gupta 2010 soil dynamics and earthquake engineering

ARTICLE IN PRESS

Soil Dynamics and Earthquake Engineering 30 (2010) 82–97

Contents lists available at ScienceDirect

Soil Dynamics and Earthquake Engineering

0267-72

doi:10.1

� Corr

E-m

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

Numerical modelling of vibrations from a Thalys high speed train in theGroene Hart tunnel

S. Gupta, H. Van den Berghe, G. Lombaert, G. Degrande �

Department of Civil Engineering, K.U.Leuven, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium

a r t i c l e i n f o

Article history:

Received 17 November 2008

Received in revised form

18 September 2009

Accepted 23 September 2009

Keywords:

High speed train induced vibrations

Groene Hart tunnel

Coupled periodic finite element–boundary

element model

Dynamic vehicle–track interaction

Dynamic track–tunnel–soil interaction

61/$ - see front matter & 2009 Elsevier Ltd. A

016/j.soildyn.2009.09.004

esponding author. Tel.: +3216 3216 67; fax:

ail address: [email protected]

a b s t r a c t

This paper presents a numerical study of vibrations due to a Thalys high speed train in the Groene Hart

tunnel, which is part of the high speed link South between Amsterdam and Antwerp and the world’s

largest bored tunnel.

A coupled periodic finite element–boundary element model is used to predict the free field response

due to the passage of a Thalys high speed train in the Groene Hart tunnel. A subdomain formulation is

used, where the track and the tunnel are modelled using a finite element method, while the soil is

modelled as a layered half space using a boundary element method. The tunnel and the soil are assumed

to be invariant in the longitudinal direction, but modelled as a periodic structure using the Floquet

transformation. A general analytical formulation to compute the response of three-dimensional periodic

media excited by moving loads is adopted.

The Groene Hart area is marshy and completely saturated. The top soil consists of layers of peat and

clay with a very low density and shear wave velocity. The numerical model allows to understand the

effect of these soft layers on vibration levels, resulting in an amplification of the horizontal response and

a large contribution of the quasi-static forces at high train speeds. Vibration levels are assessed using

the Dutch SBR guideline. It is concluded that the operation of high speed railway traffic in the Groene

Hart tunnel is not expected to cause serious vibration problems.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The Groene Hart is a relatively thinly populated area in theNetherlands enclosed by large cities of the Randstad, a verydensely populated area accommodating nearly half of the Dutchpopulation. It is a rural area that offers many green spaces withagriculture, nature and recreation as primary activities. It isthreatened by expansion of the big cities around it and other landuse which may destroy its unique landscape. With furtherexpansion and advances in the Randstad, the concern over thepreservation of the Groene Hart region has increased.

The high speed link South (HSL-Zuid) between Amsterdam andAntwerp aims to improve connectivity between importanteconomic centers of Europe. It crosses the provinces of North-and South-Holland and passes through the Groene Hart area. Itwas decided to bore a tunnel to conserve the grasslands of theGroene Hart and to leave the landscape undisturbed. A 7 km longtunnel was therefore constructed between Leiderdorp andHazerswoude-Dorp. The tunnel has been bored using the Auroratunnel boring machine, that has been manufactured for thepurpose of boring a tunnel with an external diameter of 14.5 m.

ll rights reserved.

+3216 3219 88.

(G. Degrande).

This makes the Groene Hart tunnel the largest bored tunnel in theworld. The tunnel consists of a single tube with a length of over7 km. Including the access ramps, the total length is about 8.5 km.

One of the requirements for the design of the high speed linkincluding the Groene Hart tunnel has been to comply with theDutch SBR (Building Research Foundation) guideline for measur-ing and assessing vibrations [1]. In the present paper, the problemof high speed train (HST) induced vibrations in the Groene Hartarea is examined by assessing computed free field vibration levelsusing the SBR guideline.

The environmental impact of vibrations due to railways hasrecently gained a lot of importance on account of increasing publicsensitivity to noise and vibration and stricter legislation toachieve sustainable development. Great efforts have been madein the last two decades to develop numerical models forprediction of ground-borne vibrations from railways. As far asdeterministic modelling is concerned, analytical and variousnumerical methods such as finite element, boundary elementand integral transform methods have been employed to model thecoupled track–tunnel–soil system.

The problem of vibrations from high speed railway traffic hasbeen studied by a large number of researchers. Most studies havefocused on solving the train–track–soil interaction problem,which allows to compute the vibrations from surface railways.Sheng et al. [2,3] have modelled the track–soil interaction by

www.elsevier.com/locate/soildyn

dx.doi.org/10.1016/j.soildyn.2009.09.004

mailto:[email protected]

ARTICLE IN PRESS

Table 1Dynamic soil characteristics of the Groene Hart area at location S3.

Layer Thickness (m) Cs0 (m/s) Cp0 (m/s) r (kg=m3) n0 (–) b (–)

1 3.7 50 1761 1107 0.4996 0.025

2 7.0 75 1719 1500 0.4990 0.025

3 8.3 180 1685 1970 0.4942 0.025

4 9.3 240 1715 1970 0.4900 0.025

5 1 260 1726 1970 0.4884 0.025

S. Gupta et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 82–97 83

means of a model with an infinite layered beam on top of alayered half space and have coupled a train model to this trackmodel [4,5]. Auersch [6] has coupled a finite element model for afinite part of the track to a boundary element model for the soil.This model is used for the calculation of the track compliance inthe solution of the vehicle–track interaction problem. Metrikine etal. [7] have studied the stability of a moving train bogie, modelledas a two degree of freedom system, which is coupled to a beam ofinfinite length for the track and a homogeneous half space modelfor the soil, following an approach proposed by Metrikine andPopp [8]. Lombaert and Degrande [9] and Lombaert et al. [10]apply this methodology in a boundary element formulation topredict vibrations induced by railway traffic [10,11]. The modelhas been validated by means of experiments that have beenperformed at the occasion of the homologation tests of the newHST track on the line L2 between Brussels and Koln [10,11]. Mostof these advanced numerical models assume the geometry to beinvariant in the longitudinal direction and use the Fouriertransform to efficiently formulate the problem in the frequency–wavenumber domain.

Similar models with varying degree of sophistication have alsobeen developed for predicting vibrations due to undergroundrailway traffic. Recently developed models such as the semi-analytical pipe-in-pipe model [12–14], the model based oncoupling of the finite element method and integral transformmethods [15] and the coupled finite element–boundary elementmodels [16–19] account for three-dimensional dynamic interac-tion between the track, the tunnel and the soil. These modelsexploit the invariance or periodicity of the system in thelongitudinal direction using the Fourier or the Floquet transformto formulate the problem in the frequency–wavenumber domain.Sheng et al. [20] have referred to this approach as thewavenumber finite element–boundary element method. Theyhave used this methodology to compare the transfer functions oftwo alternative tunnel designs, a single double-track tunnel and apair of single-track tunnels. Gupta et al. [21] have used thecoupled periodic finite element–boundary element approach forpredicting vibrations due to subway traffic in Beijing. The presentpaper demonstrates the applicability of this model in predictingthe vibrations from high speed trains running in tunnels.

The paper is organized in the following manner. Section 2reviews the characteristics of the soil, the tunnel, the track and thetrain. Section 3 outlines the modelling of the Groene Hart tunnelusing the coupled periodic finite element–boundary elementmodel and explains the computation of the transfer functions andthe axle loads. Section 4 presents the response of the tunnel and

Fig. 1. (a) Geotechnical profile along the Groene Hart tunnel and (b) shear wave veloci

computations (black line).

the free field due to the passage of a Thalys HST in the Groene Harttunnel. Finally, in Section 5, the predicted vibration levels arecompared to the SBR guideline to evaluate human exposure tovibrations.

2. System characteristics

2.1. Characteristics of the soil

The Groene Hart is a wet area with lot of marshes, polders andreclaimed lakes. The soil is essentially peaty and completelysaturated. The geotechnical profile along the Groene Hart tunnel isshown in Fig. 1a. The top layers mainly consist of peat and clay,while the underlying half space is sand. A shallow peat layer ofthickness 3.7 m is present on top of the clay layer up to a depth ofabout 10 m. Under the clay layer is a thin layer of peat. Under thissecond peat layer are the medium and coarse sand layers.Between PK 24 and PK 29 (project kilometers), the tunnel ismainly embedded in a layer of sand. The top part of the tunnel liesin a medium sand, while the bottom part of the tunnel lies in thecoarse sand.

The shear wave velocity of the soil has been determinedthrough seismic cone penetration tests (SCPT) [22]. These testshave been performed at three locations S1, S2 and S3 along thetunnel (Fig. 1a). The soil density is obtained from classical soilmechanics tests [22]. For the numerical calculations, the shearwave velocity profile is determined by averaging the shear wavevelocity obtained from the SCPT tests on location S3 (Fig. 1b). Thedepth of the tunnel at this location is 26.25 m with respect to thefree surface. Four layers on top of a half space are considered(Fig. 1b), whose properties are summarized in Table 1. The soil ismodelled as a horizontally layered half space, consisting ofsaturated soil layers, that are modelled using Biot’s theory asequivalent mono-phasic media or frozen mixtures with a shear

0 100 200 300 400 500−40

−30−27.94

−20

−10

−1.690

Cs [m/s]

Dep

th N

AP

[m]

PEAT

CLAY

MEDIUM SAND

COARSE SAND

ty profile obtained from the SCPT tests on location S3 (gray line) and used for the

ARTICLE IN PRESS

40

60

dB re

f 1µ

m]

S. Gupta et al. / Soil Dynamics and Earthquake Engineering 30 (2010) 82–9784

wave velocity Cs0, a longitudinal wave velocity Cp0 (and acorresponding high value of Poisson’s ratio n0) and a mixturedensity r [23–25]. As no experimental data about the hystereticmaterial damping ratio bs and bp in shear and volumetricdeformation in each layer are available, a uniform valueb¼ bs ¼ bp ¼ 0:025 is assumed for all layers.

2.2. Characteristics of the tunnel

The Groene Hart tunnel has been bored using a slurry shieldtunneling method at a depth of about 26 m. Per section, tenprefabricated concrete sections were placed to form the tunnellining and joined with bolts in the circumferential and long-itudinal direction. On the tunnel floor a technical gallery was setup for easy execution of the project and maintenance work. Thespace between the gallery and the tunnel is filled with a mixtureof sand and cement for stabilization. This is covered by a concretefloor with a thickness of 0.27 m, on which the track is installed.The tunnel hosts two tracks, which are separated by a partitionwall of thickness 0.45 m. This partition wall has been provided inview of safety and easy evacuation.

Fig. 2 shows the cross section of the tunnel. The tunnel has aninternal radius ri ¼ 6:65 m and an outer radius re ¼ 7:25 m.Additional dimensions are shown on Fig. 2. The concrete tunnellining has a Young’s modulus Et ¼ 35 000 MPa, a Poisson’s rationt ¼ 0:20, a density rt ¼ 2500 kg=m3 and a hysteretic materialdamping ratio bt ¼ 0:02. The mixture of the sand and cement has aYoung’s modulus Esc ¼ 24 000 MPa, a Poisson’s ratio nsc ¼ 0:20 anda density rsc ¼ 1850 kg=m3.

2.3. Characteristics of the track

Two tracks are installed in the tunnel, according to a tracksystem that is commonly used on high speed lines in Germanyand the Netherlands. It is a ballastless track that uses concrete bi-block sleepers with a lattice truss, which makes it easier to installand ensures an accurate gauge. The tracks are installed at 3.5 mfrom the center of the tunnel.

A geotextile layer is placed between the track and thesubstructure. The layer is fixed on the concrete floor of the tunnelwith dowels. Its main purpose is to increase the friction betweenthe track and the tunnel bed to avoid any possible separation. Thestiffness of the geotextile layer is Kg ¼ 2:23� 109 N=m3, while thedamping in the layer is Cg ¼ 2:31� 104 N s=m3. The reinforcedconcrete slab is 2.8 m wide and 0.24 m thick. Prefabricated

Fig. 2. Cross section of the Groene Hart tunnel.

concrete bi-bloc sleepers with dimensions 0:645 m� 0:292 m�0:016 m are cast into the slab. Elastic pads, base plates, and railpads discretely support the UIC 60 rails on sleepers at an intervald¼ 0:65 m. The base plate pads have a stiffness kbp ¼ 30�106 N=m and a damping cbp ¼ 8� 103 N s=m. These soft padsprovide some degree of vibration isolation. The mass of the baseplates is 12 kg. The rail pads are stiff and have a stiffness krp ¼ 1�109 N=m and a damping crp ¼ 7:77� 104 N s=m [26].

The track unevenness has been measured with a NSTOmeasurement train on the part of the high speed line South thatincludes the Groene Hart tunnel. Fig. 3 compares the one-thirdoctave band spectrum of the measured unevenness with the trackclasses 1 and 6 of the Federal Railroad Administration (FRA). FRAtrack classes have been derived on the basis of extensivemeasurements on the US railway network; six classes of trackquality are defined, class 1 being the poorest and class 6 the best[27]. Superimposed on this graph are the TSI+ [28] and ISO3095:2005 [29] limits. The unevenness measured on the presentHST track is higher than FRA track class 6 at shorter wavelengthsor higher frequencies, which can cause greater wheel-rail rollingnoise. For a train speed of v¼ 300 km=h, wavelengths between83 cm and 83 m are important for predicting ground-bornevibrations in the frequency range between 1 and 100 Hz.

2.4. Characteristics of the train

The high speed train is an articulated Thalys train with twopower cars and eight carriages; the total length of the train isequal to 200.18 m. The power cars are supported by two bogiesand have four axles. The carriages next to the power cars shareone bogie with the neighboring carriage, while the six othercarriages share both bogies with neighboring carriages. The totalnumber of bogies equals 13 and, consequently, the number ofaxles on the train is 26. The carriage length Lt, the distance Lb

between bogies, the axle distance La, the total axle mass Mt, thesprung axle mass Ms and the unsprung axle mass Mu of allcarriages are summarized in Table 2.

28321285122048−20

0

20

Une

venn

ess

[

Wavelength [cm]

TSI limit

ISO limit

Fig. 3. One-third octave band spectra of the measured unevenness along the high

speed line South (solid black line) and FRA track classes 1 (dashed gray line) and 6

(solid gray line). Superimposed on the graph are the TSI+ (dashed black line) and

ISO 3095:2005 limits (dash-dotted black line).

Table 2Geometrical and mass characteristics of the Thalys HST.

Lt (m) Lb (m) La (m) Mt (kg) Ms (kg) Mu (kg)

Power cars 22.15 14.00 3.00 17000 14937 2027

Side carriages 21.84 18.70 3.00 17000 14937 2027

Central carriages 18.70 18.70 3.00 17000 14937 2027

ARTICLE IN PRESS


3. Numerical modelling of the Groene Hart tunnel

A coupled periodic finite element–boundary element model[16,17] is used to model the response of the Groene Hart tunnelduring the passage of a Thalys high speed train at a speed of300 km/h. In the following subsections, the methodology used toderive in the frequency domain the response of periodic mediadue to moving loads, based on the transfer functions formulatedin the frequency–wavenumber domain and the frequency contentof the axle loads, will be briefly recapitulated. The reader isreferred to complementary literature [16,17,30] for a moredetailed description of the methodology.

3.1. Response of periodic media due to moving loads

The coupled track–tunnel–soil system is subjected to verticalloads moving along the longitudinal direction ey. In the fixedframe of reference, the distribution of na vertical axle loads on thecoupled track–tunnel–soil system is written as the summation ofthe product of Dirac functions that determine the time dependentposition xkðtÞ ¼ fxk0; yk0þvt; zk0g

T and the time history gkðtÞ of thek-th axle load:

rbðx; tÞ ¼Xna

k ¼ 1

dðx� xk0Þdðy� yk0 � vtÞdðz� zk0ÞgkðtÞez ð1Þ

with yk0 the initial position of the k-th axle that moves with thetrain speed v along the y-axis and ez the vertical unit vector.

A periodic structure can be analyzed using the Floquet transform[16,17] by restricting the problem domain O to a single periodic unitor reference cell ~O. If the spatial period is L, then the position x ofany point in the problem domain is decomposed as x¼ ~xþnLey,where ~x is the position in the reference cell and n is the cell number.The response of periodic domains subjected to the moving loads isgiven as follows in the frequency domain [30,31]:

uið ~xþnLey;oÞ ¼1

2pXna

k ¼ 1

Z 1�1

g kðo� kyvÞexpð�ikyðnL� yk0ÞÞ

�

Z L=2

�L=2expð�iky ~y

0Þ ~hzið ~x0; ~x;ky;oÞd ~y0 dky ð2Þ

where ky ¼ ky � 2mp=L such that kyA � � p=L;p=L½. The transferfunction ~hzið ~x

0; ~x;ky;oÞ of the coupled track–tunnel–soil system inthe frequency–wavenumber domain is the Floquet transform of thetransfer function hzið ~x

0; ~xþnLey;oÞ in the frequency–spatial domainand is defined as the displacement at ~x in the direction ei due to aunit load applied at ~x 0 in the direction ez. Expression (2)demonstrates that the response to moving loads can be calculatedif the transfer functions ~hzið ~x

0; ~x;ky;oÞ and the axle loads g kðoÞ are

Mode 5 at 5.9 Hz. Mode 9 at 29

Fig. 4. (a,b) Two in-plane modes and (c) the first out-of-pla

known. The computation of the transfer functions and the axle loadsis presented in the following subsections.

3.2. Transfer functions of the track–tunnel–soil system

A coupled periodic finite element–boundary element model isused to solve the dynamic tunnel–soil interaction problem and tocompute the transfer functions in the frequency–wavenumberdomain. A subdomain formulation is employed where the finiteelement method is used to model the track and the tunnel, whilethe boundary element method is used to model the soil as ahorizontally layered halfspace. As the reference cell of the tunnelis bounded, the tunnel displacements ~utð ~x;ky;oÞ are decomposedon a basis of tunnel modes ~Wtð ~x;kyÞ that are periodic functions ofthe second kind [16,17]. The soil displacements ~usð ~x;ky;oÞ arewritten as the superposition of waves that are radiated by thetunnel modes ~Wtð ~x;kyÞ into the soil. A weak variational formula-tion of the problem results in the following system of equations inthe frequency–wavenumber domain [16,17,32]:

½KtðkyÞ �o2MtðkyÞþKsðky;oÞ�atðky;oÞ ¼ Ftðky;oÞ ð3Þ

where KtðkyÞ and MtðkyÞ are the projection of the finite elementstiffness and mass matrix of the tunnel’s reference cell on thetunnel modes ~Wtð ~x;kyÞ. Ksðky;oÞ is the dynamic stiffness matrixof the soil and is computed using the periodic boundary elementformulation based on Green–Floquet functions defined on theperiodic structure with period L [16,17,32]. Ftðky;oÞ is theprojection of the force vector in the reference cell on the tunnelmodes ~Wtð ~x;kyÞ. Finally, atðky;oÞ are the modal coordinates thatdetermine the contribution of the modes in the kinematical basis~Wtð ~x;kyÞ.

In the present analysis, the Groene Hart tunnel is modelled as aperiodic structure with period L¼ 0:65 m, corresponding to thesleeper spacing ds. It is worth to mention that, in the lowfrequency range of interest, the tunnel could also have beenmodelled as an equivalent invariant structure using a two-and-a-half dimensional method. The periodic approach, however, has theadvantage that existing three-dimensional boundary elementtechnology for layered media can be reused, since the Green–Floquet functions have the same singularities as the three-dimensional Green’s functions [16,17]. A two-and-a-half dimen-sional approach would have necessitated the analysis of thesingularities of the two-and-a-half dimensional Green’s functions,which has now been avoided.

A finite element model of the reference cell is made usingANSYS. The circular concrete lining of the tunnel and the tunnelinvert consisting of sand/cement stabilization are modelled using8-node brick elements, including incompatible bending modes(Fig. 4). The size of the finite elements is governed by the

.75 Hz. Mode 17 at 75.03 Hz.

ne mode of the generic cell of the Groene Hart tunnel.

ARTICLE IN PRESS


boundary element mesh along the tunnel–soil interface, so thatideally a minimum of N¼ 8 elements are used per shearwavelength lmin

s ¼ Cs=fmax, with Cs ¼ 240 m=s the shear wavevelocity in the soil layer around the apex of the tunnel andfmax ¼ 100 Hz the maximum excitation frequency. This results in arecommended element length le ¼ lmin

s =N¼ 0:3 m. In the presentmodel, two elements are used in the longitudinal direction(le ¼ 0:325 m), while only 36 elements are used to model thetunnel lining in the circumferential direction (le ¼ 0:7625 m).Thus, the model is expected to be accurate up to 40 Hz, whilesome discretization errors may occur at higher frequencies. A finermesh is not used because of the long computation times involved.The partition wall and the concrete floor that rests on the sand/

Fig. 6. Observation points (a) on the slab track,

Fig. 5. Real part of the vertical displacement of the coupled track–tunnel–soil

system due to a harmonic excitation at (a) 10 Hz and (b) 40 Hz on the rails.

cement stabilization are also modelled with 8-node brickelements. The partition wall is connected to the tunnel wallwith a hinge connection; there is no connection between theconcrete floor and the tunnel wall.

As the tunnel is bounded, the displacement field ~utð ~x;ky;oÞ inthe tunnel can be decomposed on a basis of functions ~Wtð ~x;kyÞ

[16,17]. This kinematic basis is selected by performing aneigenvalue analysis of the tunnel’s reference cell. A complexeigenvalue problem is formed at each wavenumber after imposingthe periodicity condition of the second kind on the reference cellin order to comply with the definition of the Floquet transform[16,17,30]. In the present analysis, the eigenvalue problem issolved for only a limited number M of lower modes at selectedwavenumbers ½ky1;ky2; . . . ;kyq� in the range ½0;p=L�. Fig. 4 showstwo in-plane and the first out-of-plane flexible modes of thegeneric cell computed at zero wavenumber. The eigenvectorscomputed at selected wavenumbers are collected in a matrix anda singular value decomposition is applied to obtain the kinematicbasis. For the problem under consideration, a convergence studyhas demonstrated that a kinematic basis consisting of 80 vectorsguarantees good modelling accuracy [25].

The track is incorporated in the tunnel using the Craig–Bampton substructuring technique. The tunnel hosts two tracks,but only the right track is considered in the present analysis, as itis expected that the presence of the left track does not affect thedynamic train–track–tunnel–soil interaction. A finite elementmodel of the track is developed according to the track propertiesdescribed in Section 2.3. The slab is connected to the tunnel floorby discrete supports, which represent the stiffness of the

(b) on the tunnel, and (c) in the free field.

ARTICLE IN PRESS


geotextile layer. The concrete slab is modelled using plateelements, while the bi-bloc sleepers are represented using masselements and joined to the slab using rigid connections. The baseplates are modelled using mass elements, while Euler–Bernoullibeam elements are used for the rails. The base plate pads and therail pads are represented by spring elements that account for thestiffness and the damping of the resilient material. The rail padsand base plates discretely support the rail at an intervalds ¼ 0:65 m, resulting in a periodic track model. Eigenvalueanalysis of the track on a rigid base demonstrates that resonanceof the rail and the base plate on the base plate pads occurs at106.54 Hz; 25 track modes on a rigid base have been foundsufficient to obtain convergence.

The response of the tunnel in the frequency–wavenumberdomain is obtained through solution of the coupled system of Eq.(3). Once the displacements and stresses along the tunnel–soilinterface are known, the representation formula of elastody-namics is applied to compute the free field response around thetunnel. Transfer functions are finally obtained by evaluating the

Table 3Coordinates of the observation points on the track, on the tunnel, and in the free

field.

Points Position x (m) y (m) z (m)

TR1 Rail 2.635 0 �29.516

TR2 Base plate 2.635 0 �29.536

TR3 Sleeper 2.635 0 �29.672

TR4 Slab 2.537 0 �29.692

T1 Tunnel floor 3.385 0 �29.797

T2 Tunnel wall 6.650 0 �26.250

T3 Tunnel apex 0 0 �19.600

A Free field 0 0 0

B Free field 14.5 0 0

C Free field 43.5 0 0

D Free field 0 0 �10.7

E Free field 14.5 0 �10.7

F Free field 43.5 0 �10.7

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

0 50 100 150 200

0 20 40 60 80 100

−200

−180

−160

−140

−210

−190

−170

−150

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

Fig. 7. (a) Track receptance for the track on a rigid base (black line) and for the track

sleeper (TR3), and (d) the tunnel apex (T3). Superimposed on (d) is the transfer functio

inverse Floquet transformation from the wavenumber to thespatial domain [16,17,32]. As an example, Fig. 5 shows the verticalcomponent of the transfer function of the coupled track–tunnel–soil system for a unit harmonic load at excitation frequencies of 10and 40 Hz; a load of 0.5 N is applied on each rail, which isequivalent to a unit load on the system. Apart from the responseof the tunnel, the response of the soil has been computed in alimited number of output points in the reference cell, allowing tovisualize the response on two horizontal and one vertical planeafter evaluation of the inverse Floquet transformation. The wavepropagation pattern in the soil is asymmetric, as the harmonicload is applied on the right track, which is eccentrically positionedwith respect to the center of the tunnel. The presence of thetunnel significantly influences wave propagation in the soil,particularly at 40 Hz, where the wavelengths are shorter thanthe diameter of the tunnel. Computing the response on a densergrid of points in the reference cell would allow to visualize thetransfer functions in more detail in the different soil layers, butrequires substantial computational effort.

The transfer functions are subsequently discussed in moredetail for a limited number of observations points on the slabtrack, in the tunnel and in the free field, as indicated on Fig. 6; thecoordinates of these points are given in Table 3.

Fig. 7a shows the track receptance in the frequency range 1–200 Hz for the track on a rigid base. Superimposed on the samegraph is the rail receptance in the frequency range 1–100 Hz,computed with the coupled finite element–boundary elementmodel, accounting for the flexibility of the tunnel–soil system; thelatter does not substantially affect the response of the track. Thefirst peak in the response appears at the first natural frequency ofthe track at 106.54 Hz. Fig. 7b–d show the transfer function at thebase plate, the sleeper, and the tunnel apex. These transferfunctions do not exhibit any peak in the frequency range 1–100 Hz. The maximum reduction in the response is from the baseplate to the sleeper, which is due to the flexible base plate pads.The reduction in response from the rail to the base plate is notlarge as stiff rail pads are used. Superimposed on Fig. 7d is the

0 20 40 60 80 100Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

0 20 40 60 80 100

−210

−190

−170

−150

−260

−240

−220

−200

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

in the tunnel (gray line) and transfer functions at (b) the base plate (TR2), (c) the

n at the tunnel apex for the tunnel without the track (gray line).

ARTICLE IN PRESS


transfer function at the tunnel apex for the case of the tunnelwithout the track, i.e. when the load is applied directly on theconcrete floor. At low frequencies, the track does not significantlyinfluence the response at the apex, which can also be explained bythe fact that the transmittance at low frequencies is equal to one.

Figs. 8 and 9 show the horizontal and vertical component ofthe transfer function at the observation points A, B, C, D, E, and Fin the free field, for a harmonic load applied on the rails. The

0 20 40 60 80 100Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

0 20 40Freq

Dis

plac

emen

t [dB

ref 1

m/N

]

0 20 40 60 80 100

−280

−260

−240

−220

−200

−280

−260

−240

−220

−200

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

0 20 40

−280

−260

−240

−220

−200

−280

−260

−240

−220

−200

Freq

Dis

plac

emen

t [dB

ref 1

m/N

]

Fig. 8. Horizontal component of the transfer function at the points (a) A, (b) B, (c) C,

function in the free field for the tunnel without the track (gray line).

0 20 40 60 80 100Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

0 20 40Frequ

Dis

plac

emen

t [dB

ref 1

m/N

]

0 20 40 60 80 100

−280

−260

−240

−220

−200

−280

−260

−240

−220

−200

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

0 20 40

−280

−260

−240

−220

−200

−280

−260

−240

−220

−200

Frequ

Dis

plac

emen

t [dB

ref 1

m/N

]

Fig. 9. Vertical component of the transfer function at the points (a) A, (b) B, (c) C,(d) D, (e

the free field for the tunnel without the track (gray line).

points A, B, and C are located on the free surface, while points D, E,and F are located at a depth of 10.7 m, at the interface of the softclay layer and the relatively stiffer sand layer (Fig. 6c). Comparisonof the response in the points D, E, and F with the response in thepoints A, B, and C allows to appreciate if and how vibrationcomponents are amplified through the soft clay and peat layers.

Considering first the horizontal response (Fig. 8), it can beobserved that the response in the points A and D directly above

60 80 100uency [Hz]

0 20 40 60 80 100Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

60 80 100uency [Hz]

0 20 40 60 80 100

−280

−260

−240

−220

−200

−280

−260

−240

−220

−200

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

(d) D, (e) E, and (f) F in the free field. Superimposed on the graphs is the transfer

60 80 100ency [Hz]

0 20 40 60 80 100Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

60 80 100ency [Hz]

0 20 40 60 80 100

−280

−260

−240

−220

−200

−280

−260

−240

−220

−200

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

) E, and (f) F in the free field. Superimposed on the graphs is the transfer function in

ARTICLE IN PRESS


the tunnel is non-zero as the harmonic load is applied on bothrails of the track, which is eccentrically positioned with respect tothe center of the tunnel. The response in the free field ischaracterized by an oscillating behavior due to interference ofcompression and shear or Rayleigh waves. The horizontalresponse in the point B at the surface is clearly higher than theresponse in the point E at the bottom of the softer layers; a similarobservation can be made, but to a less extent, when comparingthe response in the points F and C, while no amplification isobserved between points D and A. This amplification is related tovertically propagating shear waves in the soft top layers, as will beexplained in a following paragraph. Superimposed on Fig. 8 arethe transfer functions computed without the presence of the trackwhen the harmonic load is directly applied on the tunnel invert.The track has an influence from 40 Hz onwards. The response ofthe track–tunnel–soil system is higher than that of the tunnel–soilsystem due to the resonance of the track at 106 Hz.

The vertical component of the transfer function (Fig. 9) isgenerally lower than the horizontal component and no pro-nounced amplification through the soft layers is observed.

The previous observations can be explained by visualizing thewave field around the tunnel upon harmonic excitation on thetrack. It has been mentioned, however, that the computation ofthe transfer functions of the coupled track–tunnel–soil system in

Fig. 10. Horizontal component (left), vertical component (middle) and modulus (right) o

depth of 29.5 m, corresponding to the depth of the tunnel invert, at a frequency of (a)

a large number of receivers in the reference cell is expensive.Although it has also been recognized that the tunnel importantlyinfluences the transfer functions, especially at higher frequencies,the amplification of waves in the soil can also be studied byconsidering the Green’s functions of the layered soil (withouttunnel). These Green’s functions are computed with the Elasto-Dynamics Toolbox (EDT) [33], applying a vertical unit harmonicload at a depth of 29.5 m, corresponding to the depth ofthe concrete floor of the tunnel where the load is applied inthe previous cases. The soil has the same properties as given inTable 1.

Fig. 10 shows the horizontal and vertical component, as well asthe modulus, of the Green’s function in the layered halfspace for aharmonic point load applied at a depth of 29.5 m, at an excitationfrequency of 20, 40, and 80 Hz. For the horizontal component, itcan clearly be observed how shear waves propagate upwards inthe stiffer sand layer(s) in a shear window marked by a zone thatmakes an angle from almost 303 to 603 with the horizontal axis; atthe interfaces with the softer materials (clay and peat layers), theshear waves are refracted and propagate upwards at a steeperangle within the softer material, while the amplitude increases.These figures clearly illustrate how the response is amplified byupward propagating shear waves in the soft layers. Depending onthe excitation frequency, this phenomenon is more pronounced in

f the Green’s function in a layered halfspace for a harmonic point load applied at a

20 Hz, (b) 40 Hz, and (c) 80 Hz.

ARTICLE IN PRESS


an area along the surface at a certain distance from the verticalaxis of symmetry; at larger distances along the surface, waves areattenuated due to geometrical and material damping in the soil.The vertical component is maximum in the sand layer(s) at adepth corresponding to the point of application of the load andattenuates with increasing lateral distance. No amplification ofthe vertical component occurs in the soft clay and peat layers,where the vertical component is low. The modulus of the transferfunction in the superficial layers is therefore governed by thehorizontal component and by the upward propagation of shearwaves, as can clearly be observed in Fig. 10.

3.3. Axle loads

The static and dynamic component of the axle loads areaccounted for in the following. The static component is due to theweight of the train, as summarized in Table 2. The resulting staticaxle load is equal to wk ¼ 170 kN. Dynamic axle loads are relatedto parametric and unevenness excitation.

Parametric excitation occurs at harmonics of the sleeperpassage frequency fs ¼ v=ds ¼ 128:2 Hz, with ds ¼ 0:65 m thesleeper distance and v¼ 300 km=h the speed of the train. Thesefrequencies lie outside the frequency range of interest forground-borne vibration (1–80 Hz). Due to the Doppler effect,the first harmonic of the axle load contributes to the responseat a fixed point in the free field in a frequency range½fs=ð1þv=CÞ; fs=ð1� v=CÞ�, with C the wave speed in the ground.The waves radiated behind the load can have much lowerfrequencies than fs and lie into the frequency range of interest.Higher frequencies are not considered for the prediction ofground-borne vibrations as they are significantly attenuated inthe free field due to material damping in the soil; they can be

0 50 100 150 200−200

−180

−160

−140

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

Fig. 11. The track receptance over the sleeper (dashed gray line) and at mid-span

between sleepers (solid gray line) for a train speed of 0 km/h. Superimposed on the

graph is the track receptance over the sleeper for a train speed of 300 km/h (black

line).

0 50 100 150 200−200

−180

−160

−140

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

Fig. 12. The elements (a) Ct

12ðoÞ and (b) Ct

21ðoÞ of the track compliance m

important, however, for the prediction of re-radiated noise inbuildings. The parametric excitation depends on the variation inthe track receptance along the sleeper bay. Fig. 11 shows the trackcompliance over the sleeper and at mid-span between sleepers.Since the rail and the base plate assembly are supported by softbase plate pads, the variation in the track receptance is small.Therefore, parametric excitation is not significant for the presentanalysis. Only dynamic forces due to the wheel/rail unevennessare considered.

The dynamic forces gdðoÞ due to wheel/rail unevenness can becalculated by considering the following vehicle–track interactionequation [10,21]:

½CvðoÞþ C

tðoÞ�gdðoÞ ¼ � uw=rðoÞ ð4Þ

where CvðoÞ is the vehicle compliance matrix, C

tðoÞ is the

compliance matrix of the track in the moving frame of reference,and uw=rðoÞ is the unevenness represented in the frequencydomain [10]. This approach is valid only for invariant tracks, butcan be conveniently used for the present periodic analysis as thevariation in the track receptance along the sleeper bay isnegligible. Superimposed on Fig. 11 is the diagonal elementC

t

11ðoÞ of the track compliance matrix in the moving frame ofreference for a train speed of 300 km/h. The resonance frequencynear 106 Hz shifts to lower frequencies due to the Doppler effect,but, even at a high train speed, the element C

t

11ðoÞ is not muchdifferent from the corresponding element in the fixed frame ofreference. Fig. 12 shows the off-diagonal elements C

t

12ðoÞ andC

t

21ðoÞ of the track compliance matrix in the moving and the fixedframe of reference. These elements are clearly influenced by thetrain speed. In the following, the track receptance in the movingframe of reference is therefore used to compute the wheel/railinteraction forces.

0 50 100 150 200−200

−180

−160

−140

Frequency [Hz]

Dis

plac

emen

t [dB

ref 1

m/N

]

atrix for a train speed of 0 km/h (gray line) and 300 km/h (black line).

100 101 102 10340

60

80

100

Frequency [Hz]

RM

S fo

rce

[dB

ref 1

N]

Fig. 13. One-third octave band RMS spectrum of the contact force at the front axle

of the Thalys high speed train computed with the measured unevenness (black

line) and FRA track class 6 (gray line) for a train speed of 300 km/h.

ARTICLE IN PRESS


At frequencies of more than a few Hertz, the vehicle’s primaryand secondary suspension isolate the body and the bogie from thewheelset [34]. The unsprung masses therefore are the onlycomponents that affect the vertical dynamic loads. Each axle ismodelled as a mass Mu and the vehicle compliance matrix is equalto the diagonal matrix C

vðoÞ ¼ diagf�1=ðMuo2Þg of order 26.

The measured unevenness uw=rðyÞ is transformed to thewavenumber domain and expressed in the frequency domain.Eq. (4) is subsequently used to calculate the dynamic wheel–trackinteraction forces. Fig. 13 shows the one-third octave band RMSspectrum of the train–track interaction force at the front axle ofthe Thalys HST. The frequency content of this force exhibits a clearmaximum near the wheel–track resonance frequency of about42 Hz. Superimposed on the same graph is also the dynamic forcecomputed using the FRA track class 6. The comparison betweenthe contact force for the measured unevenness and the FRA trackclass 6 shows that the unevenness on the track corresponds to thequality of the FRA track class 6. For further analysis, the dynamicforces due to the measured unevenness are used.

4. Response due to a passage of a Thalys HST in the tunnel

The static and dynamic component of the axle loads are usedto compute the response in the tunnel and in the free field. Thefrequency content of the response is calculated by means of Eq.(2), which uses the transfer functions in the frequency–wave-number domain and the frequency content of the axle loads. Thetime history is obtained by evaluating the inverse Fouriertransformation.

−3 −2 −1 0 1 2 3−0.4

−0.2

0

0.2

0.4

Time [s]

Vel

ocity

[m/s

]

0 20 40

0.01

0.02

0.03

0.04

0.05

Freq

Vel

ocity

[m/s

/Hz]

Fig. 14. (a) Time history, (b) linear frequency spectrum and (c) one-third octave band RM

at a speed of 300 km/h. Superimposed on graphs (b) and (c) is the contribution of qua

−3 −2 −1 0 1 2 3−2

−1

0

1

2x 10−3

Time [s]

Vel

ocity

[m/s

]

0 20 400

2

4

6x 10−4

Frequ

Vel

ocity

[m/s

/Hz]

Fig. 15. (a) Time history, (b) linear frequency spectrum and (c) one-third octave band R

Thalys HST at a speed of 300 km/h. Superimposed on graphs (b) and (c) is the contribu

Fig. 14 show the time history, the linear frequency spectrumand the one-third octave band RMS spectrum of the verticalvelocity on the rail (TR1). The passage of every axle can clearly beidentified from the time history that shows a nearly uniformresponse for every axle (Fig. 14a). Although the power cars and theadjacent carriages have a different axle composition than thecentral carriages, a modulation of the frequency content isobserved with peaks at the fundamental bogie passagefrequency fb ¼ v=Lb ¼ 4:46 Hz of the central carriages and itshigher harmonics, modulated at the axle passage frequencyfa ¼ v=La ¼ 27:78 Hz [35] (Fig. 14b). The contribution of thequasi-static forces is also shown (Fig. 14b and c). It can beobserved that the quasi-static response is vital for the response ofthe track in the low frequency range up to 30 Hz [10]. Thedominant frequency content at the frequencies above 30 Hz is dueto rail unevenness. Unlike the maxima near the train–trackresonance frequency in the dynamic force (Fig. 13), the maximain the response are spread over a wider frequency range aroundthe train–track resonance frequency due to the Doppler effect.

Fig. 15 shows the time history, the linear frequency spectrumand the one-third octave band RMS spectrum of the verticalvelocity in the tunnel on the concrete floor (T1). The contributionof each axle can be distinguished in the time history, but is lessclear than in the response on the rail. The contribution of quasi-static forces has diminished and shifted to lower frequencies. Thedominant frequency content above 20 Hz is due to the dynamiccomponent of the axle loads.

Figs. 16–19 show the time history and the one-third octaveband spectrum of the horizontal and the vertical velocity in thefree field at the points A, B, and C along the surface, and the points

0 60 80 100

uency [Hz]

1 2 4 8 16 31.5 63 125 25010−5

10−4

10−3

10−2

10−1

One−third octave band center frequency [Hz]

RM

S v

eloc

ity [m

/s]

S spectrum of the vertical velocity on the rail (TR1) for a passage of the Thalys HST

si-static forces (gray line).

60 80 100

ency [Hz]

1 2 4 8 16 31.5 63 125 25010−7

10−6

10−5

10−4

10−3


RM

S v

eloc

ity [m

/s]

MS spectrum of the vertical velocity on the concrete floor (T1) for a passage of the

tion of quasi-static forces (gray line).

ARTICLE IN PRESS

−3 −2 −1 0 1 2 3

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

x 10−4 x 10−4 x 10−4

x 10−4 x 10−4 x 10−4

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Time [s]

Vel

ocity

[m/s

]

Fig. 16. Time history of the horizontal velocity at points (a) A, (b) B, (c) C, (d) D, (e) E, and (f) F in the free field for a passage of the Thalys HST at a speed of 300 km/h.

1 2 4 8 16 31.5 63 125 25010−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

One−third octave bandcenter frequency [Hz]






RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

Fig. 17. One-third octave band spectrum of the horizontal velocity at points (a) A, (b) B, (c) C, (d) D, (e) E, and (f) F in the free field for a passage of the Thalys HST at a speed

of 300 km/h. Superimposed on the graphs is the contribution of quasi-static forces (gray line).


D, E, and F at a depth of 10.7 m along the interface between thesoft clay layer and the stiffer sand layer. At the point E, which isrelatively close to the tunnel, the horizontal velocity (Fig. 16e) islarger than the vertical velocity (Fig. 18e), which is due to thedominant contribution of shear waves to the horizontal response;

the vertical response is lower due to the low compressibility of themedium. The horizontal velocity is higher on the surface (points Band C, Fig. 16b and c) than at depth (points E and F, Fig. 16e and f)due to amplification of vertically polarized shear waves in the softsoil layers. Such an amplification is not observed on the vertical

ARTICLE IN PRESS

−3 −2 −1 0 1 2 3

x 10−4

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

x 10−4

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

x 10−4

x 10−4 x 10−4 x 10−4

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Time [s]

Vel

ocity

[m/s

]

−3 −2 −1 0 1 2 3

−6

−4

−2

0

2

4

6

−6

−4

−2

0

2

4

6

Time [s]

Vel

ocity

[m/s

]

Fig. 18. Time history of the vertical velocity at points (a) A, (b) B, (c) C, (d) D, (e) E, and (f) F in the free field for a passage of the Thalys HST at a speed of 300 km/h.

1 2 4 8 16 31.5 63 125 250One−third octave band center frequency [Hz]






RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

10−7

10−6

10−5

10−4

10−3

RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

1 2 4 8 16 31.5 63 125 250

RM

S v

eloc

ity [m

/s]

Fig. 19. One-third octave band spectrum of the vertical velocity at points (a) A, (b) B, (c) C, (d) D, (e) E, and (f) F in the free field for a passage of the Thalys HST at a speed of

300 km/h. Superimposed on the graphs is the contribution of quasi-static forces (gray line).


component, which is lower at the surface (Fig. 18a–c) than atdepth (Fig. 18d–f), due to the attenuation of waves as well as thereflection of waves at layer interfaces. Similar observations havebeen made in the discussion of the transfer functions (Fig. 10).These observations explain why the horizontal peak particle

velocity (PPV) along the surface is much higher than the verticalPPV; in the point B, for example, the horizontal and vertical PPVare equal to 0.60 and 0.11 mm/s, respectively.

The frequency content shifts towards lower frequenciesfor increasing distance from the track as higher frequency

ARTICLE IN PRESS


components are attenuated due to material damping in the soil(Figs. 17 and 19). The contribution of the dynamic axle loadsdominates the response at frequencies above 8 Hz. As the trainspeed is high, the contribution of quasi-static forces can also beidentified in the free field response, especially on the verticalcomponent; it has shifted to lower frequencies, as the response inthe time domain becomes broader. Metrikine et al. [36] haveshown that, for a tunnel embedded in soft soil, the critical speed isusually less than the minimum wave velocity in the soil anddepends on the depth of the tunnel. As in the present analysis theshear wave velocity of the soil around the tunnel is much higherthan the train speed, the motion is not in the super-critical regime.The prominent contribution of quasi-static forces in the verticalresponse can be explained from the transfer functions in the freefield. In Fig. 9, a difference of more than 20 dB can be observedbetween the transfer function at low frequencies below 10 Hz andthe transfer functions at higher frequencies around the wheel–track resonance frequency of 42 Hz. This large difference is alsoreflected in the response to moving loads, where a largemagnitude of the quasi-static response is observed at lowfrequencies in comparison to the dynamic response around 42 Hz.

In order to better understand the high quasi-static response inthe free field, the response at a lower train speed of 100 km/h iscompared with the response at a higher speed of 300 km/h. Fig. 20shows a contour plot of the logarithm of the vertical displacementon the free surface at the point B in the frequency–wavenumberdomain. The dominant response is confined to low wavenumbers.The extent of this region is larger for softer soils with low wavevelocities. Superimposed on the graph are the load velocity lines(ky ¼o=v) for both train speeds. This relation implies that the

20 40 60 80 1000

0.5

1

1.5

2

Frequency [Hz]

Wav

enum

ber [

rad/

m]

−45

−40

−35

−30

−25

Fig. 20. Logarithm of the vertical displacement ~uzðky ;oÞ on the free surface at

point B in the frequency–wavenumber domain. Superimposed on the graph are the

load velocity lines (ky ¼o=v) for a train speed of 100 km/h (dashed line) and

300 km/h (solid line).

2 4 6 8 100

0.2

0.4

0.6

0.8

1x 10−4

Frequency [Hz]

Vel

ocity

[m/s

/Hz]

Fig. 21. (a) Linear frequency spectrum and (b) one-third RMS spectrum of the vertical ve

a speed of 100 km/h (gray line) and 300 km/h (black line).

phase velocity Cy ¼o=ky of all waves radiated by the constantload must be equal to the velocity v of the load. For the lower trainspeed, the load velocity line lies in the region of higherwavenumbers, where the response is strongly attenuated by thetunnel–soil system. For the higher train speed, the load velocityline crosses the region with high response, causing significantamplification of the quasi-static response. This is also illustratedin Fig. 21, where the vertical velocity in the point B due to quasi-static excitation at the two considered train speeds (100 and300 km/h) is compared. At the low train speed, the displacementfield generated by the load consist of strongly attenuated waves,while, at the high train speed, waves are excited in the structurewhich can transport energy to distant points. A difference of morethan a factor of 10 is observed in the vertical velocity due to quasi-static excitation at a train speed of 100 and 300 km/h. Such anamplification is not expected in the case of moving dynamic loads,as for harmonically varying loads, waves are always radiated,irrespective of the train speed.

5. Evaluation of human exposure to vibrations

In order to assess the possibility of annoyance to people due tothe passage of a Thalys high speed train in the Groene Hart tunnel,the Dutch SBR guideline part B [1] is applied. This guideline isused for evaluating human exposure to vibrations in buildings. Inthe present analysis, however, only free field vibrations areconsidered.

The SBR guideline [1] is very similar to the German DIN 4150norm [37] and based on the assessment of the maximum Vmax ofthe running effective velocity veff ðtÞ and of the vibration level Vper

over an evaluation period (day, evening and night), defined as

Vper ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

n

Xn

i ¼ 1

v2eff ;max;30;i

vuut ð5Þ

Herein, n is the number of 30 s intervals in the evaluation periodand veff ;max;30;i is the maximum of the running effective velocityveff ðtÞ in the i-th interval of 30 s. Values of veff ;max;30;i;i lower than0.1 are set equal to zero.

The maximum vibration level Vmax and the vibration level Vper

over an evaluation period are compared to limit values A1, A2 andA3. If the maximum vibration level Vmax is less than the limitvalue A1, then the standard is satisfied. If the maximum vibrationlevel Vmax is greater than the limit value A2, then the standardis not satisfied. If the maximum vibration level Vmax lies betweenthe limit values A1 and A2, then the vibration level Vper overthe evaluation period should be calculated. The standard is not

1 2 4 8 16 31.5 63 125 25010−8

10−7

10−6

10−5

10−4


RM

S v

eloc

ity [m

/s]

locity at point B due to the quasi-static excitation for a passage of the Thalys train at

ARTICLE IN PRESS


satisfied if Vper is greater than A3. The threshold values fordifferent building classifications and for each evaluation periodare summarized in Table 4. These limit values are for repeatedoccurrence of vibrations due to railways [1].

Figs. 22 and 23 show the running effective horizontal andvertical velocity on the free surface at points A, B and C.Superimposed on these graphs are the limit values A1 and A2 forrepeated occurrence of vibrations in residential buildings duringday time, assuming that high speed train traffic through theGroene Hart tunnel will mainly be concentrated during the day.The maximum values Vmax for the vertical velocity at points A, Band C are equal to 0.0369, 0.0287 and 0.0119, respectively, andlower than the threshold value A1 ¼ 0:10; the SBR guideline issatisfied for the vertical component along the surface. Themaximum values Vmax for the horizontal velocity at points A, Band C are equal to 0.2352, 0.3621 and 0.1148, respectively. Thesevalues exceed the threshold value A1 ¼ 0:10, but do not exceed theupper threshold value A2 ¼ 0:40. A further analysis based on thevibration level Vper over the evaluation period is needed to check

Table 4Limit values for Vmax and Vper for repeated occurrence of vibrations due to railways

according to the SBR guideline part B [1].

Type of facility A1 A2 A3 A1 A2 A3

Day and evening Night

Hospitals 0.10 0.40 0.05 0.10 0.20 0.05

Residential buildings 0.10 0.40 0.05 0.10 0.20 0.05

University and office buildings 0.15 0.60 0.07 0.15 0.60 0.07

Meeting rooms 0.15 0.60 0.07 0.15 0.60 0.07

Critical work spaces 0.10 0.10 – 0.10 0.10 –

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

Time [s]

Run

ning

effe

ctiv

e ve

loci

ty

A1

A2

−3 −2 −10

0.2

0.4

0.6

Tim

Run

ning

effe

ctiv

e ve

loci

ty

Fig. 22. The running effective horizontal velocity veff ðtÞ on the free surface at points (a)

occurrence of vibrations in residential buildings due to railways (day time) according t

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

Time [s]

Run

ning

effe

ctiv

e ve

loci

ty

A1

A2

−3 −2 −10

0.2

0.4

0.6

Ti

Run

ning

effe

ctiv

e ve

loci

ty

Fig. 23. The running effective vertical velocity veff ðtÞ on the free surface at points (a) A,

occurrence of vibrations in residential buildings due to railways (day time) according t

whether the standard is satisfied or not. Since the exact number oftrain passages is not known, an inverse procedure is followed todetermine the maximum number of train passages that can beallowed such that the SBR guideline is not violated. For the sake ofsimplicity, it is assumed that only one train passage occurs perinterval of 30 s. It has been agreed by the authorities that the SBRguideline should be met in buildings located at a lateral distanceof 50 m from the Groene Hart tunnel. The farthest observationpoint considered in this paper is point C, which is located at alateral distance of 43.5 m from the tunnel. Using the limiting valueA3 ¼ 0:05 for residential buildings and the maximum vibrationlevel veff ;max;30;i ¼ 0:1148 at the point C, the number of trainpassages that can be allowed during the day is equal to 273.Likewise for the evening time, the maximum number of trainpassages that can be allowed is 91. The present frequency of theThalys train in the Groene Hart area is less than the number ofpassages that could be allowed and, therefore, the SBR guideline isnot expected to be violated. If the observation point B at a distanceof 14.5 m from the tunnel is chosen, then the maximum number oftrain passages that can be allowed during the day and evening iscalculated as 27 and 9, respectively. These numbers may becritical for buildings located closer to the tunnel as the SBRguideline is violated if the present frequency of the Thalys highspeed train in the region is slightly increased.

The analysis presented in this paper is based on free fieldvibration levels. A more rigorous analysis can be carried out bycalculating the response in buildings to check whether theguideline is satisfied or not. Due to presence of soft soil layers,the buildings in the region are likely to be founded on pilefoundations. The vibration levels on the foundation and in thebuilding can be determined by solving a dynamic soil–structureinteraction problem [38]. The vibrations at the foundation level

0 1 2 3e [s]

A1

A2

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

Time [s]

Run

ning

effe

ctiv

e ve

loci

ty

A1

A2

A, (b) B and (c) C. Superimposed on the graphs are the limits A1 and A2 for repeated

o the SBR guideline part B (gray lines).

0 1 2 3me [s]

A1

A2

−3 −2 −1 0 1 2 30

0.2

0.4

0.6

Time [s]

Run

ning

effe

ctiv

e ve

loci

ty

A1

A2

(b) B and (c) C. Superimposed on the graphs are the limits A1 and A2 for repeated

o the SBR guideline part B (gray lines).

ARTICLE IN PRESS


depend on the relative stiffness between the foundation (piles)and the soil and are expected to be less than the free fieldvibrations as the soil is very soft compared to the foundation. Theresponse in the building may be amplified due to resonances offloors and other structural elements. Nevertheless, it can beconcluded as per the SBR guideline applied at a lateral distance of43.5 m from the tunnel, that the operation of Thalys high speedtrains in the Groene Hart area is not expected to be a matter ofconcern for human exposure to vibrations.

6. Conclusion

In this paper, a coupled periodic finite element–boundaryelement model has been used to model the Groene Hart tunneland predict the response due to the passage of a Thalys high speedtrain.

The shear wave velocity of the soil has been determined fromSCPT tests, while the density has been obtained from classical soilmechanics tests. The soil is completely saturated and consists oflayers of peat, clay and sand. For numerical computations, alayered halfspace with four layers on top of a halfspace isconsidered. The top layers consisting of peat and clay are verysoft. This results in an increase of the horizontal response on thesurface due to amplification of vertically polarized shear waves, aphenomenon that is explained by analyzing the Green’s functionsof the layered halfspace.

The tunnel geometry is complex as it consists of the tunnelinvert, a technical gallery and a partition wall; it is convenientlymodelled using the finite element method. The track is efficientlyincorporated in the tunnel using the Craig–Bampton substructur-ing method.

The response due to the passage of a Thalys high speed train inthe tunnel is computed. The response on the track is due to thequasi-static as well as the dynamic axle loads. At frequenciesabove 8 Hz, the free field response is dominated by the dynamicaxle loads. The contribution of the quasi-static axle loads is alsopresent but limited to frequencies below 8 Hz. The amplitude ofthe quasi-static response increases with the train speed. More-over, the frequency content spreads to higher frequencies as thetrain speed increases. The horizontal response on the free surfaceis higher than the vertical response as it is amplified due to thepresence of soft top layers.

The SBR guideline has been applied to evaluate the humanresponse to vibrations from underground railways. Vibrationlevels are calculated at three observation points on the freesurface and compared to limit values. The vertical component ofthe velocity complies with the SBR guideline as it is lower that thelower threshold value in the SBR guideline. For the horizontalcomponent, the vibration level Vmax lies between the lower andupper threshold. Based on the maximum vibration intensity valueVmax at a lateral distance of 43.5 m from the tunnel on the freesurface, the maximum number of train passages that are possibleduring the day and in the evening without violating the SBRguideline is evaluated as 273 and 91, respectively. These numberssuggest that the operation of the Thalys high speed train in theGroene Hart region will not cause serious vibration problems.

Acknowledgments

The results presented in this paper have been obtained withinthe frame of the SBO project IWT 03175 ‘‘Structural damage due todynamic excitation: a multi-disciplinary approach’’, funded byIWT. The third author is a Postdoctoral fellow of the Research

Foundation—Flanders (FWO-Vlaanderen). The financial support ofIWT and FWO-Vlaanderen is gratefully acknowledged.

The authors are grateful to Dr. H. Stuit and Dr. W. Gardien fromMovares and to Dr. P. Holscher and Dr. V. Hopman from Deltaresfor providing useful information and input data for performingthis analysis.

References

[1] Stichting Bouwresearch. SBR deel B: Hinder voor personen in gebouwen doortrillingen: meet—en beoordelingsrichtlijn; 2002.

[2] Sheng X, Jones CJC, Petyt M. Ground vibration generated by a harmonic loadacting on a railway track. Journal of Sound and Vibration 1999;225(1):3–28.

[3] Sheng X, Jones CJC, Petyt M. Ground vibration generated by a load movingalong a railway track. Journal of Sound and Vibration 1999;228(1):129–56.

[4] Sheng X, Jones CJC, Thompson DJ. A comparison of a theoretical model forquasi-statically and dynamically induced environmental vibration from trainswith measurements. Journal of Sound and Vibration 2003;267(3):621–35.

[5] Sheng X, Jones CJC, Thompson DJ. A theoretical model for ground vibrationfrom trains generated by vertical track irregularities. Journal of Sound andVibration 2004;272(3–5):937–65.

[6] Auersch L. The excitation of ground vibration by rail traffic: theory of vehicle–track–soil interaction and measurements on high-speed lines. Journal ofSound and Vibration 2005;284(1–2):103–32.

[7] Metrikine AV, Verichev SN, Blauwendraad J. Stability of a two-mass oscillatormoving on a beam supported by a visco-elastic half-space. InternationalJournal of Solids and Structures 2005;42:1187–207.

[8] Metrikine AV, Popp K. Steady-state vibrations of an elastic beam on a visco-elastic layer under moving load. Archive of Applied Mechanics 2000;70:399–408.

[9] Lombaert G, Degrande G, Clouteau D. Numerical modelling of free field trafficinduced vibrations. Soil Dynamics and Earthquake Engineering2000;19(7):473–88.

[10] Lombaert G, Degrande G, Kogut J, Franc-ois S. The experimental validation of anumerical model for the prediction of railway induced vibrations. Journal ofSound and Vibration 2006;297(3–5):512–35.

[11] Lombaert G, Degrande G. Ground-borne vibration due to static and dynamicaxle loads of InterCity and high speed trains. Journal of Sound and Vibration2009;319(3–5):1036–66.

[12] Forrest JA, Hunt HEM. A three-dimensional tunnel model for calculation oftrain-induced ground vibration. Journal of Sound and Vibration2006;294:678–705.

[13] Forrest JA, Hunt HEM. Ground vibration generated by trains in undergroundtunnels. Journal of Sound and Vibration 2006;294:706–36.

[14] Hussein MFM, Hunt HEM. A numerical model for calculating vibration from arailway tunnel embedded in a full-space. Journal of Sound and Vibration2007;305:401–31.

[15] Muller K, Grundmann H, Lenz S. Nonlinear interaction between a movingvehicle and a plate elastically mounted on a tunnel. Journal of Sound andVibration 2008;310:558–86.

[16] Clouteau D, Arnst M, Al-Hussaini TM, Degrande G. Freefield vibrations due todynamic loading on a tunnel embedded in a stratified medium. Journal ofSound and Vibration 2005;283(1–2):173–99.

[17] Degrande G, Schevenels M, Chatterjee P, Van de Velde W, Holscher P, HopmanV, Wang A, Dadkah N. Vibrations due to a test train at variable speeds in adeep bored tunnel embedded in London clay. Journal of Sound and Vibration2006;293(3–5):626–44.

[18] Sheng X, Jones CJC, Thompson DJ. Prediction of ground vibration from trainsusing the wavenumber finite and boundary element methods. Journal ofSound and Vibration 2006;293:575–86.

[19] Andersen L, Jones CJC. Coupled boundary and finite element analysis ofvibration from railway tunnels—a comparison of two- and three-dimensionalmodels. Journal of Sound and Vibration 2006;293:611–25.

[20] Sheng X, Jones CJC, Thompson DJ. Modelling ground vibrations from railwaysusing wavenumber finite- and boundary-element methods. Proceedings ofthe Royal Society A—Mathematical Physical and Engineering Sciences2005;461:2043–70.

[21] Gupta S, Liu W, Degrande G, Lombaert G, Liu W. Prediction of vibrationsinduced by underground railway traffic in Beijing. Journal of Sound andVibration 2008;310:608–30.

[22] Hopman V, de Feijter JW. Grondonderzoek HSL, Groene Hart tunnel,seismische sonderingen. Report CO-380840/671, GeoDelft; July 2000.

[23] Biot MA. Theory of propagation of elastic waves in a fluid-saturated poroussolid. I. Low-frequency range. Journal of the Acoustical Society of America1956;28(2):168–78.

[24] Schevenels M, Degrande G, Lombaert G. The influence of the depth of theground water table on free field road traffic induced vibrations. InternationalJournal for Numerical and Analytical Methods in Geomechanics2004;28(5):395–419.

[25] Van den Berghe H. Trillingen ten gevolge van HST verkeer in de GroeneHarttunnel. Master’s thesis, Department of Civil Engineering, K.U.Leuven;2008.

ARTICLE IN PRESS


[26] N.N. Report reference B0-IVW-050020642, Movares, Geotechnical Engineer-ing and Environment Group; April 2005.

[27] Hamid A, Yang TL. Analytical description of track-geometry vibrations.Transportation Research Record 1981;838:19–26.

[28] Diehl RJ, Holm H. Roughness measurements-have the necessities changed?.Journal of Sound and Vibration 2006;293:777–83.

[29] International Organization for Standardization. ISO 3095:2005: railwayapplications—acoustics—measurement of noise emitted by railbound vehi-cles; 2005.

[30] Gupta S. Numerical modelling of subway induced vibrations. PhD thesis,Department of Civil Engineering, K.U.Leuven; 2008.

[31] Chebli H, Othman R, Clouteau D. Response of periodic structures due tomoving loads. Comptes Rendus Mecanique 2006;334:347–52.

[32] Clouteau D, Elhabre ML, Aubry D. Periodic BEM and FEM-BEM coupling:application to seismic behaviour of very long structures. ComputationalMechanics 2000;25:567–77.

[33] Schevenels M, Franc-ois S, Degrande G. EDT: an elasto dynamics toolbox FORMATLAB. Computers & Geosciences 2009;35(8):1752–4.

[34] Knothe K, Grassie SL. Modelling of railway track and vehicle/trackinteraction at high frequencies. Vehicle Systems Dynamics 1993;22:209–262.

[35] Degrande G, Lombaert G. An efficient formulation of Krylov’s predictionmodel for train induced vibrations based on the dynamic reciprocity theorem.Journal of the Acoustical Society of America 2001;110(3):1379–90.

[36] Metrikine AV, Vrouwenvelder T. Surface ground vibration due to a movingtrain in a tunnel: two-dimensional model. Journal of Sound and Vibration2000;234(1):43–66.

[37] Deutsches Institut fur Normung. DIN 4150 Teil 2: Erschutterungen imBauwesen, Einwirkungen auf Menschen in Gebauden; 1999.

[38] Fiala P, Degrande G, Augusztinovicz F. Numerical modelling of ground bornenoise and vibration in buildings due to surface rail traffic. Journal of Soundand Vibration 2007;301(3–5):718–38.

gupta 2010 soil dynamics and earthquake engineering

Documents

groene hart area

groene hart region

introductionthe groene

problem of vibrations

groene hartarea

rural area

grasslands of thegroene

high speed link south