Horizontal Vibration of a Large-Diameter Pipe Pile in Viscoelastic Soil

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. (1) The outer and inner soil layers are viscoelastic, homogeneous, and isotropic, and the material damping is of the frequency independent hysteresis type; (2) the surfaces of the outer and inner soil are free, and the bottoms of the outer and inner soil are fixed; (3) the pile is elastic, and the pile tip is clamped; (4) the deformation of the pile-soil system is small; (5) the pile has perfect contacts with the outer and inner soil; (6) 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. 𝐻 is the pile length. 𝑟₁ and 𝑟₂ are the outer and inner radii of the pile section, respectively. 𝑓₁(𝑧)𝑒^𝑖𝜔𝑡 and 𝑓₂(𝑧)𝑒^𝑖𝜔𝑡 are the soil resistances of the outer and inner soil, respectively.

Figure 1 

Computational model.

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 𝜆₁+2𝐺₁+𝑖𝜆^′₁+2𝐺^′₁𝜕𝑒𝜕𝑟¹𝑒^𝑖𝜔𝑡−1𝑟𝐺₁+𝑖𝐺^′₁𝜕𝑤𝜕𝜃¹𝑒^𝑖𝜔𝑡+𝐺₁+𝑖𝐺^′₁𝜕²𝜕𝑧²𝑢₁𝑒^𝑖𝜔𝑡=𝜌₁𝜕²𝜕𝑡²𝑢₁𝑒^𝑖𝜔𝑡,𝜆₁+2𝐺₁+𝑖𝜆^′₁+2𝐺^′₁𝜕𝑒𝑟𝜕𝜃¹𝑒^𝑖𝜔𝑡+𝐺₁+𝑖𝐺^′₁𝜕𝑤𝜕𝑟¹𝑒^𝑖𝜔𝑡+𝐺₁+𝑖𝐺^′₁𝜕²𝜕𝑧²𝑣₁𝑒^𝑖𝜔𝑡=𝜌₁𝜕²𝜕𝑡²𝑣₁𝑒^𝑖𝜔𝑡, (1)(2) where 𝑖=√−1, 𝑒₁=𝜕𝑢₁/𝜕𝑟+𝑢₁/𝑟+(1/𝑟)(𝜕𝑣₁/𝜕𝜃), and 𝑤₁=𝜕𝑣₁/𝜕𝑟+𝑣₁/𝑟−(1/𝑟)(𝜕𝑢₁/𝜕𝜃). In addition, 𝑟, 𝜃, and 𝑧 are the radial, circumferential, and vertical directions of the column coordinate, respectively; 𝑢₁ and 𝑣₁ are the amplitudes of the radial and circumferential displacements of the outer soil, respectively; 𝜆₁, 𝐺₁ and 𝜆^′₁, 𝐺^′₁ are the real and imaginary parts of the complex Lame's constants of the outer soil, respectively; 𝜌₁ is the mass density of the outer soil; 𝜔 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 𝜆₂+2𝐺₂+𝑖𝜆^′₂+2𝐺^′₂𝜕𝑒𝜕𝑟¹𝑒^𝑖𝜔𝑡−1𝑟𝐺₂+𝑖𝐺^′₂𝜕𝑤𝜕𝜃²𝑒^𝑖𝜔𝑡+𝐺₂+𝑖𝐺^′₂𝜕²𝜕𝑧²𝑢₂𝑒^𝑖𝜔𝑡=𝜌₂𝜕²𝜕𝑡²𝑢₂𝑒^𝑖𝜔𝑡,𝜆₂+2𝐺₂+𝑖𝜆^′₂+2𝐺^′₂𝜕𝑒𝑟𝜕𝜃²𝑒^𝑖𝜔𝑡+𝐺₂+𝑖𝐺^′₂𝜕𝑤𝜕𝑟²𝑒^𝑖𝜔𝑡+𝐺₂+𝑖𝐺^′₂𝜕²𝜕𝑧²𝑣₂𝑒^𝑖𝜔𝑡=𝜌₂𝜕²𝜕𝑡²𝑣₂𝑒^𝑖𝜔𝑡, (3)(4) where 𝑒₂=𝜕𝑢₂/𝜕𝑟+𝑢₂/𝑟+(1/𝑟)(𝜕𝑣₂/𝜕𝜃) and 𝑤₂=𝜕𝑣₂/𝜕𝑟+𝑣₂/𝑟−(1/𝑟)(𝜕𝑢₂/𝜕𝜃). In addition, 𝑢₂ and 𝑣₂ are the amplitudes of the radial and circumferential displacements of the inner soil, respectively; 𝜆₂, 𝐺₂ and 𝜆^′₂, 𝐺^′₂ are the real and imaginary parts of the complex Lame’s constants of the inner soil, respectively; 𝜌₂ is the mass density of the inner soil.

3.3. Dynamic Equilibrium Equation of the Pile

The horizontal displacement of the pile 𝑢_𝑝(𝑧)𝑒^𝑖𝜔𝑡 is governed by the following equation: 𝐸_𝑝𝐼_𝑝𝜕⁴𝜕𝑧⁴𝑢_𝑝𝑒^𝑖𝜔𝑡𝜕+𝑚²𝜕𝑡²𝑢_𝑝𝑒^𝑖𝜔𝑡+𝑓₁(𝑧)𝑒^𝑖𝜔𝑡+𝑓₂(𝑧)𝑒^𝑖𝜔𝑡=0, (5) where 𝐸_𝑝 is the Young’s modulus of the pile, 𝐼_𝑝 is the second moment of area of the pile section, and 𝑚 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: 𝜕𝑢₁𝜕𝑧_𝑧=0=0,𝜕𝑢₂𝜕𝑧_𝑧=0=0. (6)(7)

The boundary conditions at the bottoms of the outer and inner soil are 𝑢_1𝑧=𝐻=𝑣_1𝑧=𝐻𝑢=0,_2𝑧=𝐻=𝑣_2𝑧=𝐻=0. (8)(9)

The boundary conditions at the bottom of the pile are 𝑢_{𝑝𝑧=𝐻}=0,𝜕𝑢_𝑝𝜕𝑧_𝑧=𝐻=0. (10)

The continuity conditions of displacements on the outer interface are 𝑢₁𝑟₁,𝜃,𝑧,𝑡=𝑢_𝑝()𝑣𝑧,𝑡cos𝜃,₁𝑟₁,𝜃,𝑧,𝑡=−𝑢_𝑝(𝑧,𝑡)sin𝜃. (11)(12)

The continuity conditions of displacements on the inner interface are 𝑢₂𝑟₂,𝜃,𝑧,𝑡=𝑢_𝑝()𝑣𝑧,𝑡cos𝜃,₂𝑟₂,𝜃,𝑧,𝑡=−𝑢_𝑝(𝑧,𝑡)sin𝜃. (13)(14)

4. Solutions for the Governing Equations

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

For the amplitudes 𝑢₁ and 𝑣₁, (3) can be expressed as 𝜆₁+2𝐺₁+𝑖𝜆^′₁+2𝐺^′₁𝜕𝑒₁−1𝜕𝑟𝑟𝐺₁+𝑖𝐺^′₁𝜕𝑤₁+𝐺𝜕𝜃₁+𝑖𝐺^′₁𝜕²𝑢₁𝜕𝑧²=−𝜌₁𝜔²𝑢₁,𝜆₁+2𝐺₁+𝑖𝜆^′₁+2𝐺^′₁𝜕𝑒₁+𝐺𝑟𝜕𝜃¹+𝑖𝐺^′₁𝜕𝑤₁+𝐺𝜕𝑟₁+𝑖𝐺^′₁𝜕²𝑣₁𝜕𝑧²=−𝜌₁𝜔²𝑣₁. (15) The potential functions 𝜙₁(𝑟,𝜃,𝑧,𝑡) and 𝜓₁(𝑟,𝜃,𝑧,𝑡) are introduced as 𝑢₁=𝜕𝜙₁+1𝜕𝑟𝑟𝜕𝜓₁𝑣𝜕𝜃₁=1𝑟𝜕𝜙₁−𝜕𝜃𝜕𝜓₁.𝜕𝑟 (16) It is easily obtained that 𝑒₁=∇²𝜙₁,𝑤₁=−∇²𝜓₁. (17) Substituting (16) and (17) into (15), one obtains 𝜆₁+2𝐺₁+𝑖𝜆^′₁+2𝐺^′₁∇²𝜙₁+𝜌₁𝜔²+𝐺₁+𝑖𝐺^′₁𝜕²𝜕𝑧²𝜙₁𝐺=0,₁+𝑖𝐺^′₁∇²𝜓₁+𝜌₁𝜔²+𝐺₁+𝑖𝐺^′₁𝜕²𝜕𝑧²𝜓₁=0, (18) where ∇²=𝜕²/𝜕𝑟²+(1/𝑟)(𝜕/𝜕𝑟)+(1/𝑟²)(𝜕²/𝜕𝜃²).

Equations (18) can be written as 𝜂²₁+𝑖𝜂²₁−2𝐷_𝑣1+2𝐷_𝑠1∇²𝜙₁+𝜔𝑣_𝑠1²+1+𝑖𝐷_𝑠1𝜕²𝜕𝑧²𝜙₁=0,1+𝑖𝐷_𝑠1∇²𝜓₁+𝜔𝑣_𝑠1²+1+𝑖𝐷_𝑠1𝜕²𝜕𝑧²𝜓₁=0, (19)(20) where 𝜂=𝑣_𝑙1/𝑣_𝑠1=[(𝜆₁+2𝐺₁)/𝐺₁], 𝐷_𝑣1=𝜆^′₁/𝜆₁, and 𝐷_𝑠1=𝐺^′₁/𝐺₁. In addition, 𝑣_𝑙1 and 𝑣_𝑠1 are the longitudinal and shear wave velocities of the outer soil, respectively; 𝐷_𝑣1 and 𝐷_𝑠1 are the hysteretic damping ratios of the outer soil.

Using the method of separation of variables, given 𝜙₁=𝑅₁₁(𝑟)Θ₁₁(𝜃)𝑍₁₁(𝑧), (19) can be split into the following three equations: 1𝑅₁₁𝑑²𝑅₁₁𝑑𝑟²+1𝑟1𝑅₁₁𝑑𝑅₁₁−1𝑑𝑟𝑟²𝑚²₁=𝑞²₁₁,1𝑍₁₁𝑑²𝑍₁₁𝑑𝑧²=−𝑔²₁,1Θ₁₁𝑑²Θ₁₁𝑑𝜃²=−𝑚²₁, (21) where 𝑞²₁₁=((1+𝑖𝐷_𝑠1)𝑔²₁−(𝜔/𝑣_𝑠1)²)/(𝜂²₁+𝑖[(𝜂²₁−2)𝐷_𝑣1+2𝐷_𝑠1]).

The solutions for (21) can be easily obtained as 𝑅₁₁=𝐴₁₁𝐾_𝑚1𝑞₁₁𝑟+𝐵₁₁𝐼_𝑚1𝑞₁₁𝑟𝑍₁₁=𝐶₁₁sin𝑔₁𝑧+𝐷₁₁cos𝑔₁𝑧Θ₁₁=𝐸₁₁sin𝑚₁𝜃+𝐹₁₁cos𝑚₁𝜃, (22) where 𝐾_𝑚1(𝑞₁₁𝑟) and 𝐼_𝑚1(𝑞₁₁𝑟) are modified Bessel functions of the first and second kind of order 𝑚₁, respectively, and 𝐴₁₁, 𝐵₁₁, 𝐶₁₁, 𝐷₁₁, 𝐸₁₁, and 𝐹₁₁ are undetermined coefficients.

The potential function 𝜙₁ is expressed as 𝜙₁=𝐴₁₁𝐾_𝑚1𝑞₁₁𝑟+𝐵₁₁𝐼_𝑚1𝑞₁₁𝑟×𝐶₁₁sin𝑔₁𝑧+𝐷₁₁cos𝑔₁𝑧×𝐸₁₁sin𝑚₁𝜃+𝐹₁₁cos𝑚₁𝜃. (23) Similarly, the potential function 𝜓₁ can be obtained as 𝜓₁=𝐴₁₂𝐾_𝑚1𝑞₁₂𝑟+𝐵₁₂𝐼_𝑚1𝑞₁₂𝑟×𝐶₁₂sin𝑔₁𝑧+𝐷₁₂cos𝑔₁𝑧×𝐸₁₂sin𝑚₁𝜃+𝐹₁₂cos𝑚₁𝜃, (24) where 𝑞²₁₂=((1+𝑖𝐷_𝑠1)𝑔²₁−(𝜔/𝑣_𝑠1)²)/(1+𝑖𝐷_𝑠1) and 𝐴₁₂, 𝐵₁₂, 𝐶₁₂, 𝐷₁₂, 𝐸₁₂, and 𝐹₁₂ are undetermined coefficients.

The displacement and stress of the outer soil vanish to zero when 𝑟→∞. Hence, 𝐵₁₁=𝐵₁₂=0.

It is found from (11) and (12) that 𝑢₁ is an even function of 𝜃, and 𝑣₁ is an odd function of 𝜃. Thus, 𝐸₁₁=𝐹₁₂=0 and 𝑚₁=1.

Substituting (23) and (24) into (6) and (8), one obtains 𝐶₁₁=𝐶₁₂=0,𝑔_1𝑛=(2𝑛−1)𝜋2𝐻,𝑛=1,2,3,…. (25) Then, the potential functions 𝜙₁ and 𝜓₁ are written as 𝜙₁=^∞_𝑛=1𝐴_1𝑛𝐾₁𝑞_11𝑛𝑟cos𝑔_1𝑛𝑧𝜓cos𝜃,₁=^∞_𝑛=1𝐴_2𝑛𝐾₁𝑞_12𝑛𝑟cos𝑔_1𝑛𝑧sin𝜃. (26) Thereafter, the displacements of the outer soil can be expressed as follows: 𝑢₁=^∞_𝑛=1−𝐴_1𝑛1𝑟𝐾₁𝑞_11𝑛𝑟+𝑞_11𝑛𝐾₀𝑞_11𝑛𝑟+𝐴_2𝑛1𝑟𝐾₁𝑞_12𝑛𝑟cos𝑔_1𝑛𝑧𝑣cos𝜃_1∞𝑛=1−𝐴_1𝑛1𝑟𝐾₁𝑞_11𝑛𝑟+𝐴_2𝑛×1𝑟𝐾₁𝑞_12𝑛𝑟+𝑞_12𝑛𝐾₀𝑞_12𝑛𝑟cos𝑔_1𝑛𝑧sin𝜃. (27)(28) Substituting (27) and (28) into (11) and (12), respectively, yields ^∞_𝑛=1−𝐴_1𝑛1𝑟₁𝐾₁𝑞_11𝑛𝑟₁+𝑞_11𝑛𝐾₀𝑞_11𝑛𝑟₁+𝐴_2𝑛1𝑟₁𝐾₁𝑞_12𝑛𝑟₁cos𝑔_1𝑛𝑧=𝑢_𝑝,_∞𝑛=1−𝐴_1𝑛1𝑟₁𝐾₁𝑞_11𝑛𝑟₁+𝐴_2𝑛×1𝑟₁𝐾₁𝑞_12𝑛𝑟₁+𝑞_12𝑛𝐾₀𝑞_12𝑛𝑟₁cos𝑔_1𝑛𝑧=−𝑢_𝑝. (29)(30) It can be obtained from (29) and (30) that 𝐴_2𝑛=𝛾_1𝑛𝐴_1𝑛, (31) where 𝛾_1𝑛=((2/𝑟₁)𝐾₁(𝑞_11𝑛𝑟₁)+𝑞_11𝑛𝐾₀(𝑞_11𝑛𝑟₁))/((2/𝑟₁)𝐾₁(𝑞_12𝑛𝑟₁)+𝑞_12𝑛𝐾₀(𝑞_12𝑛𝑟₁)).

The radial displacement of the outer soil on the outer interface can be expressed as 𝑢_1𝑟=𝑟1=^∞_𝑛=1𝜂_1𝑛𝐴_1𝑛cos𝑔_1𝑛𝑧cos𝜃, (32) where 𝜂_1𝑛=−[(1/𝑟₁)𝐾₁(𝑞_11𝑛𝑟₁)+𝑞_11𝑛𝐾₀(𝑞_11𝑛𝑟₁)]+𝛾_1𝑛(1/𝑟₁)𝐾₁(𝑞_12𝑛𝑟₁).

The horizontal resistance of the outer soil can be obtained as 𝑓₁=−₀^2𝜋(𝜎_𝑟1cos𝜃−𝜏_𝑟𝜃1sin𝜃)_𝑟=𝑟1𝑟₁𝑑𝜃=𝜋𝑟₁𝐺₁^∞_𝑛=1𝜉_1𝑛𝐴_1𝑛cos𝑔_1𝑛𝑧, (33) where 𝜉_1𝑛=−{𝜂²₁+𝑖[𝐷_𝑣1(𝜂²₁−2)+2𝐷_𝑠1]}𝑞²_11𝑛𝐾₁(𝑞_11𝑛𝑟₁)−(1+𝑖𝐷_𝑠1)𝛾_1𝑛𝑞²_12𝑛𝐾₁(𝑞_12𝑛𝑟₁).

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

For the amplitudes 𝑢₂ and 𝑣₂, (4) can be expressed as 𝜆₂+2𝐺₂+𝑖𝜆^′₂+2𝐺^′₂𝜕𝑒₂−1𝜕𝑟𝑟𝐺₂+𝑖𝐺^′₂𝜕𝑤₂+𝐺𝜕𝜃₂+𝑖𝐺^′₂𝜕²𝑢₂𝜕𝑧²=−𝜌₂𝜔²𝑢₂,𝜆₂+2𝐺₂+𝑖𝜆^′₂+2𝐺^′₂𝜕𝑒₂+𝐺𝑟𝜕𝜃²+𝑖𝐺^′₂𝜕𝑤₂+𝐺𝜕𝑟₂+𝑖𝐺^′₂𝜕²𝑣₂𝜕𝑧²=−𝜌₂𝜔²𝑣₂. (34) The potential functions 𝜙₂(𝑟,𝜃,𝑧,𝑡) and 𝜓₂(𝑟,𝜃,𝑧,𝑡) are introduced as 𝑢₂=𝜕𝜙₂+1𝜕𝑟𝑟𝜕𝜓₂,𝑣𝜕𝜃₂=1𝑟𝜕𝜙₂−𝜕𝜃𝜕𝜓₂.𝜕𝑟 (35) Thus, 𝑒₂=∇²𝜙₂,𝑤₂=−∇²𝜓₂. (36) Substituting (35) and (36) into (34), (34) can be expressed as: 𝜂²₂+𝑖𝜂²₂−2𝐷_𝑣2+2𝐷_𝑠2∇²𝜙₂+𝜔𝑣_𝑠2²+1+𝑖𝐷_𝑠2𝜕²𝜕𝑧²𝜙₂=0,1+𝑖𝐷_𝑠2∇²𝜓₂+𝜔𝑣_𝑠2²+1+𝑖𝐷_𝑠2𝜕²𝜕𝑧²𝜓₂=0, (37) where 𝜂=𝑣_𝑙2/𝑣_𝑠2=[(𝜆₂+2𝐺₂)/𝐺₂], 𝐷_𝑣2=𝜆^′₂/𝜆₂, and 𝐷_𝑠2=𝐺^′₂/𝐺₂. In addition, 𝑣_𝑙2 and 𝑣_𝑠2 are the longitudinal and shear wave velocities of the inner soil, respectively; 𝐷_𝑣2 and 𝐷_𝑠2 are the hysteretic damping ratios of the inner soil.

Using the method of separation of variables, the potential functions 𝜙₂ and 𝜓₂ are obtained as 𝜙₂=𝐴₂₁𝐾_𝑚2𝑞₂₁𝑟+𝐵₂₁𝐼_𝑚2𝑞₂₁𝑟×𝐶₂₁sin𝑔₂𝑧+𝐷₂₁cos𝑔₂𝑧×𝐸₂₁sin𝑚₂𝜃+𝐹₂₁cos𝑚₂𝜃,𝜓₂=𝐴₂₂𝐾_𝑚2𝑞₂₂𝑟+𝐵₂₂𝐼_𝑚2𝑞₂₂𝑟×𝐶₂₂sin𝑔₂𝑧+𝐷₂₂cos𝑔₂𝑧×𝐸₂₂sin𝑚₂𝜃+𝐹₂₂cos𝑚₂𝜃, (38) where 𝑞²₂₁=((1+𝑖𝐷_𝑠2)𝑔²₂−(𝜔/𝑣_𝑠2)²)/(𝜂²₂+𝑖[(𝜂²₂−2)𝐷_𝑣2+2𝐷_𝑠2]) and 𝑞²₂₂=((1+𝑖𝐷_𝑠2)𝑔²₂−(𝜔/𝑣_𝑠2)²)/(1+𝑖𝐷_𝑠2).

The displacement and stress of the inner soil are limited values when 𝑟=0. Hence, 𝐴₂₁=𝐴₂₂=0.

𝑢₂ is an even function of 𝜃, and 𝑣₂ is an odd function of 𝜃. Thus, 𝐸₂₁=𝐹₂₂=0 and 𝑚₂=1.

Substituting (38) into (7) and (9), one obtains 𝐶₂₁=𝐶₂₂=0,𝑔_2𝑛=(2𝑛−1)𝜋.2𝐻 (39) Then, the potential functions 𝜙₂ and 𝜓₂ are written as 𝜙₂=^∞_𝑛=1𝐵_1𝑛𝐼₁𝑞_21𝑛𝑟cos𝑔_2𝑛𝑧𝜓cos𝜃,₂=^∞_𝑛=1𝐵_2𝑛𝐼₁𝑞_22𝑛𝑟cos𝑔_2𝑛𝑧sin𝜃. (40) Thereafter, the displacements of the inner soil can be expressed as follows: 𝑢₂=^∞_𝑛=1𝐵_1𝑛𝑞_21𝑛𝐼₀𝑞_21𝑛𝑟−1𝑟𝐼₁𝑞_21𝑛𝑟+𝐵_2𝑛1𝑟𝐼₁𝑞_22𝑛𝑟cos𝑔_2𝑛𝑧𝑣cos𝜃,₂=^∞_𝑛=1−𝐵_1𝑛1𝑟𝐼₁𝑞_21𝑛𝑟+𝐵_2𝑛×1𝑟𝐼₁𝑞_22𝑛𝑟−𝑞_22𝑛𝐼₀𝑞_22𝑛𝑟cos𝑔_2𝑛𝑧sin𝜃. (41)

Substituting (41) into (13) and (14) yields ^∞_𝑛=1𝐵_1𝑛𝑞_21𝑛𝐼₀𝑞_21𝑛𝑟₂−1𝑟₂𝐼₁𝑞_21𝑛𝑟₂+𝐵_2𝑛1𝑟₂𝐼₁𝑞_22𝑛𝑟₂cos𝑔_2𝑛𝑧=𝑢_𝑝,_∞𝑛=1−𝐵_1𝑛1𝑟₂𝐼₁𝑞_21𝑛𝑟₂+𝐵_2𝑛1𝑟₂𝐼₁𝑞_22𝑛𝑟₂−𝑞_22𝑛𝐼₀𝑞_22𝑛𝑟₂×cos𝑔_2𝑛𝑧=−𝑢_𝑝. (42) It can be obtained from (42) that 𝐵_2𝑛=𝛾_2𝑛𝐵_1𝑛, (43) where 𝛾_2𝑛=(𝑞_21𝑛𝐼₀(𝑞_21𝑛𝑟₂)−(2/𝑟₂)𝐼₁(𝑞_21𝑛𝑟₂))/(𝑞_22𝑛𝐼₀(𝑞_22𝑛𝑟₂)−(2/𝑟₂)𝐼₁(𝑞_22𝑛𝑟₂)).

The radial displacement of the inner soil on the inner interface can be expressed as 𝑢_2𝑟=𝑟2=^∞_𝑛=1𝜂_2𝑛𝐵_1𝑛cos𝑔_2𝑛𝑧cos𝜃, (44) where 𝜂_2𝑛=𝑞_21𝑛𝐼₀(𝑞_21𝑛𝑟₂)−(1/𝑟₂)𝐼₁(𝑞_21𝑛𝑟₂)+𝛾_2𝑛(1/𝑟₂)𝐼₁(𝑞_22𝑛𝑟₂).

The horizontal resistance of the inner soil can be obtained as 𝑓₂=₀^2𝜋(𝜎_𝑟2cos𝜃−𝜏_𝑟𝜃2sin𝜃)_𝑟=𝑟2𝑟₂𝑑𝜃=𝜋𝑟₂𝐺₂^∞_𝑛=1𝜉_2𝑛𝐵_1𝑛cos𝑔_2𝑛𝑧, (45) where 𝜉_2𝑛=−{𝜂²₂+𝑖[𝐷_𝑣2(𝜂²₂−2)+2𝐷_𝑠2]}𝑞²_21𝑛𝐼₁(𝑞_21𝑛𝑟₂)−(1+𝑖𝐷_𝑠2)𝛾_2𝑛𝑞²_22𝑛𝐼₁(𝑞_22𝑛𝑟₂).

4.3. Solution for the Dynamic Equilibrium Equation of the Pile

The amplitude 𝑢_𝑝 can be expressed as 𝐸_𝑝𝐼_𝑝𝑑⁴𝑢_𝑝𝑑𝑧⁴−𝑚𝜔²𝑢_𝑝=−𝑓₁−𝑓₂. (46) Substituting (33) and (45) into (46), one obtains 𝑑⁴𝑢_𝑝𝑑𝑧⁴−𝛽⁴𝑢_𝑝𝜋=−𝐸_𝑝𝐼_𝑝𝑟₁𝐺₁^∞_𝑛=1𝜉_1𝑛𝐴_1𝑛cos𝑔_1𝑛𝑧+𝑟₂𝐺₂^∞_𝑛=1𝜉_2𝑛𝐵_1𝑛cos𝑔_2𝑛𝑧, (47) where 𝛽⁴=𝑚𝜔²/𝐸_𝑝𝐼_𝑝.

The solution for (47) can be obtained as 𝑢_𝑝=𝑁₁sin𝛽𝑧+𝑁₂cos𝛽𝑧+𝑁₃sinh𝛽𝑧+𝑁₄cosh𝛽𝑧−^∞_𝑛=1𝜁_1𝑛𝐴_1𝑛cos𝑔_1𝑛𝑧−^∞_𝑛=1𝜁_2𝑛𝐵_1𝑛cos𝑔_2𝑛𝑧, (48) where 𝜁_1𝑛=𝜋𝑟₁𝐺₁𝜉_1𝑛𝐸_𝑝𝐼_𝑝𝑔⁴_1𝑛−𝛽⁴,𝜁_2𝑛=𝜋𝑟₂𝐺₂𝜉_2𝑛𝐸_𝑝𝐼_𝑝𝑔⁴_2𝑛−𝛽⁴, (49) where 𝑁₁, 𝑁₂, 𝑁₃, and 𝑁₄ are undetermined coefficients.

It can be obtained from (11), (13), (32), and (44) that 𝑁₁sin𝛽𝑧+𝑁₂cos𝛽𝑧+𝑁₃sinh𝛽𝑧+𝑁₄−cosh𝛽𝑧_∞𝑛=1𝜁_1𝑛𝐴_1𝑛cos𝑔_1𝑛𝑧−^∞_𝑛=1𝜁_2𝑛𝐵_1𝑛cos𝑔_2𝑛𝑧=^∞_𝑛=1𝜂_1𝑛𝐴_1𝑛cos𝑔_1𝑛𝑧,𝑁₁sin𝛽𝑧+𝑁₂cos𝛽𝑧+𝑁₃sinh𝛽𝑧+𝑁₄−cosh𝛽𝑧_∞𝑛=1𝜁_1𝑛𝐴_1𝑛cos𝑔_1𝑛𝑧−^∞_𝑛=1𝜁_2𝑛𝐵_1𝑛cos𝑔_2𝑛𝑧=^∞_𝑛=1𝜂_2𝑛𝐵_1𝑛cos𝑔_2𝑛𝑧. (50)(51) Equations (50) and (51) can also be expressed as ^∞_𝑛=1𝜂_1𝑛𝐴_1𝑛cos𝑔_1𝑛𝑧=^∞_𝑛=1𝜂_2𝑛𝐵_1𝑛cos𝑔_2𝑛𝑧. (52)

It is found that 𝑔_1𝑛=𝑔_2𝑛. Given 𝑔_𝑛=𝑔_1𝑛=𝑔_2𝑛, it is obtained from (52) that 𝐵_1𝑛=𝜂_1𝑛𝐴_1𝑛𝜂_2𝑛. (53)

Multiplying cos(𝑔_𝑛𝑧) on both sides of (50) and then integrating on the interval [0,𝐻], one obtains 𝜂_1𝑛𝐴_1𝑛𝐻2=^𝐻₀𝑁₁sin𝛽𝑧+𝑁₂cos𝛽𝑧+𝑁₃sinh𝛽𝑧+𝑁₄cosh𝛽𝑧cos𝑔_𝑛𝑧−𝜁𝑑𝑧_1𝑛+𝜁_2𝑛𝜂_1𝑛𝜂_2𝑛𝐴_1𝑛𝐻2. (54) Thus, 𝐴_1𝑛=𝜂_2𝑛𝛿_1𝑛𝑁₁+𝛿_2𝑛𝑁₂+𝛿_3𝑛𝑁₃+𝛿_4𝑛𝑁₄, (55) where 𝛿_1𝑛=2∫^𝐻₀sin𝛽𝑧cos𝑔_𝑛𝑧𝑑𝑧𝜂_1𝑛𝜂_2𝑛+𝜁_1𝑛𝜂_2𝑛+𝜁_2𝑛𝜂_1𝑛𝐻,𝛿_2𝑛=2∫^𝐻₀cos𝛽𝑧cos𝑔_𝑛𝑧𝑑𝑧𝜂_1𝑛𝜂_2𝑛+𝜁_1𝑛𝜂_2𝑛+𝜁_2𝑛𝜂_1𝑛𝐻,𝛿_3𝑛=2∫^𝐻₀sinh𝛽𝑧cos𝑔_𝑛𝑧𝑑𝑧𝜂_1𝑛𝜂_2𝑛+𝜁_1𝑛𝜂_2𝑛+𝜁_2𝑛𝜂_1𝑛𝐻,𝛿_4𝑛=2∫^𝐻₀cosh𝛽𝑧cos𝑔_𝑛𝑧𝑑𝑧𝜂_1𝑛𝜂_2𝑛+𝜁_1𝑛𝜂_2𝑛+𝜁_2𝑛𝜂_1𝑛𝐻. (56) Substituting (53) into (55) yields 𝐵_1𝑛=𝜂_1𝑛𝛿_1𝑛𝑁₁+𝛿_2𝑛𝑁₂+𝛿_3𝑛𝑁₃+𝛿_4𝑛𝑁₄. (57) Equation (48) can be expressed as 𝑢_𝑝=𝑁₁sin𝛽𝑧−^∞_𝑛=1𝜅_1𝑛cos𝑔_𝑛𝑧+𝑁₂cos𝛽𝑧−^∞_𝑛=1𝜅_2𝑛cos𝑔_𝑛𝑧+𝑁₃sinh𝛽𝑧−^∞_𝑛=1𝜅_3𝑛cos𝑔_𝑛𝑧+𝑁₄cosh𝛽𝑧−^∞_𝑛=1𝜅_4𝑛cos𝑔_𝑛𝑧, (58) where 𝜅_1𝑛=(𝜂_2𝑛𝜁_1𝑛+𝜂_1𝑛𝜁_2𝑛)𝛿_1𝑛, 𝜅_2𝑛=(𝜂_2𝑛𝜁_1𝑛+𝜂_1𝑛𝜁_2𝑛)𝛿_2𝑛, and 𝜅_3𝑛=(𝜂_2𝑛𝜁_1𝑛+𝜂_1𝑛𝜁_2𝑛)𝛿_3𝑛, 𝜅_4𝑛=(𝜂_2𝑛𝜁_1𝑛+𝜂_1𝑛𝜁_2𝑛)𝛿_4𝑛.

With the displacement of the pile described by (58), the angle of rotation 𝜑, the bending moment 𝑀, and the shear force 𝑄 are obtained as follows:𝑢_𝑝𝜑𝑀𝐸_𝑝𝐼_𝑝𝑄𝐸_𝑝𝐼_𝑝=^{−sin𝛽𝑧∞∑𝑛=1𝜅1𝑛𝑔cos𝑛𝑧−cos𝛽𝑧∞∑𝑛=1𝜅2𝑛𝑔cos𝑛𝑧−sinh𝛽𝑧∞∑𝑛=1𝜅3𝑛𝑔cos𝑛𝑧−cosh𝛽𝑧∞∑𝑛=1𝜅4𝑛𝑔cos𝑛𝑧+𝛽cos𝛽𝑧∞∑𝑛=1𝜅1𝑛𝑔𝑛𝑔sin𝑛𝑧+−𝛽sin𝛽𝑧∞∑𝑛=1𝜅2𝑛𝑔𝑛𝑔sin𝑛𝑧+𝛽cosh𝛽𝑧∞∑𝑛=1𝜅3𝑛𝑔𝑛𝑔sin𝑛𝑧+𝛽sinh𝛽𝑧∞∑𝑛=1𝜅4𝑛𝑔𝑛𝑔sin𝑛𝑧}_{−𝛽2+sin𝛽𝑧∞∑𝑛=1𝜅1𝑛𝑔2𝑛𝑔cos𝑛𝑧−𝛽2+cos𝛽𝑧∞∑𝑛=1𝜅2𝑛𝑔2𝑛𝑔cos𝑛𝑧𝛽2+sinh𝛽𝑧∞∑𝑛=1𝜅3𝑛𝑔2𝑛𝑔cos𝑛𝑧𝛽2+cosh𝛽𝑧∞∑𝑛=1𝜅4𝑛𝑔2𝑛𝑔cos𝑛𝑧−𝛽3−cos𝛽𝑧∞∑𝑛=1𝜅1𝑛𝑔3𝑛𝑔sin𝑛𝑧𝛽3−sin𝛽𝑧∞∑𝑛=1𝜅2𝑛𝑔3𝑛𝑔sin𝑛𝑧𝛽3−cosh𝛽𝑧∞∑𝑛=1𝜅3𝑛𝑔3𝑛𝑔sin𝑛𝑧𝛽3−sinh𝛽𝑧∞∑𝑛=1𝜅4𝑛𝑔3𝑛𝑔sin𝑛𝑧}×𝑁₁𝑁₂𝑁₃𝑁₄. (59)Assuming the displacement, the angle of rotation, the bending moment, and the shear force at the pile head as 𝑈₀, 𝜑₀, 𝑀₀, and 𝑄₀, respectively, one obtains −𝑁₁^∞_𝑛=1𝜅_1𝑛+𝑁₂1−^∞_𝑛=1𝜅_2𝑛−𝑁₃^∞_𝑛=1𝜅_3𝑛+𝑁₄1−^∞_𝑛=1𝜅_4𝑛=𝑈₀𝑁₁+𝑁₃=𝜑₀𝛽𝑁₁^∞_𝑛=1𝜅_1𝑛𝑔²_𝑛−𝑁₂𝛽²−^∞_𝑛=1𝜅_2𝑛𝑔²_𝑛+𝑁₃^∞_𝑛=1𝜅_3𝑛𝑔²_𝑛+𝑁₄𝛽²+^∞_𝑛=1𝜅_4𝑛𝑔²_𝑛=𝑀₀𝐸_𝑝𝐼_𝑝−𝑁₁+𝑁₃=𝑄₀𝐸_𝑝𝐼_𝑝𝛽³. (60) It can be obtained from (60) that 𝑁₁=𝑎₁𝜑₀−𝑎₂𝑄₀,𝑁₂=𝑎₃𝑈₀+𝑎₄𝜑₀+𝑎₅𝑀₀+𝑎₆𝑄₀,𝑁₃=𝑎₇𝜑₀+𝑎₈𝑄₀,𝑁₄=𝑎₇𝑈₀+𝑎₈𝜑₀+𝑎₉𝑀₀+𝑎₁₀𝑄₀, (61) where 𝑎₁=12𝛽;𝑎₂=12𝐸_𝑝𝐼_𝑝𝛽³𝑎₃=𝛾𝛽²+^∞_𝑛=1𝜅_4𝑛𝑔²_𝑛;𝑎₄=𝛽²+^∞_𝑛=1𝜅_4𝑛𝑔²_𝑛^∞_𝑛=1𝜅_1𝑛+^∞_𝑛=1𝜅_3𝑛+1−^∞_𝑛=1𝜅_4𝑛^∞_𝑛=1𝜅_1𝑛𝑔²_𝑛+^∞_𝑛=1𝜅_3𝑛𝑔²_𝑛𝛾𝑎₁;𝑎₅=∑^∞_𝑛=1𝜅_4𝑛−1𝛾𝐸_𝑝𝐼_𝑝;𝑎₆=𝛽²+^∞_𝑛=1𝜅_4𝑛𝑔²_𝑛^∞_𝑛=1𝜅_3𝑛−^∞_𝑛=1𝜅_1𝑛+1−^∞_𝑛=1𝜅_4𝑛^∞_𝑛=1𝜅_3𝑛𝑔²_𝑛−^∞_𝑛=1𝜅_1𝑛𝑔²_𝑛𝛾𝑎₂𝑎₇=1+∑^∞_𝑛=1𝜅_2𝑛−2𝑎₃1−∑^∞_𝑛=1𝜅_4𝑛;𝑎₈=∑^∞_𝑛=1𝜅_1𝑛+∑^∞_𝑛=1𝜅_3𝑛𝑎₁+∑^∞_𝑛=1𝜅_2𝑛−1𝑎₄1−∑^∞_𝑛=1𝜅_4𝑛;𝑎₉=∑^∞_𝑛=1𝜅_2𝑛−1𝑎₅1−∑^∞_𝑛=1𝜅_4𝑛;𝑎₁₀=∑^∞_𝑛=1𝜅_2𝑛−1𝑎₆+∑^∞_𝑛=1𝜅_3𝑛−∑^∞_𝑛=1𝜅_1𝑛𝑎₂1−∑^∞_𝑛=1𝜅_4𝑛;𝛾=11−^∞_𝑛=1𝜅_2𝑛𝛽²+^∞_𝑛=1𝜅_4𝑛𝑔²_𝑛+𝛽²−^∞_𝑛=1𝜅_2𝑛𝑔²_𝑛1−^∞_𝑛=1𝜅_4𝑛⁻¹. (62)

Substituting (61) into (58), one obtains 𝑢_𝑝=𝑈₀𝑏₁(𝑧)+𝜑₀𝑏₂(𝑧)+𝑀₀𝑏₃(𝑧)+𝑄₀𝑏₄(𝑧), (63) where 𝑏₁(𝑧)=cos𝛽𝑧−^∞_𝑛=1𝜅_2𝑛cos𝑔_𝑛𝑧𝑎₃+cosh𝛽𝑧−^∞_𝑛=1𝜅_4𝑛cos𝑔_𝑛𝑧𝑎₇,𝑏₂(𝑧)=sin𝛽𝑧−^∞_𝑛=1𝜅_1𝑛cos𝑔_𝑛𝑧−+sinh𝛽𝑧_∞𝑛=1𝜅_3𝑛cos𝑔_𝑛𝑧𝑎₁+cos𝛽𝑧−^∞_𝑛=1𝜅_2𝑛cos𝑔_𝑛𝑧𝑎₄+cosh𝛽𝑧−^∞_𝑛=1𝜅_4𝑛cos𝑔_𝑛𝑧𝑎₈,𝑏₃(𝑧)=cos𝛽𝑧−^∞_𝑛=1𝜅_2𝑛cos𝑔_𝑛𝑧𝑎₅+cosh𝛽𝑧−^∞_𝑛=1𝜅_4𝑛cos𝑔_𝑛𝑧𝑎₉,𝑏₄(𝑧)=sinh𝛽𝑧−^∞_𝑛=1𝜅_3𝑛cos𝑔_𝑛𝑧+−sin𝛽𝑧_∞𝑛=1𝜅_1𝑛cos𝑔_𝑛𝑧𝑎₂+cos𝛽𝑧−^∞_𝑛=1𝜅_2𝑛cos𝑔_𝑛𝑧𝑎₆+cosh𝛽𝑧−^∞_𝑛=1𝜅_4𝑛cos𝑔_𝑛𝑧𝑎₁₀. (64) Substituting (63) into (10), one obtains 𝑈₀𝑏₁(𝐻)+𝜑₀𝑏₂(𝐻)+𝑀₀𝑏₃(𝐻)+𝑄₀𝑏₄(𝐻)𝑈=0,₀𝑏^′₁(𝐻)+𝜑₀𝑏^′₂(𝐻)+𝑀₀𝑏^′₃(𝐻)+𝑄₀𝑏^′₄(𝐻)=0. (65) It can be obtained from (65) that 𝑀₀=𝑏₄(𝐻)𝑏^′₂(𝐻)−𝑏₂(𝐻)𝑏^′₄(𝐻)𝑏₃(𝐻)𝑏^′₄(𝐻)−𝑏₄(𝐻)𝑏^′₃(𝐻)𝜑₀+𝑏₄(𝐻)𝑏^′₁(𝐻)−𝑏₁(𝐻)𝑏^′₄(𝐻)𝑏₃(𝐻)𝑏^′₄(𝐻)−𝑏₄(𝐻)𝑏^′₃(𝐻)𝑈₀,𝑄₀=𝑏₃(𝐻)𝑏^′₁(𝐻)−𝑏₁(𝐻)𝑏^′₃(𝐻)𝑏₄(𝐻)𝑏^′₃(𝐻)−𝑏₃(𝐻)𝑏^′₄(𝐻)𝑈₀+𝑏₃(𝐻)𝑏^′₂(𝐻)−𝑏₂(𝐻)𝑏^′₃(𝐻)𝑏₄(𝐻)𝑏^′₃(𝐻)−𝑏₃(𝐻)𝑏^′₄(𝐻)𝜑₀. (66) The horizontal complex dynamic stiffness 𝐾_ℎ, rocking complex dynamic stiffness 𝐾_𝑟, and horizontal-rocking complex dynamic stiffness 𝐾_ℎ𝑟 are expressed as 𝐾_ℎ()=𝑏𝑖𝜔³(𝐻)𝑏^′₁(𝐻)−𝑏₁(𝐻)𝑏^′₃(𝐻)𝑏₄(𝐻)𝑏^′₃(𝐻)−𝑏₃(𝐻)𝑏^′₄(𝐻),𝐾_𝑟()=𝑏𝑖𝜔⁴(𝐻)𝑏^′₂(𝐻)−𝑏₂(𝐻)𝑏^′₄(𝐻)𝑏₃(𝐻)𝑏^′₄(𝐻)−𝑏₄(𝐻)𝑏^′₃(𝐻),𝐾_ℎ𝑟()=𝑏𝑖𝜔³(𝐻)𝑏^′₂(𝐻)−𝑏₂(𝐻)𝑏^′₃(𝐻)𝑏₄(𝐻)𝑏^′₃(𝐻)−𝑏₃(𝐻)𝑏^′₄(𝐻). (67)

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 𝑛 is 20. Unless otherwise specified, the following parameter values are used: 𝐻=10 m, 𝑟₁=0.5m, 𝑟₂=0.38 m, 𝐸_𝑝=25 GPa, 𝜌_𝑝=2.5 g/cm³, 𝜌₁=𝜌₂=1.8 g/cm³, 𝐺₁=𝐺₂=10 MPa, 𝐷_𝑠1=𝐷_𝑣1=𝐷_𝑠2=𝐷_𝑣2=0.02, and 𝑣₁=𝑣₂=0.3.

5.1. Verification

This solution is verified by being compared with Nogami's solution (1977) for a solid pile. Given 𝑟₂→0, 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

Figure 2 

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

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

Figure 3 

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

(a)  Real part

(b) Imaginary part

Figure 4 

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

(a)  Real part

(b) Imaginary part

Figure 5 

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

Figures 6, 7, and 8 show the influence of the pile radii on the complex stiffnesses on the pile top. With the increase of 𝑟₁ or decrease of 𝑟₂, the stiffnesses and dampings increase. When 𝑟₂=0, the stiffness and damping are close to those of the pile with 𝑟₂=0.2 m. It shows that, when 𝑟₂<0.2 m, the effect of the inner radius is negligible. Furthermore, it is seen that the stiffnesses and dampings of the pile with 𝑟₁=0.6 m and 𝑟₂=0.38 m are larger than those of the pile with 𝑟₁=0.5 m and 𝑟₂=0.2 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

Figure 6 

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

(a)  Real part

(b) Imaginary part

Figure 7 

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

(a) Real part

(b) Imaginary part

Figure 8 

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

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 𝐺₁. The horizontal and horizontal-rocking damping increase, but the rocking damping decreases with the increase of 𝐺₁. In the low-frequency range, 𝐺₂ 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 𝐺₂.

(a) Real part

(b) Imaginary part

Figure 9 

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

(a) Real part

(b) Imaginary part

Figure 10 

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

(a) Real part

(b) Imaginary part

Figure 11 

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

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 (1) 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. (2) 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. (3) 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.