^{1}

^{1}

^{2}

^{1}

^{1}

^{2}

An analytical solution is developed in this paper to investigate the horizontal dynamic response of a large-diameter pipe pile in viscoelastic soil layer. Potential functions are applied to decouple the governing equations of the outer and inner soil. The analytical solutions of the outer and inner soil are obtained by the method of separation of variables. The horizontal dynamic response and complex dynamic stiffnesses of the pipe pile are then obtained based on the continuity conditions between the pile and the outer and inner soil. To verify the validity of the solution, the derived solution in this study is compared with an existing solution for a solid pile. Numerical examples are presented to analyze the vibration characteristics of the pile and illustrate the effects of major parameters on the stiffness and damping properties.

Many studies have been devoted to the horizontal dynamic response of pile foundation in the recent years. The Winkler model is a popular method for the analysis of horizontal response of piles. Novak [

A new type of pile called cast-in-situ concrete large-diameter pipe pile (referred to as PCC pile) has been developed and widely applied in China [

The main assumptions adopted in this paper are as follow.

The computational model is shown in Figure

Computational model.

The dynamic equilibrium equations of the outer soil in polar coordinate system can be expressed as

The dynamic equilibrium equations of the inner soil in polar coordinate system can be expressed as

The horizontal displacement of the pile

The boundary conditions at the tops of the outer and inner soil are:

The boundary conditions at the bottoms of the outer and inner soil are

The boundary conditions at the bottom of the pile are

The continuity conditions of displacements on the outer interface are

The continuity conditions of displacements on the inner interface are

For the amplitudes

Equations (

Using the method of separation of variables, given

The solutions for (

The potential function

The displacement and stress of the outer soil vanish to zero when

It is found from (

Substituting (

The radial displacement of the outer soil on the outer interface can be expressed as

The horizontal resistance of the outer soil can be obtained as

For the amplitudes

Using the method of separation of variables, the potential functions

The displacement and stress of the inner soil are limited values when

Substituting (

Substituting (

The radial displacement of the inner soil on the inner interface can be expressed as

The horizontal resistance of the inner soil can be obtained as

The amplitude

The solution for (

It can be obtained from (

It is found that

Multiplying

With the displacement of the pile described by (

Substituting (

In this section, numerical results are presented to verify the validity of the solution and analyze the horizontal vibration characteristics of the pile-soil system. In the numerical procedure, the summation of ^{3}, ^{3},

This solution is verified by being compared with Nogami's solution (1977) for a solid pile. Given

Comparison of the horizontal complex stiffness of this solution with Nogami's solution (1977).

Real part

Imaginary part

The complex dynamic stiffness on the pile head is often used to analyze the vibration characteristics of the pipe pile. Three types of complex stiffnesses are given in (

Figures

Variation of horizontal complex stiffness of the pipe pile with the pile length.

Real part

Imaginary part

Variation of rocking complex stiffness of the pipe pile with the pile length.

Real part

Imaginary part

Variation of horizontal-rocking complex stiffness of the pipe pile with the pile length.

Real part

Imaginary part

Figures

Variation of horizontal complex stiffness of the pipe pile with the pile radii.

Real part

Imaginary part

Variation of rocking complex stiffness of the pipe pile with the pile radii.

Real part

Imaginary part

Variation of horizontal-rocking complex stiffness of the pipe pile with the pile radii.

Real part

Imaginary part

Figures

Variation of horizontal complex stiffness of the pipe pile with the shear modulus of soil.

Real part

Imaginary part

Variation of rocking complex stiffness of the pipe pile with the shear modulus of soil.

Real part

Imaginary part

Variation of horizontal-rocking complex stiffness of the pipe pile with the shear modulus of soil.

Real part

Imaginary part

By considering the coupled vibration between the pile and both the outer and inner soil, the analytical solution of the horizontal response of a large-diameter pipe pile in viscoelastic soil layer has been derived in this paper. The validity of the solution proposed in this study is verified by being compared with Nogami's solution for a solid pile. A parametric study has been conducted to investigate the vibration characteristics and the effects of major parameters. The calculated results reveal that

In this paper, the pile tip is clamped. However, there may be some other conditions at the pile tip, such as pinned, free, and elastic supporting. The present solution can be easily extended to other boundary conditions. Further study is needed to investigate the horizontal response of pipe pile in these boundary conditions.

The authors declare that there is no conflict of interests regarding the publication of this article.

This work was supported by the National Natural Science Joint High Speed Railway Key Program Foundation of China (Grant no. U1134207), the Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1125), and the Program for New Century Excellent Talents in University.