CN112199905B - A method for determining the axisymmetric dynamic response of two-dimensional rock-socketed piles in saturated soil - Google Patents

Abstract

The invention discloses a method for determining the axisymmetric dynamic response of a two-dimensional rock-socketed pile in saturated soil. First, based on the Hamiltonian dynamic principle, a motion equation of a two-dimensional elastic pile considering radial deformation is established and solved, and the equation containing unknown constants is obtained. Homogeneous solution of the displacement of the pile; then establish and solve the equation of motion of the soil based on the Boer porous medium theory, and obtain the general solution of the displacement and stress of the soil including the unknown constants; after uniformly determining the boundary conditions and continuity conditions of the pile-soil system, Based on the boundary conditions of the soil, the general solution of the displacement and stress of the soil with unknown constants, and the homogeneous solution of the displacement of the pile with unknown constants, the general solution of the pile foundation displacement with unknown constants is obtained; according to the boundary conditions of the pile foundation and The continuity condition between the pile and soil determines all the unknown constants in the general solution of the pile foundation displacement, obtains the fixed solution of the pile foundation displacement, and then obtains the frequency domain dynamic impedance and time domain velocity response of the pile top, and the obtained results are more reasonable and accurate.

Description

A method for determining the axisymmetric dynamic response of two-dimensional rock-socketed piles in saturated soil

technical field

The invention belongs to the technical field of pile-soil interaction research, and relates to a method for determining axisymmetric dynamic response of a two-dimensional rock-socketed pile in saturated soil considering the influence of radial deformation.

Background technique

Under the vertical harmonic load, the dynamic interaction between soil and rock-socketed pile plays an important role in geotechnical, seismic and structural engineering. Usually, the foundation is simulated as an elastic continuum, and the pile is simulated as a beam. First, the basic solution is obtained by solving the motion equation of the elastic medium, and then the pile-soil interaction problem is solved according to the boundary conditions and continuity conditions between the pile and soil. These problems are usually It is represented by the impedance function or velocity response of the pile-soil system, which can be used for dynamic design of structures and foundations or integrity testing of pile foundations.

Since Tajimi's pioneering research on the dynamic response of single piles in elastic soil layers in 1969, researchers have proposed classical processing methods such as plane strain method, radial simplification method, potential function method, Green function method, etc. The dynamic interactions of purely elastic media have been extensively studied. By assuming the soil as a saturated fluid porous medium, researchers have carried out extensive research on the axisymmetric interaction between piles and saturated soils under axial loads using the aforementioned methods. However, among these methods, the plane strain method uses a plane strain model to simulate the soil around the pile. It is assumed that the soil around the pile is in a state of plane strain, that is, the soil around the pile does not deform in the radial direction, and the deformation in the vertical direction does not follow. Depth changes, there is a large simplification. The radial simplification method ignores the radial deformation of the soil and only considers the vertical displacement of the soil, while the potential function method treats the soil as a three-dimensional continuous medium with finite thickness containing radial and vertical deformation, and uses the potential function To solve the equation, the potential function method is more rigorous than the plane strain method and the radial simplification method, but it needs to introduce a potential function for decomposition. These three methods are highly analytical and convenient for engineering practice. However, since these methods were originally proposed for the assumption of single-phase soil, some methods such as plane strain method and radial simplification method ignore the influence of radial deformation of soil. Therefore, a comparative study on the feasibility and scope of its application in saturated soil is required. The Green function method generally regards the soil as a three-dimensional elastic semi-infinite semi-space medium, and uses the integral transformation method to solve the relevant governing equations between the pile and the soil. .

On the other hand, in the above study, the rock-socketed pile is assumed to be a one-dimensional structure. Therefore, the influence of the radial deformation of the pile and the bottom reaction force on the mechanical interaction of the remaining surrounding media has not been resolved. However, scholars such as Pak and Gobert (1993), Masoumi et al. (2007), and Masoumi and Degrande (2008) have proved that in practical problems through detailed studies on piles subjected to axial loads and fully embedded in elastic media We need to consider the impact of the above factors. Although fundamental structural theories are widely used in engineering practice due to their mathematical simplicity and practical value, their application to such structure-continuum interaction problems suffers from some fundamental flaws, especially when the structures are long When the diameter ratio is not large enough. For example, the axial and radial displacements of embedded parts generally depend on the tangential and lateral boundary forces exerted on them by the surrounding medium. However, basic rod theory can only describe the axial deformation due to longitudinal loads. In practice, Due to the Poisson effect, there is also radial deformation and soil compression on the pile, which ignores the radial deformation, so the basic member theory essentially inhibits the proper lateral displacement and radial surface between the pile and soil However, the basic member theory regards the pile foundation as a one-dimensional structure with only axial deformation, and ignores the deformation of the pile foundation in the radial direction. For short piles with larger diameters, the actual pile foundation cannot be reasonably described distortion, resulting in inaccurate results. Because such a pile foundation produces significant radial deformation in addition to the axial deformation, the basic member theory cannot accurately describe the pile-soil interaction in this case. In addition to the aforementioned risks of not conforming to the laws of physics, this approximation also imposes severe limitations on the correlation analysis of important issues such as radial stress distribution and the Poisson effect on the system response.

SUMMARY OF THE INVENTION

The purpose of the embodiments of the present invention is to provide a method for determining the axisymmetric dynamic response of a rock-socketed pile in saturated soil considering the influence of radial deformation, so as to solve the existing research on the axisymmetric interaction between the pile and the saturated soil under the action of axial load. The method does not consider the radial deformation of the pile and soil, which leads to the inaccurate axisymmetric interaction between the pile and the saturated soil under the axial load.

The technical solution adopted in the embodiment of the present invention is a method for determining the axisymmetric dynamic response of a two-dimensional rock-socketed pile in saturated soil, which is performed according to the following steps:

Step S1, establish a cylindrical coordinate system, analyze the stress situation of the two-dimensional rock-socketed pile foundation in saturated soil, determine the calculation formulas of the axial force and shear force along the pile body, and establish a radial force based on the Hamilton dynamic principle. The equation of motion of the deformed two-dimensional elastic pile is then solved, and the equation of motion of the two-dimensional elastic pile considering radial deformation is solved, and the homogeneous solution of the displacement of the pile including the unknown constant is obtained;

Step S2, establishing the equation of motion of the soil based on the Boer porous medium theory, and solving the equation of motion of the soil to obtain a general solution of the displacement and stress of the soil including unknown constants;

In step S3, the boundary conditions and continuity conditions of the pile-soil system are uniformly determined, and then 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, we obtain: get a general solution for the displacement of the pile foundation with unknown constants;

Step S4: Determine all the unknown constants in the general solution of the pile foundation displacement according to the boundary conditions of the pile foundation and the continuity condition between the pile and soil, obtain the fixed solution of the pile foundation displacement, and determine the axial direction of the pile based on the fixed solution of the pile foundation displacement Definite solutions for force and shear forces and the frequency-domain dynamic impedance and time-domain velocity responses of pile tops.

The beneficial effect of the embodiment of the present invention is that a method for determining the dynamic response of a saturated soil and a rock-socketed pile considering the radial deformation of the pile and soil under the action of a vertical load is proposed, the soil is assumed to be a three-dimensional porous continuous medium, and the Boer porous medium is used. The mechanical behavior of the pile is described theoretically, the pile is regarded as a two-dimensional rod with radial and vertical deformation, and the motion equation of the pile is obtained by using the Hamiltonian variational principle; without introducing a potential function, the volumetric strain of the soil skeleton is first used and pore fluid pressure as intermediate variables to deal with the equation of motion of soil, and then further solve the equation of motion of pile and soil by separating variables; combined with the boundary conditions and continuity conditions of the pile-soil system, the displacement and stress solutions of soil and pile are obtained, The frequency domain dynamic impedance and time domain dynamic response of the pile top are further obtained. By comparing the obtained solution with the corresponding finite element model calculation results, the validity of the solution of the embodiment of the present invention is verified. Compared with the existing method, the embodiment of the present invention simultaneously considers the radial and radial sums of the saturated soil and the pile foundation. The vertical deformation can more accurately describe the actual deformation state of the pile and soil, and truly reflect the actual working state of the pile and soil, so the obtained results are more reasonable and accurate, and the calculation and application are convenient. The research method of axisymmetric interaction of soil does not consider the radial deformation of pile and soil, which leads to the inaccuracy of the axisymmetric interaction between pile and saturated soil under the action of axial load. And from the obtained results, it can be seen that the radial deformation of the pile-soil has a significant influence on the dynamic response of the pile-soil system.

Description of drawings

In order to explain the embodiments of the present invention or the technical solutions in the prior art more clearly, the following briefly introduces the accompanying drawings that need to be used in the description of the embodiments or the prior art. Obviously, the accompanying drawings in the following description are only These are some embodiments of the present invention. For those of ordinary skill in the art, other drawings can also be obtained according to these drawings without creative efforts.

Figure 1 is a schematic diagram of the mechanical model of a two-dimensional pile in saturated soil under vertical load.

Figure 2 is a schematic diagram of the mechanical model of a two-dimensional pile in saturated soil under axisymmetric loads.

Figure 3 is a schematic diagram of the axisymmetric finite element model of the pile-soil system established in ADINA.

FIG. 4 is a comparison diagram of the axisymmetric dynamic response of rock-socketed piles in saturated soil obtained by the embodiment of the invention and the finite element numerical method.

FIG. 5 is a comparison diagram of the dynamic impedance of the solution of the embodiment of the present invention, the simplified model method and the plane strain method.

FIG. 6 is a comparison diagram of the pile top velocity response of the solution of the embodiment of the present invention and the simplified model method and the plane strain method.

FIG. 7 is a static stiffness comparison diagram of the solution of the embodiment of the present invention, the simplified model method and the plane strain method.

FIG. 8 is a comparison diagram of the dynamic impedance of the solution of a two-dimensional pile and a one-dimensional pile according to an embodiment of the present invention.

Detailed ways

The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, but not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative efforts shall fall within the protection scope of the present invention.

The embodiment of the present invention provides a method for determining the axisymmetric dynamic response of a rock-socketed pile in saturated soil considering the influence of radial deformation, and the specific determination process is as follows:

1. Based on the Hamiltonian dynamics principle, establish the motion equation of the two-dimensional elastic pile considering radial deformation and vertical deformation:

P₀为外部载荷p(t)的幅值，e^iωt为时间因子，激振力圆频率ω＝2πf，f表示激振力频率，r是圆柱坐标系中的径向坐标，z是圆柱坐标系中的竖向坐标。As shown in Figure 1, an elastic pile with elastic modulus E _p , Poisson's ratio υ ^p , density ρ _p , radius r ₀ , and length L is completely embedded in a homogeneous, fluid-saturated, In the poroelastic soil layer above the bedrock, a cylindrical coordinate system is established with the center of the top of the pile as the center of the circle. The center of the top of the pile, that is, r=0 and z=0, is subjected to the vertical harmonic excitation force p(t)=P ₀ The role of e ^iωt ,

P ₀ is the amplitude of the external load p(t), e ^iωt is the time factor, the circular frequency of the exciting force is ω=2πf, f is the frequency of the exciting force, r is the radial coordinate in the cylindrical coordinate system, and z is the cylindrical coordinate vertical coordinates in the system.

In addition, the basic assumptions of this pile-soil system are as follows: (1) soil particles and pore fluid are microscopically incompressible, and there is no mass and heat exchange between soil and liquid; (2) the viscosity and gravity of pore fluid are ignored; (3) The resistance between the soil skeleton and the pore fluid is proportional to the liquid-solid relative velocity; (4) The pores in the soil are evenly distributed; (5) The vibration of the pile-soil system is small deformation vibration; (6) During the vibration process, The pile and soil are always closely connected at the interface.

Firstly, the equation of motion of the pile considering radial deformation and vertical deformation is established. As shown in Figure 1, under this load, the pile-soil system is axisymmetric about the z-axis. For convenience, the pile is separated from the pile-soil system, and the stress condition of the pile is shown in Figure 2. The pile is subjected to the radial force _tr on the pile side, the axial force t _z on the pile side, and the shear force Q on the top of the pile. (0) and the axial force N(0) at the top of the pile, the radial force _tr on the pile side and the axial force t _z on the pile side are the forces that the pile receives from the soil. Due to the axisymmetric nature, all variables are independent of the hoop coordinate θ and ignore the angular displacement, that is, the displacement along the θ direction.

For embedded piles with sufficient slenderness ratio and stiffness, mechanical behaviors such as axial compression and displacement mainly occur under the action of axial load. With this in mind, it is reasonable to express the longitudinal displacement of the pile as a function of z only. However, during the axial compression of the pile, a corresponding radial displacement field is obviously generated due to the Poisson effect. In addition, due to the lateral restraint of the surrounding soil, the surface force on the peripheral surface of the pile is generated, and the pile will be subjected to a large internal radial compression. As an initial attempt to take into account the above factors without incurring unnecessary complexity, a first non-trivial approximation of the axial displacement of a pile varying radially across its cross-section was employed. In addition to the mechanical dependencies described above, these kinematic assumptions have also been shown to be able to model radial shear phenomena, which are important in wave propagation problems. So first there are:

Among them, u _p (r, z) represents the radial displacement of any point of the pile foundation, w _p (r, z) represents the axial displacement of any point of the pile foundation, and u _p (r, z) and w _p (r, z) are functions of radial coordinate r and axial coordinate z; u _p (z) represents the radial displacement of the pile side, w _p (z) represents the axial displacement of the pile side; r ₀ represents the radius of the pile foundation. It can be seen from equation (1) that when the pile foundation is deformed by the vertical external load, the cross section of the pile foundation remains flat, so the vertical displacements on the cross section are all equal; while the radial deformation along the cross section is as follows: Linear change, that is, there is no radial displacement on the central axis of the pile foundation when r=0, the radial displacement on the side surface of the pile foundation is the largest, and the radial displacement of other points on the cross section maintains a linear relationship with the radial coordinate r . The interior of the pile foundation contains vertical displacement and radial displacement, and the vertical displacement and radial displacement are functions of the axial coordinate z and the radial coordinate r. Equation (1) is based on the vertical load of the pile foundation. The stress characteristics are modeled, and the internal displacement field of the pile foundation is processed and transformed into only the displacement of the outer surface of the pile foundation, and the displacement of any point of the pile foundation can be determined.

The axial force N(z) and shear force Q(z) along the pile body are:

是桩基横截面上的轴向应力，

是桩基横截面上的剪应力；μ^p表示桩基的Lame常数，μ^p＝E_p/2/(1+υ^p)表示桩基的Lame常数；A^p是桩基的横截面积，

J表示桩基的极惯性矩，

in,

is the axial stress on the cross section of the pile foundation,

is the shear stress on the cross section of the pile foundation; μ ^p is the Lame constant of the pile foundation, μ ^p =E _p /2/(1+υ ^p ) is the Lame constant of the pile foundation; A ^p is the cross-sectional area of the pile foundation,

J represents the polar moment of inertia of the pile foundation,

The elastic potential energy of the pile foundation is:

The kinetic energy of the pile is:

和

是具体组成部分；v^p是桩的速度矢量，位移符号上的点表示对t进行求导，即

表示桩侧面的径向速度，

表示桩侧面的轴向速度；ρ_p表示桩身密度。V为桩的体积，

表示作用在r面且沿r方向的正应力，

表示作用在r面且沿径向的正应变；

表示作用在r面且沿z方向的切应力，

表示作用在r面且沿z方向的切应变，

表示作用在θ面且沿θ方向的正应力，

表示作用在θ面且沿θ方向的正应变，

表示作用在z面上且沿z方向的正应变。where σ ^p is the stress tensor of the pile, ε ^p is the strain tensor of the pile,

and

is the specific component; v ^p is the velocity vector of the pile, and the point on the displacement symbol indicates the derivation of t, that is

is the radial velocity at the side of the pile,

represents the axial velocity of the pile side; ρ _p represents the pile body density. V is the volume of the pile,

represents the normal stress acting on the r-plane and along the r-direction,

represents the normal strain acting on the r plane and along the radial direction;

represents the shear stress acting on the r-plane and along the z-direction,

represents the shear strain acting on the r-plane and along the z-direction,

represents the normal stress acting on the θ plane and along the θ direction,

represents the normal strain acting on the θ plane and along the θ direction,

represents the normal strain acting on the z-plane and along the z-direction.

Since the pile is subjected to external forces such as the surface force of the surrounding soil, the load on the top of the pile and the reaction force at the bottom of the pile, the work done by the non-conservative force is:

Among them, ^{t p} represents the external force vector on the entire pile surface, up represents the displacement vector of the entire pile surface; _up (0) represents the radial displacement of the pile top on the side surface of the pile foundation, and _up (L) represents the pile foundation The radial displacement of the pile bottom on the side surface; Q(L) represents the pile bottom shear force.

Then the virtual work of the external load can be expressed as (Morse and Feshbach, 1953):

Among them, δ is the variation symbol, δup (z) represents the variation of u _p (z), and δw _p (z) represents the variation of w _p ( _z ).

According to Hamilton's dynamic principle:

Among them, T represents the kinetic energy of the pile, P represents the elastic potential energy of the pile, and t ₁ and t ₂ are two different time points arbitrarily taken during the movement of the pile.

According to Hamilton's dynamic principle, the equation of motion of the pile is obtained as:

Since the current consideration is the steady-state vibration that changes with the time factor e ^iωt , equation (8) can be further written as:

t_r为桩侧径向面力；

t_z为桩侧轴向面力。为简便分析，后续将统一略去时间因子e^{i ωt}。Among them, f _r (z) represents the total load caused by the inertia force of the pile foundation and the soil reaction force in the radial direction, and f _z (z) represents the total load caused by the inertia force of the pile foundation and the soil reaction force in the axial direction;

_tr is the radial force on the pile side;

t _z is the axial force on the pile side. For simplicity of analysis, the time factor e ^{i ωt} will be uniformly omitted in the following.

Suppose u _p (z)=U _p e ^ηz , w _p (z)=W _p e ^ηz , U _p is the amplitude of the distribution shape function of the radial displacement of the pile side along the axial coordinate z, and W _p is the pile side The axial displacement is the magnitude of the distribution shape function along the axial coordinate z, and η represents the eigenvalue of the distribution shape function. This expression converts the solution of the pile foundation displacement equivalently into the solution of η and U _p , W _p .

Let _tr = 0, t _z = 0, and substitute them into equation (9) to get:

If the equation (equation (10)) has a non-trivial solution, the determinant value of its coefficient matrix is required to be zero, that is:

Thus, the homogeneous solution of the displacement of the pile can be obtained as:

本发明实施例中η_j、h_j和k_j均通过MatLab等数学计算软件求得，该数学计算软件均包含求解矩阵特征值及相应特征向量的功能，此处不再赘述具体求解过程。where D _j (1≤j≤4) represents four unknown constants, the solution η _j (1≤j≤4) of equation (11) is the eigenvalue of the coefficient matrix of equation (10), h _j (1≤j ≤4) and k _j (1≤j≤4) are the elements of the eigenvector corresponding to the eigenvalue η _j , η _j contains four values, namely η ₁ , η ₂ , η ₃ , η ₄ , each value corresponds to a The eigenvector, the eigenvector is a vector containing two elements, the two elements are h _j and k _j , and h _j and k _j correspond to U _p and W _p respectively one-to-one. Substitute η _j into equation (10) to get the eigenvector

In the embodiment of the present invention, η _j , h _j and k _j are all obtained by mathematical calculation software such as MatLab. The mathematical calculation software all includes the function of solving matrix eigenvalues and corresponding eigenvectors, and the specific solving process is not repeated here.

2. Based on the Boer porous medium theory, the equation of motion of soil (soil around the pile) is established, and the equation of motion of soil is solved to obtain the general solution of soil displacement and stress including unknown parameters. The specific implementation process is as follows:

1) Based on the Boer porous media theory, the equation of motion of soil under axisymmetric conditions is established:

The _Boer porous media theory can describe the mechanical behavior of saturated soil. Therefore, the dynamic control _equation of saturated ^soil can be expressed as:

表示梯度算符，

表示散度算符；位移矢量符号上的一点表示这些符号对时间t的一次微分，位移矢量符号上的两点表示这些符号对时间t的二次微分；式(13a)表示固相即土骨架的动量平衡，图(13b)表示液相即孔隙流体的动量平衡，式(13c)代表液-固混合物即土骨架-孔隙流体系统的质量平衡。Among them, λ ^s and μ ^s represent the Lame constant of the soil skeleton, which is well known in the field, λ ^s = 2υ ^s μ ^s /(1-2υ ^s ), μ ^s = G _s , G _s is the shear modulus of the soil skeleton , υ ^s is the Poisson’s ratio of the soil skeleton; ρ ^f is the bulk density of the pore fluid, ρ ^f =ρ ^fR n ^f ; ρ ^s is the bulk density of the soil skeleton, ρ ^s =ρ ^sR n ^s , where ρ ^sR is the The true density of the soil skeleton, ρ ^fR represents the true density of the pore fluid, ^ns represents the volume fraction of the soil skeleton, and n ^f represents the volume fraction of the pore fluid; s _v is the liquid-solid coupling coefficient, s _v =n ^f ρ ^f g/ k _f , represents the interaction between the soil skeleton and pore fluid, where k _f is the Darcy permeability coefficient of the soil body, and g is the acceleration of gravity;

represents the gradient operator,

represents the divergence operator; one point on the symbol of the displacement vector represents the first derivative of these symbols with respect to time t, and two points on the symbol of the displacement vector represent the second derivative of these symbols with respect to time t; Equation (13a) represents the solid phase, the soil skeleton Figure (13b) represents the momentum balance of the liquid phase, that is, the pore fluid, and equation (13c) represents the mass balance of the liquid-solid mixture, that is, the soil framework-pore fluid system.

Under the condition of axisymmetric (pile-soil system is symmetrical about the z-axis), the field variable has nothing to do with the angular coordinate θ, and the angular displacements v _s and v _f are zero, so equation (13) can be further written in the following component form:

u_s是土骨架的径向位移(分量)，w_s是土骨架的垂直位移(分量)；u_f是孔隙流体的径向位移(分量)，w_f是孔隙流体的垂直位移(分量)；

是拉普拉斯算子，

where _es is the volumetric strain of the soil skeleton,

u _s is the radial displacement (component) of the soil skeleton, _ws is the vertical displacement (component) of the soil skeleton; u _f is the radial displacement (component) of the pore fluid, and w _f is the vertical displacement (component) of the pore fluid;

is the Laplace operator,

2) Take the volume strain of the soil skeleton and the pore fluid pressure as the intermediate variables to deal with the equation of motion of the soil, and use the variable separation method to solve the equation of motion of the soil, and obtain the general solution of the displacement and stress of the soil with unknown parameters:

From formulas (14c) to (14d), we can get:

并将式(15a)～(15b)代入其中，推得含有土骨架的体积应变和孔隙流体压力的式(16)：The simultaneous process between equations (14a) to (14b) can be abbreviated as

Substitute equations (15a) to (15b) into it, and derive equation (16) containing the volumetric strain and pore fluid pressure of the soil skeleton:

Among them, the intermediate variable

并将式(14e)代入其中，推得含有土骨架的体积应变和孔隙流体压力的式(17)：Similarly, the simultaneous process between equations (15a) and (15b) can be abbreviated as

Substituting Equation (14e) into it, Equation (17) containing the volumetric strain and pore fluid pressure of the soil skeleton is derived:

Among them, the intermediate variable

Substituting equation (17) into equation (16), we can get:

Among them, the intermediate variable α ² =a ₂ -a ₁ a ₃ .

Using the method of separating variables, let _es =R(r)Z(z), that is, to separate variables for _es , R(r) and Z(z) are unknown functions, which need to be determined by the following derivation, and substituted into Equation (18), we get:

Among them, b ₁ , b ₂ are unknown constants and satisfy

The solution of equation (19) is:

Among them, A ₁ , A ₂ , B ₁ and B ₂ are unknown constants, I ₀ (·) represents the zero-order deformed Bessel function of the first kind, and K ₀ (·) represents the zero-order deformed Bessel function of the second kind.

but:

e _s =[A ₁ K ₀ (b ₁ r)+A ₂ I ₀ (b ₁ r)][B ₁ sin(b ₂ z)+B ₂ cos(b ₂ z)]; (21)

From formula (17), it can be deduced that:

Equation (22) is an inhomogeneous equation, and its general solution includes the general solution of the homogeneous equation and the specific solution of the inhomogeneous equation. The homogeneous solution of Equation (22) is obtained by using the separation variable method:

p ^fh =[A ₃ K ₀ (b ₃ r)+A ₄ I ₀ (b ₃ r)][B ₃ sin(b ₃ z)+B ₄ cos(b ₃ z)]; (23)

where b ₃ , A ₃ , A ₄ , B ₃ and B ₄ are unknown constants.

Let the special solution of equation (22) be:

p ^ft =T ₁ e _s =T ₁ [A ₁ K ₀ (b ₁ r)+A ₂ I ₀ (b ₁ r)][B ₁ sin(b ₂ z)+B ₂ cos(b ₂ z)] ; (twenty four)

Equation (24) is defined because the special solution form of p ^f needs to be consistent with _es to ensure that equation (22) holds. Equation (24) indicates that the special solution of p ^f is consistent with the form of _es , but the size is different, and the intermediate parameter T ₁ quantifies the difference in size.

Substituting equation (24) into equation (22), we get:

So there are:

The volumetric strain _es and pore fluid pressure p ^f of the soil skeleton have been obtained, and the volumetric strain _es and pore fluid pressure p ^f in equation (14) can be moved to the right side of the equation, so that equation (14) is the Homogeneous equations, the aforementioned has explained how to solve the general solution of the non-homogeneous equation (first use the separation of variables method to obtain the homogeneous solution, and then add the special solution), solve equation (14) in a similar way to solve p ^f , you can get The general solution of all field variables in the equation of motion in saturated soil (substituting the solution of the volume strain _es and pore fluid pressure p ^f of the soil skeleton obtained above and equation (15a) into equation (14a), the radial displacement of the soil skeleton can be obtained. the inhomogeneous differential equation of u _s , and then the solution of u _s can be obtained by solving the inhomogeneous differential equation of soil skeleton volume strain es and pore fluid pressure p ^f _; The solutions of e _s and pore fluid pressure p ^f and equation (15b) are substituted into equation (14b) to get the inhomogeneous differential equation about the vertical displacement _ws of the soil skeleton, and then the volumetric strain _es and pore fluid of the soil skeleton are solved by similar The solution of ws can be solved by the method of the inhomogeneous differential equation of the pressure p ^f ; the solutions of the obtained pore fluid pressure p ^f and the radial displacement u _s of the soil skeleton and the vertical displacement _{ws s of} the soil skeleton are respectively substituted into the formula ( 15a-15b) Arrange the radial u _f of the available pore fluid and the vertical displacement w _f ) of the pore fluid, and the obtained results include:

(1) Soil displacement component:

(2) Stress component in soil

表示作用在r平面即垂直于径向坐标轴r轴的平面然后指向r方向的土骨架有效应力分量；

表示作用在z平面即垂直于竖向坐标轴z轴的平面然后指向z方向的土骨架有效应力分量；

表示作用在r平面然后指向z方向的土骨架有效应力分量；

表示作用在z平面然后指向r方向的土骨架有效应力分量。在本发明实施例中采用了线弹性的应力应变关系模型描述土骨架的力学行为，线弹性应力应变关系模型为力学领域内公知，具体表达式可写为σ^s＝2μ^sE_s+λ^se_sI，其中σ^s表示土骨架应力张量，在柱坐标系下包含分量

比如上述

即表示i＝r和j＝r时的分量，其它分量同理；I为单位对角矩阵，为数学领域内公知，E_s为土骨架应变张量，且

其中u_s表示土骨架位移向量，在轴对称条件下，u_s只有径向和竖向两个位移分量，即u_s＝(u_s,w_s)，其中右上角的符号T表示矩阵的转置。在已经得到了土骨架和孔隙流体的位移解的基础上，只需将这些位移解代入前述线弹性的应力应变关系模型中即可得到

b₁～b₇,A₁～A₈和B₁～B₈是常数，b₁～b₇,A₁～A₈和B₁～B₈均通过桩和土的边界条件以及连续性条件求得。I₁(·)表示一阶第一类变形贝塞尔函数，K₁(·)分别表示一阶第二类变形贝塞尔函数。

从边界和连续性条件(28a)～(28g)中可以确定b₅、b₇，从而根据此处确定b₄、b₆；中间变量

in,

Represents the effective stress component of the soil skeleton acting on the r plane, that is, the plane perpendicular to the radial coordinate axis r axis and then pointing in the r direction;

Represents the effective stress component of the soil skeleton acting on the z plane, that is, the plane perpendicular to the z axis of the vertical coordinate axis and then pointing in the z direction;

represents the effective stress component of the soil skeleton acting on the r plane and then pointing in the z direction;

represents the effective stress component of the soil skeleton acting on the z plane and then pointing in the r direction. In the embodiment of the present invention, the linear elastic stress-strain relationship model is used to describe the mechanical behavior of the soil skeleton. The linear elastic stress-strain relationship model is well known in the field of mechanics, and the specific expression can be written as σ ^s =2μ ^s E _s +λ ^s e _s I, where σ ^s represents the soil skeleton stress tensor, containing the components in the cylindrical coordinate system

such as the above

That is, it represents the component when i=r and j=r, and the other components are the same; I is a unit diagonal matrix, which is well known in the field of mathematics, E _s is the soil skeleton strain tensor, and

where u _s represents the displacement vector of the soil skeleton. Under the condition of axisymmetric, u _s has only two displacement components, radial and vertical, namely u _s =(u _s ,w _s ), and the symbol T in the upper right corner represents the rotation of the matrix set. On the basis that the displacement solutions of soil skeleton and pore fluid have been obtained, it is only necessary to substitute these displacement solutions into the aforementioned linear elastic stress-strain relationship model to obtain

b ₁ ～b ₇ , A ₁ ～A ₈ and B ₁ ～B ₈ are constants, and b ₁ ～b ₇ , A ₁ ～A ₈ and B ₁ ～B ₈ are all obtained by the boundary conditions and continuity conditions of piles and soils have to. I ₁ (·) represents a first-order deformed Bessel function of the first type, and K ₁ (·) respectively represents a first-order deformed Bessel function of the second type.

From the boundary and continuity conditions (28a) to (28g), b ₅ and b ₇ can be determined, so that b ₄ and b ₆ can be determined according to this; intermediate variables

3. Determine the boundary conditions and continuity conditions of the pile-soil system, and based on the boundary conditions of the soil body, the general solution of the soil displacement and stress, and the homogeneous solution of the displacement of the pile containing the unknown constant, obtain the pile containing the unknown constant. General solution for basis displacement.

As shown in Figure 1, the boundary and continuity conditions of the pile-soil system are as follows:

All field variables decay to zero as r→∞, that is:

The normal stress of the saturated soil surface (r≥r ₀ ) is zero and the surface is permeable, namely:

The displacement of the saturated soil layer on the rigid bedrock is zero, that is:

w _s (r, z=L, t)=0, w _f (r, z=L, t)=0; (27c)

The pile is impermeable, so the radial displacement of the pore fluid at the side surface of the pile foundation is equal to the radial displacement of the pile, namely:

u _f (r ₀ ,z,t) ₌ up (z,t); (27d)

u _f (r ₀ , z, t) represents the radial displacement of the pore fluid on the side surface of the pile foundation, that is, at r=r ₀ .

The pile and soil are completely bonded at the contact surface, and the displacement is continuous, that is:

u _s (r ₀ ,z,t)=up (z,t),w _s (r ₀ ,z,t) _{=w p (} z,t); (27e)

The position of the pile-soil contact surface is r=r ₀ . The pile and the soil are completely bonded at the contact surface, that is, they deform together and do not detach, so their displacements are equal, and the displacement is continuous, that is, the radial displacement of the soil skeleton at that location. and vertical displacement are respectively equal to the radial displacement and vertical displacement of the side surface of the pile foundation, u _s (r ₀ , z, t) is the radial displacement of the soil skeleton, _ws (r ₀ , z, t) is the soil skeleton The vertical displacement of , u _p (z, t) is the radial displacement of the side surface of the pile foundation, and w _p (z, t) is the vertical displacement of the side surface of the pile foundation.

The load p(t) acts on the top of the pile, and the surface of the top of the pile is smooth, that is:

N(z=0,t)=p(t), Q(z=0,t)=0; (27f)

N(z=0,t) represents the axial force at the top of the pile at z=0, and Q(z=0,t) represents the shear force at the top of the pile at z=0.

The displacement of the rock-socketed pile at the bedrock should be zero, namely:

up (z=L,t) ₌ 0, _wp (z=L,t)=0; (27g).

In the embodiment of the present invention, when the steady-state load is considered, each variable can be expressed in the form of f(r,z,t)=f(r,z)e ^iωt , so that the time factor e ^iωt becomes a common factor and can be eliminated , so the time no longer appears in the formula; if the load is a transient load, the time t cannot be eliminated, so there is t in the above formula.

和边界条件(27a)～(27b)，可推得场变量的解为：The unknown constants in the general solutions of the equations of motion of the pile and soil are determined according to the boundary conditions and continuity conditions of the pile-soil system, that is, equations (27a) to (27g). First, substitute the vertical displacement equation of the soil skeleton (Equation (A2)) into the boundary conditions, that is, substitute Equation (A2) into Equation (27c) to obtain the characteristic equation cos(b _n L)=0, and then use the volumetric strain equation

and boundary conditions (27a)～(27b), the solution of the field variable can be deduced as:

in,

Then use the general solution of soil stress, that is, equations (28e) to (28g) to obtain the pile side radial force _tr and the pile side axial force t _z , and according to the pile side radial force _tr , the pile side axis The general solution for the displacement of the pile foundation is obtained by applying the force t _z to the surface.

The stress at the pile-soil interface is continuous, so the surface force on the pile side can be obtained:

为作用在桩基侧表面上的径向总应力分量，

其是根据有效应力原理得到的。in,

is the radial total stress component acting on the side surface of the pile foundation,

It is obtained according to the principle of effective stress.

By substituting equation (29) into equation (9) and combining equations (12) and (28e) to (28g), the displacement of the pile can be deduced as:

Among them, C _1n to C _3n are unknown constants determined by boundary conditions and continuity conditions, β _1n to β _6n are intermediate variables, and

△_3n＝c_2nc_3n+c_1nc_4n，而

Δ _3n =c _2n c _3n +c _1n c _4n , and

4. According to the boundary conditions of the pile foundation and the continuity conditions between the pile and soil, determine all the unknown constants in the general solution of the pile foundation displacement, and obtain the fixed solution of the pile foundation displacement, which is based on the pile foundation boundary conditions and the continuity conditions between the pile and soil. Solve the unknown constants C _1n , C _2n and C _3n , get the unknown constants C _1n , C _2n and C _3n , and put them into equation (30) to get the definite solution of pile foundation displacement and the definite solution of soil displacement and stress , the obtained C _1n , C _2n and C _3n are specifically:

P₀为外荷载p(t)的幅值，

D₁～D₄和C_1n～C_3n这7个未知常数是通过边界条件和连续性条件式(27d)～(27g)，以及函数sin(b_nz)和cos(b_nz)的正交特性

推得的。由桩基边界条件式(27f)和(27g)所得的桩基位移解中系数的中间变量

为：

其中，X₁＝h₁η₁+h₁M_q1-k₁M_q2，X₂＝h₂η₂+h₂M_q3-k₂M_q4，X₃＝h₃η₃+h₃M_q5-k₃M_q6，X₄＝h₄η₄+h₄M_q7-k₄M_q8，

由桩土间连续性条件式(27d)和(27e)所得的桩基位移解中系数的中间变量△_1n＝β_1n(γ_5nγ_9n+γ_6nγ_8n)-β_2n(γ_6nγ_7n-γ_4nγ_9n)-β_3n(γ_4nγ_8n+γ_5nγ_7n)，△_2n＝β_1n(γ_3nγ_5n+γ_2nγ_6n)+β_2n(γ_3nγ_4n-γ_1nγ_6n)-β_3n(γ_2nγ_4n+γ_1nγ_5n)，其中，γ_1n＝β_1n-T₂b_1nK₁(b_1nr₀)，γ_2n＝β_2n-T₃b_nK₁(b_nr₀)，γ_3n＝β_3n-K₁(b_4nr₀)，γ_4n＝(ρ^fω²T₂α²+n^fa₃)b_1nK₁(b_1nr₀)，γ_5n＝α²(n^f-ρ^fω²T₃)b_nK₁(b_nr₀)，γ_6n＝α²ρ^fω²K₁(b_4nr₀)，γ_7n＝T₄b_nK₀(b_1nr₀)+β_4n，γ_8n＝T₅b_nK₀(b_nr₀)+β_5n，

R_n＝(γ_2nγ_4n+γ_1nγ_5n)(γ_6nγ_7n-γ_4nγ_9n)+(γ_3nγ_4n-γ_1nγ_6n)(γ_4nγ_8n+γ_5nγ_7n)；

in,

P ₀ is the amplitude of the external load p(t),

The seven unknown constants D ₁ ～D ₄ and C _1n ～C _3n are obtained through the boundary conditions and continuity conditional expressions (27d)～(27g), and the positive values of the functions sin( _bn z) and cos( _bn z). cross characteristic

push. Intermediate variables of the coefficients in the pile foundation displacement solutions obtained from the pile foundation boundary condition equations (27f) and (27g)

for:

Wherein, X ₁ =h ₁ η ₁ +h ₁ M _q1 -k ₁ M _q2 , X ₂ =h ₂ η ₂ +h ₂ M _q3 -k ₂ M _q4 , X ₃ =h ₃ η ₃ +h ₃ M _q5 -k ₃ M _q6 , X ₄ =h ₄ η ₄ +h ₄ M _q7 -k ₄ M _q8 ,

Intermediate variables of the coefficients in the pile foundation displacement solution obtained from the pile-soil continuity conditions (27d) and (27e) △ _1n =β _1n (γ _5n γ _9n +γ _6n γ _8n )-β _2n (γ _6n γ _7n -γ _4n γ _9n )-β _3n (γ _4n γ _8n +γ _5n γ _7n ), Δ _2n =β _1n (γ _3n γ _5n +γ _2n γ _6n )+β _2n (γ _3n γ _4n -γ _1n γ _6n )-β _3n (γ _2n γ _4n +γ _1n γ _5n ), where γ _1n =β _1n -T ₂ b _1n K ₁ (b _1n r ₀ ), γ _2n =β _2n -T ₃ b _n K ₁ ( b _n r ₀ ), γ _3n =β _3n -K ₁ (b _4n r ₀ ), γ _4n =(ρ ^f ω ² T ₂ α ² +n ^f a ₃ )b _1n K ₁ (b _1n r ₀ ), γ _5n =α ² (n ^f -ρ ^f ω ² T ₃ )b _n K ₁ (b _n r ₀ ), γ _6n =α ² ρ ^f ω ² K ₁ (b _4n r ₀ ), γ _7n =T ₄ b _n K ₀ (b _1n r ₀ )+β _4n , γ _8n =T ₅ b _n K ₀ (b _n r ₀ )+β _5n ,

R _n =(γ _2n γ _4n +γ _1n γ _5n )(γ _6n γ _7n -γ _4n γ _9n )+(γ _3n γ _4n -γ _1n γ _6n )(γ _4n γ _8n +γ _5n γ _7n );

5. Determine the fixed solutions of the axial force and shear force along the pile body and the frequency domain dynamic impedance and time domain velocity response of the pile top based on the fixed solution of the pile foundation displacement.

According to formula (2) and formula (30), the axial force and shear force along the pile body can be obtained as:

Among them, the intermediate variable

In pile foundation engineering, the dynamic impedance and velocity response of the pile top play a crucial role. For example, dynamic impedance is often used to evaluate the bearing capacity and energy dissipation capacity of pile-soil systems, while velocity responses can be used to calculate stress wave velocities, examine test data, and detect pile shaft defects in non-destructive testing of piles.

The frequency domain dynamic impedance K _d (ω) of the pile is defined as:

Among them, K _R (ω) represents the real dynamic stiffness and the ability of the pile-soil system to resist axial strain, K _R (ω)=real(K _d (ω)); C _I represents the vibration radiation damping of the pile-soil system and The damping caused by the relative motion of the pore fluid and the soil skeleton reflects the energy dissipation of the pile-soil system, C _I = imag(K _d (ω)).

In the low-strain measurement of the pile foundation, the pile top excitation (hammering load, which is a transient load) can be simplified to a half-sine pulse load, namely:

Among them, T is the pulse width, and Q _max is the pulse amplitude.

为：Then the frequency domain response of the pile top velocity under the action of unit strength load

for:

Therefore, by convolving the time domain response of the pile top load and the unit top velocity time domain, the time domain semi-analytical solution of the pile top velocity can be expressed as follows:

6. Numerical Results and Discussion

Using the pile-soil system and load parameters shown in Table 1, through numerical calculation, the rationality of the solution in the embodiment of the present invention is verified, and the dynamic characteristics of the pile-soil system are studied. It is worth noting that the loading parameters used for the transient response analysis correspond to the excitation force generated by a polyethylene hammer in the pile integrity test.

Table 1 Pile-soil system and load parameters

(1) Numerical verification

In order to verify the correctness of the method proposed in the embodiment of the present invention, an axisymmetric finite element model of the pile-soil system in FIG. 1 is established by using ADINA software, and the model is shown in FIG. 3 . Fig. 4 shows the comparison results of the method proposed in the embodiment of the present invention and the finite element method with respect to the displacement of the pile body and the velocity response of the top of the pile. In this finite element model, the soil is assumed to be a poroelastic material, and the soil is simulated by a 9-node rectangular element. The left side of the model is set as an axisymmetric boundary, the surface of the soil layer is set as a free boundary, and the bottom of the model is an impermeable fixed boundary , the right side is set as a fixed boundary without pore pressure to simulate infinite boundary conditions, which is consistent with the boundary conditions specified in Figure 1. It is worth noting that in this example, the model width of 50m can already obtain the steady-state response amplitude, and the boundary effect on the right side is well eliminated. It can be seen from the comparison in FIG. 4 that the dynamic response solution of the embodiment of the present invention is in good agreement with the finite element simulation result, thereby verifying the rationality of the method.

(2) Comparison of solutions

Figure 5 shows the comparison between the dynamic impedance obtained by the method of the embodiment of the present invention and the simplified model solution, including the radial simplified solution obtained by ignoring the radial displacement of the soil body, and the plane strain solution obtained by using the plane strain assumption for the soil body. It can be seen from the figure that when the excitation frequency f>5Hz, the dynamic impedance of the embodiment of the present invention is in good agreement with the dynamic impedance obtained by the plane strain model; when f≤5Hz, the solution of the embodiment of the present invention is better than the plane strain model. The dynamic stiffness of the obtained solution is large, and the solution of the embodiment of the present invention can reflect the "cutoff frequency" behavior, which means that the pile-soil system has no damping. The dynamic impedance of the radial simplified solution is obviously different from the other two methods; in single-phase soil, the dynamic impedance of the embodiment of the present invention and the radial simplified model are basically the same, and the comparison of the dynamic impedance of the plane strain model and the embodiment of the present invention The results are similar to those in saturated soil. These conclusions can be further confirmed by comparing the pile top velocity response and static impedance of the different methods, as shown in Figures 6 and 7. In addition, it can be seen from Fig. 7 that the static stiffness of the pile eventually tends to a stable value with the increase of the pile length, which indicates that the effect of the pile length on the static impedance 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 present invention and the plane strain model gradually becomes smaller. plane strain assumption.

Both the radial simplification method and the plane strain method ignore the influence of the radial deformation of the soil, while the radial and vertical deformation of the pile and soil are considered in the embodiment of the present invention. It can be seen from Figures 5-6 that the radial simplification method is In the case of single-phase soil, the solution is relatively close to the solution of the embodiment of the present invention, but in the case of saturated soil, there will be a big difference and cannot be used. Although the plane strain method compares well with the results of the embodiment of the present invention for single-phase soil and saturated soil in the high frequency stage, it cannot reflect the physical phenomenon of the cut-off frequency of the pile-soil system, and has defects, and it is not consistent with the embodiment of the present invention. The solutions are significantly different in the low frequency stage, which means that the plane strain method cannot give enough accurate results in the low frequency stage.

As shown in Figure 8, the dynamic impedance of the two-dimensional pile obtained in the embodiment of the present invention was compared with Liu, H.L., Zheng, C.J., Ding, X.M., Qin, H.Y., 2014. Vertical dynamic response of a pipe pile in saturatedsoil layer.Comput .Geotech., 61, 57-66. Compared with the dynamic impedance of the one-dimensional pile obtained, it can be seen that the radial deformation of the pile has a significant impact on the impedance function of the pile. The static stiffness of 2D piles is smaller than that of 1D piles because the lateral boundaries of 1D piles are fixed. However, the peak value of dynamic impedance of two-dimensional piles is larger than that of one-dimensional piles, which means that assuming the piles as one-dimensional members actually underestimate the maximum dynamic impedance of the pile-soil system.

The above descriptions are only preferred embodiments of the present invention, and are not intended to limit the protection scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention are included in 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:

wherein p is^fIs pore fluid pressure, λ^sAnd mu^sLame constant, lambda, representing the earth skeleton^s＝2υ^sμ^s/(1-2υ^s)，μ^s＝G_s，G_sIs the shear modulus, upsilon, of the soil skeleton^sIs the poisson's ratio of the soil framework; rho^fDenotes the bulk density, ρ, of the pore fluid^f＝ρ^fRn^f，ρ^fRRepresenting the true density of the pore fluid, n^fRepresents the volume fraction of pore fluid; rho^sRepresenting the bulk density, ρ, of the soil skeleton^s＝ρ^sRn^s，ρ^sRRepresenting the true density of the soil skeleton, n^sRepresenting the volume fraction of the soil skeleton; s_vIs the liquid-solid coupling coefficient, s_v＝n^fρ^fg/k_fDenotes the interaction of the soil skeleton and pore fluid, wherein k_fThe soil body Darcy permeability coefficient is adopted, and g is the gravity acceleration; e.g. of a cylinder_sIs the volume strain of the soil framework,

u_sis the radial displacement, w, of the soil skeleton_sIs the vertical displacement of the soil skeleton; u. u_fIs the radial displacement of the pore fluid, w_fIs the vertical displacement of the pore fluid; v²Is a function of the laplacian of arithmetic,

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:

the simultaneous process between the expressions (14a) to (14b) is abbreviated

Substituting the formulas (15a) to (15b) into the formula (16) to obtain the volume strain and pore fluid pressure of the soil skeleton:

▽²e_s-a₁▽²p^f+a₂e_s＝0； (16)

wherein the intermediate variable

Similarly, the simultaneous processes between the expressions (15a) to (15b) are abbreviated as

Substituting the formula (14e) into the formula (17), wherein the volume strain and the pore fluid pressure of the soil framework are obtained:

▽²p^f-a₃e_s＝0； (17)

wherein the intermediate variable

Substituting formula (17) for formula (16) by:

▽²e_s+α²e_s＝0； (18)

wherein the intermediate variable alpha²＝a₂-a₁a₃；

By a separation variable method, let e_sR (r) Z (z), i.e. to e_sPerforming 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:

wherein, b₁、b₂Is an unknown constant and satisfies

The solution of equation (19) is:

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:

e_s＝[A₁K₀(b₁r)+A₂I₀(b₁r)][B₁sin(b₂z)+B₂cos(b₂z)]； (21)

it can be derived from equation (17):

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:

p^fh＝[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 satisfied^fSpecial solution form of and e_sIn agreement, therefore let the special solution of equation (22) be:

p^ft＝T₁e_s＝T₁[A₁K₀(b₁r)+A₂I₀(b₁r)][B₁sin(b₂z)+B₂cos(b₂z)]； (24)

substituting equation (24) into equation (22) yields an intermediate variable T₁：

Therefore, there are:

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 E_pPoisson ratio is upsilon^pDensity of rho_pRadius 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 P₀e^iωtSeparating the pile from the pile-soil system, and analyzing the stress condition of the pile, wherein

P₀The magnitude of the external load p (t); e.g. of the type^iωtThe 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:

wherein u is_p(z) denotes radial displacement of the pile flanks, w_p(z) represents axial displacement of the pile sides; mu.s^pLame constant, mu, of pile foundation^p＝E_p/2/(1+υ^p)；A^pIs the cross-sectional area of the pile foundation,

j represents the polar moment of inertia of the pile foundation,

f_r(z) represents the total load in the radial direction due to the pile foundation inertia and the soil reaction force, f_z(z) represents the total load generated by pile foundation inertia force and soil reaction force in the axial direction;

t_rthe pile side radial surface force is obtained;

t_zis pile side axial force;

the axial force and the shearing force of the pile are calculated according to the formula:

wherein N (z) is the axial force of the pile, Q (z) is the shear force of the pile,

is the axial stress on the cross section of the pile foundation,

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 u_p(z)＝U_pe^ηz，w_p(z)＝W_pe^ηz，U_pAmplitude, W, of a distribution form function of radial displacement of the pile side along an axial coordinate z_pThe 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 t_r＝0、t_z0, and substituting them into formula (9):

if equation (10) has a non-trivial solution, its coefficient matrix determinant value is required to be zero, i.e.:

a homogeneous solution of the displacement of the pile containing unknown constants can thus be obtained as:

wherein D_jRepresents 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 k_jIs a corresponding characteristic value η_jElement in the feature vector of (1), the feature value η_jThe feature vector of

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 framework_sAnd pore fluid pressure p^fThe solution of (2) and the formula (15a) are substituted into the formula (14a) to obtain the radial displacement u of the soil framework_sThe non-homogeneous differential equation of (a); then solving the volume strain e of the soil framework_sAnd pore fluid pressure p^fThe non-homogeneous differential equation can be solved to obtain u_sThe solution of (1); then the volume strain e of the obtained soil framework_sAnd pore fluid pressure p^fThe solution of (a) and the formula (15b) are substituted into the formula (14b) to obtain the vertical displacement w of the soil framework_sThen solving the volume strain e of the soil framework_sAnd pore fluid pressure p^fThe non-homogeneous differential equation can be solved to obtain w_sThe solution of (1); finally, the obtained pore fluid pressure p^fRadial displacement u of earth skeleton_sVertical displacement w of soil skeleton_sThe solution of (A) is respectively substituted into the formulas (15a) to (15b) to obtain the radial displacement u of the pore fluid_fAnd vertical displacement w of pore fluid_fThe 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 framework_sThe solution of (A) is as follows:

vertical displacement w of the soil framework_sThe solution of (a) is:

radial displacement u of the pore fluid_fThe solution of (a) is:

vertical displacement w of the pore fluid_fThe solution of (a) is:

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

Comprises the following steps:

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

Comprises the following steps:

effective stress component of soil framework acting on r plane and then pointing to z direction

And effective stress component of soil framework acting on z plane and pointing to r direction

Comprises the following steps:

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

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.:

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.:

the displacement of the saturated soil layer on the rigid bedrock is zero, namely:

w_s(r,z＝L,t)＝0，w_f(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.:

u_f(r₀,z,t)＝u_p(z,t)； (27d)

wherein u is_f(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:

u_s(r₀,z,t)＝u_p(z,t)，w_s(r₀,z,t)＝w_p(z,t)； (27e)

wherein u is_s(r₀Z, t) is the radial displacement of the soil skeleton, w_s(r₀Z, t) is the vertical displacement of the soil skeleton, u_p(z, t) is the radial displacement of the pile flank, w_p(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.:

u_p(z＝L,t)＝0，w_p(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

And boundary conditions (27a) to (27b), which are derived as:

wherein the intermediate variable

Then, since the stress at the pile-soil interface is continuous, the pile side radial surface force t can be obtained_rAnd pile flank axial force t_zComprises the following steps:

wherein,

representing the total radial stress component acting on the lateral surface of the pile foundation, according to the effective stress principle

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:

wherein, C_1n、C_2nAnd C_3nAre all unknown constants, β_1n～β_6nAre all intermediate variables, and

Δ_3n＝c_2nc_3n+c_1nc_4n，

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 soils_1n、C_2nAnd C_3nAnd solving to obtain an unknown constant C_1n、C_2nAnd C_3nThen 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 solving_1n、C_2nAnd C_3nThe method specifically comprises the following steps:

wherein,

P₀the amplitude of the external load p (t),

D₁～D₄and C_1n～C_3nThe 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

The result is obtained; intermediate variables of coefficients in pile foundation displacement solution obtained from pile foundation boundary condition equations (27f) and (27g)

Comprises the following steps:

X₁＝h₁η₁+h₁M_q1-k₁M_q2，X₂＝h₂η₂+h₂M_q3-k₂M_q4，X₃＝h₃η₃+h₃M_q5-k₃M_q6，

X₄＝h₄η₄+h₄M_q7-k₄M_q8，

intermediate variable Delta of coefficient in pile foundation displacement solution obtained from pile-soil continuity conditional expressions (27d) and (27e)_1n＝β_1n(γ_5nγ_9n+γ_6nγ_8n)-β_2n(γ_6nγ_7n-γ_4nγ_9n)-β_3n(γ_4nγ_8n+γ_5nγ_7n)，Δ_2n＝β_1n(γ_3nγ_5n+γ_2nγ_6n)+β_2n(γ_3nγ_4n-γ_1nγ_6n)-β_3n(γ_2nγ_4n+γ_1nγ_5n) Wherein γ is_1n＝β_n1-Tb_nK₂(b_1nr)，γ_2n＝β_2n-T₃b_nK₁(b_nr₀)，γ_3n＝β_3n-K₁(b_4nr₀)，γ_4n＝(ρ^fω²T₂α²+n^fa₃)b_1nK₁(b_1nr₀)，γ_5n＝α²(n^f-ρ^fω²T₃)b_nK₁(b_nr₀)，γ_6n＝α²ρ^fω²K₁(b_4nr₀)，γ_7n＝T₄b_nK₀(b_1nr₀)+β_4n，γ_8n＝T₅b_nK₀(b_nr₀)+β_5n，

R_n＝(γ_2nγ_4n+γ_1nγ_5n)(γ_6nγ_7n-γ_4nγ_9n)+(γ_3nγ_4n-γ_1nγ_6n)(γ_4nγ_8n+γ_5nγ_7n)；

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:

wherein the intermediate variable

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 displacement_d(ω) is:

wherein, K_R(ω) represents the true dynamic stiffness and ability of the pile-soil system to resist axial strain, K_R(ω)＝real(K_d(ω))；C_IRepresenting 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, C_I＝imag(K_d(ω))；

Further, the time domain semi-analytic solution expression of the pile top velocity V (t) is obtained as follows:

wherein T is the pulse width of pile top excitation during low strain dynamic measurement of pile foundation, Q_maxThe pulse amplitude of pile top excitation during low-strain dynamic measurement of the pile foundation is obtained.

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