DYNAMICAL RESPONSE OF AN ELASTIC SUPPORTING PILE EMBEDDED IN SATURATED SOIL UNDER HORIZONTAL VIBRATION

Using the principles of soil dynamics and structural dynamics, an analytical solution for an elastic supporting pile under horizontal vibration was formulated, based on an improved beam-ondynamic-Winkler-foundation model and taking into consideration the change along the soil depth and the inertia phase of the fluid. The degenerated impedance on the top of the elastic supporting pile was proved to be correct. The parameters influencing the horizontal vibration characteristics of the elastic supporting pile were analysed in detail, and the influence of the elastic support on the impedance on the pile-top was discussed.


INTRODUCTION
The mechanism of pile-soil interaction in horizontal vibration of an elastic supporting pile during action of the machine is an important issue for geotechnical workers.Numerous studies on the dynamic response of pile foundations under vibration have been conducted.Novak [1], Nogami and Novak [2], and Novak and Aboul-Ella [3] have systematically studied pile-soil interactions in single-phase soil under vibration.Fixation of the underground pile may not be completely rigid when the soil around the pile is soft and the pile is not embedded deep into bedrock or when the bedrock is soft.Novak's plane strain model uses a series of simplified algorithms of radiation damping to calculate when the pile is horizontally vibrating in saturated soil [4][5][6].Some researchers have simulated the vertical dynamic response of a pipe pile, based on Biot's 3D poroelastic theory of porous media in saturated soil, by separating the variables and the potential function to provide an analytic solution for the impedance in a pile-tip, when the pile is embedded in saturated soil [7,8].The results will inevitably be inaccurate if the fixation of the pile is assumed to be completely rigid.In previous studies, the supporting role of the pile-tip soil against the pile has often been simplified to rigid support, but studies on the bearing contact of the pile-tip are rare.Low-strain integrity testing has been widely used in pile foundation quality detection.In this, the tubular structure is subjected to a transient point loading, and analytical solutions are obtained [9][10][11][12].---------------------------------------------------------------------------------------------------------------- Currently, no studies have been reported on the dynamical characteristics of an elastic supporting pile embedded in saturated soil under horizontal vibration.This paper considers the components of displacement along the depth and fluid inertia of the pile.Based on the dynamic consolidation theory proposed by Biot, the dynamic characteristics of an elastic supporting pile embedded in saturated soil under horizontal vibration are obtained.The solution has broad applicability and can meet the demand of projects.

Basic assumptions
The analyses in this paper are based on the following basic assumptions.The pile is assumed to be a cylinder of radius 0 r , height H , and constant cross-sectional dimensions (Figure 1).The soil around the pile is saturated and is both isotropic and homogeneous.The pile of soil is isotropic, is of uniform quality, and has a constant radius.Based on an improved beam-on-dynamic-Winklerfoundation model, the gradient varies and the inertia phase of the fluid along the depth of volume stress, and the displacement components in the soil around the pile are considered.When horizontal vibration occurs, the pile and soil are only displaced laterally and vertical displacement is negligible.The bottom support of the pile is the elastic support, only can obtain motion in horizontal direction but not relative rotation.During vibration, the pile and soil are in full contact; therefore, slip and detachment do not occur.The spring anti-horizontal displacement comprises a horizontal resistance coefficient of soil

Boundary Conditions
The five boundary conditions are described below: (1) Boundary conditions for the soil at the pile hole wall: (2) Horizontal distance continues infinitely.Displacement is attenuated to 0.
(3) Upper and lower boundary conditions for the soil: (4) Boundary conditions for the pile: (5) Coordination conditions of the pile and soil:

Governing Equation
According to the analysis of Zienkiewicz [13], for general soil dynamics problems, Biot's wave equation in saturated soil can be written as follows: The equation of motion of the fluid is as follows: Assuming that the soil particles and fluids are not compressible, the seepage flow equation of continuity in cylindrical coordinates is given as follows: where r u and u  denote the displacement of the soil skeleton along the directions of r and  , respectively.
n is the porosity of the soil.
 represents the second order operator of Laplace, such that .  and M are the parameters that reflect the compressibility of soil, such that 01   and 0 M    , when M  and 1   , which means that the particles and pore fluid are not compressible.

Derivations of the equations
The following section introduces the scalar potential function By substituting the potential function into Equations.( 6) to (10) and separating the vector and scalar, the following equations are obtained: It can be seen that 11 ( , , ) ( ) By substituting Equations ( 16) and (17) into Equations ( 12) to (15), we arrive at the following equations: .14311/CEJ.2017.02.0010 103 where From the boundary conditions of the elastic supporting pile, Equation (3) yields the following: can be calculated by programming.
To give Equations ( 18) and ( 19) an nontrivial solution, the coefficient determinant must be made 0 and then should be simplified as follows: where Using the operator decomposition theory, the solution of Equation ( 26) is obtained by separation of variables: DOI 10.14311/CEJ.2017.02.0010 104 where    27) and ( 28) can be simplified as follows: ) Substituting Equations (29) to (32) into Equations ( 12) to (15) yields the following: ) From the stress-strain relationship of saturated soil, the expressions of u into a series form results in the following equation: By contacting the boundary conditions (1) and the impervious conditions of the pile-soil contact surfaces, the following is obtained: 3n equations can be got by simultaneous Equations ( 38) to (40), and simultaneous every three equations corresponding to each n-order mode, the corresponding variables of each equation (

C c c c c c c c c c c c c c c c c c c c c c c c c c c U
where The transverse dynamic reaction force 1 u f of the pile-side soil to the pile can be calculated.The value of 1 u f is positive when its direction coincides with the direction of movement.Then, 1 () u fz can be represented as follows:

SOLVING THE DIFFERENTIAL EQUATION
The differential equation of the single pile under horizontal vibration can be built by considering the dynamic equilibrium conditions when the thickness of thin layer element in the pile is .For steady-state vibration, the horizontal displacement of the pile can be expressed as . Substituting this into Equation ( 46) and simplifying yields, the following equation is obtained: where For ease of calculation, t i e  is omitted.The solution of this fourth order ordinary differential equation is as follows: According to the contact condition of the pile and soil, the following equation is obtained: The cosh function () n ch J z is an orthogonal function system in [0, ] H , means the following: Multiplying each side of Equation ( 50) by () m ch J z then integrating over [0, H gives the following:

[ sin( ) cos( ) ( ) ( )] ( )
Taking the integral of Equation ( 3) to (51) gives the following equation: where Then n U can be obtained as follows: where The rotation amplitude  , the bending moment amplitude () Mz, the displacement amplitude U , and the shear amplitude () Qz can be obtained using the relations . By setting the displacement and internal forces of the pile-top for     The impedance in the pile-tip can be calculated as follows: , and 41_ 2 f are coefficients when unit horizontal displacement occurs in the piletop, and can be also solved by a boundary condition as follows: ( ) 0, ( ) 0 The impedance in the pile-tip can be calculated using the following equation: Equations ( 62), ( 63), (66), and (67) are impedances in the pile-top.For ease of calculation, the dynamic factors of an elastic supporting pile under horizontal vibration can be defined as follows:

MODEL VALIDATION
To validate the rationality of the theory in this paper, the lateral complex impedance factors were used as a sample, with Figure 2 shows that when the horizontal stiffness coefficient of the pile bottom is reasonable, the model described in this paper produces similar results to the model of a rigid supporting pile.This means that if the stiffness in the tip of an elastic supporting pile is large enough, it is reasonable to simplify it as a rigid supporting pile.

Influence of the lateral resistance coefficient of the pile-tip on lateral complex impedance of the pile-top
The relationship curve between the lateral resistance coefficient of the pile-tip on the lateral complex impedance and the frequency of the pile-top is shown in Figure 3 .The horizontal resistance coefficient ratio in the pile-top * p k reflects the bearing capacity of the pile-bottom medium to pile-tip.The greater is its value, the stronger is the ability of the pile-tip to resist horizontal displacement and the larger is the horizontal displacement stiffness of the pile bottom.When * p k is infinite, the elastic supporting pile can be equivalent to a rigid supporting pile.
Figure 3 depicts that the horizontal complex stiffness factor in the pile-tip appears to have more obvious peaks and troughs.The smaller is the horizontal resistance coefficient in the pile-tip, the larger is the wave amplitude of the horizontal complex stiffness in the pile-top and the more obvious is the influence of the horizontal resistance coefficient ratio in the pile-tip.
When the ratio of the length to diameter is small, the changes of * p k have a large impact on the complex stiffness in the pile-tip and will generate large fluctuations in the horizontal, swing, or horizontal-swing complex impedance amplitude.With an increase in the ratio of length to diameter, the horizontal resistance in the pile-tip becomes less obvious, and the volatility in the corresponding vibration frequency grows.This means that the complex impedance in a pile-tip with a small ratio of length to diameter is more susceptible to the influence of the horizontal resistance coefficient in the pile bottom.
For the pile of larger diameter, when the frequency is low (especially the low-frequency range relevant to power foundation design), the horizontal resistance coefficient in the pile-tip has almost no effect on the horizontal complex impedance in the pile-top.As the frequency increases, the influence of the lateral resistance coefficient of the pile-tip on lateral complex impedance of the pile-top increases.
-  In the physical mechanism, under the conditions of low frequency and soft soil, wave propagation follows the horizontal direction.The horizontal connection condition of the pile-tip (horizontal reaction coefficient of soil around the pile bottom) has an influence on the complex impedance factor in the pile-top.Under conditions of high frequency (the vibration frequency is greater than 0.35 Hz) this effect is not obvious.

CONCLUSION
This work derived the analytical solution of an elastic supporting pile embedded in saturated soil under horizontal vibration.We substantiated the validity of this method by retrogressed verification of the model, and analyzed the influence of elastic support for lateral complex impedance in a piletop.

sk
and a horizontal resistance coefficient of pile-tip p k , which means that the shear resistance of the soil and the pile-tip need to resist a unit horizontal displacement.

Fig. 1 -
Fig. 1 -Structure diagram of an elastic supporting pile embedded in saturated soil under horizontal vibration

fp
is the excess pore water pressure.
the dynamic permeability coefficient of the soil, where d k is the permeability coefficient of the soil and g is the gravitational acceleration.e is the volume strain of the soil skeleton, where 1

1  and 2  , the vector potential function 1  and 2 
, the displacement vector of the saturated soil skeleton u , and the relative displacement vector of the fluid w :

2 
have the same form of the potential function, then 11 ( , , ) ( )


and

C
) can be shown as n U , and then all coefficients contain only one variable.n U can be determined from the condition of pile-soil interaction.Below, is the horizontal displacement of the pile at soil depth of z , p E and p I are the elastic moduli and the rotational moment of inertia of the pile, respectively, and p m is the mass of the pile per unit length.

Q
, and substituting them into Equation (61), the coefficient matrix [T(z)] can be calculated.
it is assumed that unit horizontal displacement occurs in the pile-top, which can be solved by the boundary condition: 14311/CEJ.2017.02.0010 108 to calculate the lateral complex impedance in the pile-tip using an elastic supporting pile and a rigid supporting pile.Custom frequency ratio

DOI 10 .Fig. 2 -
Fig. 2 -Comparison of present model (in this work) and the model of a rigid supporting pile

Fig. 3 -
Influence of lateral resistance coefficient ratio of the pile-tip on lateral complex impedance factors in the pile-topInfluence of lateral resistance coefficient of soil on the lateral complex impedance of the pile-top

Figure 4 .Fig. 4 -
Figure 4 depicts the relationship curve when the lateral resistance coefficient ratio of soil is*