Abstract

A numerical method for evaluating the vertical vibration isolation effect of pile rows embedded in a viscoelastic half space subjected to a moving load is developed in this paper on the basis of the Cole-Cole model and Muki’s method. Based on the proposed method, the influence of various parameters on the vibration isolation effect of pile rows embedded in the viscoelastic half space is investigated numerically.

1. Introduction

Vibration induced by railway traffic is a major concern for civil engineers as it causes annoyance to residents or even damage to adjacent structures. Generally, the effects of ground vibrations can be mitigated by two kinds of vibration isolation methods: the active and the passive vibration isolation methods. The active isolation system is often used to reduce the ground vibration near the source. It is usually installed either around the vibration source or at a close distance to the source. The passive isolation system, on the other hand, usually is far away from the source and surrounds the protected structure. Normally, there are two passive vibration isolation methods: the trench (open or infilled) isolation method and the pile (pile rows or sheet piles) isolation method. To date, many studies concerning vibration isolation using trenches or piles have been conducted. For example, Emad and Manolis [1] utilized the boundary element method (BEM) with constant elements to examine the efficiency of vibration reduction by open trench with a rectangular or a circular cross-section. Considering the coupling effects between the soil skeleton and underground water, Cao et al. [2] proposed an analytical model to investigate the screening efficiency of trenches to moving-load induced ground vibrations based on Biot’s dynamic poroelastic theory. Cai et al. [3] investigated the vibration isolation effect of pile rows embedded in a poroelastic medium by using the wave function expansion method. Kattis et al. [4, 5] used 3D BEM to calculate the screening effectiveness of a pile row in the frequency domain. Also, by means of the frequency domain BEM, the screening effectiveness of four types of circular piles in a row against the vibration due to a massless square foundation subjected to a harmonic vertical loading is studied by Tsai et al. [6]. Besides, by means of the fictitious pile method developed by Muki and Sternberg [7] and the direct superposition method, Lu et al. [8, 9] analyzed the vibration isolation effect of pile rows.

It is noticed that previous studies concern the vibration isolation modelling in a generalized standard linear viscoelastic solid. For the generalized standard linear viscoelastic solid, the complex modulus belongs to the rational function. However, both creep tests [10] and vibration tests [11] suggest that the derivatives of stress relaxation functions have an asymptotic behaviour const. As a result, the convolution in the constitutive relation of the heterogeneous viscoelastic medium cannot be eliminated by the method used in the generalized standard linear viscoelastic solid. Unlike the standard linear solid, the complex modulus of the Cole-Cole viscoelastic medium is not the rational function, a viscoelastic kernel having a singularity const at , where , is used to characterize the viscoelastic medium [12]. The Cole-Cole model has been validated experimentally in both quasi-static and vibrations experiments [10]. It has been pointed out in several studies, for example Bagley and Torvik [13], Lu and Hanyga [14], and Soula et al. [15], that the Cole-Cole model fits experimental data over several decades of frequency. Experimental evidences in rock physics [16, 17] also point out to the Cole-Cole type behavior. Furthermore, the Cole-Cole relaxation model has many advantages over other models. In particular, the Cole-Cole relaxation model is compatible with a finite speed of wave propagation. It is very important to conduct accurate analysis of the wave attenuation and dispersion when moving loads are of high velocities.

In this study, a numerical method for evaluating the vertical vibration isolation effect of pile rows embedded in a viscoelastic half space subjected to a moving load is developed on the basis of the Cole-Cole model [12] and Muki’s method [7]. Based on the proposed method, the influence of various parameters on the vibration isolation effect of pile rows embedded in the viscoelastic half space is investigated numerically. It is noted that the proposed method in this study belongs to the semianalytical category. Thus, compared with conventional domain discretization methods such as the finite element method (FEM) and the boundary element method (BEM), it significantly reduces the computational time.

2. The Free Wave Field Solution and the Fundamental Solution for a Circular Uniform Patch Load

Usually, for a linear isotropic viscoelastic medium described by the Cole-Cole model, the and the wave modes satisfy the Cole-Cole relaxation law with different parameters. Consequently, in terms of the Cole-Cole models, the complex moduli for the and the waves in the frequency domain have the following forms:where a bar over the function denotes the Fourier transform for time frequency . The subscripts and denote the and waves in a viscoelastic medium and and are the complex modulus corresponding to the two Lame constants. is the limit of the complex modulus for , is the value of the complex modulus for , is a characteristic relaxation time, and controls the width of the transition zone between and . Besides, the conditions and follow from thermodynamics argument [12].

In this study, the free wave field solution is defined as the solution of the moving load in the absence of the pile rows. For a moving load, axisymmetry is lost due to the orientation of the load speed; thus, it is more convenient to consider the moving load problem in the Cartesian coordinate system (Figure 1). A moving load with a constant speed and an oscillating frequency is applied on the surface of the viscoelastic half space. The load moves along the negative direction of the y-axis and the distance between the load and y-axis is (Figure 1). For the moving load applied over a rectangular area , the boundary conditions in the time-space domain are as follows:where is the intensity of the distributed load, is the frequency of moving load, is the Heaviside step function, and is the y coordinate of the center of the distributed load at .

Figure 1 

Piles rows embedded in a viscoelastic half space as the vibration isolation system against the vibration due to moving loads.

Performing a triple Fourier transform with respect to time and the two horizontal coordinates on ((2a), (2b), and (2c)), respectively, the following boundary conditions are derived in the frequency wavenumber domain:in which and represent the two horizontal wave numbers corresponding to x- and y-coordinates, respectively, and is the Dirac delta function.

For a moving point load, the boundary conditions for are the same as those for the moving rectangular distributed load, while the boundary condition for is as follows Likewise, the boundary condition for in the frequency wavenumber domain is given by

Using ((3a), (3b), and (3c)) or (5) as well as the expressions for the stress in the frequency wavenumber domain, the free wave field due to the surface moving load can be determined. In view of (3a), (3b), and (3c), all the variables in the frequency wavenumber domain due to the moving distributed rectangular load can be expressed in the following form: where is the solution of a variable corresponding to a unit boundary value in (3c). The moving point load has similar expression

Performing the inverse Fourier transform with respect to the two horizontal wavenumbers and using the property of the delta function, the frequency domain free field solution for the moving distributed rectangular load has the form:

For the moving point load, the frequency domain free field solution for all variables is represented by

To establish the integral equations for the pile rows, the frequency domain fundamental solution for a uniform vertical patch load applied in the viscoelastic half space is required. As the problem for a viscoelastic half space subjected to a uniform vertical patch load over a circular area with a radius (Figure 2) is axisymmetric with respect to the center of the circular area, thus, it is more convenient to consider the problem in the cylindrical coordinate system ().

Figure 2 

The model for decomposition of the pile-half space system. (a) an extended poro-elastic half-space (b) a fictitious pile.

For a uniform vertical patch load applied in the viscoelastic half space, the surface of the half space is stress-free. Therefore, the boundary conditions for the half-space surface are as follows: where denotes the mth order first kind of Bessel function and denotes the Hankel transform parameter.

It is assumed that the circular patch load is applied at the depth . Accordingly, the half space is divided by the plane into an upper and a lower region and, therefore, the continuity conditions at the plane are as follows:

Using the general expressions for the displacement and the stress and (10) and (11), the arbitrary constants involved in the expressions for the potentials, the displacement, and the stress can be determined. The frequency fundamental solution can be obtained by performing inverse Hankel transform on the solutions for the displacement, the stress in the frequency-wavenumber domain [18].

3. Fredholm Integral Equations Describing Dynamic Interaction between Piles and the Viscoelastic Half Space

As shown in Figure 1, the pile rows embedded in the viscoelastic half space are used to isolate the vibration generated by a moving load. The number of the total pile is , where and denote the number of pile rows and the number of the piles in the kth row. The spacing between two neighboring piles in each pile row is denoted by s. The spacing between two neighboring pile rows is . Also, it is assumed that each pile has the same diameter d () and the same length L (). A moving load with a constant speed c and an oscillating frequency is acting on the surface of the viscoelastic half space and moves along the negative direction of the y-axis (Figure 1).

When the pile-half-space system is subjected to a vertical moving load, generally, the pile will undergo both vertical and horizontal response. However, as the influence of the horizontal interaction between the pile-soil system is relatively smaller, the horizontal interaction between piles and the half space is neglected in this paper. Thus, only the vertical interaction between the piles and the half space is considered in the paper.

Following Muki and Sternberg [7] and Pak and Jennings [19], the current problem is decomposed into two subproblems: an extended viscoelastic half space and the multiple fictitious piles. The extended viscoelastic half space is described by the continuum dynamic theory for a viscoelastic medium, while the fictitious piles are described by the 1D bar vibration theory. The decomposition procedure is illustrated in Figure 2 by the ith pile of the pile rows.

It is assumed that the axial force of the ith fictitious pile is and the vertical distributed load along the ith fictitious pile is (Figure 2(b)). The top and the bottom of the ith fictitious pile are subjected to forces and , respectively. The viscoelastic half space is subjected to the following loads (Figure 2(a)): which is distributed over the region occupied by the th pile; and which are applied to the circular areas and , respectively. Note that denotes the cross-section area of the ith pile.

For the ith fictitious pile, the displacement , the distributed vertical force , and the axial force satisfy the following relations:in which is the vertical displacement of the ith pile.

The vertical strain of the extended half space along the axis of the ith pile is composed of two parts: the first part is due to the free wave field, while the second part is due to the force applied to the extended half space by the fictitious piles. Thus, the vertical strain of the extended half space along the axis of the ith pile can be written as In (13), the superscript and subscript i and j denote the ith and the jth pile, respectively, is the free field vertical strain at the axis of the ith pile, which is determined by the free field frequency domain solution for the moving load, and represents the vertical strain at the center of due to a unit patch load applied at (Figure 2(a)) and is the horizontal distance between the axis of the ith and jth pile. It is worth noting that, for the case is vanishing.

Using (12a) and (12b) and (13), the following relation is obtained where and denote the vertical strain of the viscoelastic half space at the center of of the ith pile when the patch load approaches from up and down side, respectively.

In this study, the compatibility condition between the ith pile and the viscoelastic half space is fulfilled by requiring the vertical strain of the ith fictitious pile and that of the extended half space along the axis of the ith fictitious pile to be equal to where represents the vertical strain of the ith fictitious pile.

Using (12a) and (12b), (14) and (15), the Fredholm integral equation in the frequency domain describing the vertical interaction between the ith pile and the half space has the form where

Following the similar procedures, the surface vertical displacement for the viscoelastic half space in the presence of the pile rows can be calculated as follows: where represents the free field vertical displacement, denotes the vertical displacement at the surface point () due to a unit patch load applied at , and is the horizontal distance between the surface point and the axis of the jth pile.

In (11), the vertical displacement of the ith pile top is also unknown. The unknown can be represented by the axial force of the fictitious piles if the vertical displacement of the ith pile top and the surface vertical displacement of the extended half space at the ith pile top are assumed to be equal; that is, . Note that can be obtained via (18) by setting coinciding with the center of the ith pile. Thus, using (12a) and (12b) and (18), the following supplementary equations for are derived: where is the Kronecker delta and

4. Numerical Results and Discussions

The integral equations in the frequency domain accounting for the vertical interaction between pile rows and the half space can be solved numerically. The methodology for solving integral equation (15) was detailed in [19]. After discretization of (16) and (19), the following linear algebraic equations in the frequency domain are obtained: where is the coefficient matrix determined by discrete integral equations which is associated with the fundamental solution, is the right-handed term which is determined by the free field solution, such as , and is the discrete unknowns of the integral equations.

In order to recover the solution in the time domain, a series of frequency domain solutions at discrete sample points need to be determined first. Assuming the number of the frequency domain sample points is , then (21) for the sample points has the following form: where is the frequency increment for the sample points in the frequency domain and is given by

Due to the vibration frequency of the moving load in (2a), (2b), and (2c) and (4), the right-handed term for the sample points should be determined by the following equation:

After numerical solution of the integral equation (16) and (19) for the sample points , all the variables in the frequency domain are obtained. The time domain solution for the variables can be obtained by performing inverse Fourier transform on the corresponding frequency domain solutions, which is implemented by the FFT method in this study [20].

For verification purposes, Section 4.1 demonstrates the comparison between the solution of a special case from the proposed method and published results. In Section 4.2, some numerical examples and corresponding analysis are presented.

4.1. Comparison of Our Results with Known Results

In this section, the method developed in this study is justified by comparing a special case of our model with existing results. As shown in Figure 1, the vibration source is a moving distributed load with a constant speed c in the negative direction of the -axis. The intensity of the load is , and it is uniformly distributed over a rectangle region with a vibration frequency . A single 8-pile row with circular cross-sections is used as the passive isolation vibration system. All piles are featured by a diameter of , a length of , Young’s modulus of , and a density of . The net spacing between two neighboring piles is . The distance between y-axis and the moving load is .

According to [14], if the parameters () for the viscoelastic half space are assumed to tend to 1, then the viscoelastic half space is reduced to an elastic half space. Moreover, if the speed of the vertical moving distributed load approaches zero, the moving vibration load is reduced to a fixed time-harmonic force. In this paper, the parameter for the viscoelastic is , , , and . The moving load speed is . The corresponding vibration isolation against the fixed time-harmonic force using pile rows in an elastic medium was reported by Kattis et al. [5]. In calculation, the wavelength for the Rayleigh wave of the reduced elastic medium is .

To assess the vibration isolation effect of pile rows, the amplitude reduction ratio at point , which is the ratio between the amplitude of the surface vertical displacement of the half space in the presence of the pile rows and that of the free field solution, is defined as follows: where is the amplitude of the vertical displacement of the half space in the presence of the pile rows and is the amplitude of the vertical displacement of the soil given by the free field solution.

Woods [21] proposed an average amplitude reduction ratio for the evaluation of vibration isolation effect, which is defined as follows: where A is the rectangle with its width and length determined by a reference length and the width of a pile row. In this study, , where is the width of the first pile row and is the wavelength for the Rayleigh wave of the reduced elastic medium.

Figure 3 shows the contour of the amplitude reduction ratio for the single pile row embedded in the elastic half space according to the present method when the moving load with a speed is located at the point . It can be observed that the amplitude reduction ratio right behind the pile rows is much smaller than those at other areas. According to the calculation in this study, the average amplitude reduction ratio for the elastic half space is 0.718, while the result of Kattis et al. [5] for the elastic half space is 0.712. The difference between the present solution and that of Kattis et al. [5] is only 0.842%.

Figure 3 

The contour of the amplitude reduction ratio for a single pile row with eight piles as vibration isolation system for the reduced elastic half space.

4.2. Numerical Results for the Vibration Isolation of Pile Rows

In this section, the influences of the moving load speeds (c), the number of the pile rows (K), Young’s modulus () of the pile, the pile length (), and the net spacing (s) between two neighboring piles in a pile row as well as the spacing between neighboring pile rows () on the vibration isolation effect will be discussed.

Typical values of the parameters for the viscoelastic half space are ,  ,  , , , and . The reference length is . The vibration source is a moving distributed load with a constant speed c in the negative direction of the -axis. The intensity of the load is , and it is uniformly distributed over a rectangle with a vibration frequency . The load moves along a line parallel to the -axis at a constant speed in the negative direction of the -axis, where . The distance between the center of the first rows piles and the center of the distributed load is . Each pile has the same diameter and the same density . Each pile has the same length and the same Young’s modulus (). Also, the net spacing between two neighboring piles is and the spacing between the two adjacent pile rows is .

Note that when the influence of one parameter is examined, all the other parameters will take the typical values as given above.

4.2.1. Effects of the Speed of the Moving Load

In this section, the influences of the speed of the moving load (c) on the vibration isolation effect of the pile rows will be examined. In the calculation, four different moving load speeds , , , and are considered, where , while the remainder parameters for the vibration source and the viscoelastic half space take the typical values as given above. Also, two piles rows embedded in the viscoelastic half space are used as the passive vibration isolation system. Thus, the number of pile rows is and the numbers of piles in the pile rows are and  , respectively.