
j represents the polar moment of inertia of the pile foundation,
Figure FDA0003643450540000045
fr(z) represents the total load in the radial direction due to the pile foundation inertia and the soil reaction force, fz(z) represents the total load generated by pile foundation inertia force and soil reaction force in the axial direction;
Figure FDA0003643450540000046
trthe pile side radial surface force is obtained;
Figure FDA0003643450540000047
tzis pile side axial force;
the axial force and the shearing force of the pile are calculated according to the formula:
Figure FDA0003643450540000048
wherein N (z) is the axial force of the pile, Q (z) is the shear force of the pile,
Figure FDA0003643450540000051
is the axial stress on the cross section of the pile foundation,
Figure FDA0003643450540000052
shear stress on the cross section of the pile foundation;
step S14, solving a motion equation of the two-dimensional elastic pile considering radial deformation, and specifically realizing the following process:
suppose up(z)＝Upeηz，wp(z)＝Wpeηz，UpAmplitude, W, of a distribution form function of radial displacement of the pile side along an axial coordinate zpThe amplitude of the distribution form function of the axial displacement of the pile side along the axial coordinate z, eta represents the characteristic value of the distribution form function, let tr＝0、tz0, and substituting them into formula (9):
Figure FDA0003643450540000053
if equation (10) has a non-trivial solution, its coefficient matrix determinant value is required to be zero, i.e.:
Figure FDA0003643450540000054
a homogeneous solution of the displacement of the pile containing unknown constants can thus be obtained as:
Figure FDA0003643450540000055
wherein DjRepresents 4 unknown constants, j is more than or equal to 1 and less than or equal to 4; solution η of equation (11)jIs the eigenvalue, h, of the coefficient matrix of equation (10)jAnd kjIs a corresponding characteristic value ηjElement in the feature vector of (1), the feature value ηjThe feature vector of
Figure FDA0003643450540000056
1≤j≤4。
3. The method for determining the axisymmetric dynamic response of the two-dimensional socketed pile in the saturated soil as claimed in claim 2, wherein the step S23 is implemented as follows:
firstly, the volume strain e of the obtained soil frameworksAnd pore fluid pressure pfThe solution of (2) and the formula (15a) are substituted into the formula (14a) to obtain the radial displacement u of the soil frameworksThe non-homogeneous differential equation of (a); then solving the volume strain e of the soil frameworksAnd pore fluid pressure pfThe non-homogeneous differential equation can be solved to obtain usThe solution of (1); then the volume strain e of the obtained soil frameworksAnd pore fluid pressure pfThe solution of (a) and the formula (15b) are substituted into the formula (14b) to obtain the vertical displacement w of the soil frameworksThen solving the volume strain e of the soil frameworksAnd pore fluid pressure pfThe non-homogeneous differential equation can be solved to obtain wsThe solution of (1); finally, the obtained pore fluid pressure pfRadial displacement u of earth skeletonsVertical displacement w of soil skeletonsThe solution of (A) is respectively substituted into the formulas (15a) to (15b) to obtain the radial displacement u of the pore fluidfAnd vertical displacement w of pore fluidfThe solution of (1).
4. The method of claim 2 or 3The method for determining the axisymmetric dynamic response of the two-dimensional socketed pile in the saturated soil is characterized in that the radial displacement u of the soil frameworksThe solution of (A) is as follows:
Figure FDA0003643450540000061
vertical displacement w of the soil frameworksThe solution of (a) is:
Figure FDA0003643450540000062
radial displacement u of the pore fluidfThe solution of (a) is:
Figure FDA0003643450540000063
vertical displacement w of the pore fluidfThe solution of (a) is:
Figure FDA0003643450540000064
further, it is found that:
effective stress component of soil framework acting on plane perpendicular to r axis of radial coordinate axis, namely r plane and then pointing to r direction
Figure FDA0003643450540000065
Comprises the following steps:
Figure FDA0003643450540000071
effective stress component of soil framework acting on a plane perpendicular to the z axis of the vertical coordinate axis, namely the z plane and then pointing to the z direction
Figure FDA0003643450540000072
Comprises the following steps:
Figure FDA0003643450540000073
effective stress component of soil framework acting on r plane and then pointing to z direction
Figure FDA0003643450540000074
And effective stress component of soil framework acting on z plane and pointing to r direction
Figure FDA0003643450540000075
Comprises the following steps:
Figure FDA0003643450540000076
wherein, b1～b7，A1～A8And B1～B8Is an unknown constant, and b1～b7,A1～A8And B1～B8All the parameters are obtained according to boundary conditions and continuity conditions of piles and soil; i is0(. represents a zero-order transformed Bessel function of the first kind, K0(. cndot.) represents a zeroth order transformed Bessel function of the second type; i is1(. represents a first order deformed Bessel function of the first kind, K1(. cndot.) denotes a first-order transformed Bessel function of the second kind, respectively; intermediate variables
Figure FDA0003643450540000077
Figure FDA0003643450540000078
5. The method for determining the two-dimensional socketed pile axisymmetric dynamic response in saturated soil according to claim 4, wherein the boundary conditions and continuity conditions of the pile-soil system determined in the step S3 are as follows:
all field variables decay to zero at r → ∞, i.e.:
Figure FDA0003643450540000081
the surface of the saturated soil layer, namely r is more than or equal to r0In time, the normal stress is zero and the surface is permeable, i.e.:
Figure FDA0003643450540000082
the displacement of the saturated soil layer on the rigid bedrock is zero, namely:
ws(r,z＝L,t)＝0，wf(r,z＝L,t)＝0； (27c)
the pile is water impermeable, so the radial displacement of the pore fluid at the pile foundation side surface is equal to the radial displacement of the pile, i.e.:
uf(r0,z,t)＝up(z,t)； (27d)
wherein u isf(r0Z, t) on the pile-side surface, i.e. r-r0Radial displacement of pore fluid;
the pile and the soil are completely bonded at the contact surface, and the displacement is continuous, namely:
us(r0,z,t)＝up(z,t)，ws(r0,z,t)＝wp(z,t)； (27e)
wherein u iss(r0Z, t) is the radial displacement of the soil skeleton, ws(r0Z, t) is the vertical displacement of the soil skeleton, up(z, t) is the radial displacement of the pile flank, wp(z, t) is the axial displacement of the pile side;
the external load p (t) acts on the pile top, and the surface of the pile top is smooth, namely:
N(z＝0,t)＝p(t)，Q(z＝0,t)＝0； (27f)
n (z is 0, t) represents an axial force at the pile top z is 0, and Q (z is 0, t) represents a shearing force at the pile top z is 0;
the displacement of the socketed pile at the bedrock should be zero, i.e.:
up(z＝L,t)＝0，wp(z＝L,t)＝0 (27g)。
6. the method for determining the axisymmetric dynamic response of the two-dimensional socketed pile in the saturated soil according to claim 5, wherein the concrete implementation process of obtaining the general solution of the pile foundation displacement containing the unknown constant based on the boundary conditions of the soil body, the general solution of the displacement and stress of the soil containing the unknown constant and the homogeneous solution of the displacement of the pile containing the unknown constant in the step S3 is as follows:
first, the equation (a2) which is the solution of the vertical displacement of the soil skeleton including the unknown number is substituted into the equation (27c) to obtain the characteristic equation cos (b)nL) is 0, then the volume strain equation is used
Figure FDA0003643450540000083
And boundary conditions (27a) to (27b), which are derived as:
Figure FDA0003643450540000084
Figure FDA0003643450540000091
Figure FDA0003643450540000092
Figure FDA0003643450540000093
Figure FDA0003643450540000094
Figure FDA0003643450540000095
Figure FDA0003643450540000096
wherein the intermediate variable
Figure FDA0003643450540000097
Figure FDA0003643450540000098
Then, since the stress at the pile-soil interface is continuous, the pile side radial surface force t can be obtainedrAnd pile flank axial force tzComprises the following steps:
Figure FDA0003643450540000099
wherein,
Figure FDA00036434505400000910
representing the total radial stress component acting on the lateral surface of the pile foundation, according to the effective stress principle
Figure FDA00036434505400000911
By substituting equation (29) for equation (9) and combining the homogeneous solution of the pile displacement and equations (28e) to (28g), the general solution of the pile displacement is:
Figure FDA0003643450540000101
wherein, C1n、C2nAnd C3nAre all unknown constants, β1n～β6nAre all intermediate variables, and
Figure FDA0003643450540000102
Figure FDA0003643450540000103
Figure FDA0003643450540000104
Figure FDA0003643450540000105
Figure FDA0003643450540000106
Figure FDA0003643450540000107
Δ3n＝c2nc3n+c1nc4n，
Figure FDA0003643450540000108
Figure FDA0003643450540000109
7. the method for determining the two-dimensional socketed pile axisymmetric dynamic response in saturated soil as claimed in claim 6, wherein said step S4 is based on the boundary condition of the pile foundation and the continuity strip between the piles and soilDetermining all unknown constants in the pile foundation displacement general solution by the parts to obtain the solution of the pile foundation displacement, and solving the unknown constant C according to the pile foundation boundary condition and the continuity condition between piles and soils1n、C2nAnd C3nAnd solving to obtain an unknown constant C1n、C2nAnd C3nThen the displacement of the pile foundation is brought into the formula (30) to obtain the definite solution of the displacement of the pile foundation, and the C obtained by solving1n、C2nAnd C3nThe method specifically comprises the following steps:
Figure FDA00036434505400001010
Figure FDA0003643450540000111
Figure FDA0003643450540000112
wherein,
Figure FDA0003643450540000113
P0the amplitude of the external load p (t),
Figure FDA0003643450540000114
Figure FDA0003643450540000115
D1～D4and C1n～C3nThe 7 unknown constants are the boundary conditions of the pile foundation and the continuity conditions (27d) - (27g) between the piles and the soil, and the function sin (b)nz) and cos (b)nz) orthogonality property
Figure FDA0003643450540000116
The result is obtained; intermediate variables of coefficients in pile foundation displacement solution obtained from pile foundation boundary condition equations (27f) and (27g)
Figure FDA0003643450540000117
Comprises the following steps:
Figure FDA0003643450540000118
Figure FDA0003643450540000119
Figure FDA00036434505400001110
X1＝h1η1+h1Mq1-k1Mq2，X2＝h2η2+h2Mq3-k2Mq4，X3＝h3η3+h3Mq5-k3Mq6，
X4＝h4η4+h4Mq7-k4Mq8，
Figure FDA00036434505400001111
Figure FDA00036434505400001112
Figure FDA00036434505400001113
Figure FDA00036434505400001114
Figure FDA00036434505400001115
Figure FDA0003643450540000121
Figure FDA0003643450540000122
Figure FDA0003643450540000123
Figure FDA0003643450540000124
intermediate variable Delta of coefficient in pile foundation displacement solution obtained from pile-soil continuity conditional expressions (27d) and (27e)1n＝β1n(γ5nγ9n+γ6nγ8n)-β2n(γ6nγ7n-γ4nγ9n)-β3n(γ4nγ8n+γ5nγ7n)，Δ2n＝β1n(γ3nγ5n+γ2nγ6n)+β2n(γ3nγ4n-γ1nγ6n)-β3n(γ2nγ4n+γ1nγ5n) Wherein γ is1n＝βn1-TbnK2(b1nr)，γ2n＝β2n-T3bnK1(bnr0)，γ3n＝β3n-K1(b4nr0)，γ4n＝(ρfω2T2α2+nfa3)b1nK1(b1nr0)，γ5n＝α2(nf-ρfω2T3)bnK1(bnr0)，γ6n＝α2ρfω2K1(b4nr0)，γ7n＝T4bnK0(b1nr0)+β4n，γ8n＝T5bnK0(bnr0)+β5n，
Figure FDA0003643450540000125
Rn＝(γ2nγ4n+γ1nγ5n)(γ6nγ7n-γ4nγ9n)+(γ3nγ4n-γ1nγ6n)(γ4nγ8n+γ5nγ7n)；
Figure FDA0003643450540000126
Figure FDA0003643450540000127
Figure FDA0003643450540000128
8. The method for determining the axial-symmetric dynamic response of the two-dimensional socketed pile in the saturated soil as claimed in claim 7, wherein the determination of the axial force and the shear force of the pile based on the determination of the pile foundation displacement in the step S4 is determined according to the following formula (2) and formula (30), and the determination of the axial force and the shear force of the pile is obtained as follows:
Figure FDA0003643450540000129
Figure FDA00036434505400001210
wherein the intermediate variable
Figure FDA0003643450540000131
Figure FDA0003643450540000132
9. The method for determining the axisymmetric dynamic response of the two-dimensional socketed pile in the saturated soil as claimed in claim 7 or 8, wherein the step S4 is to determine the frequency domain dynamic impedance K of the pile top based on the solution of the pile foundation displacementd(ω) is:
Figure FDA0003643450540000133
wherein, KR(ω) represents the true dynamic stiffness and ability of the pile-soil system to resist axial strain, KR(ω)＝real(Kd(ω))；CIRepresenting the damping of the vibration radiation of the pile-soil system and the damping produced by the relative movement of the pore fluid and the soil framework, CI＝imag(Kd(ω))；
Further, the time domain semi-analytic solution expression of the pile top velocity V (t) is obtained as follows:
Figure FDA0003643450540000134
wherein T is the pulse width of pile top excitation during low strain dynamic measurement of pile foundation, QmaxThe pulse amplitude of pile top excitation during low-strain dynamic measurement of the pile foundation is obtained.
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