#### Abstract

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.

#### 1. **Introduction**

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 [1] studied the dynamic response of pile subjected to horizontal, vertical, and rocking loads for the frequency domain. Based on Novak’s frequency domain solution, Nogami and Konagai [2] developed a time domain Winkler model consisting of series of springs, dashpots, and masses units with frequency independent parameters. Yao and Nogami [3] proposed an analytical solution for the low-frequency cyclic response of a pile in a linear viscoelastic Winkler subgrade. A nonlinear Winkler model was developed by Nogami and Chen [4] in the frequency domain and by Nogami et al. [5] in the time domain for pile lateral response. The Winkler model theory is simple and practical, but it neglects the coupled vibration between the pile and soil, and the impedance parameters of the model are hard to determine. Assuming the soil as an elastic half space and the pile as a virtual rod, Rajapakse and Shah [6, 7] proposed the integral solution of the soil and pile considering the real pile-soil reaction. However, this method is only suitable for a floating pile, and the corresponding numerical calculation requires considerable computational effort. Nogami and Novak [8, 9] developed a method in which the soil around the pile was considered as a homogeneous and isotropic layer. A closed-form solution of an end-bearing pile was proposed based on the continuity assumption of the displacement and stress between the pile and soil layer.

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 [10–13]. Besides PCC piles, many other large-diameter pipe piles are also widely used in practical engineering, such as prestressed concrete pipe piles and large-diameter steel pipe piles [14–16]. For large-diameter pipe piles, apart from the soil around the pile, soil exists inside the pile as well. The way in which the inner soil interacts with the pipe pile is worth studying. Only the interaction between the outer soil and the pile was considered in previous studies, and therefore their solutions cannot be used to analyze the dynamic response of large-diameter pipe piles. In this paper, an analytical solution for the horizontal dynamic response of a large-diameter pipe pile in viscoelastic soil layer is developed with the interaction of the pipe pile between both the outer and inner soil being taken into account. Numerical results are presented to analyze the vibration characteristics of the pile-soil system.

#### 2. **Basic Assumptions and Computational Model**

The main assumptions adopted in this paper are as follow. The outer and inner soil layers are viscoelastic, homogeneous, and isotropic, and the material damping is of the frequency independent hysteresis type; the surfaces of the outer and inner soil are free, and the bottoms of the outer and inner soil are fixed; the pile is elastic, and the pile tip is clamped; the deformation of the pile-soil system is small; the pile has perfect contacts with the outer and inner soil; the vertical displacements of the pile and soil are zero.

The computational model is shown in Figure 1. The pile is subjected to a time-harmonic horizontal force or rocking moment at the pile head. is the pile length. and are the outer and inner radii of the pile section, respectively. and are the soil resistances of the outer and inner soil, respectively.

#### 3. Governing Equations and Boundary Conditions

##### 3.1. Dynamic Equilibrium Equations of the Outer Soil

The dynamic equilibrium equations of the outer soil in polar coordinate system can be expressed as where , , and . In addition, , , and are the radial, circumferential, and vertical directions of the column coordinate, respectively; and are the amplitudes of the radial and circumferential displacements of the outer soil, respectively; , and , are the real and imaginary parts of the complex Lame's constants of the outer soil, respectively; is the mass density of the outer soil; is the excitation frequency.

##### 3.2. Dynamic Equilibrium Equations of the Inner Soil

The dynamic equilibrium equations of the inner soil in polar coordinate system can be expressed as where and . In addition, and are the amplitudes of the radial and circumferential displacements of the inner soil, respectively; , and , are the real and imaginary parts of the complex Lame’s constants of the inner soil, respectively; is the mass density of the inner soil.

##### 3.3. Dynamic Equilibrium Equation of the Pile

The horizontal displacement of the pile is governed by the following equation: where is the Young’s modulus of the pile, is the second moment of area of the pile section, and is the mass of the pile per unit length.

##### 3.4. Boundary Conditions and Initial Conditions

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

#### 4. **Solutions for the Governing Equations**

##### 4.1. Solutions for the Dynamic Equilibrium Equations of the Outer Soil

For the amplitudes and , (3) can be expressed as The potential functions and are introduced as It is easily obtained that Substituting (16) and (17) into (15), one obtains where .

Equations (18) can be written as where , , and . In addition, and are the longitudinal and shear wave velocities of the outer soil, respectively; and are the hysteretic damping ratios of the outer soil.

Using the method of separation of variables, given , (19) can be split into the following three equations: where .

The solutions for (21) can be easily obtained as where and are modified Bessel functions of the first and second kind of order , respectively, and , , , , , and are undetermined coefficients.

The potential function is expressed as Similarly, the potential function can be obtained as where and , , , , , and are undetermined coefficients.

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

It is found from (11) and (12) that is an even function of , and is an odd function of . Thus, and .

Substituting (23) and (24) into (6) and (8), one obtains Then, the potential functions and are written as Thereafter, the displacements of the outer soil can be expressed as follows: Substituting (27) and (28) into (11) and (12), respectively, yields It can be obtained from (29) and (30) that where .

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

The horizontal resistance of the outer soil can be obtained as where .

##### 4.2. Solutions for the Dynamic Equilibrium Equations of the Inner Soil

For the amplitudes and , (4) can be expressed as The potential functions and are introduced as Thus, Substituting (35) and (36) into (34), (34) can be expressed as: where , , and . In addition, and are the longitudinal and shear wave velocities of the inner soil, respectively; and are the hysteretic damping ratios of the inner soil.

Using the method of separation of variables, the potential functions and are obtained as where and .

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

is an even function of , and is an odd function of . Thus, and .

Substituting (38) into (7) and (9), one obtains Then, the potential functions and are written as Thereafter, the displacements of the inner soil can be expressed as follows:

Substituting (41) into (13) and (14) yields It can be obtained from (42) that where .

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

The horizontal resistance of the inner soil can be obtained as where .

##### 4.3. Solution for the Dynamic Equilibrium Equation of the Pile

The amplitude can be expressed as Substituting (33) and (45) into (46), one obtains where .

The solution for (47) can be obtained as where where , , , and are undetermined coefficients.

It can be obtained from (11), (13), (32), and (44) that Equations (50) and (51) can also be expressed as

It is found that . Given , it is obtained from (52) that

Multiplying on both sides of (50) and then integrating on the interval , one obtains Thus, where Substituting (53) into (55) yields Equation (48) can be expressed as where , , and , .

With the displacement of the pile described by (58), the angle of rotation , the bending moment , and the shear force are obtained as follows:Assuming the displacement, the angle of rotation, the bending moment, and the shear force at the pile head as , , , and , respectively, one obtains It can be obtained from (60) that where

Substituting (61) into (58), one obtains where Substituting (63) into (10), one obtains It can be obtained from (65) that The horizontal complex dynamic stiffness , rocking complex dynamic stiffness , and horizontal-rocking complex dynamic stiffness are expressed as

#### 5. **Numerical Results and Analysis**

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 is 20. Unless otherwise specified, the following parameter values are used: m, , m, GPa, g/cm^{3}, g/cm^{3}, MPa, , and .

##### 5.1. Verification

This solution is verified by being compared with Nogami's solution (1977) for a solid pile. Given , the solution for the pipe pile of this study is simplified to that of a solid pile. Figure 2 indicates that the simplified solution of this study agrees well with the solution proposed by Nogami (1977) for the horizontal response of a solid pile. The stiffness oscillates in the low-frequency range and then diminishes to zero and attains negative values in the high-frequency range. The damping approaches to zero at first and increases almost linearly with the frequency in the high-frequency range.

**(a) Real part**

**(b) Imaginary part**

##### 5.2. Analysis of the Vibration Characteristics of the Pile-Soil System

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 (67). The real parts of the complex stiffnesses represent the real stiffness, while the imaginary parts reflect the damping of the pile-soil system. The complex stiffnesses are influenced by many parameters such as the pile length, radii of pile section, and shear modulus of soil.

Figures 3, 4, and 5 show the influence of the pile length on the complex stiffnesses on the pile top. The stiffnesses decrease in low-frequency range (about 0~150 Hz) but increase in high-frequency range (150~300 Hz) as the pile length increases. In the low-frequency range, the dynamic stiffness of pile mainly depends on the static stiffness of pile. The static stiffness of pile decreases with the increase of the pile length since the pile tip is clamped. With increasing frequency, the pile-soil coupled vibration provides more soil resistances to the pile, so the stiffnesses become higher as the pile length increases in high-frequency range. The dampings increase with the increase of the pile length in the whole frequency range. However, when the pile length reaches a critical value, the stiffnesses and dampings show little change.

**(a) Real part**

**(b) Imaginary part**

**(a) Real part**

**(b) Imaginary part**

**(a) Real part**

**(b) Imaginary part**

Figures 6, 7, and 8 show the influence of the pile radii on the complex stiffnesses on the pile top. With the increase of or decrease of , the stiffnesses and dampings increase. When , the stiffness and damping are close to those of the pile with m. It shows that, when m, the effect of the inner radius is negligible. Furthermore, it is seen that the stiffnesses and dampings of the pile with m and m are larger than those of the pile with m and m, while the section areas of the two piles are approximately equal. It shows that the dynamic stiffness of a pipe pile increases with the increase of the average radius of the pile section.

**(a) Real part**

**(b) Imaginary part**

**(a) Real part**

**(b) Imaginary part**

**(a) Real part**

**(b) Imaginary part**

Figures 9, 10 and 11 show the influence of the shear modulus of soil on the complex stiffnesses on the pile top. The stiffnesses increase steeply with the increase of . The horizontal and horizontal-rocking damping increase, but the rocking damping decreases with the increase of . In the low-frequency range, has negligible influence on the stiffnesses and dampings. In the high-frequency range (about 200~300 Hz), the stiffnesses increase, but the dampings decrease slightly with the increase of .

**(a) Real part**

**(b) Imaginary part**

**(a) Real part**

**(b) Imaginary part**

**(a) Real part**

**(b) Imaginary part**

#### 6. **Conclusions**

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 the stiffnesses decrease in low-frequency range but increase in high-frequency range with the increase of the pile length. The dampings increase with the pile length in the whole frequency range. However, when the pile length reaches a critical value, the stiffnesses and dampings with different pile lengths have a little difference. With the increase of the outer radius or decrease of the inner radius, the stiffnesses and dampings all increase. However, when the inner radius is smaller than 0.2 m, the effect of the inner radius is negligible. Moreover, the dynamic stiffness of a pipe pile increases with the increase of the average radius of the pile section. The stiffnesses and the horizontal and horizontal-rocking damping increase, but the rocking damping decreases with the increase of the shear modulus of the outer soil. In the low-frequency range, the shear modulus of the inner soil has negligible influence on the stiffnesses and dampings. In the high-frequency range, the stiffnesses increase but the dampings decrease slightly with the increase of the shear modulus of the inner soil.

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.

#### Conflict of Interests

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

#### Acknowledgments

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.