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This paper presents a theoretical investigation on the lateral force of pile in liquefaction site. Regarded liquefied soil as fluid, the vector method can be used to analyze the liquefaction velocity field and solve the analytical solution of the dynamic field by making use of the principle of fluid mechanics. In addition, by solving the velocity field with vector symbol operation method, the analytical expression of the lateral force in the liquefied flow field is obtained, and the sensitivity of the parameters in the analytical expression is analyzed. The results show that the stress field of the pile contains both the pressure resistance caused by surface pressure and the friction resistance caused by shear stress, when the liquefied soil flows laterally. The lateral forces on the pile are mainly composed of inertial forces and damping forces and are related to density, fluid viscosity, pile radius, and vibration frequency. With the increase of density, fluid viscosity, and pile radius, the added mass and added damping increase gradually. In a certain range, added mass and added damping are sensitive to vibration frequency.

Sand liquefaction caused by almost every earthquake is accompanied by large deformation, and the lateral displacement caused by seismic liquefaction causes great damage to the pile foundation [

The researchers mainly used the solid mechanics concept of pile-soil interaction to explain the failure of pile foundation in liquefiable site. It is believed that the inertial force of the upper structure is the main reason for the strength failure of the pile foundation. Ting [

In recent years, lateral displacement caused by seismic liquefaction has caused great losses to pile. Due to the significant difference between the mechanical properties of the liquefaction soil and the solid soil, such as the low shear resistance, high fluidity, and other fluid properties, it is a novel research idea to study the liquefaction of saturated sand soil on the basis of fluid mechanics. The basic idea of adopting fluid mechanics to study the liquefaction effect is to view the liquefied sand as a kind of fluid and then to study the liquefaction effects such as flow deformation or soil-structure interaction by using the computational methods of mechanics.

Relevant scholars have done some researches and analysis and have obtained some useful research conclusions. Towhata et al. [

In this paper, the liquefied soil is regarded as fluid, and the dynamic field analysis of liquefaction site is carried out based on the principle of fluid mechanics and the vector symbol operation method, and the analytical solution of the kinetic force field is solved. After obtaining the analytical expression of lateral force of pile considering liquefaction effect, it is known that the lateral force of the liquefaction site on pile is mainly composed of inertia force and damping force and is related to fluid density, fluid viscosity, pile radius, and vibration frequency. Finally, the sensitivity analysis of the parameters in the analytical solution is carried out to obtain the sensitivity degree of different parameters.

In the process of soil liquefaction, the soil changes from solid state to liquid state due to the increase of pore water pressure and the decrease of effective stress [

Schematic diagram of velocity field of the liquefaction site.

Assume the radius of cross section of the pile is

The velocity of the pile in the liquefied flow field is

The liquefied soil can be viewed as incompressible fluid and can be obtained according to the following continuous equation:

According to the vector theory, the polar vector

Therefore, the velocity field caused by the vibration of the pile at the velocity

Since the liquefied soil is an incompressible fluid, the momentum equation can be obtained as follows:

Since the fluid motion caused by soil liquefaction belongs to a uniform flow with low Reynolds number, the inertia term

Take the curl from left to right of the above equation, and because

Substitute equation (

Then

It is known from the boundary conditions at infinity that

Velocity field can be obtained by using the vector symbol operation method in the cylindrical coordinate system:

According to the boundary condition

According to equation (

Since flowing liquefied soil is an incompressible fluid, according to the Navier–Stokes equation and stress tensor formula, equations (

Therefore, the pressure distribution around the pile is

Thus, the stress field around the pile can be obtained as

According to the calculation results,

The lateral force applied to the pile is parallel to the velocity of soil liquefaction flow. Therefore, all surface forces acting on the surface of the pile are projected to the direction of fluid velocity, and integrate along the whole cylinder of the pile. The lateral force acting on the unit length pile is

On the pile cylinder

According to formula (

Formula (

Finally, the lateral force of liquefaction site on the pile can be expressed as

Sensitivity is defined as the derivative of a function with respect to a parameter; at the parameter

The sensitivity of density, fluid viscosity, pile radius, and vibration frequency is analyzed below, and take the reference value of each parameter as density of liquefied soil

The curves of the added mass and added damping with the density of liquefied soil are shown in Figure

Influence of density on the added mass (a) and added damping (b).

The curves of added mass and added damping with the viscosity of liquefied soil are shown in Figure

Influence of fluid viscosity on the added mass (a) and added damping (b).

The curves of added mass and added damping with pile radius are shown in Figure

Influence of pile on the added mass (a) and added damping (b).

The curves of the added mass and added damping with the vibration frequency of the pile are shown in Figure

Influence of vibration frequency on the added mass (a) and added damping (b).

Using the CFD fluid module of ABAQUS finite element platform, Abaqus/CFD provides advanced computational fluid dynamics capabilities with extensive support for preprocessing and postprocessing provided in Abaqus/CAE. These scalable parallel CFD simulation capabilities address a broad range of nonlinear coupled fluid-structural problems. Abaqus/CFD can solve the following types of incompressible flow problems: the internal or external flows that are steady-state or transient, span a broad Reynolds number range, and involve complex geometry that may be simulated with Abaqus/CFD. This includes flow problems induced by spatially varying distributed body forces.

A fluid finite element model is established to carry out numerical calculation of liquefied dynamic field, based on the principle of minimum potential energy and basic control equation of flow. A site is adopted to simulate the flow effect of the liquefaction site. The schematic diagram of the site is shown in Figure

Schematic diagram of the liquefied site. (a) Sectional view. (b) Vertical view.

For the boundary conditions, in order to better simulate the infinite dynamic flow field, in the whole liquefied soil flow area, all the boundaries except the fluid outlet are set as the velocity inlet. Some parameters used in numerical calculation are as follows: the CFD model is adopted for liquefied soil mass, and thus,

The comparison of the analytical and numerical solutions of the stress distribution field around the pile is shown in Figure

Analytical solution and numerical solution of the stress field around the pile (the section at 10 m depth).

Regarding the liquefied soil as fluid, the dynamic field analysis of liquefaction site is carried out based on the principle of fluid mechanics and the vector symbol operation method, and the analytical solution of the kinetic force field is solved. The analytical expression of the lateral force of pile is obtained considering the liquefaction effect, the sensitivity analysis of the parameters in the analytical solution is carried out, and then the influence of different parameters on the lateral force of pile is obtained. The main conclusions are as follows:

According to the analytical solution calculated by the liquefaction dynamic field theory, it can be known that when the liquefied soil flows laterally, the stress field around the pile includes the pressure resistance caused by the surface pressure and the friction resistance caused by the shear stress.

The lateral force of liquefied soil on pile is mainly composed of added inertia and added damping, which are related to density, fluid viscosity, pile radius, and vibration frequency.

With the increase of density, fluid viscosity, and pile radius, the added mass and added damping are gradually increasing. The influence of density and fluid viscosity on the added damping is greater, and the pile radius is only sensitive to the additional mass. Vibration frequency is particularly sensitive to added mass and added damping within a certain range.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors gratefully acknowledge the financial support for this study from the project of the National Natural Science Foundation of China (Grant no. 51408281), the Natural Science Foundation of Jiangsu Province (Grant no. BK20140108), and Qing Lan Project (2016).