A Numerical Model for Re-radiated Noise in Buildings from Underground Railways

Abstract

A numerical prediction model is developed to quantify vibrations and re-radiated noise due to underground railways. A coupled
FE-BE model is used to compute the incident ground vibrations due to the passage of a train in the tunnel. This source model
accounts for three-dimensional dynamic interaction between the track, tunnel and soil. The incident wave field is used to
solve the dynamic soil-structure interaction problem on the receiver side and to determine the vibration levels along the
essential structural elements of the building. The soil-structure interaction problem is solved by means of a 3D boundary
element method for the soil coupled to a 3D finite element method for the structural part. An acoustic 3D spectral finite
element method is used to predict the acoustic response. The Bakerloo line tunnel of London Underground has been modelled
using the coupled periodic FE-BE approach. The free-field response and the re-radiated noise in a portal frame office building
is predicted.

Full text

Page 1

A numerical model for re-radiated noise in buildings from
underground railways
P. Fiala1,2, S. Gupta2, G. Degrande2, F. Augusztinovicz1
1Vibroacoustics and Audio Laboratory, Budapest University of Technology and Economics, Magyar
tudósok körútja 2. H-1117, Budapest, Hungary,
2K.U.Leuven, Department of Civil Engineering, Kasteelpark Arenberg 40, B-3001, Leuven, Belgium
Tel: +36 1 463 2543, Email: fiala@hit.bme.hu
Abstract
A numerical prediction model is developed to quantify vibrations and re-radiated noise due to
underground railways.
A coupled FE-BE model is used to compute the incident ground vibrations due to the passage of a
train in the tunnel. This source model accounts for three dimensional dynamic interaction between the
track, tunnel and soil. The incident wave field is used to solve the dynamic soil-structure interaction
problem on the receiver side, to determine the vibration levels along the essential structural elements
of the building. The soil-structure interaction problem is solved by means of a 3D boundary element
method for the soil coupled to a 3D finite element method for the structural part. An acoustic 3D
spectral finite element method is used to predict the acoustic response.
The Bakerloo line tunnel of London Underground has been modelled using the coupled periodic
FE-BE approach. The free-field response and the re-radiated noise in a portal frame office building is
predicted.Thenumericalsourcemodelaccountsformovingloadsandvariousexcitationmechanisms,
including quasi-static loads, random loads due to the rail and wheel unevenness and impact excitation
due to the rail joints. The effect of wall absorption on the internal pressure levels is investigated, and
the problem of modelling wall openings by absorbing impedance boundary conditions is dealt with.
1Introduction
Ground-borne vibrations induced by underground railways are a major environmental concern in
urban areas. These vibrations propagate through the tunnel and the surrounding soil into nearby build-
ings, causing annoyance to people. Humans in buildings are affected both by vibrations of the struc-
ture (1-80 Hz) and through the re-radiated noise (> 80 Hz) from the walls and ceilings of the rooms.
For the prediction of ground-borne vibrations and re-radiated noise in buildings, a modular architec-
ture is adopted, which consists of the following subproblems: the dynamic vehicle-track-tunnel-soil
interaction problem, the dynamic soil-structure interaction problem, and finally the prediction of vi-
brations and primary re-radiated noise in the structures. The first two subproblems can be assumed
weakly coupled, if the distance between the source and the receiver is much larger than the dominant
wavelengths in the soil. The second and third subproblems are weakly coupled too, as the mechan-
ical impedance of the air inside the rooms is much smaller than the mechanical impedance of the
building’s walls.
In the first subproblem, the free field vibrations are predicted, by computing the contact force
generated by the wheel/track interaction and then solving the dynamic track-tunnel-soil interaction.
Fiala, Gupta, Degrande, Augusztinovicz
1

Page 2

The dynamic track-tunnel-soil interaction problem is tackled using the coupled periodic FE-BE model
developed within the framework of CONVURT [1]. The Floquet transform is used to exploit the
periodicity of the problem, which allows to model wave propagation over multiples of wavelengths
both in the longitudinal and transverse directions of the tunnel, which would have been impossible
with standard three-dimensional finite element and boundary element formulations. A finite element
method is used to model a periodic unit of the tunnel, while a boundary element method is used to
model the soil as a horizontally layered elastic half space [2].
Once the incident wave field in the soil has been determined, the dynamic response of a three-
dimensional building due to this incident wave field is computed. Similarly to the tunnel-soil interac-
tion, a subdomain formulation is employed where a finite element method is used for the structure and
boundary element formulation is used for the soil. This approach allows to investigate the influence of
dynamic soil-structure interaction on the structural response [3]. The kinematics of the structure are
described by its rigid and flexible modes. The Craig-Bampton substructuring is used to differentiate
between the foundation modes and their quasi-static transmission into the building, and the modes of
the superstructure on rigid base.
In the third subproblem, the computed structural displacements are used as a vibration input for the
computation of ground-borne noise in the building’s enclosures. For typical room dimensions in office
buildings or family dwellings, the targeted frequency range is relative low. Therefore, deterministic
3D methods can be used to solve the acoustic wave equation inside the enclosure. A spectral finite
element method is applied to the acoustic problem, which, for the case of low wall absorption, can
lead to a direct integral representation of the internal sound pressure.
To demonstrate the efficiency of the approach, the tunnel on the Bakerloo line of London Under-
ground is modelled using a coupled periodic FE-BE approach. The free field response is predicted in
the frequency range 1-150 Hz, and subsequently the re-radiated noise in a hypothetic nearby multi-
story portal frame office building is estimated (Fig 1).
uinc
pa
vs
x
z
Figure 1. Problem outline.
2The incident wave field
The model of a moving vehicle on a track periodic or invariant in the longitudinal direction is used
to compute the incident wave field. The periodicity of the tunnel and the soil in the longitudinal y
Fiala, Gupta, Degrande, Augusztinovicz
2

Page 3

directionisexploitedusingtheFloquettransform,limitingthediscretizationefforttoasinglebounded
reference cell and formulate the problem in the frequency-wavenumber domain [4,2].
The vehicle is modelled as a set of concentrated masses representing the train’s unsprung masses,
connected to the track by Hertzian springs. The reference cell of the track and the tunnel is modelled
by means of a finite element method. The tunnel is embedded in a horizontally layered soil, charac-
terized by its shear wave velocity Cs, dilatational wave velocity Cp, material density ρ and hysteretic
material damping ratio β for each homogeneous layer.
First, the vehicle-track interaction problem is solved. The compliance matrices of the vehicle
ˆCv(ω) and the track-soil systemˆCt(ω) are used to compute the dynamic axle loads ˆ g(ω) arising
due to the rail unevenness ˆ uw/r(ω) as
[ˆCv(ω) +ˆCt(ω)]ˆ g(ω) = ˆ uw/r(ω)
(1)
The soil’s dynamic stiffness, which is incorporated in the track-soil compliance matrix, is computed
with a periodic boundary element formulation with Green-Floquet functions defined on the periodic
structure with period L along the tunnel [5,6,4].
In a second phase, the free-field soil vibrations ˆ ui(˜ x + nLey,ω) due to a set of axle loads ˆ gk(ω)
moving in a periodic domain are computed as [7]:
ˆ ui(˜ x + nLey,ω) =
1
2π
?∞
−∞ˆ gk(ω − kyv)e−iky(nL−yk)
?L/2
−L/2e−iky˜ y′˜ˆhzi(˜ y′,˜ x,κ0,ω)d˜ y′dky
(2)
where˜ˆhzi(˜ y′,˜ x,κ0,ω) is the Floquet transform of the transfer function in the frequency-wavenumber
domain relating the displacement at ˜ x to a unit load at the position {˜ x = 0, ˜ y′, ˜ z = 0}.
The transfer functions are computed by means of the coupled periodic FE-BE model using the
classical domain decomposition approach based on the finite element method for the tunnel and the
boundary element method for the soil [4,2,5,6].
(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
of the reference cell.
The Bakerloo line tunnel of London Underground is a deep bored tunnel with a cast iron lining and
asingletrack,embeddedinLondonclayatadepthof28m.Thetunnelhasaninternalradiusof1.83m
and a wall thickness of 0.022 m (Fig. 2). There are six longitudinal stiffeners and one circumferential
stiffener at an interval of 0.508 m, resulting in a periodic structure. Dynamic soil characteristics have
been determined by in situ and laboratory testing [8]. For the numerical calculations, the tunnel is
assumed to be embedded in a layered soil consisting of a single shallow layer with a thickness of 5
Fiala, Gupta, Degrande, Augusztinovicz
3

Page 4

m on top of a homogeneous half space consisting of clay. The top layer has a shear wave velocity
Cs= 275 m/s, a longitudinal wave velocity Cp= 1964 m/s, a density ρs= 1980kg/m3and a material
damping ratio βs= 0.042. The underlying half space has a shear wave velocity Cs = 220 m/s, a
longitudinal wave velocity Cp= 1571 m/s, a density ρs= 1980kg/m3and a material damping ratio
βs= 0.039.
The track is a non-ballasted concrete slab track with Bull head rail supported on hard wooded
sleepers nominally spaced at 0.95 m with cast iron chairs. Both ends of a sleeper are concreted into
the invert and the space between the sleepers is filled with shingle. Resilience is mainly provided
by the timber sleepers, as the rails are not supported by rail pads. The rails are modelled as Euler-
Bernoulli beams and are attached to the invert via a continuous elastic support. The smeared stiffness
of the support is assumed as¯ktr= 80MN/m2, while a value of 12 × 104Ns/m2is assumed for the
distributed damping in the support ¯ ctr.
The train consists of seven cars: a driving motor car, a trailer car, two non-driving motor cars, two
trailer cars and a driving motor 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
distance between the first and the last axle of the train is 108.33 m. The wheels are of the monobloc
type and have a diameter of about 0.70 m. The tare mass of a motor car is 15330 kg, while the bogie
mass is 6690 kg and the mass of wheel set 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 wheel set is 950 kg. For the computations, only the
unsprung mass of the cars is considered to act on the rails. The speed of the train is 50 km/h.
The rail roughness is expressed as a stochastic process characterized by a single-sided power spec-
tral density (PSD)˜Sw/r(ky), written as a function of the wavenumber ky= ω/v = 2π/λy[9]:
˜Sw/r(ky) =˜Sw/r(ky0)
?ky
ky0
?−w
(3)
The parameters˜Sw/r(ky0) and w depend on the quality of the rail. For rail unevenness, ky0= 1 rad/m
and w = 3.5 are commonly assumed. An artificial profile uw/r(y) is generated from the PSD-curve
based on the superposition of simple random processes with known statistical properties. The PSD
ˆSw/r(ω) of the unevenness in the frequency domain is found as 1/v˜Sw/r(−ω/v) and increases with
the vehicle speed proportional to v2.5, where w = 3.5 in equation (3). Thus, an increase in speed is
expected to give rise to higher dynamic axle loads and higher vibration levels.
(a)
−10 −8 −6 −4 −2 0 2 4 6 8 10
Time [s]
−5
0
5
10
x 10
−5
Velocity [m/s]
(b)
0 50
Frequency [Hz]
100150
0
0.5
1
1.5
2
2.5
3
3.5x 10
−5
Velocity [m/s/Hz]
Figure 3. (a) Time history and (b) frequency content of the vertical (blue) and horizontal (red) component of
the incident wave field in the soil at coordinates {−5 m,−7.5 m,0 m}.
Fig. 3 shows the horizontal (x) and vertical component of the free-field incident velocity computed
Fiala, Gupta, Degrande, Augusztinovicz
4

Page 5

in the corner point of the building at coordinates {−5 m,−7.5 m,0 m}. The dominating part of the
frequency content is between 20 Hz and 80 Hz with a peak at 50 Hz, which corresponds to the
wheel-track resonance frequency. The bogie passages are not clearly visible in the time history as the
tunnel is situated at a considerable depth, however, due to the specific train composition, the observed
velocity spectrum is quasi-discrete. The computed incident wave field has been validated by means
of the experiments performed on the Bakerloo line [8,10].
3Dynamic soil-structure interaction
A weak coupling between the incident wave field and the structure is assumed, meaning that the
presence of the building has no effect on the vibration generation mechanism and the free field dis-
placements are applied as an excitation on the coupled structure-soil model.
The decomposition method proposed by Aubry et al. [11] and Clouteau [12] is used to formu-
late the dynamic soil-structure interaction problem. The finite structure is modelled in the frequency
domain by a 3D structural finite element method. The equation of motion of the building is:





Kss
Ksb
KbsKbb+ˆKg
bb


 − ω2


MssMsb
MbsMbb











ˆ us
ˆ ub




=





0
ˆfb





(4)
where the structural displacements are separated to the displacement DOF of the foundation ˆ uband
the remaining DOF of the superstructure ˆ us. M and K denote the finite element mass and stiffness
matrices, andˆKg
the impedance of the soil are computed using a 3D boundary element method [12] in the frequency
domain, using the Green’s functions of a layered half-space.
bbstands for the frequency dependent impedance matrix of the soil. The forceˆfband
Figure 4. Finite element mesh of the office building.
The portal frame office building has the dimensions 15 m × 10 m × 9.6 m and is symmetrically
placed on the free surface above the tunnel. The finite element mesh of the modelled portal frame
structure is shown in Fig. 4. The three story superstructure is supported by a 0.3 m thick reinforced
concrete raft foundation. The basic structure consists of a reinforced concrete portal frame structure
Fiala, Gupta, Degrande, Augusztinovicz
5