Parameter calibration in the modelling of railway traffic induced
vibrations by use of Barkan
F.M.B. Galanti, A. Koopman & G. Esposito
TNO Building and Construction Research, Delft, The Netherlands
ABSTRACT: Reliable prediction of vibration levels in the surroundings of railway links is dependent on an
accurate estimation of model parameters. An important modelling parameter is damping, but because of difficulties in its estimation, particularly in the case of soils, calibration by use of in-situ vibration tests is often
wanted. In the case of FEM, such an optimization is not straightforward. As an intermediate step, the Barkan
formula, which explicitly handles damping, can be used as a common ground to which both measurements
and FEM modelling results are translated. In this article such a side step is investigated.
1
INTRODUCTION
Due to increased demands in reliable surface transport, there has been a considerable increase in the
past two decades in the number of high speed railway links. A direct drawback of such railway links,
especially in dense urban environments, is the generation of noise and vibration not only during the
exploitation phase but also during the prior construction phase. Consequently, it has become necessary to
estimate the levels of noise and vibration in the vicinity of (future) railway lines. Based on the results
of such estimates, alternatives may need to be selected for the construction method and for the design
of the railway line, whereby in the latter, noise and
vibration mitigation measures may be taken into
consideration.
Considering only the vibration problem, estimates
or predictions of vibration levels can be made using
a variety of analysis tools ranging from empirical
models to advanced models such as the finite element method. In a recent Delft Cluster project, the
accuracy of the existing prediction strategies was investigated. The results of the project highlighted that
vibration predictions are characterized by a large uncertainty, de Wit & Waarts (2003). The project also
demonstrated that the uncertainty does not significantly decrease with the complexity of the used prediction tool. Sophisticated multi-dimensional numerical models seem to yield the same level of
uncertainty as that associated with engineering
judgment. Nevertheless, advanced models probably
offer the best modelling options, but may be difficult
to tune due to the necessity to make a number of
simplifying assumptions and due to the large number
of modelling parameters.
In literature, explicit treatment of the reliability of
vibration prediction models has received so far almost no attention. Details as to total model uncertainty or as to what part of a model leads to the
greatest uncertainty are not available. Vibration prediction models therefore can only be used to provide
a rough indication of the response of the structures
in the surroundings of a railway line.
In order to promote further application of vibration
prediction models in engineering practice it is necessary to gain insight as to the reliability which can be
expected from the use of a specific model. As a first
step in this direction, this study will focus on the estimation of the value of damping to be assumed in
vibration prediction models, since this is one of the
most important parameters and also one of the most
difficult to determine. The underlying model is
based on an an approach consists whereby separate
modules for the vibration source, the transmission
path and the receiver, are used. Any mechanical interaction between the modules is neglected, see Fig.
1.
Source
Path
Receiver
Figure 1. Typical components of a vibration prediction model.
The present study will focus on the response of
the transmission path model, due to the passage of a
train, without considering the response of the receiver. The practical case which is considered is that
of a section of high speed rail track in the vicinity of
Waremme, Belgium. This section of the high speed
rail track has been the subject of a an extensive vibration measurement campaign, consisting of falling
weight tests and measurements of field vibrations
during several passages of a Thalys TGV, Esposito
(2003) and Esposito (2004). A procedure will be
outlined with which damping parameters can be
calibrated on the basis of a comparison of finite element models of the transmission path and field
measurements.
2
CALIBRATION PROCEDURE
The modelling scheme under investigation consists of a source model for trains (called ‘TRINT’),
FEM transfer models (either 2D plane strain or 3D),
and a procedure to combine source with transfer, as
described by Koopman & Courage (2001).
A considerable source of modelling uncertainty
are the soil parameters needed for the transfer models. Of these, material damping is the most important. Damping is both hard to determine from geotechnical investigations (like SPT’s) and therefore
often not available, and at the same time very influential on the model outcome. In this article, therefore, the focus lies on dealing with material damping.
When modelling ground-borne vibration, often
there is a need and an opportunity to update the
modelling with some sort of in-situ vibration measurements. In those cases, the question arises how to
perform an update and how to set up a measurement
for such a tuning. Given, for example, a falling
weight experiment, both from ‘reality’ as from FEM
modelling so much vibration output can be derived,
and so many discrepancies between model en measurement can be expected, that an update scheme will
never be straightforward, especially given the large
number of input variables that can be tuned.
Of course there are various numerical update procedures available for many-input many-output problems. However, such procedures often are black
boxes in character. What is necessary, in any specific project and for the advancement of this field in
general, is an update scheme based on physical insight. Then, further steps after tuning, like the design of mitigation measures, will also benefit.
Given a model with a given level of detail, a
more abstract model can form the basis for physical
insight in the results. For ground-borne vibration,
the Barkan model often fulfils the role of the more
abstract model. In essence, it describes vibration as
propagating wave energy and only uses geometry
and damping for the energy balance. Relating this to
FEM, the geometry factor mainly links to the soil
layering in the FEM model and the damping factor
links directly to material damping in the FEM model
while slightly less directly to the other material parameters.
A possible update scheme is using Barkan as a
mediator between measurement and FEM model.
As stated, damping is often a crucial parameter in a
FEM model and Barkan handles it explicitly. The
update then revolves around the relevant parameter
and physical insight can be gained.
First the two Barkan parameters, geometry and
material damping, are fitted to the measurements.
Obviously such a fit, although not straightforward, is
nevertheless considerably less demanding than a fit
of a FEM model to measurements.
Next the FEM model is fitted to the Barkan
model by tuning the material damping properties.
The whole update procedure should be performed
per frequency band (like octave bands) in order to
appreciate the frequency dependency of the physics
involved. The updating can then be enhanced further by also involving the other material parameters
and/or the soil layering, through exploitation of their
differences in frequency dependency effects. In
such an updating care should be taken to keep the
parameters within physical limits, preferably within
the bandwidth of uncertainty established preparing
the first modelling phase.
A special case form the FEM codes that use
Rayleigh damping, which actually most do.
Rayleigh damping, although numerically advantageous, imposes a frequency dependency on material
damping that is hardly physical. Compensation by
an update that involves more than only the material
damping then is hardly avoidable.
Further enhancement can be gained by performing different types of measurements. For instance, if
possible, apart from a falling weight also a train passage by can be used as a source. One then has a
point source (falling weight) as well as a line source
(train). In such a case first a 2D plane strain FEM
model is tuned to a Barkan fit of the train passage.
The geometrical Barkan parameter will then be
small and full attention can be paid to the damping
parameter. Next a 2D axial symmetric or a 3D
model is fit to the falling weight test, using the formerly found damping parameter and focussing on
the geometrical parameter. Even better would be a
coupled approach, having two parameters and using
two equations. The extra dimensions that frequency
dependency and possibly other material parameters
introduce will impede such approach however.
3
TRAIN, TRACK AND SOIL MODEL
The section of interest is a double track inside a
semi-submerged open tunnel with a concrete base
floor, 0.8 m thick, resting on sheet piles and concrete
piles, 13 m in length. The soil consists of a top layer
of sandy loam about 12.5 m thick resting on a chalk
layer. The track and the soil comprise the transmission module which is simulated using both two-
dimensional and three-dimensional finite element
models with the use of silent boundaries.
The model for the soil is used to determine the
transfer function between a point on the track and a
series of points along a line perpendicular to the
track at the surface level. Subsequently, the response
due to a train passage could calculated by combining
the transfer functions with modelling or measurements of the train loading.
A diagram of the two-dimensional model is given
in Fig. 2. In this model, a total of 8527 quadrilateral
8-node plane strain elements are used. The model
extends over a cross section of 150 m containing the
track and surrounding soil up to a depth of 23 m. A
half sine unit impulsive load with a duration of 5 ms
is applied at in the middle of the rightmost track. A
Newmark-beta time integration scheme is used
(γ=1/2, β=1/4, unconditionally stable) with a time
step of 1 ms. The total simulation time is 0.6 s. Silent boundaries have been used along all the edges
of the model. In order to avoid relying too much on
the silent boundary’s capability to absorb incident
waves, the size of the model has been selected in
such a way that the total travel distance of waves
over the simulation time (including the travel distance after reflection off the boundary) would be too
small to affect adversely the response of the area of
interest. In this case the area of interest is the free
field surface to the right of the open tunnel at a distance between 5 and 50 m from the source.
Material parameters have been selected on the basis of SCPT’s and borings carried out at the site.
From these tests the elastic modulus and Poisson ratio of the two soil layers have been estimated. The
tests, however, gave little indication as to the material damping. This property, which determines part
F
Z
X
+3.4 m
-2.5
-9.5
-19.5
11.5
50
15.7
84.3 m
Figure 2. Two dimensional model. The bottom of the concrete
tunnel floor is at z=–2.5 m. The black line denotes the boundary between the soft top layer (loamy sand) and the bottom
chalk layer is at z=–9.5 m.
of the attenuation of vibrations generated at the
source, had to be estimated. The top soil layer has a
damping ratio of 3.5%, whilst the bottom layer and
all other materials have a damping ratio of 1.2%.
From these damping ratios, equivalent Rayleigh
damping ratios have been extrapolated in the frequency range between 3 and 100 Hz. The final material parameters used in the analysis are given in Table 1.
Table 1. Material properties used in the simulations.
Property
Young's
modulus
Material
Upper
Lower
Concrete Steel
soil layer soil layer
[MPa] 200
500
25800
210000
0.3
0.15
0.3
[kg/m3] 1800
2000
2540
7800
a [s-1]
1.24
0.41
0.41
0.41
b [s]
1.12E-04 3.7E-05
3.7E-05
3.7E-05
188
-
-
Poisson ratio [-]
Density
Rayleigh
damping
Shear wave
[m/s]
speed
0.38
287
In order to compare different modelling options, a
similar analysis has been carried out using a three
dimensional model, see Fig. 3. This model is simply
an extrusion of the two-dimensional model, however, with a slightly rougher mesh. The mesh is
formed by a total of 7077, 20 node brick elements,
and extends by 120 m in depth. Silent boundary
elements are placed along the edges of the model
except for the edge at z=0, which, out of symmetry
considerations, is fixed in the z direction (the load
being applied at this edge).
120
Y
X
Z
Figure 3. Three dimensional model with silent boundaries.
The first step of the analysis will be that of comparing the response obtained from the two models
with the response measured from falling weight
tests. From the comparison of the attenuation in the
experiment and in the models, the accuracy of the
initial estimate of the damping in the soil will be
evaluated. As a final step the response of the models
due to the passage of a train will be compared with
measurements.
4
COMPARISON OF MODEL RESULTS WITH
FALLING WEIGHT TESTS
In order to compare the attenuation of vibration in
the models with that which is observed in the experiments, frequency response functions at various
locations along the surface have been evaluated.
Plots of the mobility against frequency are given in
Fig. 4 for two locations on the free field at 6 m and
48 m distance from the source. From the graphs, it
can be observed that the response of the 2D model is
less than two orders of magnitude greater than that
of the 3D model and that the measured response lies
in between that of the two models. The 2D model response is greater than that of the 3D model since the
2D model effectively represents a line loading with
infinite extension on an elastic half-space, whilst in
the 3D model the loading consists of a concentrated
force. It is interesting to note that the model spectra
and that of the experiment all show a similar trend at
all distances.
10
FF01Z - ch. 13
-6
response functions, damping parameters r and α can
be estimated. Subsequently the material damping parameter α can be used to determine the damping ratio of the soil, given the following relationship:
α = α( f ) =
2πfζ ( f )
VR
(2)
where f is the frequency, VR is the Rayleigh wave
speed and ζ is the damping ratio. Although frequency dependent, it will be be assumed that the
Rayleigh wave speed is constant.
10
50
10
10
10
mobility [s/kg]
10
10
10
25
-8
13
2D model
3D model
experiment
-9
0
20
40
60
frequency [Hz]
80
one-third octave - centre frequency [Hz]
mobility [s/kg]
2D model
100
-7
100
FF06Z - ch. 22
-6
-7
-8
-9
6
8
12 16 32 48
3D model
100
-100
50
-120
25
-140
13
6
8
12 16 32 48
experiment
6
8 12 16 32 48
distance [m]
-160
100
50
10
10
-10
2D model
3D model
experiment
-11
0
20
40
60
frequency [Hz]
80
25
13
100
Figure 4. Mobility compared between different models and experiment.
4.1 Estimation of damping
Attenuation of waves traveling from the source is
caused by radiation damping and by material damping and is typically described by Barkan’s formula:
u(x ) § x0 ·
= ¨ ¸ exp[− α ( x − x0 )]
u ( x0 ) © x ¹
r
(1)
where x is the distance from the source and r and α
are respectively the geometrical and material damping parameters. Frequency response functions taken
at various distances from the source can be used to
evaluate damping, since the amount by which a signal attenuates solely due to material damping is frequency dependent. This dependency can be observed
in Fig. 5 which shows the variation of inertance
spectra with the distance from the source. By fitting
the Barkan model for each single frequency in the
Figure 5. Attenuation of inertance frequency response functions for the vertical direction as a function of distance from
the source: experiment and 2D model.
The first attempts to use the fitting procedure led
to results which were rather inconclusive. It was
noted that the geometrical damping parameter varied
erratically with frequency and that very little attenuation occurred at frequencies below 30 Hz. The
latter can be explained by the fact that signals were
measured within a section of the surface between 5
and 50 m distance from the source. Since planar
Rayleigh waves form only beyond a wavelength’s
distance from the source and recalling a Rayleigh
wave speed for the site of 185 m/s, only Rayleigh
waves with a frequency above 37 Hz should be considered. Waves with a frequency below this value
(or with a wavelength greater than 5 m) therefore are
not fully formed and will not be detected properly.
Damping ratio
ingspectrum that 2D plane strain FEM modelling
yields for this case. Therefore, tuning the material
damping through the 2D model to the train passage
measurements is disregarded at this stage.
Damping ratio
0.055
0.05
0.045
0.04
0.035
ζ
With respect to the first problem, the two damping parameters have a similar effect on the attenuation in the Barkan model, hence small variations in
the data can easily lead to an interchange of attenuation values between the two parameters. If an assumption is made as to the amount of geometrical
damping and if only frequencies corresponding to
wavelengths smaller than 5 m are considered, then
more realistic material damping values are obtained.
In Fig. 6 average damping ratios over one-third octave bands for the experiment and the 3D model are
presented. For the 3D model and the falling weight
experiment, the exponent for the geometrical damping, r=0.5. Considering the frequency range between
20 and 100 Hz, average damping ratios of 2.7% for
the experiment, and 2.8% for the 3D model are obtained.
0.03
0.025
2D Model
Avg. 2D model
train passage
Avg. train passage
0.02
0.015
0.01
20
25
32
40
50
63
one-third octave centre frequency [Hz]
79
100
Figure 7. Damping ratio evaluated per one-third octave from
data relating to a high speed train passage
0.06
0.05
ζ
0.04
0.03
0.02
3D Model
Avg. 3D model
experiment
Avg. experiment
0.01
0
20
25
32
40
50
63
one-third octave centre frequency [Hz]
79
100
Figure 6. Damping ratio evaluated per one-third octave for the
2D model (top) and the 3D model (bottom).
Nonetheless, no particular trend in the variation
of damping ratio with frequency is to be observed
both in the model and in the experiment. Variations
in damping ratio with frequency, both in the model
and the experiment could be associated with the special geometry of the situation which includes the
open tunnel. Because of its presence, the open tunnel
could lead to a wave field which attenuates more
like that caused by a line load.
4.2 Updating
Fitting Barkan on the train passage measurements
and the related 2D plane strain FEM model, assuming r=0, yields the results shown in Figure 7. On average, the chosen FEM parameters seem to give rise
to too much damping, compared to the measurements. Remarkably, this FEM damping is also too
high compared to the material damping that was put
into the model and to which it is supposed to be
closely related. Also, the characteristic frequency
dependency of Rayleigh damping ('the bathtubspectrum') is not reproduced by the Barkan fit. As
of yet, no explanation has been found for this damp-
The next step is fitting Barkan on the falling
weight test and the related 3D FEM model (Figure
6). Initially, the damping found in the 2D case
would be used as given and the geometry factor
would be fitted. However, in this case the geometry
factor must be considered the least uncertain and set
to 0.5 while fitting the damping once again.
Measurement and modelling resemble each other
much more now, especially in the frequency range
up to 50 Hz, and further tuning of the model might
even be considered superfluous. The damping found
is also much closer to the material damping put into
the model. Considering that the geometry of the
situation is not quite axial symmetric (as stated before) and the set geometry (r=0.5) is therefore
unlikely to be very accurate or frequency independent, no bathtub shaped damping spectrum could be
expected.
Now, the next step would be to combine the traintrack source model with the now verified 3D FEM
model to make predictions. A further verification
step, which would mainly verify the source model
and the way the two models are coupled, would be
to compare the results with the train passage measurements again. This step, however, lies outside the
scope of this article.
5
CONCLUSIONS
Accurate assessment of situations which may lead to
the production of noise and vibration requires prediction models which can represent reliably the frequency and amplitude of vibrations. To gain reliability, models can be tuned with in-situ vibration
measurements. To facilitate tuning of FEM models,
more general models such as the Barkan formula
might be of use.
- Comparison between measurement and modeling on the basis of the Barkan fits they produce
turns out to be quite satisfactory for 3D FEM
modeling, for the case under investigation.
- Given that damping is a crucially uncertain parameter this makes it possible to fit material
damping input for FEM models to in-situ vibration measurements by use of the Barkan formula.
- The Barkan fit did not work well in the 2D case,
yielding damping ratio’s that seem to be too
high.
- An updating scheme involving both 2D and 3D
measurements and modelling as coupled tuning
problems could therefore not be put to the test.
The actual quality of the investigated updating
scheme can only be determined when an updated
model is validated by separate measurements, preferably in the changed, future situation for which the
modelling was needed. Such a validation is left to
be undertaken.
ACKNOWLEDGEMENT
C.B.M. Blom
Holland Railconsult / HSL Zuid. Delft, The Netherlands
REFERENCES
M.S. de Wit & P.H. Waarts. 2003. Reliability of vibration
predictions in civil engineering applications. ESREL 2003.
G. Esposito. 2003. Experimental Determination of the Impedance of a RC Railway Construction in Waremme(BE), High
Speed line Brussels – Cologne. TNO-report 2003-CIR0066.
G. Esposito. 2004. Experimental Determination of Vibrations
Induced by IC and HS Train Passages. TNO-report 2004CI-R0072.
A. Koopman & W.C. Courage. 2001. Modelling of soil vibrations from railway tunnels – Part 1 : TNO. Proceedings of
7th IWRN.
H.R. Stuit, W. Gardien, A. Koopman, R.J. Aartsen. 2001.
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