ECCOMAS Thematic Conference on Computational Methodsin Structural Dynamics and Earthquake Engineering

M. Papadrakakis, D.C. Charmpis, N.D. Lagaros, Y. Tsompanakis (eds.)Rethymno, Crete, Greece, 13-16 June 2007

EXPERIMENTAL VALIDATION OF A NUMERICAL MODEL FORSUBWAY INDUCED VIBRATIONS

S. Gupta, G. Degrande and G. Lombaert

Department of Civil Engineering, K.U. LeuvenB-3001 Leuven, Belgium

e-mail: shashank.gupta@bwk.kuleuven.be

Keywords: subway induced vibrations, periodic FE-BE method, dynamictrack-tunnel-soilinteraction, dynamic train-track interaction, ground vibration.

Abstract. This paper presents the experimental validation of a numerical model for the pre-diction of subway induced vibrations. The model fully accounts for the dynamic interactionbetween the train, the track, the tunnel and the soil. The three-dimensional dynamic tunnel-soilinteraction problem is solved with a subdomain formulation, using a finite element formulationfor the tunnel and a boundary element method for the soil, modelled as a horizontally layeredelastic half space. The periodicity of the tunnel and the soil in the longitudinal direction isexploited using the Floquet transform, limiting the discretization effort to a single bounded ref-erence cell. The track-tunnel-soil interaction problem issolved in the frequency-wavenumberdomain and the wave field radiated into the soil is computed.

A general analytical formulation is used to compute the response of three-dimensional in-variant or periodic media that are excited by moving loads. The numerical model is capableof dealing with various types of excitations, including quasi-static loads, random loads due torail and wheel unevenness, impact excitation due to rail joints and wheel flats, and parametricexcitation due to sleeper periodicity.

The numerical model is validated by means of several experiments that have been performedat a site in Regent’s Park on the Bakerloo line of London Underground. Vibration measure-ments have been performed on the axle boxes of the train, on the rail, the tunnel invert and thetunnel wall, and in the free field, both at the surface and at a depth of 15 m. Prior to these vi-bration measurements, the dynamic soil characteristics and the track characteristics have beendetermined. The Bakerloo line tunnel of London Undergroundhas been modelled using thecoupled periodic FE-BE approach and free field vibrations due to the passage of a train havebeen predicted and compared to the measurements.

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S. Gupta, G. Degrande and G. Lombaert

1 INTRODUCTION

Ground-borne vibrations induced by underground railways are a major environmental con-cern in urban areas. These vibrations propagate through thetunnel and the surrounding soilinto nearby buildings, causing annoyance to people. Vibrations are perceived directly or theyare sensed indirectly as re-radiated noise. The frequency range of interest for subway inducedvibrations is 1-80 Hz and for the re-radiated noise it is 1-200 Hz. To quantify these vibrations,great efforts have been made in recent years to develop the prediction models [1, 2, 3, 4, 5, 6]that account for the three dimensional dynamic track-tunnel-soil interaction. These advancedmodels take the advantage of the invariance (or the periodicity) of the geometry along the tunnelaxis using a Fourier transformation (or a Floquet transformation).

This paper concentrates on the coupled periodic finite element-boundary element (FE-BE)model [1, 2] that was developed within the frame of the EC-growth project CONVURT [7].Within the frame of the CONVURT project elaborate in situ vibration measurements have beenperformed at a site in Regent’s Park situated above the north- and south-bound Bakerloo linetunnels of London Underground [8]. These tunnels are deep-bored segmented tunnels with acast iron lining and a single track, embedded in London clay at a depth of 28 m. The vibrationmeasurements have been carried out on a straight section of the north-bound Bakerloo line.The reference section is situated at kilometer post 46.306 (figure 1), which is 581 m west ofRegent’s Park station or approximately 200 m east of Baker Street station. The measurementsite is surrounded by a boating lake in the north and by the Outer Circle on the south. A row ofRegency houses is built along York Terrace West, parallel tothe Outer Circle, at a distance ofabout 70 m from the north-bound Bakerloo line tunnel.

Figure 1: Plan of the measurement site in Regent’s Park.

Vibration measurements have been performed during engineering hours at night for 35 pas-sages of a test train in the north-bound Bakerloo line tunnelat a speed between 20 and 50 km/h.In addition, rail and wheel roughness have been measured, while the track characteristics havebeen determined by rail receptance measurements [9]. The dynamic soil characteristics havebeen determined by in situ tests (SCPT, SASW) and by laboratory testing on undisturbed sam-

2

S. Gupta, G. Degrande and G. Lombaert

ples [10]. The results of the vibration measurements are presently used to validate the coupledperiodic FE-BE model.

The modelling of the Bakerloo line tunnel of London Underground using the coupled peri-odic FE-BE approach is described and the results are compared to the in situ vibration measure-ments. This study demonstrates the applicability of the numerical model for the prediction ofvibrations from underground railways.

2 THE NUMERICAL METHOD

Within the frame of the CONVURT project [7], a coupled periodic FE-BE model has beendeveloped that exploits the longitudinal invariance or periodicity of the track-tunnel-soil system[1, 2]. The three-dimensional dynamic tunnel-soil interaction problem is solved with a subdo-main formulation, using a finite element method for the tunnel and a boundary element methodfor the soil, modelled as a horizontally layered elastic half space. The response to movingloads is deduced from the frequency content of the axle loadsand the transfer functions in thefrequency-wavenumber domain.

2.1 Response due to moving loads

In the fixed frame of reference, the distribution ofn vertical axle loads moving in the longi-tudinal directioney on the coupled track-tunnel-soil system is written as the summation of theproduct of Dirac functions that determine the time-dependent positionxk = xk0, yk0+vt, zk0

T

and the time historygk(t) of thek-th axle load:

ρb(x, t) =n∑

k=1

δ(x− xk0)δ(y − yk0 − vt)δ(z − zk0)gk(t)ez (1)

yk0 is the initial position of thek-th axle that moves with the train speedv along they-axis andez denotes the vertical unit vector.

An infinite periodic structure can be analyzed using the Floquet transform method [1, 2] byrestricting the problem domainΩ to a single periodic unitΩ (reference cell). If the spatial periodis L, then the positionx of any point in the problem domain is decomposed asx = x + nLey,wherex is the position in the reference cell andn is the cell number. The response to movingloads in case of periodic domains is given by [11]:

ui(x + nyLey, ω) =1

2π

n∑

k=1

∫

∞

−∞

gk(ω − kyv) exp [−iky(nyL− yk0)]

×∫ L/2

−L/2

exp (−iky yk)˜hzi(x

′, x, κy, ω) dy′ dky (2)

whereκy = ky − 2mπ/L and ky = (ω − ω)/v. The transfer function in the frequency-

wavenumber domain˜hzi(x′, x, κy, ω) is the Floquet transform of the transfer function in the

frequency-spatial domainhzi(x′, x + nyLey, ω).

It can be seen from equation (2) that the transfer function˜hzi(x

′, x, κy, ω) and the frequencycontent of the axle loadgk(ω) are needed to compute the response to moving loads.

2.2 Transfer functions

The transfer functions are computed by means of the coupled periodic FE-BE model usingthe classical domain decomposition approach based on the finite element method for the tunnel

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S. Gupta, G. Degrande and G. Lombaert

and the boundary element method for the soil. The Floquet transform is used to exploit theperiodicity of geometry and to restrict the problem domain to a single bounded reference cell.

(a)

rp 1.88m

ri 1.829m

re 1.953 m

ts 0.022m

hb 0.102m

(b)

Figure 2: (a) Cross section of the metro tunnel on the Bakerloo line at Regent’s Park. (b) Finite element model ofthe reference cell.

The Bakerloo line tunnel of London Underground is a deep bored tunnel with a cast ironlining and a single track, embedded in London clay at a depth of 28 m. The tunnel has aninternal radius of 1.83 m and a wall thickness of 0.022 m (figure 2). There are six longitudinalstiffeners and one circumferential stiffener at an interval of 0.508 m, resulting in a periodicstructure. Figure 2b shows the finite element model of the tunnel’s reference cell, where shellelements have been used for the cast iron lining, while the longitudinal and circumferentialstiffeners are modelled using the beam elements. The concrete on the tunnel invert has beenmodelled using the 8-node volume elements with incompatible bending modes.

Dynamic soil characteristics have been determined by in situ and laboratory testing [8]. Thetesting revealed that the tunnel is embedded in a layered soil consisting of a single shallow layerwith a thickness of 5 m on top of a homogeneous half space consisting of London clay. The toplayer has a shear wave velocityCs = 275 m/s, a longitudinal wave velocityCp = 1964 m/s, adensityρs = 1980 kg/m3 and a material damping ratioβs = 0.042. The underlying half spacehas a shear wave velocityCs = 220 m/s, a longitudinal wave velocityCp = 1571 m/s, a densityρs = 1980 kg/m3 and a material damping ratioβs = 0.039.

The track is a non-ballasted concrete slab track with Bullhead rail supported on hard woodensleepers nominally spaced atd = 0.95 m with cast iron chairs. Both ends of a sleeper areconcreted into the invert and the space between the sleepersis filled with shingle. The rails havea mass per unit lengthρrAr = 47 kg/m and a bending stiffnessErIr = 3.04×106 Nm2. The railsare not supported by rail pads and the resilience is mainly provided by the timber sleepers, whichhave a varying stiffness depending on its moisture content.The track is modelled as infinitebeams on continuous supports. The model consists of two infinite Euler beams representing therails and the mass elements representing the sleepers. The mass of the sleepers is distributedin the longitudinal direction with a mass per unit lengthms = Ms/d = 52.63 kg/m. Asthere are no rail pads, a stiff connection is assumed betweenthe rails and the sleepers, whilethe sleepers are continuously supported on the tunnel invert with springs of vertical stiffnessks = ks/d = 80 MN/m2. The continuous elastic support below the sleepers accounts for theresilience of the timber sleepers. A high value of the damping cs = 12× 104 Ns/m2 is assumedin the elastic support to suppress the resonance, as no resonance has been observed in the railreceptance measurements within 200 Hz.

As the reference cell of the tunnel is bounded, the tunnel displacementsut(x, κ, ω) are de-

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S. Gupta, G. Degrande and G. Lombaert

200 400 600−200

−180

−160

−140

−120

Dis

plac

emen

t [dB

ref m

/N]

Frequency [Hz]

Figure 3: Measured and computed vertical rail receptance ofthe rail. The graph displays the receptance measuredover the sleeper (red dashed line) and at mid span between thesleepers (red solid line) and the rail receptancecomputed, accounting for the track-tunnel-soil interaction (blue line) and for the track on a rigid base (black line).

composed on a basis of functionsψm(x, κ), while the soil displacementsus(x, κ, ω) are writtenas the superposition of waves that are radiated by the tunnelinto the soil.

The weak or variational formulation of the problem results in the following system of equa-tions in the frequency-wavenumber domain [1, 2, 12]:

[

Kt(κ) − ω2Mt(κ) + Ks(κ, ω)]

α(κ, ω) = Ft(κ, ω) (3)

whereKt(κ) andMt(κ) are the projection of the finite element stiffness matrix andmass matrixof the tunnel’s reference cell on the tunnel modesψm(x, κ), while Ks(κ, ω) is the dynamicstiffness matrix of the soil calculated with a periodic boundary element formulation with Green-Floquet functions defined on the periodic structure with periodL along the tunnel [1, 12].

Equation (3) is solved to obtain the displacement field in thereference cell in the frequency-wavenumber domain. The displacements in the frequency-spatial domain are obtained usingthe inverse Floquet transform[1, 2].

The Craig-Bampton substructuring technique is used to efficiently incorporate a track in thetunnel. The advantage of this approach is that the soil impedance only depends on the periodictunnel modes and does not change when alternative track structures are considered.

Figure 3 shows the measured and the computed receptance of the rail. The vertical rail recep-tance has been calculated accounting for the full track-tunnel-soil interaction in the frequencyrange 1-150 Hz and has been compared to the rail receptance for the case of the track on a rigidbase. The rail receptance in both cases is the same in the frequency range 1-150 Hz and there-fore, the measured receptance has been compared to the receptance of the track on a rigid baseat frequencies above 150 Hz. The computed receptance is close to the measured receptance atfrequencies above 200 Hz. The receptance measured at mid span between the sleepers shows apinned-pinned resonance frequency of 380 Hz, which is not visible in the calculated results, asa homogeneous model of the track has been assumed.

The wave field radiated into the soil is computed using the dynamic representation theoremin the unbounded soil domain corresponding to the referencecell. This corresponds to thetransfer function in the frequency-wavenumber domain thatis used in equation (2) to computethe incident wave field due to a moving train.

In the following, the transfer functions are calculated with 0.5 N load on each rail to accountfor the equal distribution of the train load on both rails. Figure 4 shows the transfer function atthe tunnel invert, and in the free field at the surface and at a depth of 15 m. The observationpoint at the surface and at the depth is at a lateral distance of 23.5 m from the center of thetunnel. 84 periodic modes of the second kindψm(x, κ) are used for the computation, which

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S. Gupta, G. Degrande and G. Lombaert

(a)0 20 40 60 80 100

−210

−190

−170

−150D

ispl

acem

ent [

dBre

f m/N

]

Frequency [Hz] (b)0 20 40 60 80 100

−260

−240

−220

−200

Dis

plac

emen

t [dB

ref m

/N]

Frequency [Hz] (c)0 20 40 60 80 100

−260

−240

−220

−200

Dis

plac

emen

t [dB

ref m

/N]

Frequency [Hz]

Figure 4: (a) Vertical transfer function at the tunnel invert and in the free field (b) at the surface and (c) at a depthof 15 m.

include 60 free tunnel modes and 24 track modes on a rigid base. The transfer function at thetunnel invert does not show any marked resonance due to the dynamic tunnel-soil interactionand the dissipation of energy due to material and geometrical damping in the soil. The transferfunctions in the free field show undulations, which is due to the interference of compressionand shear or Rayleigh waves. The experimental transfer functions are not available from thismeasurement campaign and thus, the experimental validation of the transfer functions is notpossible.

2.3 The train-track interaction forces

There are various excitation mechanisms responsible for generating vibrations due to movingtrains. For the experimental validation of the numerical model, three excitation mechanismsare considered: the quasi-static excitation, the unevenness excitation due to wheel and railroughness, and the impact excitation due to rail joints.

The test train employed for vibration measurements on the Bakerloo line consisted of sevencars: a driving motor car, a trailer car, two non-driving motor cars, two trailer cars and a drivingmotor car. The length of a motor car is 16.09 m, while the length of the trailer car is 15.98 m.The bogie and axle distances on all cars are 10.34 m and 1.91 m,respectively. The distancebetween the first and the last axle of the train is 108.33 m. Thewheels are of the monobloc typeand have a diameter of about 0.70 m. The tare mass of a motor caris 15330 kg, while the bogiemass is 6690 kg and the mass of wheelset is 1210 kg. The tare mass of a trailer car is 10600 kg,while the bogie mass is 4170 kg and the mass of a wheelset is 950kg.

The quasi-static excitation occurs when successive axles of the train pass over the track andcan be modelled as constant forces moving along the track with the train speedv. The timehistorygk(t) of the constant load is equal to the total load of the train distributed over the trainaxlesPk. The Fourier transformgk(ω) equals2πPkδ(ω). The frequency content of the free fielddisplacements is obtained by substitutinggk(ω) in equation (2).

2.3.1 Unevenness excitation

Roughness is the main generation source of vibrations from moving trains. For the simplecase of the vertical interaction between the wheel and the rail, the contact forcesg(ω) at thetrain’s axles in the frequency domain, generated due to an uneven profile are given as [13]:

[Cv(ω) + Ct(ω)]g(ω) = uw/r(ω) (4)

whereuw/r(ω) is the relative displacement (roughness) between the wheeland the rail, whileCv(ω) andCt(ω) are the compliance of the vehicle and the track, respectively. The frequency

6

S. Gupta, G. Degrande and G. Lombaert

content of the rail unevennessuw/r(ω) is calculated from the wavenumber domain representa-tion uw/r(ky) of the unevennessuw/r(y).

The train can be well represented with the vehicle’s unsprung mass, as the vehicle’s primaryand secondary suspensions isolate the body and the bogie from the wheel set at frequenciesabove a few Hertz. In this case, the vehicle compliance matrix is equal to the diagonal matrixCv(ω) = diag−1/Muω

2 of order 28. The track complianceCt(ω) is calculated in the fixedframe of reference as the train speed considered is 47.6 km/h, which is much less than thecritical speed of the track-tunnel-soil system. Like the transfer functions, the track complianceis also computed for 0.5 N load on each of the two rails. The element Ct

lk(ω) of the trackcompliance matrix represents the track response at the position of axlel due to the applied loadat axlek.

Theoretically, rail roughnessuw/r(y) is expressed as a stochastic process characterized by asingle-sided power spectral density (PSD)Sw/r(ky), written as a function of the wavenumberky = ω/v = 2π/λy [14]:

Sw/r(ky) = Sw/r(ky0)

(

ky

ky0

)

−w

(5)

The parametersSw/r(ky0) andw depend on the quality of the rail, whileky0 = 1 rad/m andw = 3.5 are commonly assumed. An artificial profileuw/r(y) is generated from the PSD-curvebased on the superposition of simple random processes with known statistical properties.

Rail roughness have been measured on the site using Muller BBM rail roughness measure-ment equipment (RM1200E). Measurements were taken on both rails over 50 m of the track atseveral positions. The data were processed to produce spatial profiles and averaged one-thirdoctave band spectra. It is seen that the rail roughness dominates at low wavenumbers, whilethe wheel roughness dominates at high wavenumbers [8]. The low wavenumbers correspondto longer wavelengths and are responsible for excitation inthe frequency range of interest (1-80 Hz); higher wavenumbers are important for higher frequency excitation that may give riseto re-radiated noise in buildings in the frequency range upto 200 Hz. In these measurements,no rail unevenness with wavelengths longer than 0.1 m could be measured with the availableequipment, thus restricting the analysis to frequencies above 130 Hz for a train speed of 47.6km/h.

More information about the roughness can be derived from theaxle box vibrations. Axlebox vibrations have been measured on six axle boxes during the whole journey of the test trainon the section between Regent’s Park and Baker Street stations. Figure 5a shows the timehistory of the acceleration of axle box 3 for a period of time corresponding to the passage ofa train over the test section at a speed of 47.6 km/h. The axle box response is affected bythe combined rail and wheel roughness, although the peaks inthe response are clearly due tothe passage of the axle over joints in the rail. One-third octave band spectra of the axle boxdisplacements can be obtained from the axle box accelerations. The axle box displacementswill be approximately equal to the combined rail and wheel roughness in the lower frequencyrange (below the wheel-track resonance frequency), where the vehicle compliance dominatesthe track compliance. However, it has been observed here that, even at high frequencies, theaxle box displacements correspond well with the measured roughness. Figure 5b shows theroughness derived from axle box accelerations and the measured roughness as a function offrequency for train speed 47.6 km/h. The roughness from the PSD curve (cfr. subsection 2.3.1)has been fitted against the roughness derived from the measurements. The parameterSw/r(ky0)

7

S. Gupta, G. Degrande and G. Lombaert

is found to be equal to1.2 × 10−5 and corresponds to a very poor quality of the rail. This isalso confirmed by the observation on the site that the rails were indeed heavily corrugated. Itshould be mentioned that the roughness from the axle box acceleration does not give a goodrepresentation at very low frequencies below 5 Hz. This is because the wavelengths are verylong at such low frequencies and are not represented well in the signal, moreover a band passfilter with a high-pass frequency of 3 Hz has been used in the processing of the recorded data.

(a)50 52 54 56 58

−200

−100

0

100

200

Time [s]

Acc

ele

ratio

n [m

/s2]

(b)10

010

110

210

310

4−20

0

20

40

60

80

Frequency [Hz]

Rou

ghne

ss [d

Bre

f µ m

]

Figure 5: (a) Time history of the vertical acceleration on axle box 3 for a period of time corresponding to thepassage of a train length over the test section at a train speed of 47.6 km/h. (b) One-third octave band roughnessspectra from the axle box accelerations (red line), from thePSD curve (black line) and from the measurements onthe left (blue line) and right (green line) rail and wheel.

Visual inspection of the track also revealed that there werea number of rail joints in thevicinity of the reference section, which could generate significant impact forces at the wheel-railinterface. Three rail joints were identified in the immediate vicinity of the reference sectionsat positionsy = 31 m, y = 19 m andy = −10 m. The height of the three rail joints isassumed asH = 1 mm. In the numerical model, these rail joints have been considered onboth rails at the same location, as the model accounts for identical inputs on the two rails. Thisis, however, different from the actual measurement conditions, where the rail joints were atdifferent positions on both rails.

The simplest way to predict the excitation force due to rail joints is to determine the relativedisplacementuw/r(y) between the wheel and the rail, when the wheel rolls over the rail jointon a rigid track and subsequently solve a dynamic wheel-railinteraction problem (4). In thesimplest situation, the trajectoryuw/r(y) of the center of the wheel while passing over the jointcan be given as:

uw/r(y) = H − y2/(2R) (6)

whereR is the radius of the wheel andy is measured from the rail joint in the positive direction.This expression represents a step-up joint. For a step-downjoint the expression is still valid butfor the oppositey−direction. Furthermore, it has been assumed that the heightdifferenceH ismuch less that the radius of the wheelR (H ≪ R).

It should be noted that this model is linear and the analysis is performed in the frequency-domain. The non-linear effects like separation of the wheeland the rail are not accounted for,and therefore this model can only be used for the trains running at very low speeds. The modelsdeveloped by Wu and Thompson [15] account for the non-lineareffect and simulate the wheel-rail impact in the time domain. However, the approach described here yields satisfactory resultsand is followed in this paper.

The train-track interaction forces are calculated from equation (4) with total roughnessuw/r(ω)equal to the sum of the roughness from the PSD curve and equivalent roughness for the joints.

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S. Gupta, G. Degrande and G. Lombaert

(a)0 50 100 150

−50

0

50

100

150

Frequency [Hz]

Rou

ghne

ss [d

Bre

f µ m

]

(b)0 50 100 150

0

5000

10000

15000

Frequency [Hz]

Con

tact

forc

e [N

/Hz]

Figure 6: (a) Magnitude of the roughness and (b) the contact force as a function of the frequency for a train speedof 47.6 km/h.

Figure 6 shows the total roughness and the contact force at the front axle of the train for a trainspeed of 47.6 km/h. The frequency content of these forces exhibits a clear maximum near thetrain-track resonance frequency between 50 and 60 Hz.

3 RESPONSE DURING THE PASSAGE OF A TRAIN IN THE BAKERLOO LINETUNNEL

The response can be calculated by adding the contribution ofthe dynamic forces and thequasi-static forces in the frequency domain that follows from equation (2).

Firstly, the response in the tunnel is compared to the experimental results on the rail [9] andthe tunnel invert [9]. Figures 7 and 8 compare the predicted and measured vertical velocity onthe rail (A1) and tunnel invert (A6) during the passage of thetest train at a speed of47.6 km/h.On the rail, the contribution of each axle can clearly be distinguished, resulting in a quasi-discrete spectrum at low frequencies governed by the bogie and axle distances and the trainspeed. A peak corresponding to the axle passage frequency offa = v/La = 6.92 Hz (La =1.91 m) is observed in the predicted as well as the measured spectra.The dominant frequencycontent is situated in the frequency range above 40 Hz. The predicted results do not showa strong influence of the rail joints in the reference section, which is about 10.8 m from thenearest rail joint. The vibration levels are maximum on the rail and decrease on the tunnel

(a)−10 −5 0 5 10

−0.2

0

0.2

Time [s]

Vel

ocity

[m/s

]

A1

(b)−10 −5 0 5 10

−0.02

0

0.02

Time [s]

Vel

ocity

[m/s

]

A6

Figure 7: The experimental (red line) and computed (black line) time history of the response on (a) the rail (A1)and (b) the tunnel invert (A6) during the passage of a test train at a speeds 47.6 km/h.

9

S. Gupta, G. Degrande and G. Lombaert

(a)0 50 100 150

0

0.005

0.01

0.015

0.02

Frequency [Hz]

Vel

ocity

[m/s

/Hz]

A1

(b)0 50 100 150

0

1

2

3x 10

−3

Frequency [Hz]

Vel

ocity

[m/s

/Hz]

A6

Figure 8: The experimental (red line) and computed (black line) frequency content of the response on (a) the rail(A1) and (b) the tunnel invert (A6) during the passage of a test train at a speeds 47.6 km/h.

invert. The response at the tunnel invert is underestimatedby the numerical model, in particularat frequencies above 100 Hz. This could be due to the underestimation of the impact forces atthe joints. The critical speed for a tramway, above which thewheel and rail separation occursduring the passage of a wheel over the rail joint is around 30-40 km/h and the simple modellingof the rail joints in the frequency domain is not accurate at high speeds.

Free field vibration measurements have been performed in Regent’s Park above the Bakerlooline tunnels. The vibrations measurements have been performed on the surface as well as at adepth of 15 m, where tri-axial accelerometers have been installed in a seismic cone [10]. In thispaper, the computed vertical response is compared to the measurements on the surface (FF02z)and at a depth of 15 m (FF03z), at a distance of 23.5 m from the tunnel.

(a)−10 −5 0 5 10−2

0

2x 10

−4

Time [s]

Vel

ocity

[m/s

]

FF02z

(b)−10 −5 0 5 10−2

0

2x 10

−4

Time [s]

Vel

ocity

[m/s

]

FF03z

Figure 9: The experimental (red line) and computed (black line) time history of the free field vibration (a) on thesurface (FF02) and (b) at the depth (FF03) during the passageof a test train at a speed of 47.6 km/h.

Figures 9 and 10 compare the experimental and computed time history and frequency con-tent of the vertical free field vibration at points FF02 and FF03 during the passage of a test trainat a speed of 47.6 km/h. Both the experimental and numerical results show that the dominantfrequency content is around the wheel-track resonance frequency of about 50 Hz and is relatedto the wheel and rail unevenness. A relatively good agreement between the experimental and

10

S. Gupta, G. Degrande and G. Lombaert

(a)20 40 60 80 100

0

2

4

6

x 10−5

Frequency [Hz]

Vel

ocity

[m/s

/Hz]

FF02z

(b)20 40 60 80 100

0

2

4

6

x 10−5

Frequency [Hz]

Vel

ocity

[m/s

/Hz]

FF03z

Figure 10: The experimental (red line) and computed (black line) frequency content of the free field vibration (a)on the surface (FF02) and (b) at the depth (FF03) during the passage of a test train at a speed of 47.6 km/h.

numerical results is observed for the response in the free field. The bogie passages are notclearly visible in the time history of the response in the free field as the tunnel is situated at aconsiderable depth. The vertical response at the surface (FF02z) has approximately the samemagnitude as the vertical component at depth at the same location (FF03z). Furthermore, itcan be observed that the amplitude decrease for increasing distance from the tunnel due to geo-metrical damping, while higher frequency components at increasing distance are significantlyattenuated by material damping in the soil.

4 CONCLUSIONS

In this paper, the experimental validation of a numerical model for the prediction of subwayinduced vibrations has been presented. The coupled periodic FE-BE model fully accounts forthe dynamic interaction between the train, the track, the tunnel and the soil. The response tomoving loads (trains) is computed by first considering the wheel-track interaction to estimatethe excitation forces and then solving the track-tunnel-soil interaction problem to compute thevibrations in the free field. Three excitation mechanisms have been considered in the computa-tions, however, the excitation due to rail (and wheel) unevenness is found as the most importantsource of vibrations in the free field.

An elaborate measurement campaign has been conducted at a site in Regent’s Park situatedabove the Bakerloo line tunnels of London Underground to validate the numerical model. In situvibration measurements have been performed on the axle boxes of the test train, in the tunneland in the free field. Apart from these measurements, other tests and measurements have alsobeen performed to determine the soil properties and track characteristics. It has been discussedin the paper how the preliminary data for the modelling of thetrain-track-tunnel-soil system canbe derived from these tests and subsequently the response tomoving trains can be calculated.The free field vibrations for the passage of a test train in theBakerloo line tunnel have beenpredicted and validated. The correspondence between the predicted and experimental results isreasonably good, given the large number of modelling uncertainties.

ACKNOWLEDGEMENTS

The results presented in this paper have been obtained within the frame of the SBO projectIWT 03175 ” Structural damage due to dynamic excitation: a multi-disciplinary approach”,funded by IWT Vlaanderen. Their financial support is gratefully acknowledged.

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S. Gupta, G. Degrande and G. Lombaert

REFERENCES

[1] D. Clouteau, M. Arnst, T.M. Al-Hussaini, and G. Degrande. Free field vibrations due to dynamicloading on a tunnel embedded in a stratified medium.Journal of Sound and Vibration, 283(1–2):173–199, 2005.

[2] G. Degrande, D. Clouteau, R. Othman, M. Arnst, H. Chebli,R. Klein, P. Chatterjee, andB. Janssens. A numerical model for ground-borne vibrationsfrom underground railway trafficbased on a periodic finite element - boundary element formulation. Journal of Sound and Vibra-tion, 293(3-5):645–666, 2006. Proceedings of the 8th International Workshop on Railway Noise,Buxton, U.K., 8-11 September 2004.

[3] J.A. Forrest and H.E.M. Hunt. A three-dimensional tunnel model for calculation of train-inducedground vibration.Journal of Sound and Vibration, 294(4-5):706–736, 2006.

[4] M.F.M. Hussein. Vibration from underground railways. PhD thesis, Department of Engineering,University of Cambridge, 2004.

[5] X. Sheng, C.J.C. Jones, and D.J. Thompson. Prediction ofground vibration from trains using thewavenumber finite and boundary element methods.Journal of Sound and Vibration, 293:575–586,2006.

[6] L. Andersen and C.J.C. Jones. Coupled boundary and finiteelement analysis of vibration from rail-way tunnels-a comparison of two- and three-dimensional models. Journal of Sound and Vibration,293:611–625, 2006.

[7] http://www.convurt.com, 2003.

[8] G. Degrande, M. Schevenels, P. Chatterjee, W. Van de Velde, P. Holscher, V. Hopman, A. Wang,and N. Dadkah. Vibrations due to a test train at variable speeds in a deep bored tunnel embeddedin London clay.Journal of Sound and Vibration, 293(3-5):626–644, 2006. Proceedings of the 8thInternational Workshop on Railway Noise, Buxton, U.K., 8-11 September 2004.

[9] A. Wang. Track measurements on London Underground Bakerloo Line. Report 16487-2, Pandrol,May 2003. CONVURT EC-Growth Project G3RD-CT-2000-00381.

[10] P. Holscher and V. Hopman. Test site Regent’s Park London. Soil description. Report 381540-104,Version 2, GeoDelft, December 2003. CONVURT EC-Growth Project G3RD-CT-2000-00381.

[11] S. Gupta, G. Degrande, H. Chebli, D. Clouteau, M.F.M. Hussein, and H.E.M. Hunt. A coupled pe-riodic fe-be model for ground-borne vibrations from underground railways. In C.A. Mota Soares,editor,Proceedings of the 3th European Conference on Computational Mechanics, Lisbon, Portu-gal, June 2006.

[12] D. Clouteau, M.L. Elhabre, and D. Aubry. Periodic BEM and FEM-BEM coupling: application toseismic behaviour of very long structures.Computational Mechanics, 25:567–577, 2000.

[13] G. Lombaert, G. Degrande, J. Kogut, and S. Francois. The experimental validation of a numericalmodel for the prediction of railway induced vibrations.Journal of Sound and Vibration, 297(3-5):512–535, 2006.

[14] H. Braun and T. Hellenbroich. Messergebnisse von Strassenunebenheiten.VDI Berichte, 877:47–80, 1991.

[15] T.X. Wu and D.J. Thompson. On the impact noise generation due to a wheel passing over railjoints. Journal of Sound and Vibration, 267:485–496, 2003.

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