Soil-Pile Interaction in the Pile Vertical Vibration Based on Fictitious Soil-Pile Model

Abstract

By introducing the fictitious soil-pile model, the soil-pile interaction in the pile vertical vibration is investigated. Firstly, assuming the surrounding soil of pile to be viscoelastic material and considering its vertical wave effect, the governing equations of soil-pile system subjected to arbitrary harmonic dynamic force are founded based on the Euler-Bernoulli rod theory. Secondly, the analytical solution of velocity response in frequency domain and its corresponding semianalytical solution of velocity response in time domain are derived by means of Laplace transform technique and separation of variables technique. Based on the obtained solutions, the influence of parameters of pile end soil on the dynamic response is studied in detail for different designing parameters of pile. Lastly, the fictitious soil-pile model and other pile end soil supporting models are compared. It is shown that the dynamic response obtained by the fictitious soil-pile model is among the dynamic responses obtained by other existing models if there are appropriate material parameters and thickness of pile end soil for the fictitious soil-pile model.

1. Introduction

The soil-pile dynamic interaction is a complicated and multifaceted contact problem. During vibration, the pile not only interacts with the surrounding soil of pile, but also interacts with the pile end soil. Thus, the soil-pile dynamic interaction has been widely studied and many kinds of dynamic interaction models on this topic have been presented in the existing literatures. From the viewpoint of surrounding soil-pile dynamic interaction, the existing models can be classified into three categories, that is, dynamic Winkler model [1–6], Plane-Strain model [7–11], and various three-dimensional axisymmetric continuum models [12–16]. It can be seen that extensive and in-depth research work has been done in the field of surrounding soil-pile dynamic interaction.

However, for the pile end soil-pile dynamic interaction, the existing research findings on this topic are not sufficient due to the difficulty to build rigorous coupling model for the dynamic interaction of pile end soil and pile. In general, there are two basic approaches to deal with this contact problem. In the first approach, the dynamic support condition of pile end soil on the pile is assumed to be rigid boundary [3, 10, 16]. This simplified method can be used in the condition that the pile is end bearing, but it is not sufficiently accurate for most floating piles. The other approach is that the support condition of pile end soil is supposed to be elastic boundary (modeled by a linear spring) [2, 11, 14, 15] or viscoelastic boundary (modeled by a linear spring and a dashpot connected in parallel) [1, 5, 6]. Although these methods are amenable to practical application in pile nondestructive testing, the parameters of spring and dashpot cannot be gained by means of routine soil tests. There are also many kinds of empirical methods to obtain the two parameters, for instance, simulating formula method [17], constant value method [1], Q-z curve equation method [18], ultimate bearing capacity method [19], and so on. However, all the aforementioned methods cannot take into account the frequency of load. In order to overcome the above shortcomings, Yang and Wang [20] and Wu et al. [21] presented a simplified model to simulate the dynamic interaction between the pile end soil and pile, namely, fictitious soil-pile model. Soon afterwards, Wu et al. [22] verified the accuracy of this model by comparing it with the model test results. The parameters of fictitious soil-pile model are closely related to the parameters of pile end soil. Meanwhile, this model can take into consideration the layered characteristics and construction effect of pile end soil. Therefore, the fictitious soil-pile model is an effective model.

In light of this, in order to extend the application of the fictitious soil-pile model, this paper has introduced it to study the vertical dynamic response of pile embedded in viscoelastic soil. Then, the analytical solution of velocity response in frequency domain and its corresponding semianalytical solution of velocity response in time domain are derived. Based on these solutions, the influence of parameters of pile end soil on the pile dynamic response is investigated.

2. Mathematical Model and Assumptions

2.1. Computational Model

The problem studied here is the vertical vibration of pile embedded in viscoelastic soil considering its vertical wave effect. The diagram of soil-pile interaction model is shown in Figure 1. 𝐻_𝑝 and 𝑟₀ represent the length and radius of pile, respectively. 𝑞(𝑡) is the arbitrary harmonic force acting on the pile head. The thickness of foundation soil and pile end soil is denoted by 𝐻 and 𝐻_𝑠, respectively. Based on the fictitious soil-pile model, the soil-pile system can be divided into two parts, that is, the pile end soil-pile system and the surrounding soil-pile system, which are numbered by 1 and 2, respectively.

Figure 1 

Schematic of pile-soil interaction model.

The main assumptions adopted in this paper are as follows.(1)The soil is a homogeneous, isotropic, and viscoelastic layer. The damping force of soil is proportional to the strain rate, and the coefficient of proportionality is denoted by 𝜂_𝑠.(2)The top surface of soil layer is assumed to be free boundary with no normal and shear stresses, and the bottom surface of soil layer is assumed to be rigid boundary.(3)The soil-pile system is subjected to small deformations and strains during vibration.(4)The pile and fictitious soil pile are perfectly elastic and vertical with a uniform circular cross-section and have perfect contact with the surrounding soil during vibration.

2.2. Governing Equations of Soil-Pile System

The axisymmetric problem of vertical harmonic oscillations in linearly elastic soil is taken into account in this paper. Denoting 𝑤=𝑤(𝑟,𝑧,𝑡) to be the vertical displacement of soil, the dynamic equilibrium equation of soil motion allowing for its vertical wave effect can be expressed as 𝜆_𝑠+2𝐺_𝑠𝜕²𝑤𝜕𝑧²+𝐺_𝑠1𝑟𝜕𝑤+𝜕𝜕𝑟²𝑤𝜕𝑟²+𝜂_𝑠𝜕𝜕𝜕𝑡²𝑤𝜕𝑧²+𝜂_𝑠𝜕1𝜕𝑡𝑟𝜕𝑤+𝜕𝜕𝑟²𝑤𝜕𝑟²=𝜌_𝑠𝜕²𝑤𝜕𝑡², (1) where 𝜆_𝑠 and 𝐺_𝑠 represent the Lame constants of soil which should satisfy 𝜆_𝑠=𝐸_𝑠𝜇_𝑠/[(1+𝜇_𝑠)(1−2𝜇_𝑠)] and 𝐺_𝑠=𝜌_𝑠𝑉²_𝑠. 𝐸_𝑠, 𝜇_𝑠, 𝑉_𝑠, and 𝜌_𝑠 denote the elastic modulus, Poisson’s ratio, shear wave velocity, and density of soil, respectively.

Based on the viscoelasticity theory, the shear stress at any point in the soil layer can be expressed as 𝜏_{𝑠,𝑟𝑧}=𝐺_𝑠𝜕𝑤𝜕𝑟+𝜂_𝑠𝜕²𝑤.𝜕𝑡𝜕𝑟 (2)

Denoting 𝑢_𝑗=𝑢_𝑗(𝑧,𝑡) to be the vertical displacement of pile (including fictitious soil pile) and according to the Euler-Bernoulli rod theory, the dynamic equilibrium equation of pile can be written as 𝐸_𝑝,𝑗𝐴_𝑝,𝑗𝜕²𝑢_𝑗𝜕𝑧²−𝑚_𝑝,𝑗𝜕²𝑢_𝑗𝜕𝑡²−2𝜋𝑟₀𝜏_{𝑠,𝑟𝑧}𝑟₀,,𝑧,𝑡=0𝑗=1,2 (3) where 𝐸_𝑝,𝑗, 𝐴_𝑝,𝑗, and 𝑚_𝑝,𝑗 represent the elastic modulus, cross-sectional area, and mass per unit length of pile (including fictitious soil pile), respectively. When 𝑗=1 and 𝑗=2, these above parameters refer to the parameters of fictitious soil pile and pile, respectively.

Combining with the assumptions, the boundary and initial conditions of soil-pile system can be established as follows.(1)Boundary conditions of soil are as follows. At the top surface of soil,𝜕𝑤𝜕𝑧_𝑧=0=0. (4a) At the bottom surface of soil, 𝑤|_𝑧=𝐻=0. (4b) At an infinite radial distance of soil, 𝜎(∞,𝑧)=0,𝑤(∞,𝑧)=0. (4c)(2)Boundary conditions of pile are as follows:𝜕𝑢₂𝜕𝑧_𝑧=0𝑞(𝑡)=−𝐸_𝑝,2𝐴_𝑝,2,𝑢_1𝑧=𝐻=0,𝑢_{1𝑧=𝐻𝑝}=𝑢_{2𝑧=𝐻𝑝},𝐸_𝑝,1𝐴_𝑝,1𝜕𝑢₁𝜕𝑧_{𝑧=𝐻𝑝}=𝐸_𝑝,2𝐴_𝑝,2𝜕𝑢₂𝜕𝑧_{𝑧=𝐻𝑝}. (5a)(5b)(5c)(5d)(3)At the interface of soil-pile system,𝑤𝑟₀,𝑧,𝑡=𝑢_𝑗(𝑧,𝑡),𝑗=1,2. (6)(4)Initial conditions of soil-pile system are as follows. Initial conditions of soil are𝑤|_𝑡=0=0,𝜕𝑤𝜕𝑡_𝑡=0𝜕=0,²𝑤𝜕𝑡²_𝑡=0=0. (7a) Initial conditions of pile are 𝑢_𝑗𝑡=0=0,𝜕𝑢_𝑗𝜕𝑡_𝑡=0=0. (7b)

3. Solutions of the Equations

3.1. Vibrations of the Soil Layer

Denote 𝑊(𝑟,𝑧,𝑠) to be the Laplace transform with respect to time of 𝑤(𝑟,𝑧,𝑡). Associating with the initial conditions (7a), (1) is transformed by using the Laplace transform technique and can be further written as 𝜆_𝑠+2𝐺_𝑠+𝜂_𝑠1⋅𝑠𝑍(𝑧)𝜕²𝑍(𝑧)𝜕𝑧²+𝐺_𝑠+𝜂_𝑠1⋅𝑠𝑅(𝑟)1𝑟𝜕𝑅(𝑟)+𝜕𝜕𝑟²𝑅(𝑟)𝜕𝑟²=𝜌_𝑠𝑠², (8) where 𝑊(𝑟,𝑧,𝑠)=∫₀^+∞𝑤(𝑟,𝑧,𝑡)𝑒^−𝑠𝑡d𝑡. 𝑠 is the Laplace transform parameter which should satisfy 𝑠=𝑖𝜔 during the numerical analysis.

By means of the separation of variables technique, (8) can be decoupled as follows: d²𝑅(𝑟)d𝑟²+1𝑟d𝑅(𝑟)d𝑟−𝜉²𝑅(𝑟)d=0,₂𝑍(𝑧)d𝑧²+𝛽²𝑍(𝑧)=0, (9) where constants 𝛽 and 𝜉 must satisfy the following relationship: 𝜉²=𝜆_𝑠+2𝐺_𝑠+𝜂_𝑠⋅𝑠𝛽²+𝜌_𝑠𝑠²𝐺_𝑠+𝜂_𝑠.⋅𝑠 (10)

Equation (9) represents the Bessel equation and ordinary differential equation of second order whose general solutions can be easily obtained. Associating with these general solutions, the vertical displacement of soil, 𝑊(𝑟,𝑧,𝑠), can be derived as 𝑊(𝑟,𝑧,𝑠)=𝐴𝐾₀(𝜉𝑟)+𝐵𝐼₀(×,𝜉𝑟)𝐶sin𝛽𝑧+𝐷cos𝛽𝑧 (11) where 𝐼₀(⋅) and 𝐾₀(⋅) represent the modified Bessel functions of order zero of the first and second kinds, respectively. 𝐴, 𝐵, 𝐶, and 𝐷 are constants determined by the boundary conditions.

According to the properties of the modified Bessel functions, that is, if 𝑟→∞, then 𝐼_𝑛(⋅)→∞ and 𝐾_𝑛(⋅)→0, it can be obtained that 𝐵=0 and 𝐶=0 from boundary conditions (4c) and (4a). By virtue of boundary condition (4b), the following equation can be obtained: cos𝛽𝐻=0. (12)

By solving (12), it can be obtained that 𝛽_𝑛=(𝜋/2𝐻)(2𝑛−1)  (𝑛=1,2,3,…). Substituting this formula into (11), 𝑊(𝑟,𝑧,𝑠) can be rewritten as 𝑊(𝑟,𝑧,𝑠)=^∞_𝑛=1𝐴_𝑛𝐾₀𝜉_𝑛𝑟cos𝛽_𝑛𝑧, (13) where 𝐴_𝑛 are a series of constants which can be obtained from the boundary conditions. 𝜉_𝑛 can be obtained by substituting 𝛽=𝛽_𝑛 into (10).

3.2. Vibrations of the Pile

Denoting 𝑈(𝑧,𝑠) to be the Laplace transform with respect to time of 𝑢(𝑧,𝑡), (3) is transformed by using the Laplace transform technique and can be rewritten combining (2) and (13): 𝑉²_𝑝,𝑗𝜕²𝑈_𝑗𝜕𝑧²−𝑠²𝑈_𝑗−2𝜋𝑟₀𝜌_𝑝,𝑗𝐴_𝑝,𝑗𝐺_𝑠+𝜂_𝑠×⋅𝑠_∞𝑛=1𝐴_𝑛𝜉_𝑛𝐾₁𝜉_𝑛𝑟₀cos𝛽_𝑛𝑧=0,𝑗=1,2, (14) where 𝑉_𝑝,𝑗=𝐸_𝑝,𝑗/𝜌_𝑝,𝑗 represents the elastic longitudinal wave velocity of pile (including fictitious soil pile). Letting 𝜆²=−𝑠², the general solution of (14) can be derived as 𝑈_𝑗=𝑀_𝑗𝜆cos𝑉_𝑝,𝑗𝑧+𝑁_𝑗𝜆sin𝑉_𝑝,𝑗𝑧+^∞_𝑛=1𝜙_𝑗𝑛cos𝛽_𝑛𝑧, (15) where 𝜙_𝑗𝑛 =  −2𝜋𝑟₀(𝐺_𝑠+𝜂_𝑠⋅𝑠)𝐴_𝑛𝜉_𝑛𝐾₁(𝜉_𝑛𝑟₀)/𝜌_𝑝,𝑗𝐴_𝑝,𝑗[(𝛽_𝑛𝑉_𝑝,𝑗)²+𝑠²] and 𝐾₁(⋅) represents the modified Bessel function of order one of the second kind.

Based on the fictitious soil-pile model, the impedance function at the head of fictitious soil pile can be obtained. Then, substituting the impedance function into the governing equation of pile, the solution of pile dynamic response can be obtained. Therefore, the governing equation of pile can be solved through two steps.

Step I (deriving the displacement impedance function at the head of fictitious soil pile). Taking the Laplace transform of (6) and substituting the vertical displacements of fictitious soil pile and its surrounding soil into (6), the following equation can be obtained by means of orthogonality of the inherent function system cos(𝛽_𝑛𝑧)in the interval [0,𝐻_𝑠]: 𝑀₁2sin𝜆/𝑉_𝑝,1+𝛽_𝑛𝐻_𝑠𝜆/𝑉_𝑝,1+𝛽_𝑛+sin𝜆/𝑉_𝑝,1−𝛽_𝑛𝐻_𝑠𝜆/𝑉_𝑝,1−𝛽_𝑛−𝑁₁2cos𝜆/𝑉_𝑝,1+𝛽_𝑛𝐻_𝑠−1𝜆/𝑉_𝑝,1+𝛽_𝑛+cos𝜆/𝑉_𝑝,1−𝛽_𝑛𝐻_𝑠−1𝜆/𝑉_𝑝,1−𝛽_𝑛=𝐴_𝑛𝐾₀𝜉_𝑛𝑟₀+2𝜋𝑟₀𝐺_𝑠+𝜂_𝑠⋅𝑠𝜉_𝑛𝐾₁𝜉_𝑛𝑟₀𝜌_𝑝,1𝐴_𝑝,1𝛽_𝑛𝑉_𝑝,1²+𝑠²×^𝐻𝑠₀cos²𝛽_𝑛𝑧d𝑧. (16)

Substituting (16) into (15), the vertical displacement at the head of fictitious soil pile can be derived as 𝑈₁=𝑀₁𝜆cos𝑉_𝑝,1𝑧+^∞_𝑛=1𝜒^′_1𝑛cos𝛽_𝑛𝑧+𝑁₁𝜆sin𝑉_𝑝,1𝑧−^∞_𝑛=1𝜒^′′_1𝑛cos𝛽_𝑛𝑧, (17) where 𝜒^′_1𝑛=𝜒_1𝑛sin𝜆/𝑉_𝑝,1+𝛽_𝑛𝐻_𝑠𝜆/𝑉_𝑝,1+𝛽_𝑛+sin𝜆/𝑉_𝑝,1−𝛽_𝑛𝐻_𝑠𝜆/𝑉_𝑝,1−𝛽_𝑛,𝜒^′′_1𝑛=𝜒_1𝑛cos𝜆/𝑉_𝑝,1+𝛽_𝑛𝐻_𝑠−1𝜆/𝑉_𝑝,1+𝛽_𝑛+cos𝜆/𝑉_𝑝,1−𝛽_𝑛𝐻_𝑠−1𝜆/𝑉_𝑝,1−𝛽_𝑛,𝜒_1𝑛=−𝜋𝑟₀𝐺_𝑠+𝜂_𝑠⋅𝑠𝜉_𝑛𝐾₁𝜉_𝑛𝑟₀𝜌_𝑝,1𝐴_𝑝,1𝛽_𝑛𝑉_𝑝,1²+𝑠²𝜑_1𝑛𝐿_1𝑛,𝐿_1𝑛=^𝐻𝑠₀cos²𝛽_𝑛𝑧d𝑧. (18)

Taking the Laplace transform of boundary condition (5c) and substituting (17) into boundary condition (5c), the following equation can be obtained: 𝑀₁𝑁₁=−sin𝜆/𝑉_𝑝,1𝐻−∑^∞_𝑛=1𝜒^′′_1𝑛cos𝛽_𝑛𝐻cos𝜆/𝑉_𝑝,1𝐻+∑^∞_𝑛=1𝜒^′_1𝑛cos𝛽_𝑛𝐻. (19)

Then, the displacement impedance function at the head of fictitious soil pile can be derived as 𝑍₁(𝑠)=−𝐸_𝑝,1𝐴_𝑝,1(𝜕𝑈₁/𝜕𝑧)_𝑧′=0𝑈_1𝑧′=0=−𝐸_𝑝,1𝐴_𝑝,1×−𝑀₁𝑁₁𝜆𝑉_𝑝,1𝜆sin𝑉_𝑝,1𝐻_𝑝+^∞_𝑛=1𝜒^′_1𝑛𝛽_𝑛sin𝛽_𝑛𝐻_𝑝+𝜆𝑉_𝑝,1𝜆cos𝑉_𝑝,1𝐻_𝑝+^∞_𝑛=1𝜒^′′_1𝑛𝛽_𝑛sin𝛽_𝑛𝐻_𝑝×𝑀₁𝑁₁𝜆cos𝑉_𝑝,1𝐻_𝑝+^∞_𝑛=1𝜒^′_1𝑛cos𝛽_𝑛𝐻_𝑝+𝜆sin𝑉_𝑝,1𝐻_𝑝−^∞_𝑛=1𝜒^′′_1𝑛cos𝛽_𝑛𝐻_𝑝⁻¹. (20)Step II (deriving the displacement impedance function at the head of pile). In the same solving manner of (16), the following equation can be obtained: 𝑀₂2sin𝜆/𝑉_𝑝,2+𝛽_𝑛𝐻_𝑝𝜆/𝑉_𝑝,2+𝛽_𝑛+sin𝜆/𝑉_𝑝,2−𝛽_𝑛𝐻_𝑝𝜆/𝑉_𝑝,2−𝛽_𝑛−𝑁₂2cos𝜆/𝑉_𝑝,2+𝛽_𝑛𝐻_𝑝−1𝜆/𝑉_𝑝,2+𝛽_𝑛+cos𝜆/𝑉_𝑝,2−𝛽_𝑛𝐻_𝑝−1𝜆/𝑉_𝑝,2−𝛽_𝑛=𝐴_𝑛𝐾₀𝜉_𝑛𝑟₀+2𝜋𝑟₀𝐺_𝑠+𝜂_𝑠⋅𝑠𝜉_𝑛𝐾₁𝜉_𝑛𝑟₀𝜌_𝑝,2𝐴_𝑝,2𝛽_𝑛𝑉_𝑝,2²+𝑠²×^𝐻𝑝₀cos²𝛽_𝑛𝑧d𝑧. (21)

Substituting (21) into (15), the vertical displacement at the head of pile can be derived as 𝑈₂=𝑀₂𝜆cos𝑉_𝑝,2𝑧+^∞_𝑛=1𝜒^′_2𝑛cos𝛽_𝑛𝑧+𝑁₂𝜆sin𝑉_𝑝,2𝑧−^∞_𝑛=1𝜒^′′_2𝑛cos𝛽_𝑛𝑧, (22) where 𝜒^′_2𝑛=𝜒_2𝑛sin𝜆/𝑉_𝑝,2+𝛽_𝑛𝐻_𝑝𝜆/𝑉_𝑝,2+𝛽_𝑛+sin𝜆/𝑉_𝑝,2−𝛽_𝑛𝐻_𝑝𝜆/𝑉_𝑝,2−𝛽_𝑛,𝜒^′′_2𝑛=𝜒_2𝑛cos𝜆/𝑉_𝑝,2+𝛽_𝑛𝐻_𝑝−1𝜆/𝑉_𝑝,2+𝛽_𝑛+cos𝜆/𝑉_𝑝,2−𝛽_𝑛𝐻_𝑝−1𝜆/𝑉_𝑝,2−𝛽_𝑛,𝜒_2𝑛=−𝜋𝑟₀𝐺_𝑠+𝜂_𝑠⋅𝑠𝜉_𝑛𝐾₁𝜉_𝑛𝑟₀𝜌_𝑝,2𝐴_𝑝,2𝛽_𝑛𝑉_𝑝,2²+𝑠²𝜑_2𝑛𝐿_2𝑛,𝜑_2𝑛=𝐾₀𝜉_𝑛𝑟₀+2𝜋𝑟₀𝐺_𝑠+𝜂_𝑠⋅𝑠𝜉_𝑛𝐾₁𝜉_𝑛𝑟₀𝜌_𝑝,2𝐴_𝑝,2𝛽_𝑛𝑉_𝑝,2²+𝑠²,𝐿_2𝑛=^𝐻𝑝₀cos²𝛽_𝑛𝑧d𝑧. (23)

Substituting (20) into the governing equation of pile and combining the boundary conditions (5c) and (5d), the following equation can be derived: 𝑀₂𝑁₂=𝜆𝑉_𝑝,2𝜆cos𝑉_𝑝,2𝐻_𝑝+^∞_𝑛=1𝜒^′′_2𝑛𝛽_𝑛sin𝛽_𝑛𝐻_𝑝+𝑍₁(𝑠)𝐸_𝑝,2𝐴_𝑝,2𝜆sin𝑉_𝑝,2𝐻_𝑝−^∞_𝑛=1𝜒^′′_2𝑛cos𝛽_𝑛𝐻_𝑝×𝜆𝑉_𝑝,2𝜆sin𝑉_𝑝,2𝐻_𝑝+^∞_𝑛=1𝜒^′_2𝑛𝛽_𝑛sin𝛽_𝑛𝐻_𝑝−𝑍₁(𝑠)𝐸_𝑝,2𝐴_𝑝,2𝜆cos𝑉_𝑝,2𝐻_𝑝+^∞_𝑛=1𝜒^′_2𝑛cos𝛽_𝑛𝐻_𝑝⁻¹. (24)

Then, the displacement impedance function at the head of pile can be derived as 𝑍₂(𝑠)=−𝐸_𝑝,2𝐴_𝑝,2(𝜕𝑈₂/𝜕𝑧)_𝑧=0𝑈_2𝑧=0𝐸=−^𝑝,2𝐴_𝑝,2𝐻_𝑝𝜆𝑀₂/𝑁₂1+∑^∞_𝑛=1𝜒^′_2𝑛−∑^∞_𝑛=1𝜒^′′_2𝑛, (25) where 𝜆=𝜆𝑇_𝑐 is dimensionless quantity. 𝑇_𝑐=𝐻_𝑝/𝑉_𝑝,2 represents the propagation time of elastic longitudinal wave propagating from the pile head to pile end.

The velocity transit function at the head of pile can be derived as 𝐺_𝑣(𝑠)=𝑠𝑍₂(𝑠)=−𝑠𝐻_𝑝𝐸_𝑝,2𝐴_𝑝,2𝜆𝑀₂𝑁₂1+^∞_𝑛=1𝜒^′_2𝑛−^∞_𝑛=1𝜒^′′_2𝑛. (26)

Letting 𝑠=𝑖𝜔 and substituting it into (26) yield the velocity response at pile head in frequency domain: 𝐻_𝑣(𝑖𝜔)=𝑖𝜔𝑍₂()1𝑖𝜔=−𝜌_𝑝,2𝐴_𝑝,2𝑉_𝑝,2𝐻^′_𝑣, (27) where |𝐻^′_𝑣| is the dimensionless velocity admittance at pile head which can be expressed as 𝐻^′_𝑣=𝑖𝑀₂𝑁₂1+^∞_𝑛=1𝜒^′_2𝑛−^∞_𝑛=1𝜒^′′_2𝑛. (28)

By means of the inverse Fourier transform, the velocity response in time domain at pile head can be obtained as 𝑉(𝑡)=IFT[𝑄(𝜔)𝐻_𝑣(𝑖𝜔)], where 𝑄(𝜔) denotes the Fourier transform of 𝑞(𝑡), which is the vertical excitation acting on the pile head.

In particular, the excitation can be regarded as a half-sine pulse in the nondestructive detection of pile foundation as follows: 𝑞(𝑡)=𝑄_max𝜋sin𝑇𝑡,𝑡∈(0,𝑇),0,𝑡≥𝑇, (29) where 𝑇 and 𝑄_max denote the duration of the impulse or impulse width and the maximum amplitude of the vertical excitation, respectively. Then, the velocity response in time domain of the pile head can be expressed as 𝑉(𝑡)=IFT𝑄(𝑖𝜔)⋅𝐻_𝑣(𝑄𝑖𝜔)=−^max𝜌_𝑝,2𝐴_𝑝,2𝑉_𝑝,2𝑉^′_𝑣, (30) where 𝑉^′_𝑣 is the dimensionless velocity response which can be expressed as 𝑉^′_𝑣=12^∞_−∞𝑖𝑀₂𝑁₂1+^∞_𝑛=1𝜒^′_2𝑛−^∞_𝑛=1𝜒^′′_2𝑛×𝑇𝜋²−𝑇²𝜔²⋅1+𝑒^{−𝑖𝜔𝑇}𝑒^𝑖𝜔𝑡d𝜔, (31) where 𝜔=𝑇_𝑐𝜔 represents the dimensionless frequency. 𝑇 represents the dimensionless pulse width which should satisfy 𝑇=𝑇/𝑇_𝑐. 𝑡 represents the dimensional time variable which should satisfy 𝑡=𝑡/𝑇_𝑐.

4. Analysis of Vibration Characteristics

Based on the solutions, a parametric study is conducted to investigate the influence of thickness of pile end soil on the dynamic response at pile head. Unless otherwise specified, parameters of homogeneous foundation used here are given as follows: the density, shear wave velocity, Poisson’s ratio and viscous damping coefficient of soil are 1800kg/m³, 180m/s, 0.4 and 1000 N⋅m⁻³⋅s, respectively.

4.1. Influence of Thickness of Pile End Soil on the Dynamic Response at Pile Head for Different Pile Lengths

Pile length is an important parameter in the design of pile bearing capacity, and the existing researches have shown that pile length also has significant influence on the dynamic response of pile. Therefore, it is needed to discuss the influence of thickness of pile end soil on the dynamic response at pile head for different pile lengths. Parameters used here are as follows. The pile lengths are 10 m and 20 m, respectively. The radius, density, and elastic longitudinal wave velocity of pile are 0.5 m, 2500kg/m³, and 3800m/s, respectively. Denoting 𝑑 to be the diameter of pile, then the thickness of pile end soil is 𝐻_𝑠=0.5𝑑,1𝑑,3𝑑,5𝑑,10𝑑, respectively.

Figures 2 and 3 show the influence of thickness of pile end soil on the dynamic response at pile head for different pile lengths. As shown in Figures 2(a) and 3(a), it can be seen that the velocity admittance curves oscillate about the mean amplitude as the frequency increases and the shorter the pile is, the more remarkable the change regulation is. It can also be noted that the velocity admittance curves tend to be convergent as the thickness of pile end soil increases. If the thickness of pile end soil is beyond a certain value, say, 𝐻_𝑠=5𝑑 in this case, the further increase of thickness of pile end soil no longer affects the velocity admittance, which means that there is a critical influence thickness of pile end soil. As shown in Figures 2(b) and 3(b), the pile head will receive the reverse reflected signal back from the bedrock when the pile end soil is thin enough, and the shorter the pile is, the stronger the reflected wave signal is. It can be noted that the reflected wave signal curves also tend to be convergent as the thickness of pile end soil increases. If the thickness of pile end soil is beyond a certain value, the further increase of thickness of pile end soil has little influence on the reflected wave signal curves.

(a) Velocity admittance curves

(b) Reflected wave signal curves

Figure 2 

Influence of thickness of pile end soil on the dynamic response at pile head when 𝐻𝑝=10m.

(a) Velocity admittance curves

(b) Reflected wave signal curves

Figure 3 

Influence of thickness of pile end soil on the dynamic response at pile head when 𝐻𝑝=20m.

The above results show that when the thickness is beyond this critical influence thickness, further increase of the thickness of the pile end soil has little influence on the dynamic response of pile. Nevertheless, when the thickness of the pile end soil is less than this critical influence thickness, it has significant influence on the dynamic response of pile.

4.2. Influence of Thickness of Pile End Soil on the Dynamic Response at Pile Head for Different Pile Radii

In this section, the influence of thickness of pile end soil on the dynamic response at pile head for different pile radii is studied. Parameters used here are as follows. The pile radii are 0.3 m and 0.5 m, respectively. The length, density, and elastic longitudinal wave velocity of pile are 15 m, 2500 kg/m³, and 3800 m/s, respectively. The thickness of pile end soil is 𝐻_𝑠=0.5𝑑,1𝑑,3𝑑,5𝑑,10𝑑, respectively.

Figures 4 and 5 show the influence of thickness of pile end soil on the dynamic response at pile head for different pile radii. As shown in Figures 4(a) and 5(a), the amplitude of resonance peaks gradually decreases as the thickness of pile end soil increases, but the resonance frequency of velocity admittance curves basically remains unchanged. It can also be seen that the velocity admittance curves tend to be convergent if the thickness of pile end soil is beyond the critical influence thickness. As shown in Figures 4(b) and 5(b), the reflected wave signal curves also tend to be convergent as the thickness of pile end soil increases. Figures 4 and 5 also show that the larger the pile radius is, the more remarkable the above change regulation is.

(a) Velocity admittance curves

(b) Reflected wave signal curves

Figure 4 

Influence of thickness of pile end soil on the dynamic response at pile head when 𝑟0=0.3m.

(a) Velocity admittance curves

(b) Reflected wave signal curves

Figure 5 

Influence of thickness of pile end soil on the dynamic response at pile head when 𝑟0=0.5m.

4.3. Influence of Thickness of Pile End Soil on the Dynamic Response at Pile Head for Different Elastic Longitudinal Wave Velocities of Pile

In this section, the influence of thickness of pile end soil on the dynamic response at pile head for different elastic longitudinal wave velocities of pile is studied. Parameters used here are as follows. The elastic longitudinal wave velocities of pile are 3600 m/s and 4000 m/s, respectively. The length, radius, and density of pile are 15 m, 0.5 m, and 2500 kg/m³, respectively. The thickness of pile end soil is 𝐻_𝑠=0.5𝑑,1𝑑,3𝑑,5𝑑,10𝑑, respectively.

Figures 6 and 7 show the influence of thickness of pile end soil on the dynamic response at pile head for different elastic longitudinal wave velocities of pile. As shown in Figures 6(a) and 7(a), with the increase of thickness of pile end soil, the resonance frequency of velocity admittance curves basically remains unchanged, but the amplitude of resonance peaks gradually decreases. It can also be noted that the velocity admittance curves tend to be convergent if the thickness of pile end soil is beyond the critical influence thickness. As shown in Figures 6(b) and 7(b), it can be noted that the reflected wave signal curves also tend to be convergent as the thickness of pile end soil increases. Figures 6 and 7 also show that the larger the elastic longitudinal wave velocity of pile is, the more remarkable the above change regulation is.

(a) Velocity admittance curves

(b) Reflected wave signal curves

Figure 6 

Influence of thickness of pile end soil on the dynamic response at pile head when 𝑉𝑝,2=3600m/s.

(a) Velocity admittance curves

(b) Reflected wave signal curves

Figure 7 

Influence of thickness of pile end soil on the dynamic response at pile head when 𝑉𝑝,2=4000m/s.

5. Comparison with Other Models

In this section, the fictitious soil-pile model is compared with other models mentioned in the introduction. No matter the support condition of pile end is rigid or viscoelastic, it can be described by the complex impedance of pile end which can be expressed as 𝑍_𝑏=𝑘_𝑏+𝜂_𝑏⋅𝑖𝜔, (32) where 𝑘_𝑏 and 𝜂_𝑏 represent the spring constant and damping coefficient of the support condition of pile end. Then, the various support condition of pile end can be reflected by adjusting these two parameters. Parameters used here are the length, radius, density, and elastic longitudinal wave velocity of pile which are 15 m, 0.5 m, 2500 kg/m³, and 3800 m/s, respectively. The thickness of pile end soil is 2𝑑. The support condition of pile end can be regarded to be free if 𝑘_𝑏=0, 𝜂_𝑏=0 and rigid if 𝑘_𝑏=∞, 𝜂_𝑏=∞ and viscoelastic otherwise.

Figure 8 presents the comparison of the fictitious soil-pile model with other models. As shown in Figure 8(a), it can be noted that the case of free support is out of phase opposite to the case of rigid support; that is, the phase angle difference of these two cases is close to 180^∘. The phase angle difference varies from 0^∘ to 180^∘ when the values of 𝑘_𝑏 and 𝜂_𝑏 are other constants. It can also be noted that the amplitude of resonance peaks gradually decreases firstly and then gradually increases when the values of 𝑘_𝑏 and 𝜂_𝑏 vary from zero to infinity. As shown in Figure 8(b), it can be noted that the amplitude of reflected wave signals at pile end gradually decreases firstly and then increases when the values of 𝑘_𝑏 and 𝜂_𝑏 vary from zero to infinite. The reflected wave signal is orthokinetic when the complex impedance of pile end is small; for example, 𝑘_𝑏=0, 𝜂_𝑏=0 and 𝑘_𝑏=0.1, 𝜂_𝑏=0.06. Meanwhile, the reflected wave signal is reverse when the complex impedance of pile end is large; for example, 𝑘_𝑏=15, 𝜂_𝑏=15 and 𝑘_𝑏=∞, 𝜂_𝑏=∞. The amplitude of reflected wave signal for the case of fictitious soil-pile model is between that of free support and rigid support. As can be seen from the results of the analysis, it is shown that the fictitious soil-pile model is reliable and can be simulated in other support models by varying parameters of pile end soil.

(a) Velocity admittance curves

(b) Reflected wave signal curves

Figure 8 

Comparison of the fictitious soil-pile model with other support models.

6. Conclusions

(1) By introducing the fictitious soil-pile model, this paper derived the analytical solution of velocity response at pile head in frequency domain and its corresponding semianalytical solution of velocity response at pile head in time domain. Whereafter, the influence of parameters of pile end soil on the dynamic response at pile head is investigated in detail.

(2) The finite pile end soil layer adjacent to the pile end has significant influence on the dynamic response at pile head, and there is a critical influence thickness for pile end soil. If the thickness of pile end soil is within the critical influence thickness, the increase of the thickness of pile end soil can dramatically affect the dynamic response at pile head. If the thickness of pile end soil is beyond the critical influence thickness, the further increase of the thickness of pile end soil has little influence on the dynamic response at pile head.

(3) The influence of parameters of pile end soil on the dynamic response at pile head may be more remarkable with the decrease of pile length and with the increase of pile radius and elastic longitudinal wave velocity of pile.

Conflict of Interests

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

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Grant nos. 51309207 and 51378514), the China Postdoctoral Science Foundation Funded Project (Grant nos. 2012M521495 and 2013T60759), and the Fundamental Research Funds for the Central Universities (Grant no. CUG120821).