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Experimental validation of a numerical model for
the ground vibration from trains in tunnels
Qiyun Jin; David Thompson; Daniel Lurcock; Martin Toward; Evangelos Ntotsios;
Samuel Koroma
Institute of Sound and Vibration Research, University of Southampton, Southampton, UK
Mohammed Hussein
Civil & Architectural Engineering Department, College of Engineering, Qatar University,
P.O. Box 2713, Qatar
Summary
Ground vibration and ground-borne noise from trains in tunnels are attracting increasing attention
from researchers and engineers. They are important environmental issues related with the
operation of underground networks in intensively-populated urban areas. An accurate prediction
for this train-induced vibration can be very helpful in the implementation of countermeasures to
achieve the control of vibration or noise levels. In this paper, a numerical model is introduced
based on the 2.5D Finite Element / Boundary Element methodology. The part of the metro line
concerned is built with a cast-iron tunnel lining. The tunnel structure and the track are modelled
with finite elements while the ground is modelled using boundary elements. Then the 2.5D track-
tunnel-ground model is coupled with a multiple-rigid body vehicle model to determine the
response caused by the passage of a train. To validate the prediction results, measurements have
been carried out of the vibration of the rail, tunnel invert, tunnel wall and ground surface when the
train is passing by and these are compared with the predictions with good agreement.
PACS no. 43.50.Lj, 43.40.At
1. Introduction*
Ground vibration caused by trains in tunnels is
transmitted into nearby buildings and radiated as
ground-borne noise, which may be disturbing for
the inhabitants of these buildings. This is an
important environmental problem for both existing
metros and planned new lines. Empirical,
analytical and numerical prediction methods can
be considered.
The empirical approach avoids the need for the
detailed input data and theoretical models required
for the calculation approaches, but the realisation
of this method depends on obtaining suitable
measurement data appropriate to the situation
considered. In [1] a procedure is described that
uses measured transfer functions and source terms
which are combined to predict the vibration for a
new line. Additionally, empirical prediction
models can be established using databases of field
measurements. For example, in [2] two calculation
procedures were proposed based on the analysis of
more than 3000 measurements.
Analytical approaches have the advantage of being
rapid and computationally efficient. In [3, 4]
analytical models were successfully developed
which give promising prediction results of the
ground vibration induced by a passing train.
However, these models are limited to simple
circular tunnel structures which cannot be too
close to the ground surface.
For more complex situations, a numerical
approach is useful either to verify the simpler tools
or as a prediction method itself. The finite element
(FE) and boundary element (BE) methods are
most commonly used. With the development of
high-performance computers, complex
transmission problems can be solved more
effectively and realistically. Coupled FE-BE
models in 2D and 3D were proposed and
compared in [5]. It was found that the 3D results
are closer to the absolute vibration levels, although
at a much higher computational cost. As a
compromise, the so-called 2.5D FE-BE method
can provide 3D results but with significantly
reduced model size for situations where the
structure is invariant in the third direction. In this
approach a 2D FE-BE mesh is used and the
1
problem is solved for a range of wavenumbers in
the third dimension. The full 3D solution can be
recovered from an inverse Fourier transform over
wavenumber. Applications of this approach can be
found in [6, 7].
In [8] an alternative approach was used based on a
periodic FE-BE model, where the Floquet
transform was employed to form the geometry of
the tunnel allowing for the periodicity of the
tunnel segments. However this is computationally
less efficient.
In this paper, a numerical model and the coupling
method used for the prediction of train-induced
vibration will be introduced. In this model, the
tunnel and ground are modelled using a 2.5D FE-
BE method, that is, the cross-section of the tunnel
and ground is modelled by finite elements and
boundary elements while the third direction is
represented in the wavenumber domain. This
model is coupled with a train represented by a
series of vehicles, each given by a 10 degree-of-
freedom (DOF) multi-body model. The
unevenness of the rail and wheel is used as the
excitation to determine the contact forces between
them. Then the train-induced vibration is
determined by using these forces in combination
with the transfer functions obtained from the
tunnel-ground model.
The results of the numerical model are compared
with measurements obtained as part of the MOTIV
project (http://motivproject.co.uk) at a location in
London. These include the vibration of the rail, the
tunnel and the ground surface during the passage
of trains. At the measurement location the train
speed is 48 km/h and the tunnel depth is about
20 m (from the surface to the tunnel crown).
2. Model description
The prediction model used here includes three
parts: the train, the tunnel/track system and the
ground. The train is modelled by four vehicle
units, each represented by a multi-body vehicle
model. The track-tunnel-ground model is
assembled using the WANDS software [9]. The
cross-section is represented as a 2D FE-BE model,
while the third dimension is modelled in the
wavenumber domain. The track and tunnel are
represented using finite elements and the ground is
modelled using boundary elements. The
unevenness of the rail is used as the excitation of
the coupled system. The detailed model and
coupling method will be explained in the
following sub-sections.
2.1. Train model
The train is modelled as a four-vehicle train
consisting of two driving cars at the ends and two
trailer cars in the middle. Although shorter than
the actual seven-coach train, this is sufficient to
give an estimate of the average level during a
pass-by. Each vehicle is represented as a 10-DOF
multi-body model including the pitching motion of
the car-body and bogie.
Each vehicle is 16 m in length. The distance
between bogie centres is 10.4 m and the bogie
wheelbase is 1.9 m. The wheel diameter is
0.787 m. The parameters of a vehicle unit are
listed in Table I. The car body mass corresponds
to the loading when the seats are fully occupied.
2.2. Tunnel, track and ground model
The tunnel, shown in Figure 1, is a cast-iron
structure made up of seven segments in each
section. The segments are bolted together at their
end flanges. These flanges provide additional
stiffeners in both the circumferential and the
longitudinal directions. The stiffeners in the
longitudinal direction are modelled by beam
elements. The ones around the circumference are
discretely distributed in the longitudinal direction
and cannot be realised in WANDS, but can only
be simulated in an average sense by increasing the
lining thickness. However, it has been found that
increasing the lining thickness does not have a
large effect on the ground response.
In WANDS, the cast-iron lining is modelled by 8-
noded solid elements. The invert is modelled as
shown in Figure 2 in which the gravel fillings and
Table I Parameters of vehicle model
Car
body
mass
18156 kg
pitching moment of
inertia
2.0
u106
kg.m2
Bogie
mass
2096 kg
pitching moment of
inertia
6000 kg.m2
Unsprung
wheelset mass
Trailer car
1220 kg
Motor car
2308 kg
Pr
imary
suspension
stiffness
1.346 MN/m
viscous damping
21.4 kNs/m
Secondary
suspension
stiffness
5.655 MN/m
viscous damping
22.5 kNs/m
Contact stiffness (two wheels)
2.926 GN/m
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the sleepers under the rail are included as part of
the concrete invert. The two rails are modelled as
beam elements, while the rail pads are modelled as
orthotropic material to simulate the fact that they
are actually discretely distributed along the track.
Figure 1 Cross section of the metro tunnel
Figure 2 FE model of tunnel and track
All the material properties of track and tunnel are
listed in Table II. The properties of the rail pad are
averaged over the sleeper spacing. An equal force
is applied on each rail.
The soil is formed by boundary elements. The soil
properties used in the ground model are based on
London clay, with a shear wave speed of 220 m/s
and longitudinal wave speed of 1571 m/s [8]. The
soil density is 1980 kg/m3 and loss factor is 0.078.
2.3. Coupling method
The coupling between the train model and tunnel-
ground model occurs at the contact between the
wheel and rail, as shown in Figure 3. The
measured unevenness of the rail, shown in Figure
4, is used as dynamic excitation.
Figure 3 Train model and the coupling with track
As the wheels of the train are running on the same
rails, the excitation of all the wheelsets is related
by the speed of the train and the distances between
them. The equation of motion is given by
>@
^` ^`
44
rwc r
nn
YY Y F Tir
Z
u
(1)
where F is the force amplitude, r is the unevenness
amplitude, and
Z
is the circular frequency. Yr, Yw
and Yc are the mobility of the rail, wheel and
contact spring respectively. Yr and Yw are obtained
from the tunnel-ground model and vehicle model
separately while Yc=iω/kHz in which kHz is the
stiffness of a Hertzian spring, given in Table I.
A time delay vector Tr is introduced, given by
00 0
(1)/ (2)/ ( )/
{}
w
ix v ix v ix n vT
r
Te e e
ZZ Z
0
(
0
(
(
0
(2)
Table II Material properties of the rail, rail pad and tunnel lining
Rail
Vertical bending
stiffness
Lateral bending
stiffness
Mass per unit length
per rail
Damping
loss factor
4.86u106 Nm2
0.96u106 Nm2
56 kg/m
0.01
Rail pad
Stiffness per unit length
Damping loss factor
Pad distance
-
2.62u108 N/m2
0.12
0.915 m
-
Cast Iron
(Grade 20)
Density
Young’s Modulus
Poisson’s ratio
Loss factor
7150 kg/m
100 GPa
0.3
0.01
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where x0 is the vector of wheelset positions, (1, 2
… nw indicate the number of wheelsets), and v is
the speed of the train.
Figure 4 Unevenness spectrum in 1/3 octave bands
3. Result analysis and comparison
3.1. Measurements
A series of measurements was carried out at a
location in London. Coordinated measurements
took place in the tunnel and on the ground surface.
The in-tunnel measurements included: i) the track
mobility and decay rate, ii) the rail unevenness as
shown in Figure 4, and iii) the train-induced
vibration on the rail and tunnel structure (invert
and wall). The in-tunnel pass-by accelerations
were recorded for both the vertical and lateral
directions at two cross-sections. Only the vertical
results are shown and these are averaged over the
measurements at the two locations. The above-
ground measurements consisted of the vibration
on the surface of the pavement. Indoor vibration
and ground-borne noise measurements were also
performed in two buildings (not shown here).
In this paper, the vibration levels to be compared
are those measured on the rail, tunnel invert,
tunnel wall and ground surface during the passage
of train. All the results shown below are expressed
as averages over the train length and converted
into 1/3 octave band spectra.
3.2. Rail vibration
Figure 5 shows the mobility of the 10DOF vehicle
(1st wheel), as well as the mobility of the rail
obtained from the tunnel/ground model. The wheel
mobility is given for the trailer and motor cars
which have different unsprung masses (as shown
in Table I). The crossing point between the
mobility of the wheel and rail corresponds to the
resonance frequency of the coupled vehicle / track
system. It can be seen that the mobility of the
trailer car is higher than that of the motor car, and
consequently the resonance frequency for the
trailer car occurs at a slightly higher frequency
than for the motor car. All subsequent results are
the average of the two types of vehicle.
Figure 5 Mobility of wheel and rail
Figure 6 Comparison of rail vibration
Figure 6 compares the rail vibration predicted by
the 2.5D FE-BE model with the measurement. The
figure shows both the dynamic component of
vibration (due to the train-track interaction caused
by the wheel/rail unevenness) and the total
vibration level, when a train passes at a speed of
48 km/h. The latter includes the quasi-static
component of vibration which is also obtained
from the WANDS model by setting the excitation
frequency of the moving load to zero Hz. Figure 6
shows that, generally speaking, the rail vibration
can be well predicted by the 2.5D FE-BE model.
The fluctuations at low frequencies (below 8 Hz)
are due to the axle spacings, which cannot be
10-3
10-2
10-1
100
-20
-10
0
10
20
30
40
50
Wavelength [m]
Roughness level [dB re 1
P
m]
Average of two rails
1 2 4 8 16 31.5 63 125 250
10
-9
10
-8
10
-7
10
-6
10
-5
10
-4
Frequency [Hz]
Mobility [m/s/N]
Wheel (trailer)
Wheel (motor)
Rail
1 2 4 8 16 31.5 63 125 250
10
-5
10
-4
10
-3
10
-2
Rail vibration
1/3 octave band frequency [Hz]
RMS velocity [m/s]
Measurement
WANDS (Dyn)
WANDS (Total)
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detected in the measurements due to the influence
of background noise. At frequencies above 63 Hz,
the prediction is about 10 dB lower than the
measurements. This is believed to be due to the
omission of wheel unevenness.
3.3. Vibration on the tunnel structure
The vertical vibration measured on the tunnel
invert and tunnel wall is compared with the
prediction results in Figures 7 and 8. The
calculation results agree well with the
measurements for frequencies between 8 Hz and
63 Hz. The differences at frequencies above 63 Hz
are the same as those shown for the rail vibration.
Figure 7 Comparison of vibration on tunnel invert
Figure 8 Comparison of vibration on tunnel wall
3.4 Ground vibration
The ground vibration was measured on the
pavement above the tunnel. By synchronizing the
measurements with those in the tunnel the
vibration associated with trains could be extracted.
For the far field vibration, the quasi-static
component of vibration decays rapidly. Therefore,
the ground vibration is dominated by the dynamic
component. Figure 9 shows a comparison of the
vibration on the ground surface above the tunnel
given by the WANDS model and measurements.
The background vibration is also shown for
comparison. It can be seen that the predictions are
generally lower than the measurements. However,
the shape of the spectra is quite similar and both
have a peak at around 63 Hz.
Figure 9 Comparison of ground vibration
Figure 10 Ratio of the vibration on invert / tunnel wall /
ground surface to the vibration on rail
(Thin lines: WANDS; Thick lines: Measurements)
As for the vibration on the rail and the tunnel
structure (in Figures 6, 7 and 8), the predicted
ground vibration drops quicker than the
measurements at high frequency. To demonstrate
the consistency of the calculation results, the
ratios of vibration on tunnel structure and ground
to that on the rail are plotted in Figure 10. It can
be seen that the vibration ratios from the
prediction results and measurement data agree
with each other over the whole frequency range.
16 31.5 63 125 250
10
-6
10
-5
10
-4
10
-3
Vibration on the tunnel invert
1/3 octave band frequency [Hz]
RMS velocity [m/s]
Measurement
WANDS (Dyn)
16 31.5 63 125 250
10-6
10-5
10-4 Vibration on the tunnel wall
1/3 octave band frequency [Hz]
RMS velocity [m/s]
Measurement
WANDS (Dyn)
16 31.5 63 125 250
10
-8
10
-7
10
-6
10
-5
Ground vibration
1/3 octave band frequency [Hz]
RMS velocity [m/s]
Measurement
WANDS (Dyn)
Background Noise
16 31.5 63 125 250
10
-4
10
-3
10
-2
10
-1
10
0
1/3 octave band frequency [Hz]
Rati o of RMS veloci ty
Grou nd s urface / Rail
Invert / Rail
Tunnel wall / Rail
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This suggests that the differences between
measurements and predictions at higher
frequencies are due to the unevenness spectrum
used which has not included the wheel roughness.
Figure 11 Rail vibration at different rail pad stiffness
Figure 12 Ground vibration at different rail pad
stiffness
3.5 Effect of rail pad stiffness
In Figure 11 and Figure 12, the vibration levels on
the rail and the ground surface are shown when
three different values are used for the rail pad
stiffness. In addition to the conventional rail pad,
these values correspond to a soft rail pad and a
resilient baseplate. The peaks in the responses, at
31.5, 50 and 63 Hz for rail pads with different
stiffness, are the respective resonance frequencies
controlled by the mobility of the vehicle and the
rail. Below the resonance frequency, the vibration
level on the rail increases with the reduction of rail
pad stiffness. However, as expected, the soft rail
pad gives a reduction in the ground vibration at
frequencies above the resonance frequency.
4. Conclusions
A prediction procedure has been proposed for the
vibration induced by a train running in a tunnel,
based on the 2.5D FE-BE methodology. Both the
quasi-static and dynamic components of vibration
can be evaluated. Using the coupled train-tunnel-
ground model, the vibration levels on the rail and
ground surface caused by the train are predicted
and compared with the measurements. The results
show a good agreement. The effect from varying
the rail pad stiffness on the vibration of rail and
ground surface is studied, which indicates a
practical way of controlling the vibration.
Acknowledgements
The first author is supported by a scholarship from
the China Scholarship Council. The experimental
work has been supported by the EPSRC research
grant EP/K006002/1, “MOTIV: Modelling of
Train Induced Vibration”. The authors are grateful
to London Underground for their assistance.
References
[1] Transit noise and vibration impact assessment. Office of
Planning and Environment, Federal Transit
Administration (FTA), FTA-VA-90-1003-06; 2006
[2] R. A. Hood, et al. The calculation and assessment of
ground-borne noise and perceptible vibration from
trains in tunnels. Journal of Sound and Vibration. 193.1
(1996) 215-225.
[3] J. A. Forrest, H. E. M. Hunt: A three-dimensional tunnel
model for calculation of train-induced ground vibration.
Journal of Sound and Vibration 294.4 (2006): 678-705.
[4] M. F. M. Hussein, et al. Using the PiP model for fast
calculation of vibration from a railway tunnel in a multi-
layered half-space. Noise and Vibration Mitigation for
Rail transportation systems. Springer, 2008. 136-142.
[5] L. Andersen, C. J. C. Jones. Coupled boundary and
finite element analysis of vibration from railway
tunnels—a comparison of two-and three-dimensional
models. Journal of Sound and Vibration 293.3 (2006)
611-625.
[6] G. Lombaert, G. Degrande. Ground-borne vibration due
to static and dynamic axle loads of InterCity and high-
speed trains. Journal of Sound and Vibration 319.3
(2009): 1036-1066.
[7] X. Sheng, C. J. C. Jones, D. J. Thompson: Modelling
ground vibration from railways using wavenumber
finite- and boundary-element methods. Proceedings of
the Royal Society A 461.2059 (2005): 2043-2070.
[8] G. Degrande et al., A numerical model for ground-
borne vibrations from underground railway traffic based
on a periodic finite element-boundary element
formulation. Journal of Sound and Vibration 293.3
(2006): 645-666.
[9] C. M. Nilsson and C. J. C. Jones, Theory manual for
WANDS 2.1, ISVR Technical Memorandum 975,
University of Southampton, (2007).
16 31.5 63 125 250
10
-5
10
-4
10
-3
10
-2
Rail vibration
1/3 octave band frequency [Hz]
RMS velocity [m/s]
k=262 MN/m
k= 90 MN/m
k= 25 MN/m
16 31.5 63 125 250
10
-8
10
-7
10
-6
10
-5
Ground vibration
1/3 octave band frequency [Hz]
RMS velocity [m/s]
k=262 MN/m
k= 90 MN/m
k= 25 MN/m
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