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Journal of Sound and Vibration

Journal of Sound and Vibration 332 (2013) 4423–4439

0022-46http://d

n CorrE-m

journal homepage: www.elsevier.com/locate/jsvi

Attenuation zones of periodic pile barriers and its applicationin vibration reduction for plane waves

Jiankun Huang, Zhifei Shi n

School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China

a r t i c l e i n f o

Article history:Received 15 September 2012Received in revised form19 December 2012Accepted 23 March 2013

Handling Editor: L.G. Tham

attenuation zone are discussed. The screening effectiveness of finite periodic pile barriers

Available online 1 May 2013

0X/$ - see front matter & 2013 Elsevier Ltd.x.doi.org/10.1016/j.jsv.2013.03.028

esponding author. Tel.: +86 10 51688367.ail addresses: jiankunhuang@yahoo.com (J.K

a b s t r a c t

The periodic theory of solid-state physics is introduced to study the reduction character-istics of periodic pile barriers. The attenuation zones of a two-dimensional infiniteperiodic pile barrier subjected to plane waves are analyzed by plane wave expansionmethod. Influences of soil parameters and pile configurations on the first no-directional

is simulated by the finite element method. The present theoretical results are in wellagreement with experimental data, which validates the existence of attenuation zones inthe periodic structures. The results show that vibrations with frequencies in theattenuation zones can be reduced significantly. The present investigation provides anew concept for designing pile barriers to block mid-frequency vibration.

& 2013 Elsevier Ltd. All rights reserved.

1. Introduction

Vibrations induced by traffic loads, construction blasting and machinery foundations not only affect sensitive equipment,damage structures, but also distress the daily life of residents. Therefore, reduction or isolation of vibrations for requiredarea is a major concern for engineers and researchers. One proven method of vibration isolation is to set up a barrierbetween the vibration source and the protected structure. The continuous barriers such as open or in-filled trenches areconsidered to be the appropriate ways due to their effectiveness and low cost. However, there are limitations to these trenchmethods, such as soil instability and the obvious requirement to excavate deep trenches. The discontinuous barrier method,such as rows of piles, is considered to be a superior method of vibration isolation. For instance, piles can be driven verydeeply into the soil and can be arranged in any desirable configuration to create wave barriers.

To date, many researches concerning vibration isolation by piles have been conducted. Woods et al. [1] experimentallyinvestigated the problem of the screening effectiveness of piles and proposed some preliminary criteria for the design of arow of piles used as a wave barrier. Liao and Sangrey [2] employed a two-dimensional (2D) acoustic P-wave model tosimulate the passive isolation of foundations from Rayleigh wave disturbances. By using a plane strain model and an anti-plane strain model, Avilés and Sánchez-Sesma [3,4] reached a conclusion that rigid piles performed more effectively thanflexible piles after the studies on the scattering of plane waves (P-wave, SH-wave, SV-wave) in an infinitely long pile system.Boroomand and Kaynia [5] examined the efficiency of vibration reduction by a row of piles by utilizing an approximateanalytical model. In the recent past, Kattis et al. [6,7] developed a three-dimensional (3D) frequency-domain boundaryelement method (BEM) to study the vibration isolation by a single row of piles. Kattis et al. [6] considered the pile–soilsystem as a periodic structure characterized by a unit cell, and treated a row of piles as an equivalent trench for simplicity.

All rights reserved.

. Huang), zfshi178@bjtu.edu.cn, zfshi178@sohu.com (Z.F. Shi).

zyx

Fig. 1. Vibration generated by the railway and the periodic pile barrier. Incidence of plane waves.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–44394424

Tsai et al. [8] also adopted a 3D frequency domain BEM to analyze four types of piles, and obtained that screeningeffectiveness was related to both length and material properties of the piles. Gao et al. [9] calculated 3D multi-rows ofsquare piles as a barrier by means of an integral equation for Rayleigh wave scattering based on Green's solution of Lambproblem. Xia et al. [10] studied an arbitrary configuration of piles as barriers by a new theoretical way called “multiplescattering” technique. Lu et al. [11,12] investigated the isolation of moving-load induced vibrations using pile rowsembedded in a poroelastic half-space or a layered poroelastic half-space. In the investigation of vibration due to Rayleighwaves, Xu et al. [13] showed that for the same vibration source, the same pile rows can produce a better vibration isolationeffect for the poroelastic medium than for a single phase elastic medium. Cai et al. [14,15] developed 2D plane strainanalytical solutions of scattering coefficients for incident plane P and S waves by a row of piles.

Most of previous researchers focused their attention on the development of different theoretical or numericalmethodologies to analyze the performance of discontinuous pile barriers. In most cases, only a single row of piles wasinvestigated [1,3–8,11,14,15]. Though rows of piles are investigated in the previous works [2,9,10,12,13], the periodic natureof rows of piles was not considered and the concept of dispersion curves was not proposed. The pile–soil system with thepiles arranged in a periodic configuration forms a type of periodic structures, as shown in Fig. 1. Here, pile barriers with pilesarranged in a periodic configuration are called as “periodic pile barriers” or “periodic structures”. In fact, if the dispersioncurves are obtained, the attenuation zones (AZs) of a periodic structure can be determined. Elastic waves with frequencies inthe range of the AZs cannot propagate in the media [16]. Therefore, the AZs can be utilized to reduce the dynamic responseof structures located behind periodic pile barriers, and to explain the vibration reduction mechanism of waves due to pilerows. Based on the concept of periodic structures, Shi and his coworkers [17–20] studied the possible applications of one-dimensional (1D) and two-dimensional (2D) periodic structures in seismic isolation or vibration isolation in the field of civilengineering. Gupta et al. [21] presented a periodic approach to assess the vibration isolation efficiency of continuous anddiscontinuous floating slab tracks, and the dispersion curves of periodic floating slab tracks are displayed. They cited that thefrequency range of interest for subway induced vibrations is 1–80 Hz, which is also the frequency range discussed inthe paper.

In the present paper, investigations are conducted for an infinitely extended periodic pile–soil system, with a square ortriangular array of identical piles. Some concepts usually used in solid-state physics are introduced into civil engineering andthe AZs are used to control the capacity of vibration reduction. So, one of the major works of the present paper is to studythe characteristic of the AZs. Plane wave expansion method is introduced to calculate the AZs of infinite periodic pilebarriers. The factors influencing on the AZs are discussed comprehensively. To verify the feasibility of the finite rows ofperiodic pile barriers, the scattering of the incident SV wave is considered by finite element method (FEM).

This paper is structured as follows. In Section 2, the theoretical method called plane wave expansion (PWE) method isintroduced to calculate the AZs of the periodic structure and the concepts of typical cell, Brillouin zone, and the Bloch wavevector are presented. In Section 3, the influences of certain parameters on the AZs of an infinite periodic pile barrier areinvestigated. In Section 4, a FEM analysis is used to investigate the dynamic response of the periodic pile barrier in both thefrequency domain and the time domain. For periodic pile barriers, the numerical results show that vibrations are reducedsignificantly in the range of the calculated AZs. The screening effectiveness due to different pile parameters is also discussed.This investigation also shows that periodic pile barriers can be beneficial in blocking vibrations.

2. Theoretical formulation

2.1. Plane wave expansion method

Periodic structure is a type of repeated system, in which the materials and configurations are arranged periodically.Periodic structure can be simplified as a typical cell based on the Bloch theory. This typical cell is not the same as the unitcell mentioned in Ref. [6], in which the composite is replaced by a homogeneous but anisotropic medium. Herein the typical

Table 1[22] Waves for various sources.

Physical sources Type of sources Wave Monitor location

Highway/rail footing array Line Surface SurfaceBody Surface

Car in pothole, single ‘el’ footing Point Rayleigh SurfaceBody Surface

Tunnel Buried line Body InteriorBuried explosion Buried point Body Interior

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–4439 4425

cells are not homogenized. The typical cell can represent the whole periodic structure and be used to analyze the dispersioncurves. So the complexity of the problem and the computation time can be reduced significantly. Plane wave expansionmethod, which has been widely used in studying the dispersion curves of periodic structures and crystals in solid statephysics, is adopted in the present paper.

In the present investigation, it is assumed that the elastic wave propagates within the x−y plane perpendicular to thepiles and the piles are infinite in the z direction. Thus, the polarization modes of the elastic waves can be decoupled into thein-plane (x−y) mode and the anti-plane (z) mode. In fact, 2D plane strain model and 2D anti-plane strain model are oftenused in the investigation of pile barriers by other authors [3,4,10,14,15]. Although this is an approximated approach, theproposed formulation corresponds to the cases in which the surface boundary conditions are not important. Besides, varioussources of vibrations generate different types of waves, including body waves (P-wave, SH-wave and SV-wave) and surfacewaves (Rayleigh wave and Lovewave). The types of the propagating wave, source type and location are shown in Table 1 [22]. Foran interior load, the body waves are dominant. The contribution of Rayleigh waves to the free field response depends on thedepth of the source. An increase in the tunnel depth decreases the frequency range where Rayleigh waves are important [23]. Forexample, for a blast point located at a depth of 28 m below the ground surface, the typical time domain signals were measuredby a 3D geophone at a depth of 7.5 m. Depending on the orientations of transducers, either P or S wave energy was dominant[24]. If it is buried deep enough, the tunnel excites principally body waves. Unless the source is very shallow, the surface wavereflected from a free surface is relatively insignificant [22]. Generally speaking, if the interior load is at a depth of 0.7 timewavelength of shear wave in the half-space, the body wave is dominant, and the surface component is small [25].

In addition, both soil and concrete are assumed to be homogenous, linearly elastic and perfectly bonded at the interface.The simplest material model, i.e. the elastic model is adopted in the present investigation. This model runs fast and onlycontains two material parameters [26]. In the following analysis, the elastic parameters, Poisson's ratio and Young's modulusof the soil, are assumed to be constants. The modulus of elasticity and Poisson's ratio for different types of soils given by Das[27] and some software manuals [26] are adopted in the present numerical simulation. In practical engineering, it may bebetter that the soil is described as an elastic–plastic material. However, the soil behaves completely elastic at very smalldeformation [28]. Although the constitutive relation of soils under static and dynamic loads may not be strictly consideredas linear elastic, solutions based on the linear elastic theory are also meaningful and can be utilized to estimate thedistribution of stress and strain in the field [29]. On the other hand, lower damping (such as the scaled parameters pscale¼0and qscale¼0.5 in damping matrix in Ref. [30]; and λ¼λe−0.1iλe and μ¼μe−0.1iμe in Ref. [31], in which subscript e refers to theelastic Lame constant) has a slight effect on the dispersion curves [30] and attenuation [31]. The linear elastic model hasbeen proven to be a good approximation to real field behavior [32]. Therefore, we apply the linear elastic model to describethe wave propagation in the present paper.

For the harmonic incident plane wave (P–SH waves, SV waves), the displacement component ui (i¼x,y) and uz satisfiesthe following two equations:

ρðrÞ ∂2ui

∂t2¼∇T μðrÞ∇Tui½ � þ∇T μðrÞ ∂uT

∂xi

� �þ ∂

∂xiλðrÞ∇TuT½ � (1)

where, uT ¼ uxex þ uyey and ∇T ¼ ∂=∂xxex þ ∂=∂xyey.

ρ∂2uz

∂t2¼ μ

∂2uz

∂x2þ ∂2uz

∂y2

� �(2)

where, ρ, λ and μ are the density and Lame's parameters of the medium, respectively. Due to periodicity, the density ρ,Lame's parameters λ and μ can be expanded as a Fourier series [33]:

FðrÞ ¼∑G1FðG1Þ⋅eiG1r; ðF ¼ ρ; λ; μÞ (3)

where, i¼ffiffiffiffiffiffiffi−1

p, r¼ ðx; yÞ is the position vector. The symbol G1 is the Fourier wave-vectors of the density and Lame's

parameters as defined in Section 2.2.From the Bloch theory, the solutions of the wave equation have the following form [33]:

uðr; tÞ ¼ e−iωt∑G2uK ðG2Þ⋅eiðKþG2Þr (4)

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–44394426

where, K is the Bloch wave vector and is determined from the Fourier wave-vectors, and ω is the circular frequency of thewave. The symbol G2 is the Fourier wave-vectors of the displacement u as defined in Section 2.2. One problem is that theconvergence rate is very slow once the systems are with either very high or very low filling fraction, or large elasticmismatch. In order to improve the convergence of the method, the inverse Laurant's rule is used in the following formula [34].Substituting Eqs. (3) and (4) into Eqs. (1) and (2), the governing equation of the system becomes

ω2

∑G2

ρG3−G20

0 ∑G2

ρG3−G2

2664

3775 ux

K ðG2ÞuyK ðG2Þ

" #¼

Kxx Kxy

Kyx Kyy

" #uxK ðG2Þ

uyK ðG2Þ

" #(5)

ω2∑G2ρðG3−G2Þuz

KðG2Þ ¼∑G2

1μðG3−G2Þ

� �−1ðKþG2ÞðKþG3Þuz

KðG2Þ (6)

where, G3 ¼ G1 þ G2, and

Kjl ¼∑G2

1μG3−G2

" #−1

∑l½ðK þ G2ÞlðK þ G3Þl�⋅δjl þ

1λG3−G2

� �−1ðK þ G2ÞlðK þ G3Þj þ

1μG3−G2

" #−1

ðK þ G2ÞjðK þ G3Þl

24

35

8<:

9=;; j; l¼ x; y

alternatively, Eqs. (5) and (6) can be rewritten in the form of a standard eigenvalue problem.The structure function FðGÞ is evaluated by

FðGÞ ¼ S−1Z

FðrÞe−iG⋅rd2r (7)

where the integration is over the unit cell and S is its area. The symbol G is the Fourier wave-vectors as defined in Section2.2. For solid piles

FðGÞ ¼Fpf f þ Fsð1−f f Þ G¼ 0ðFp−FsÞPðGÞ G≠0

((8)

where PðGÞ ¼ S−1RSpile

e−iG⋅rd2r¼ ð2f f J1ðjGj⋅RÞÞ=ðjGj⋅RÞ, R is the radius of the solid pile. Fp and Fs represent the parameterfunction FðrÞ of the pile and soil, respectively. J1 is the first kind Bessel function. The filling fraction of the solid pile isf f ¼ πR2=S.

For hollow piles

FðGÞ ¼Fpðf f−f f hÞ þ Fsð1−f f Þ G¼ 0ðFp−FsÞPðGÞ−FpPhðGÞ G≠0

((9)

where PhðGÞ ¼ 2f f hJ1ðjGj⋅rÞ=ðjGj⋅rÞ. The filling fraction of the hollow pile is f f h ¼ πr2=S, r is the inner radius of the hollow pile.

2.2. Typical cell, Brillouin zone and Bloch wave vector

For a periodic pile barrier system, the periodic theory is introduced and the piles and the surrounding soil are consideredas a fiber-reinforced composite material characterized by a typical cell. Figs. 2 and 3 show two sets of 2D periodic pilebarriers in a square configuration and in a triangular configuration, respectively. The two base vectors are R1ð0; aÞ andR2ða;0Þ. The periodic constant a is the size of a unit cell.

The complete set of reciprocal base vectors is written as G¼N1b1+N2b2, where N1 and N2 are integers.The corresponding reciprocal base vectors b1 and b2 can be obtained by [35]

b1 ¼ 2πR2 � R3

R1⋅ðR2 � R3Þ; (10a)

pile

soilR1

R2

hollow

pilesoil

R

ra

Fig. 2. The periodic pile barrier in a square configuration. (a) Top view of the periodic pile barrier using solid piles and (b) typical cell using hollow pile.

R2R1

R

hollow

pilesoil

r

R

Fig. 3. The periodic pile barrier in a triangular configuration. (a) Top view of the periodic pile barrier using solid piles and (b) typical cell using hollow pile.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–4439 4427

b2 ¼ 2πR3 � R1

R1⋅ðR2 � R3Þ; (10b)

where, R3 is the unit vector in the direction z. The Fourier wave-vector space is defined by all Fourier wave-vectors G, wherethe first Brillouin zone is the primitive cell. The Bloch wave vector K includes every point in the first Brillouin zone.For example, if the expansion of Eq. (6) is truncated by choosing N Fourier wave-vectors, it becomes a set of equations withN�N. For a given value of the Bloch wave vector K, the dispersion curves ω(K) can be obtained. Due to symmetry, it issufficient to consider the triangular area, the irreducible Brillouin zone “ҐXMҐ”, and to consider K only on the edge of theirreducible Brillouin zone (as shown in the triangle ҐXM in the inset in Fig. 4).

2.3. Numerical verification

To verify the proposed theoretical method for calculating the AZs, the dispersion curves of two-dimensional periodicstructures with Duralumin cylindrical inclusions embedded in an epoxy matrix are calculated and compared with bothVasseur's experimental and numerical results [36]. The cylinders have a diameter d¼16 mm, and the periodic constanta¼20 mm. The density and the elastic constants C11 and C44 of epoxy are 1142 kg/m3, and 7.54 GPa, and 1.48 GPa,respectively, and those for Duralumin are 2799 kg/m3, and 112.6 GPa and 26.81 GPa, respectively. The experimentaltransmission spectrum and the calculated densities of states is shown in Fig. 4(a) and the theoretical dispersion curvescalculated by the present method (N¼5) are shown in Fig. 4(b). Fig. 4(b) shows a no-direction attenuation zone between54.82 kHz and 84.24 kHz. It can be seen in Fig. 4(a) that there is a large region of null density of states between 58 kHz and90 kHz for Vasseur's numerical results. The low transmission region in the experimental spectrum and calculated spectrumoverlaps the theoretical AZ. This is a small low transmission region near 120 kHz in the calculated and experimental resultswhich are consistent with a small no-direction attenuation zone. For a large contrast material parameter, the dispersioncurves of two-dimensional periodic structures with Pb cylindrical inclusions embedded in a rubber matrix (N¼5) arereconsidered. The periodic constant a is 15.5 mm and the radius is 5 mm. The density, Lame's constants λ and μ for Pb are11,630 kg/m3, and 42.3 GPa, and 14.9 GPa, respectively. The density, Lame's constants λ and μ for rubber are 1300 kg/m3, and0.7 MPa, and 0.06MGPa, respectively. The results are plotted in Fig. 5. It is found that the measured and calculated frequencyresponses agree well with the corresponding direction attenuation zones [37]. In this paper, the hollow piles are utilized toform pile barriers. The experiments of periodic hollow piles or hollow cylinders are rare. Here, we calculate the dispersioncurves of the hollow cylinders periodically arranged in air and compared with Vasseur's experiment [38]. The equations ofacoustic attenuation zone are similar with those of the z mode of present solid/solid model. The only difference of thefundamental equation between acoustics and z mode of elastic wave is that μ is replaced by C11 ¼ ρc2l . cl Is the speedof longitudinal wave. The periodic constant a is 30 mm. The inner and outer radii are 13 mm and 14 mm, respectively.The density and speed of longitudinal wave cl for Copper are 8950 kg/m3, 4330 m/s2, respectively. The density and speed oflongitudinal wave cl for air are 1.3 kg/m3 and 340 m/s2, respectively. The experimental results are shown in Fig. 6(a) and thetheoretical results by present analysis (N¼5) are shown in Fig. 6(b). One can note that the measured low transmissioncoefficient zones agree well with the corresponding direction attenuation zones.

The above mentioned experiments are based on the x−y mode and acoustic P-wave mode, because these experimentsare relatively simple. Although the theoretical analysis is common, the experiment of pure SV-wave mode is too difficult tobe implemented. For example, Although the AZs of a 2D periodic structure [34] consisting of Aluminum alloy cylinders in aNickel alloy matrix are investigated. The elastic constants used in the calculation are C44¼75.4 GPa for Ni and C44¼27.9 GPafor Al. The mass densities are ρ¼8936 kg/m3 for Ni and ρ¼2697 kg/m3 for Al. The filling fraction ff is 0.75, and a factor ofN¼ 10 was used as shown in Ref. [34]. These results are plotted in Fig. 6, which agree very well with Cao's results [34]. Fig. 7also shows that it is possible to obtain a no-directional attenuation zone. Thus, SV waves with frequencies that have nocorresponding wave vector K cannot propagate in any direction in the plane.

Furthermore, Biwa et al. [39] showed that the wave ceases to propagate in the composite in certain frequency bandsfor the periodic fiber-reinforced composite materials. The wave fields in such frequency bands are shown to possess a

0

50

100

150

200

0 10 20 30 40

Transitted Power (TP, arbitrary units)

Vasseur's Calculation (DOS) Vasseur's experiment (TP)Fr

eque

ncy

(kH

z)

Density of States (DOS, arbitrary units)M X M

0.00 0.05 0.10

AZ

AZ

X

Fig. 4. Spectrum and dispersion curves of steel cylinders in epoxy matrix.

-30 -20 -10 0 100

200

400

600

800

1000

Wen's Calculation Wen's experiment

Freq

uenc

y (H

z)

FRF (dB)

(a)

AZ

(M) X M

(b)

Γ

Fig. 5. Spectrum and dispersion curves of Pb cylinders in rubber matrix.

0.0 0.3 0.6 0.90.0

5.0k

10.0k

15.0k Vasseur' experiment

Freq

uenc

y (H

z)

Transmission

IPWE

AZ

AZ

AZ

Γ X M Γ

Fig. 6. Spectrum and dispersion curves of hollow cylinders in air.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–44394428

0.0

0.5

1.0

1.5

2.0 Present analysis Cao et al. (2004)

a/2

Ct

X MM

AZ

Fig. 7. Dispersion curves of Al alloy cylinders in a Ni alloy matrix.

0.0 0.5 1.00.0

40.0M

80.0M

120.0M

160.0M Biwa et al. (2004)

Freq

uenc

y (H

z)

Normarized energe flowX

AZ

AZ

Fig. 8. Normalized energy transmission and dispersion curves.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–4439 4429

standing-wave nature and involve spatially decaying amplitudes and vanishing energy flow in the propagation direction.The energy transmission for frequency divisions is calculated, but without giving out the dispersion curves ω(K). However,they expected that the whole Brillouin zone presented by Kushwaha et al. [33] would be more appropriate to explain thisphenomenon. For comparison, the model considered by Biwa is studied once more. The elastic constants used in thiscalculation are C44¼45 GPa for Ti-alloy matrix, and C44¼177 GPa for SiC fibers. The densities are ρ¼5400 kg/m3 for Ti-alloymatrix and ρ¼3200 kg/m3 for SiC fibers. The geometrical parameter R¼0.071 mm and a¼0.25 mm were used in Biwa'sexample. The energy transmission obtained by Biwa is shown in Fig. 8(a). The direction attenuation zones along ҐX obtainedby the present method (N¼5) are plotted in Fig. 8(b). The propagation characteristics of the incident wave (SV-wave) cantherefore be explained by the theory of AZs in the anti-plane strain model. In other words, AZs can be utilized to analyze thepropagation characteristics of plane waves, and design an effective wave barrier. These comparisons with experiment andsimulation validate the accuracy of the present theoretical method and the existence of attenuation zones in periodicstructures.

3. Parameter investigation

As mentioned before, waves with frequencies in the range of the AZs cannot propagate in the medium. So, we can designperiodic structures producing appropriate AZs to block the propagation of waves. The AZs are different when differentmaterials and configurations of piles are proposed. Therefore, it is necessary to analyze the influences of factors on the AZs.In this section, a comprehensive investigation of factors influencing the first no-directional attenuation zone is studied.The lower bound frequency (LBF), upper bound frequency (UBF) and the width of attenuation zone (WAZ) are presented.The factors studied herein include the material properties such as the modulus and density of soil, and the pile geometricconfigurations such as pile radius, periodic constant, pile arrangement and the effect of hollow piles. The periodic constantand pile radius are a¼2 m and R¼0.65 m, respectively. The material properties of the soil and piles are shown in Table 2.In the following parametric discussion, when one of the parameters changes, all the other parameters remain unchanged.

Table 2Material properties (density ρ, elastic modulus E, and Poisson's ratio υ).

Material ρ (kg/m3) E (GPa) υ

Soil 1800 2�10−2 0.35Concrete 2500 30 0.2

0 5 10 15 20 25 30 350

5

10

15

20

25

30

35

Freq

uenc

y (H

z)

E (MPa)

LBF UBF WAZ

Fig. 9. The first no-direction AZ versus the elastic modulus of soil.

1700 1800 1900 2000 2100 22000

5

10

15

20

25

30

Freq

uenc

y (H

z)

Kg/m3

LBF UBF WAZ

Fig. 10. The first no-direction AZ versus the soil density.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–44394430

In the following discussion, we focus our attention on the SV wave only because the lower bound frequency of AZ of SV waveis relatively low, and the vertical displacement response is taken as an example. The analysis of the P–SH wave is similarwith the SV wave. The characteristic of hollow piles is firstly introduced into periodic pile barriers.

3.1. Elastic modulus and density of soil

The modulus and density of the soil are two important parameters in design of pile barriers. They vary significantly dueto different site conditions and different soil types, especially the soil modulus (from 1 MPa to 1000 MPa) [40]. Forsimplicity, the elastic modulus of soil is chosen in the range of 1–30 MPa, which can represent a type of soft soils, such asloose uniform sand, stiff clay, and soft clay according to the laboratory-scale [27]. The density of soil is in the range of 1700–2200 kg/m3 in this paper. Fig. 9 shows that the LBF, UBF and WAZ increase with the increase of soil elastic modulus. Thissuggests that the softer soil (or stiffer piles) tends to achieve low-frequency vibration isolation. However, Fig. 10 shows that

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–4439 4431

the LBF, UBF and WAZ decrease almost linearly with the increase of soil density. It can be seen that the influence of soildensity is much smaller.

3.2. Pile radius

For a square unit cell, keeping a¼2 m as a constant, the influence of pile radius (filling fraction) on the first no-directionAZ is shown in Fig. 11. As expected, the WAZ increases rapidly with the increase of pile radii. A wider WAZ can be obtainedby simply increasing the pile radii. As shown in Fig. 12, for a triangular typical cell, the WAZ is much larger than that ofa square typical cell. The WAZ increases rapidly with the increase of pile radii. For example, when the pile radius is 0.75 m,a complete band of frequency gap ranging from 23.167 Hz to 48.795 Hz is obtained. This corresponds to the middledominant frequencies of vibrations for rail traffic. Thus, commonly available pile diameters have potential applications inground vibration reduction.

3.3. Screening effectiveness of hollow piles

Figs. 13 and 14 show the first no-direction AZ in a square and a triangular configuration versus the inner radii of thehollow piles, respectively. The inner radii of the hollow piles have virtually no influence on the UBF. However, the LBFincreases with the increase of inner radii of the hollow piles. Therefore, WAZ decreases with the increasing inner radii of thehollow piles. Additionally, when the inner radii of the hollow piles are relatively small, less than 0.5R, the increase of LBF is

0.55 0.60 0.65 0.70 0.750

5

10

15

20

25

30

35

Freq

uenc

y (H

z)

R (m)

LBF UBF WAZ

Fig. 11. The first no-direction AZ in a square configuration versus pile radius.

0.55 0.60 0.65 0.70 0.750

10

20

30

40

50

LBF UBF WAZ

Freq

uenc

y (H

z)

R (m)

Fig. 12. The first no-direction AZ in a triangular configuration versus pile radius.

0.0 0.1 0.2 0.3 0.4 0.55

10

15

20

25

30

35

40

LBF UBF WAZ

Freq

uenc

y (H

z)

r (m)

Fig. 14. The first no-direction AZ in a triangular configuration versus the inner radii of the hollow piles.

0.0 0.1 0.2 0.3 0.4 0.50

5

10

15

20

25

30

LBF UBF WAZ

Freq

uenc

y (H

z)

r (m)

Fig. 13. The first no-direction AZ in a square configuration versus the inner radii of the hollow piles.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–44394432

not significant and the WAZ can be kept as a constant at a high level. The selection of appropriate hollow piles, therefore, cansave the material and reduce cost.

4. Numerical simulation

Based on above theoretical analysis, we can use the AZs to block the wave propagation in infinite, umdamped, periodicpile barriers. Theoretically speaking, for an infinite and perfectly periodic structure without damping, perfect filteringproperties exist in the complete AZs. However, for a finite number of piles, the frequency response function (FRF) can beused to describe the properties of finite periodic structures subjected to periodic loading effectively [41]. The FRF curves areobtained by solving the full set of linear Eq. (13) for each frequency f¼ω/2π. The governing equations for small-amplitudedisplacements are given as

M €uþKu¼ PðωÞeiωt ; (11)

where M and K are the mass- and stiffness-matrix, respectively, and P is a vector of loading of circular frequency ω.Furthermore, the displacement vector is assumed as

uðtÞ ¼Aeiωt (12)

substituting Eq. (12) into Eq. (11), the linear set of equations to be solved for the amplitude A is obtained as follows:

ð−ω2M þ KÞA¼ PðωÞ (13)

adding Viscous-spring boundary element

adding Viscous-spring boundary element

addingsymmetric boundary

x

y

0

pile

vibration source

Fig. 15. Schematic of a row of piles.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–4439 4433

In this section, the FRF of a field with or without pile barriers is studied by using the finite element method. The effects ofpile radii, number of rows and the inner radii of the hollow piles on the FRF are taken into account. The time dependent loadresponse of a periodic pile barrier in a triangular configuration is then presented. The decay characteristic of the periodicpile barrier indicates that the periodic theory is appropriate for the design of a pile barrier system.

4.1. Model verification

The FRF of a finite pile–soil system is obtained by using ANSYS and is compared with the AZs obtained from thetheoretical solution. The out-of-plane displacement is the unknown degree of freedom in an anti-plane model, in which theSV wave is considered only. A numerical model has been used by Chen and Wang [42] to obtain the solution of an isotropicsector under anti-plane shear loadings. In order to indicate the method, a numerical model is considered and is shown inFig. 15. Due to the symmetric boundary condition, the size of the semi-model is 93(length)×48(width) ×1(thickness, in z-direction), which consists of a row of 4 piles. The material parameters of the piles and soil are the same as used by Aviles [4].To account for the non-reflecting boundary conditions, viscous-spring boundary elements [43] are used at the top, bottomand the right side of the boundaries. The soil and piles are modeled by 8-node brick elements. Symmetric boundaryconditions are applied to the left side of the boundaries. The computation time can be decreased considerably due to thesimplified finite element model. It is noted that the simplified model is accurate and feasible in the investigation of theincident SV wave scattered by pile barriers, which is validated in the following.

To simulate the incident plane wave, a displacement load parallel to the z-direction with a section of 0.375(width)�1(thickness, in z-direction) is considered. The vibration source can be simplified as a point source relative to the full field.The anti-plane wave propagates in the x−y plane in the type of a cylindrical wave. However, if the source distance (SD) islarge enough, the wave achieved at the front of the barrier approximates a plane wave and propagates in the x−y planealong the ‘ҐX’ direction.

The response |uz/uz0| and comparisons near the barrier are shown in Fig. 16. The normalized frequency is defined asη¼Rω/πcs, in which cs is the transverse speed of the wave in the soil. It is found that it is accurate enough to use SD¼5L toapproximately obtain an incident plane wave. Thus, SD¼5L is adopted for the subsequent numerical analysis. SD isthe distance from the vibration source to the center of the first row of piles, and L is the distance from the left boundary tothe outer boundary of the fourth pile, as shown in Figs. 15 and 17. The results, as shown in Fig. 16, validate the accuracy ofthe finite element model, and this model is used to calculate the FRF of different periodic pile barriers in the following.

-10 -5 0 5 100.0

0.4

0.8

1.2

1.6

2.0

u / u

z0

y / R

Aviles et al. (1988) SD=5L SD=3L SD=L

=0.4 x=10.5

-10 -5 0 5 100.0

0.4

0.8

1.2

1.6

2.0

y / R

u / u

z0

Aviles et al. (1988) SD=5L SD=3L SD=L

= 0.4 x=1.5

Fig. 16. Normalized displacement amplitude for a/R¼3, ρp/ρs¼1.35, μp/μs¼1000 and normalized frequency η¼0.4. Incidence of plane SV wave. (a) x¼10.5and (b) x¼1.5.

referencepoint B

L

a

a

1.5l

1.5l

3l

l

referencepoint A

pile

Fig. 17. Schematic of three rows of piles in a triangular configuration.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–44394434

4.2. Frequency response of periodic pile barriers

Aviles [5] studied the response of a single row of piles. In the present paper, the response of periodic pile barriersperiodically arranged as shown in Fig. 17 is studied. To evaluate the screening effectiveness, define an amplitude reductionratio Ary as

Ary ¼Amplitude with the pile barrier

Amplitude without the pile barrier(14)

furthermore, the averaged amplitude reduction radio Ary proposed by Woods [44] can be obtained:

Ary ¼1A

ZAArydA (15)

in which A is the rectangular area right behind the piles, the shaded part in Fig. 17.The FRF is defined as FRF ½dB� ¼ 20log10Ary, and the averaged FRF is defined as FRF½dB� ¼ 20log10Ary .Fig. 18(a) gives the FRF and the averaged FRF for the system as shown in Fig. 17. Fig. 18(b) shows the analytical results

about AZs. It is found that the frequency attenuation region agrees very well with the AZs. This means that the magnitude ofvibration with frequencies in the range of the AZs can be reduced significantly, while the magnitude of vibration withfrequencies outside the range of the BFGs cannot be attenuated. In the following analysis, the averaged FRF is used.

Fig. 19 shows the effect of the number of rows of piles on vibration reduction. An important parameter in theperformance of AZs is the number of rows of piles. If a few rows or typical cells are used only, the potential attenuation ofvibrations in the AZs is lower than that when more rows of piles are used. Frequency responses for a periodic barrier withdifferent rows of piles are shown in Fig. 19. For comparison, the directional attenuation zones corresponding to the directionof wave propagation, which are calculated for the infinite periodic pile barrier, are also shown in Fig. 19 with shadow areas.It can be obtained that reduction of vibration in the AZs becomes more pronounced with the increase of the number of rows.

-40 -20 00

10

20

30

40

50

60

70

Averaged FRF Reference point A Reference point B

Freq

uenc

y (H

z)

FRF (dB)

X

(a)

M

(b)

1st AZ

2nd AZ

xM

Γ Γ

Fig. 18. Frequency responses for three rows of piles.

0 10 20 30 40 50 60 70-30

-25

-20

-15

-10

-5

0

5

4 rows 3 rows 2 rows

FRF

(dB

)

Frequency (Hz)

1st AZ 2nd AZ

Fig. 19. Frequency responses for different rows.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–4439 4435

When the number of rows is larger than three, vibrations with frequencies in the AZs are reduced in excess of 20 dB.Moreover, if the number of rows is two or three, the response is reduced substantially inside the directional attenuationzones, which means two or three rows of piles arranged in a periodic configuration can perform the properties of AZs.In addition, it also implies that the application of periodic pile barriers with only two or three rows of piles is feasiblein practice engineering. In the following discussions, three rows of piles arranged as shown in Fig. 17 are considered.

Fig. 20 shows the effect of radii of solid piles on vibration reduction. The decay region disappears rapidly with thedecrease of pile radii. When the radii of solid piles are 0.45 m, two theoretical AZs in the “ГX” direction ranged from18.21 Hz to 25.83 Hz and from 38.23 Hz to 40.44 Hz are obtained. However, when the pile radii are 0.25 m, neitherdirectional attenuation zone nor no-directional attenuation zone is found. So the FRF is almost equal to zero. This meansthat this pile barrier has almost no contribution to block plane waves in the investigated frequency range.

Fig. 21 shows the effect of hollow piles on vibration reduction. When the inner radii of the hollow piles are 0.4R, hollowpiles have a slight influence on the averaged FRF. However, if the inner radii of the hollow piles are too large, such asr¼0.8 R, the first attenuation range of the averaged FRF (corresponding to the analytical WAZ from 24.83 Hz to 37.61 Hz in“ҐX” direction) decreases rapidly and the second one disappears because there are no AZs in this range. The FEM analysisshows that hollow piles can save material and have only a slight influence on the AZs.

4.3. Time domain analysis of periodic pile barriers

According to the theoretical and experimental works of British Rail Technology center, the vertical wheel rail forcesappear mainly in three frequency ranges: low-frequency range (0.5–10 Hz), mid-frequency range (30–60 Hz) and high-frequency range (100–400 Hz) [45]. Since high-frequency vibration induced by train traffic decays quickly, low-frequency

0 10 20 30 40 50 60 70-25

-20

-15

-10

-5

0

5

10

FRF

(dB

)

Frequency (Hz)

R=0.25m R=0.45m R=0.65m

Fig. 20. Frequency responses for different radii of solid piles.

0 10 20 30 40 50 60 70-25

-20

-15

-10

-5

0

5

10

FRF

(dB

)

Frequency (Hz)

r=0.8Rr=0.4Rr=0

Fig. 21. Frequency responses for different inner radii of the hollow piles.

0 1 2 3 4 5-1.0

-0.5

0.0

0.5

1.0

u z(m

m)

t (s)0 14 28 42 56 70

0.000

0.075

0.150

0.225

0.300

Frequency (Hz)

Am

plitu

de(m

m/H

z)

Fig. 22. The vertical displacement of vibration source. (a) Time history and (b) frequency contents.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–44394436

and mid-frequency vibrations have received significant attention in recent years. In this part, the problem of vibrationisolation based on AZs addressed in the present paper is solved by FEM in time domains. The aim of this study is to furtherillustrate that only vibrations with frequencies inside AZs can be reduced significantly.

0 1 2 3 4 5-0.10

-0.05

0.00

0.05

0.10u z(m

m)

t (s)

without barrier with barrier

reference point A

0 10 20 30 40 50 60 700.000

0.001

0.002

0.003

0.004

0.005

Am

plitu

de(m

m/H

z)

Frequency (Hz)

without barrier with barrier

reference point A

Fig. 23. The vertical displacement at point A. (a) Time history and (b) frequency contents.

0 1 2 3 4 5-0.10

-0.05

0.00

0.05

0.10

u z(m

m)

t (s)

without barrier with barrier

reference point B

0 10 20 30 40 50 60 700.000

0.001

0.002

0.003

0.004

0.005

Am

plitu

de(m

m/H

z)

Frequency (Hz)

without barrier with barrier

reference point B

Fig. 24. The vertical displacement at point B. (a) Time history and (b) frequency contents.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–4439 4437

The following displacement excitation function is used to simulate the vibration source:

uz ¼ 16

∑6

n ¼ 1sin½2π⋅f ðnÞ⋅t� ðmmÞ (16)

where, f(n)¼[5 Hz, 15 Hz, 25 Hz, 35 Hz, 50 Hz, 57 Hz]. In which, 5 Hz, 15 Hz, 57 Hz are outside the range of AZs, and allothers are in the range of AZs. The vertical displacement–time history and the frequency contents of vibration source areshown in Fig. 22.

Figs. 23 and 24 show the displacement–time response and frequency contents at reference point A and point B,respectively. It can be seen that the displacement amplitudes at point A and point B are reduced by almost 50 percent whenthe periodic pile barrier exists. Moreover, Figs. 23(b) and 24(b) imply that the magnitudes of the vibrations with frequencycontents around 25 Hz, 35 Hz, and 50 Hz decrease significantly; however, almost no reduction is found around 5 Hz, 15 Hzand 57 Hz. These also imply that the incident wave energy is significantly reduced if the vibration frequencies are in therange of the AZs. If the incident wave frequencies are out of the range of the BFGs, the vibration can propagate across thepile barrier without reduction.

5. Conclusions

The purpose of this work is to introduce the concept called attenuation zones from the field of solid-state physics to thefield of civil engineering. Based on the periodic theory, the reduction characteristics of periodic pile barriers subjected toplane waves are studied. The effects of material and geometrical parameters on the AZs are investigated and the FRF behindthe pile barrier are analyzed. Based on the present investigation, the following conclusions are drawn:

1.

A periodic pile barrier with softer and high-density soil can produce lower LBF. Furthermore, larger filling fraction andperiodic pile barriers in a triangular configuration are useful to obtain a wider WAZ.

J.K. Huang, Z.F. Shi / Journal of Sound and Vibration 332 (2013) 4423–44394438

2.

The effects of AZs become obvious with the increase of the number of rows. When the number of rows is more thanthree, vibrations with frequencies in the range of the AZs are reduced in excess of 20 dB.

3.

When the inner radius of the hollow piles is less than 0.5R, the influence of the inner radius is small. 4. Pile diameters usually used in engineering can be used to obtain AZs corresponding to the mid-frequency range of rail

traffic. Vibrations with frequencies in the range of the AZs are attenuated significantly. However, vibration withfrequencies outside of the range of the AZs cannot be reduced.

Acknowledgment

This work is supported by the National Natural Science Foundation of China (Grant no. 51178036), the 111 Project(B13002), and the Research Fund of Beijing Jiaotong University (2011YJS042).

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