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Stiff wave barriers for the mitigation of railway induced vibrations

P. Coulier1, A. Dijckmans1, J. Jiang2, D.J. Thompson2, G. Degrande1 and G. Lombaert1

1 KU Leuven, Department of Civil Engineering,

Kasteelpark Arenberg 40, 3001 Leuven, Belgium

Tel: +32 16 32 16 75, Fax: +32 16 32 19 88, E-mail: pieter.coulier@bwk.kuleuven.be 2 Institute of Sound and Vibration Research, University of Southampton,

Southampton SO17 1BJ, United Kingdom

Summary This paper studies the efficiency of stiff wave barriers for the mitigation of railway induced vibrations. Coupled finite element–boundary element models developed at KU Leuven and ISVR are employed; these models have been cross–validated within the EU FP7 project RIVAS [1]. A first mitigation measure consists of a block of stiffened soil included in a halfspace that acts as a wave impeding barrier. The existence of a critical frequency from which this mitigation measure starts to be effective, as well as a critical angle delimiting the area where the vibration levels are reduced, is demonstrated. Next, a sheet piling wall is considered, accounting for the orthotropic behaviour of this wall. Calculations show that the reduction of vibration levels is entirely due to the relatively high axial stiffness and vertical bending stiffness, while the longitudinal bending stiffness is too low to affect the transmission of vibrations. Field tests are being carried out in Spain and Sweden to confirm the conclusions of these numerical computations.

1 Introduction Railway induced vibrations are an important source of annoyance in the built environment, that can cause the malfunction of sensitive equipment and nuisance to people. Various measures can be taken to mitigate vibrations, either at the source (railway track), in the propagation path between source and receiver, or at the receiver (surrounding buildings) [2].

An open trench in the soil is a typical example of a mitigation measure in the transmission path. Such trenches aim to reflect the impinging waves and are very effective for trench depths of greater than about one Rayleigh wavelength [2]. The construction of open trenches is, however, limited to shallow depths for stability

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reasons; the use of either soft or stiff in–fill materials allows for an increase of depth. If a soft in–fill material is used, the behaviour of a filled trench resembles that of an open trench. The use of a stiff in–fill material, however, fundamentally alters the physical mechanism by which vibration levels are reduced.

This paper focuses on the application of stiff wave barriers to mitigate railway induced vibrations. Two such mitigation measures are presented: subgrade stiffening next to the track and a sheet pile wall. The study is part of the EU FP7 project RIVAS (Railway Induced Vibration Abatement Solutions) [1].

2 Methodology For the prediction of railway induced ground vibration, project partners KU Leuven and ISVR make use of different two–and–a–half–dimensional (2.5D) coupled finite element–boundary element (FE–BE) models [4,5]. In these models, the geometry of the track–soil system is assumed to be invariant in the longitudinal direction. First, independent analysis was conducted by both project partners for a benchmark case of a 12 m deep barrier in a homogeneous halfspace, characterized by a shear wave velocity Cs = 250 m/s, a dilatational wave velocity Cp = 1470 m/s, a density ρ = 1945 kg/m³, and material damping ratios βs = βp = 0.025 in both deviatoric and volumetric deformation. The wall barrier has a thickness t = 0.396 m, a modulus of elasticity E = 6.99 x 109 Pa, a Poisson's ratio ν = 0.30, and a density ρ = 286.6 kg/m³. The track is disregarded. At KU Leuven, the wall is modeled with 2–noded shell elements with a mesh size of 0.309 m. The soil at both sides of the sheet pile wall is modeled with a conforming BE mesh, assuring 8 elements per shear wave length up to 100 Hz. In the model of ISVR, the wall is modeled with 3–noded elastic solid elements. The node to node distance was set to 0.1 m in the regions within 1 m from the top and bottom of the wall and 0.25 m everywhere else.

(a) (b)

Fig. 1. (a) Benchmark study for a stiff wave barrier in a homogeneous halfspace. (b) Vertical transfer mobility at a distance of 8 m (blue), 16 m (red), 32 m (green) and 64 m (magenta) from a vertical point source. KU Leuven results (solid lines) and ISVR results (circles).

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Transfer mobilities for a vertical point source are shown in Fig. 1. The agreement between both models is excellent. The largest discrepancies are found at the highest frequencies. In the next sections, results of a parametric study for stiff wave barriers are presented.

3 Subgrade stiffening The first vibration mitigation measure consists of subgrade stiffening next to the track, where a block of stiffened soil is implemented in the soil (e.g. by means of jet grouting). Throughout this section, a homogeneous halfspace characterized by a shear wave velocity s 200 m/sC = , a dilatational wave velocity p 400 m/sC = , a density 32000 kg/mρ = , and material damping ratios s p 0.025β β= = in both deviatoric and volumetric deformation is considered. The vibration reduction efficiency of a block of stiffened soil with a width 2 mw = and a depth 2 mh = will be investigated. This stiff wave barrier has a shear wave velocity s 550 m/sC = and a dilatational wave velocity p 950 m/sC = ; the same density and material damping ratios as in the halfspace are used. In order to facilitate physical interpretation, an incident wavefield is generated by the application of a unit vertical harmonic point load at the surface of the halfspace; the presence of the track is thus neglected.

3.1 The free field response in the frequency–spatial domain Fig. 2a and 2b show the real part of the vertical displacement $ ( ),zu ωx at a frequency of 45 Hz in the reference case (without barrier) and in the case a block of stiffened soil is included, respectively.

(a) (b)

(c)

Fig. 2. Real part of the vertical displacement $ ( ),zu ωx due to a unit harmonic vertical

excitation at the origin of the coordinate system at 45 Hz (a) on a homogeneous halfspace and (b) when a block of stiffened soil is included. The vertical insertion loss is shown in (c). A cylindrical wavefront with a Rayleigh wavelength R R2 / 4.1mCλ π ω= = can clearly be observed in the reference case. The wavefield is considerably perturbed, however, in case the stiff wave barrier is included in the halfspace. The vibration

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reduction efficiency of the barrier can be quantified through the vertical insertion loss ( )IL x,z ω :

( )( )( )

ref

10

ˆ ,IL , 20log [dB]

ˆ ,z

zz

u

u

ωω

ω=

xx

x (3.1)

where ( )refˆ ,zu ωx and ( )ˆ ,zu ωx represent the vertical free field displacement in the reference case (without barrier) and in case the barrier is included, respectively. Positive values of the insertion loss indicate a reduction of the vertical free field vibrations. The insertion loss ( )IL x,z ω is shown in Fig. 2c. A delimited area where significant reduction of vibration levels is achieved can be observed. The insertion loss reaches values of 10 dB and more in this region. The reduction is not only obtained at the surface of the halfspace, but also at depth, although some localized areas can be identified with increased vibration levels with respect to the reference case. Lines of constructive and destructive interference between direct and reflected Rayleigh waves can furthermore be observed at the opposite side of the track (i.e. where no soil stiffening is applied). The physical interpretation for these observations will be formulated in the subsequent subsection.

3.2 The free field response in the frequency–wavenumber domain In order to explain the behaviour of a stiff barrier, the cylindrical wavefield should be decomposed into a superposition of plane waves. Each of the plane waves is characterized by a dimensionless longitudinal wavenumber yk , defined as s s/ /y y yk k C C Cω= = , with sC the shear wave velocity of the halfspace and yC the phase velocity of the waves. The efficiency of the stiff wave barrier can hence also be quantified in the frequency–wavenumber domain through the vertical insertion loss ± ( )IL , , ,z yx k z ω , where a tilde above a variable denotes its representation in the

frequency–wavenumber domain. Fig. 3 shows the insertion loss ± ( )IL 8m, , 0m,z yx k z ω= = in a range R0 yk k≤ ≤ , where the dispersion

curve R s R/yk k C C= = corresponds to a Rayleigh wave propagating in the

y–direction. A clearly delimited area in the ( ), ykω –domain corresponding to a

significant insertion loss can be distinguished in this figure.

Fig. 3 can be interpreted by considering the interaction of the Rayleigh wave in the soil and bending waves in the block of stiffened soil [6]. Superimposed on Fig. 3 is the dispersion curve byk k= of a free bending wave in an infinitely long Timoshenko beam with the same properties as the block of stiffened soil. The region where a substantial insertion loss is obtained in the ( ), ykω –domain is

clearly bounded by the Rayleigh wave dispersion curve Ryk k= and the free bending wave dispersion curve byk k= . At low frequencies, the wavenumber bk is

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larger than the wavenumber Rk , and the Rayleigh wave propagates unhindered through the block of stiffened soil. From a critical frequency on, the wavefield contains propagating plane waves with a wavenumber yk larger than bk (i.e. with a wavelength yλ smaller than bλ ). The transmission of these plane waves is impeded by the block of stiffened soil, as the admittance of a beam of infinite length is then dominated by its bending stiffness and decreases proportionally to

4yk−

at a given radial frequencyω . This explains why the zone of significant insertion loss in Fig. 3 is delimited by the free bending wave dispersion curve.

Fig. 3. Vertical insertion loss ± ( )IL 8m, , 0m,z yx k z ω= = in case a block of stiffened soil is

included in a homogeneous halfspace. Superimposed are the dispersion curve of a Rayleigh wave in the y–direction (solid black line) and the free bending wave dispersion curve in an infinitely long beam (dashed black line).

The critical radial frequency cω from which the block of stiffened soil can act as a wave impeding barrier is determined by the intersection of the Rayleigh wave and the free bending wave dispersion curves [6]:

( )( )2

c R 2 2R R

A ECEI E C Cρ µκ

ωρ µκ ρ

=− −

(3.2)

which equals 2 12 Hzπ × in the actual case. E is the Young's modulus, µ the shear modulus, ρ the density, A the cross-sectional area and I the moment of inertia of the block of stiffened soil. As a square cross section is considered, the area moment of inertia is the same for bending around the x- and the z-axis, respectively; this is in general not the case. Numerical simulations indicate that bending around the x-axis is dominant, provided that the block of stiffened soil has a minimum width in order to ensure that it behaves as a beam [6]. It is clear from the discussion above that the mitigation measure can only be effective for frequencies above cω . Equation (3.2) reveals that the critical frequency strongly depends on the stiffness contrast between the soil and the block of stiffened soil,

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indicating that this mitigation measure will be more effective at sites with a soft soil. Equation (3.2) is of great practical importance, as it provides a simple design guideline to assess the expected vibration reduction efficiency of soil stiffening next to the track, without the need of an extensive FE–BE calculation.

The propagating plane waves are characterized by a wave propagation direction ( )1

Rsin /yk kθ −= . This angle is situated between 0θ = (plane wave

propagation in the x–direction) and / 2θ π= (plane wave propagation in the y–direction). As a result, a reduction of vibration levels will only be obtained in an area delimited by a critical angle ( ) ( )1

b Rc sin /k kθ ω −= , defined as:

( )( ) ( )22

2

c R

4

sin2

E AE E

IC

E

µκµκ µκ

ρ ωθ ω ρ

µκ

+ + − +

= (3.3)

The physical mechanism outlined above explains the pattern of insertion loss that is observed in Fig. 2c. As the considered frequency of 45 Hz is above the critical frequency of 12 Hz, a significant reduction of vibration levels is achieved, but only in a limited area due to the existence of a critical angle.

Within the frame of the EU FP7 project RIVAS [1], a field test is being carried out in Spain to confirm the findings of the numerical simulations. A jet grouting wall with a width of 1 m, a depth of 7.5 m and a length of 60 m will be constructed at a site with relatively soft soil layers.

4 Sheet pile wall At Furet, Sweden, a sheet pile wall has been installed next to the track to reduce train induced vibrations in several buildings close to the track [7]. The sheet pile wall consists of VL 603-K profiles (figure 4). The depth of the sheet piles is 12 m with every fourth pile extended to 18 m. The soil profile at Furet consists of very soft clayey silt up to a depth of approximately 12 m. Here, the effectiveness of a 12 m deep sheet pile wall installed in a homogeneous halfspace is investigated, to facilitate the physical interpretation. The soil characteristics as given in Sec. 2 are used in this example. An equivalent orthotropic plate model of the sheet pile wall was adopted in the 2.5D calculations to account for the fact that the bending stiffness in the vertical direction (along the profiles) is much larger than the bending stiffness in the longitudinal direction (perpendicular to the profiles). In the frequency range of interest (0-100 Hz), the bending wavelength in the sheet pile wall is much larger than the repetition distance of the sheet pile wall. Therefore, the profiling of the plate can be disregarded. The thickness, density, moduli of elasticity, and Poisson's ratios of the equivalent orthotropic plate are chosen such that the plate has approximately the same mass, bending stiffness in two directions as well as vertical axial stiffness as the VL 603-K profile.

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Fig. 4. Cross section of the sheet pile wall (VL 603-K profile)

Fig. 5a shows the vertical insertion loss at 25 Hz. Fig. 5b shows the vertical insertion loss at a distance of 32 m in the frequency-wavenumber domain. Very little vibration reduction is seen below 20 Hz. The frequency above which a reduction in vibration levels can be expected, is in the first place determined by the depth of the wall with respect to the Rayleigh wavelength [3]. At low frequencies, the penetration depth of the Rayleigh waves is very large and a significant amount of the vibrational energy passes underneath the sheet pile wall. Increasing the depth improves the insertion loss at low frequencies [7].

(a) (b) Fig. 5. (a) Vertical insertion loss at 25 Hz and (b) vertical insertion loss ± ( )IL 32m, , 0m,z yx k z ω= = in case a 12 m deep sheet pile wall is included in a homogeneous

halfspace. Superimposed is the dispersion curve of a Rayleigh wave in the y–direction.

At higher frequencies, the insertion loss is determined by the reflection and transmission properties of the sheet pile wall. As explained in Sec. 3, the effectiveness of a stiff wave barrier is determined by the contrast in stiffness between the barrier and the soil. Contrary to the results for subgrade stiffening, the reduction of vibration is entirely due to the relatively high axial stiffness and high bending stiffness in the z-direction. The wave impeding effect caused by the longitudinal bending stiffness, which leads to a delimited area with high insertion loss, is not seen for the sheet pile wall. Longitudinal bending waves for bending around the x-axis are not excited due to the limited width of the wall. The longitudinal bending stiffness for bending around the z-axis is too low to affect the transmission of vibration in the frequency of interest. At 25 Hz, the sheet pile wall reduces the vibration levels at the surface with insertion loss values around 3 dB.

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The insertion loss is fairly homogeneous at the surface behind the sheet pile wall. The reduction in vibration levels, however, is restricted to the top five meters of soil. In the present case, the efficiency of the sheet pile wall is relatively low. A larger reduction in vibration levels is predicted for sites with a soft soil, like the test site in Furet [7].

5 Conclusions Numerical simulations show that a stiff buried wall barrier like a block of stiffened soil or a sheet pile wall can act as an effective wave impeding barrier. The mitigation measure will be more effective at sites with a relatively soft soil. It is beneficial to increase the depth of the buried wall barrier to obtain a reduction at low frequencies. Field tests are being carried out in Spain and Sweden to confirm these results.

Acknowledgments The results presented in this paper have been obtained within the frame of the EU FP7 project RIVAS (Railway Induced Vibration Abatement Solutions) under grant agreement No. 265754 [1]. The first author is a doctoral fellow of the Research Foundation Flanders (FWO). The financial support is gratefully acknowledged.

References [1] http://www.rivas-project.eu [2] L. Andersen, S.R.K. Nielsen. Reduction of ground vibration by means of

barriers or soil improvement along a railway track, Soil Dynamics and Earthquake Engineering, 25, 701–716 (2005).

[3] R.D. Woods. Screening of surface waves in soils, Journal of the Soil Mechanics and Foundation Division, Proceedings of the ASCE, 94(SM4), 951–979 (1968).

[4] S. François, M. Schevenels, G. Lombaert, P. Galvín, G. Degrande. A 2.5D coupled FE–BE methodology for the dynamic interaction between longitudinally invariant structures and a layered halfspace, Computer Methods in Applied Mechanics and Engineering, 199(23–24), 1536–1548 (2010).

[5] C.-M. Nilsson and C.J.C. Jones, Theory manual for WANDS 2.1, ISVR Technical Memorandum No. 975. University of Southampton (2007).

[6] P. Coulier, S. François, G. Degrande, G. Lombaert. Subgrade stiffening next to the track as a wave impeding barrier for railway induced vibrations, Soil Dynamics and Earthquake Engineering, 48, 119-131 (2013).

[7] A. Dijckmans, A. Ekblad, A. Smekal, G. Degrande, G. Lombaert: A sheet piling wall as a wave barrier for train induced vibrations, Proceedings of COMPDYN 2013, Kos Island (Greece).

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