Verification of an empirical prediction method for railway induced vibrations by means of numerical simulations

Journal of Sound and Vibration 330 (2011) 1692–1703

Contents lists available at ScienceDirect

Journal of Sound and Vibration journal homepage: www.elsevier.com/locate/jsvi

Verification of an empirical prediction method for railway induced vibrations by means of numerical simulations H. Verbraken , G. Lombaert, G. Degrande Department of Civil Engineering, K.U.Leuven, Kasteelpark Arenberg 40, B-3001 Leuven, Belgium

a r t i c l e in fo

abstract

Article history: Received 23 September 2009 Received in revised form 3 September 2010 Accepted 22 October 2010 Handling Editor: M.P. Cartmell Available online 24 November 2010

Vibrations induced by the passage of trains are a major environmental concern in urban areas. In practice, vibrations are often predicted using empirical methods such as the detailed vibration assessment procedure of the Federal Railroad Administration (FRA) of the U.S. Department of Transportation. This procedure allows predicting ground surface vibrations and re-radiated noise in buildings. Ground vibrations are calculated based on force densities, measured when a vehicle is running over a track, and line source transfer mobilities, measured on site to account for the effect of the local geology on wave propagation. Compared to parametric models, the advantage of this approach is that it inherently takes into account all important parameters. It can only be used, however, when an appropriate estimation of the force density is available. In this paper, analytical expressions are derived for the force density and the line source transfer mobility of the FRA procedure. The derivation of these expressions is verified using a coupled finite element–boundary element method. & 2010 Elsevier Ltd. All rights reserved.

1. Introduction Ground-borne vibrations induced by railway traffic are a major environmental concern in urban areas, particularly when new rail infrastructure is built. Ensuring the environmental friendliness of new railway lines requires an accurate assessment of vibration and re-radiated noise, so that cost-effective mitigation measures can be taken. Several numerical models have been developed for the prediction of ground vibrations due to railway traffic at grade [1–4] and in tunnels [5–7]. An accurate estimation of the vibration levels requires the knowledge of input parameters, such as the dynamic track and soil characteristics, for which in situ testing may be required. In practice, simplified empirical prediction methods are often used. The Federal Railroad Administration (FRA) and the Federal Transit Administration (FTA) of the U.S. Department of Transportation have developed a set of empirical procedures to predict vibration levels due to railway traffic [8,9]. Three different levels of assessment are described: the screening procedure, the generalized vibration assessment and the detailed vibration assessment. The first two levels are used for general screening purposes. The third level is based on a prediction technique developed by Bovey [10] and Nelson and Saurenman [11] and presents a more elaborate method for the prediction of the vibration velocity levels in the free field, based on field measurements. The detailed vibration assessment predicts the vibration velocity level Lv [dB ref 10  8 m/s] as the root mean square (RMS) value vRMS of the vibration velocity during the stationary part of a train passage in one-third octave bands, using the

 Corresponding author. Tel.: + 3216321668; fax: + 3216321988.

E-mail address: [email protected] (H. Verbraken). 0022-460X/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2010.10.026

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

1693

following equation: (1) Lv ¼ LF þTML pffiffiffiffiffi The first term in Eq. (1) is the force density LF ½dB ref N= m and is a measure for the power per unit length radiated by the pffiffiffiffiffi source. The second term is the line source transfer mobility TML ½dB ref 108 ðm=sÞ=ðN= mÞ and is a measure for the vibration energy that is transmitted through the soil relative to the power per unit length radiated by the source. The line source transfer mobility TML is determined by means of in situ experiments. Fig. 1 shows a measurement setup for vibration propagation tests with a track at grade (Fig. 1a) and in a tunnel (Fig. 1b). Impacts are given at equally spaced points along the track and the resulting vibration velocity is measured along a line perpendicular to the track. The transfer function between the applied force in a single impact point k and the velocity in the free field is called the point source transfer mobility TMPk ½dB ref 108 ðm=sÞ=N and is determined in one-third octave bands. An estimate of the line source transfer mobility TML is obtained by the superposition of point source transfer mobilities: ! n X 10TMPk =10 TML ¼ 10 log10 h (2) k¼1

where h is the interval between impact points and n is the total number of impact points. In the case where the tunnel has not been constructed yet, several boreholes are required to perform the impacts at depth according to the setup in Fig. 1b. To avoid the need for multiple boreholes, an alternative setup is presented in the FRA procedure (Fig. 2). In this setup, optimal use is made of a single borehole by using several measurement lines in a radial pattern with the origin at the borehole location. This setup is not further considered in the present paper. The force density due to a train passage is determined using Eq. (1) as follows. The vibration velocity level Lv due to the train passage is measured in a point x. Also, the line source transfer mobility TML is determined in the same point x using the previously described experimental procedure. The force density LF is then obtained as the difference Lv  TML. As the response close to the track is dominated by quasi-static excitation, i.e. the response to the moving static axle loads, the receiver point x should be located outside the influence zone of the moving static axle loads, at a sufficiently large distance from the track. In the case where a prediction is required for a site where new railway infrastructure is introduced, the following two-step approach is followed. First, a site is selected (site 1) where a track is present and where the characteristics of the track and the rolling stock are similar to the new railway infrastructure. The response due to a train passage is recorded and Eq. (1) is applied to estimate the force density LF. Second, the line source transfer mobility TML is measured at the site where a new track will be built (site 2), and Eq. (1) is applied once more to predict the vibration velocity level due to the new railway infrastructure, using the estimated force density LF and the measured line source transfer mobility TML. The vibration velocity level due to a train passage, and therefore also the force density, can only be determined on a site where the track is already present. As no track is present on site 2, the line source transfer mobility can only be determined by means of impacts at the soil’s surface. The line source transfer mobility at site 1 should therefore be determined in a similar Rail alignment Impact locations

z x

Impact locations

Measurement line

z y

Measurement lines

y x Rail alignment

Fig. 1. Setup for vibration propagation tests (a) at grade and (b) in tunnels.

Measurement lines

Impact location

Fig. 2. Alternative setup for vibration propagation tests using a radial pattern around a borehole.

1694

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

way with impacts at the soil’s surface adjacent to the track. In order to account for the distance to the center of the track, the impact points at site 2 should be located at the same distance from the track centerline. When the force density is determined in this way, it no longer represents the actual force transmitted by the wheel/track system, but an equivalent force that has to be applied adjacent to the track in order to obtain the same vibration velocity as the one recorded during the train passage. In the FRA procedure, the local transfer of vibrations through the soil is correctly accounted for due to the experimental determination of the transfer mobility TML. This is an advantage compared to numerical prediction models where the model parameters need to be determined for an accurate prediction. The accuracy of the empirical prediction is largely dependent, however, on the availability of an appropriate force density LF. In this paper, analytical expressions for the force density LF and the line source transfer mobility TML are derived based on the analytical expressions used in numerical predictions [12]. Section 2 derives analytical expressions for the force density LF and the line source transfer mobility TML. In Section 3, the assumptions required in the derivation of the analytical expressions are verified in a numerical example where the case of vibrations due to a train running in a tunnel is considered [13]. Calculations are performed with a coupled finite element–boundary (FE–BE) element method [6,14]. The conclusions of this study are presented in Section 4. 2. Derivation of the analytical expressions for the detailed vibration assessment of the FRA In order to derive Eq. (1) of the FRA procedure, the velocity vðxu,tÞ at a point xu of the coupled track–soil system due to an arbitrary body force rbðx,tÞ is computed as follows [12]: vðxu,tÞ ¼

Zt Z

Hðxu,x,ttÞrbðx, tÞ dx dt

(3)

1 O

where O is the domain of the coupled track–soil system. Each element hij ðxu,x,tÞ of the matrix Hðxu,x,tÞ represents the velocity at a point x in the direction ej at time t due to an impulsive load at a point xu in the direction ei at time t= 0. In the following, dynamic reciprocity is used to replace the matrix Hðxu,x,tÞ by HT ðx,xu,tÞ where the superscript T denotes the matrix transpose. For na axle loads gk(t) moving at a constant speed v in the direction ey (Fig. 1), the body force rbðx,tÞ is equal to

rbðx,tÞ ¼

na X

dðxxk ðtÞÞgk ðtÞ

(4)

k¼1

where xk ðtÞ ¼ xk0 þ vtey is the time-dependent position of the kth axle load, xk0 is the position at the time t=0 and gk(t) is a vector that contains the time histories of the kth axle load in each direction. The time history gk(t) can be decomposed into the static component gsk and a dynamic component gdk(t). The contribution of the static component to the free field response is referred to as the quasi-static excitation [12]. For a train moving at a speed v that is small compared to the Rayleigh wave velocity CR in the soil, the quasi-static excitation only contributes significantly at low frequencies close to the track. The free field response is therefore dominated by the contribution of the dynamic axle loads. The dynamic axle loads are determined by a large number of excitation mechanisms such as parametric excitation due to the spatial variation of the track support stiffness, e.g. the discrete position of the sleepers, transient excitation due to the rail joints and wheel flats, and excitation due to wheel and rail roughness and track unevenness [15]. In the present study, only dynamic excitation due to random track unevenness is considered. Introducing expression (4) in Eq. (3), the response due to the moving loads is calculated as follows: vðxu,tÞ ¼

t na Z X

HT ðxk ðtÞ,xu,ttÞgk ðtÞ dt

(5)

k ¼ 1 1

In the following, the prime denoting the receiver position is omitted. An expression for the mean square vibration velocity v2RMS(x) can be derived based on the second-order statistical characteristics of the vibration velocity. As the time history of the vibration velocity in a point x has a transient character, its second-order statistical characteristics are described by the nonstationary auto-correlation function Rv (x,t1,t2) [(m/s)2]: Rv ðx,t1 ,t2 Þ ¼ E½vðx,t1 ÞvT ðx,t2 Þ

(6)

where E½  denotes the expected value operator. This expression is further elaborated by introducing Eq. (5) for the time history of the vibration velocity: 2t t 3 Z1 Z2 na X na X T T 4 Rv ðx,t1 ,t2 Þ ¼ (7) H ðxk ðt1 Þ,x,t1 t1 ÞE½gk ðt1 Þgl ðt2 ÞHðxl ðt2 Þ,x,t2 t2 Þ dt2 dt1 5 k¼1l¼1

1 1

where it is assumed that the dynamic axle loads are the only random processes, so that the expected value operator can be restricted to the product of the time histories gk ðt1 Þ and gTl ðt2 Þ of the dynamic loads at axles k and l. As both random processes are stationary, the averaged product E½gk ðt1 ÞgTl ðt2 Þ is equal to the 3 by 3 cross-correlation function Rgkl ðt1 t2 Þ [N2] of the dynamic load at two axles k and l, that depends on the time lag t1 2t2 . According to the Wiener–Kintsjin theorem, the cross-correlation function Rgkl ðt1 2t2 Þ can be written as the inverse Fourier transform of the (single-sided) cross-power

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

1695

spectral density (PSD) matrix S^ gkl ðoÞ [N2/(rad/s)]: Rgkl ðt1 t2 Þ ¼

þ Z1

S^ gkl ðoÞexp½ioðt1 t2 Þ do

(8)

0

The cross-correlation function Rgkl ðt1 t2 Þ in Eq. (8) is now introduced in Eq. (7) for the auto-correlation function Rv(x,t1,t2) of the vibration velocity. By rearranging the order of integration and collecting the terms depending on t1 and t2 , the following expression is obtained: 2t 3 þ Z1 Z1 X Zt2 na X na T ^ 4 H ðxk ðt1 Þ,x,t1 t1 Þexpðiot1 Þ dt1 S gkl ðoÞ Rv ðx,t1 ,t2 Þ ¼ Hðxl ðt2 Þ,x,t2 t2 Þexpðiot2 Þ dt2 5do (9) 0

k¼1l¼1

1

1

2.1. Assumption of fixed point loads During the passage of a relatively long train composed of carriages with similar characteristics, the response in the free field contains a nearly stationary part. An attempt can therefore be made to estimate the stationary part of the response by assuming that the dynamic axle loads are applied at fixed positions. An estimation of the auto-correlation function Rv(x,t1,t2) in Eq. (9) is obtained by omitting the time-dependency of the source positions xk ðt1 Þ and xl ðt2 Þ: 2t 3 þ Z1 X Zt2 Z1 na X na T ^ 4 H ðxk ,x,t1 t1 Þexpðiot1 Þ dt1 S gkl ðoÞ Rv ðx,t1 t2 Þ ¼ Hðxl ,x,t2 t2 Þexpðiot2 Þ dt2 5 do (10) 0

k¼1l¼1

1

1

When the axles are at a fixed position, the response is stationary and the auto-correlation function Rv(x,t1 t2) in Eq. (10) only depends on the time lag t1–t2. Due to causality of the transfer function the limits of the integrations with respect to t1 and t2 can be extended from t1 and t2 to þ1. The Fourier transformations with respect to t1 and t2 can then be performed straightforwardly by taking into account the time shifts t1 and t2: # þ Z1 " X na X na ^ T ðxk ,x, oÞexpðiot1 ÞS^ gkl ðoÞH ^  ðxl ,x, oÞexpðiot2 Þ do H (11) Rv ðx,t1 t2 Þ ¼ 0

k¼1l¼1

As the stationary auto-correlation function Rv(x,t1 t2) is the inverse Fourier transform of the PSD matrix S^ v ðx, oÞ according to the Wiener–Kintsjin theorem, the PSD matrix S^ v ðx, oÞ can be derived from Eq. (11) as follows: S^ v ðx, oÞ ¼

na X na X

^  ðxl ,x, oÞ ^ T ðxk ,x, oÞS^ gkl ðoÞH H

(12)

k¼1l¼1

This expression could also have been obtained directly from classical random vibration theory for the case of axle loads at a fixed position. In the following, only vertical loads gzk(t) with cross-PSDs S^ gkl ðoÞ are considered. 2.2. Assumption of incoherent and equal point loads The mean square value v2RMS(x) of the vertical velocity vz(x,t) in the frequency band ½o1 , o2  is obtained by integrating the PSD S^ vz ðx, oÞ of the response in this frequency band: v2RMS ðxÞ ¼

o2 Z na X na X

 h^ zz ðxk ,x, oÞS^ gkl ðoÞh^ zz ðxl ,x, oÞ do

(13)

o1 k ¼ 1 l ¼ 1

The cross-PSDs S^ gkl ðoÞ in Eq. (13) allow accounting for the coherence between axle loads and are determined by the time delay between two axles k and l. Wu and Thompson [16] have shown that the coherence between different loads can be neglected if the bandwidth is large enough. The terms in Eq. (13) containing the cross-PSDs are therefore omitted in the following. Furthermore, it is assumed that all PSDs S^ gkk ðoÞ for each axle k in Eq. (13) can be replaced by a mean value S^ g ðoÞ for all axles, resulting in

v2RMS ðxÞ ¼

o2 Z na X o1 k ¼ 1 o2 Z

 h^ zz ðxk ,x, oÞS^ gkk ðoÞh^ zz ðxk ,x, oÞ do

S^ g ðoÞ

¼ o1

na X k¼1

jh^ zz ðxk ,x, oÞj2 do

(14)

1696

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

2.3. Assumption of a frequency-averaged value of the transfer function The vibration velocity level Lv in the FRA procedure is defined as 10 times the logarithm to base 10 of the mean square value v2RMS(x) over the stationary part of the vibration velocity determined in one-third octave bands, with respect to a reference value v0 = 10  8 m/s: Lv ¼ 10 log10 ðv2RMS Þ10 log10 ðv20 Þ 2 3 o2 Z na X 6 7 ¼ 10 log10 4 S^ g ðoÞ jh^ zz ðxk ,x, oÞj2 do5Lv0

(15)

k¼1

o1

In the following, the reference value Lv0 is omitted. In order to obtain a force density per unit length and a line source transfer mobility, Eq. (15) is rewritten as 2 3 o2 Z na ^ X ð o Þ S g 6 7 La jh^ zz ðxk ,x, oÞj2 do5 (16) Lv ¼ 10 log10 4 La k¼1 o1

where La is a characteristic length over which each axle radiates energy into the soil, defined as the length L of the train divided by the number of axles na. In the FRA procedure, the influence of the source and the vibration transfer through the soil are conveniently separated. This is achieved by approximating Eq. (16) as follows: 20 10 13 o2 o2 Z Z na X 6B S^ g ðoÞ CB 1 C7 2 (17) doA@ La jh^ zz ðxk ,x, oÞj doA5 Lv ¼ 10 log10 4@ La Do k¼1 o1

o1

Pna

2 ^ k ¼ 1 jh zz ðxk ,x, oÞj is approximated by its average over the frequency band ½o1 , o2  with bandwidth Do. The ^ integrated PSD S g ðoÞ of the load reduces to the RMS value g2RMS in the corresponding one-third octave band. The vibration velocity level Lv is finally calculated as " # R o2 ^  2  2 na X g o1 jh zz ðxk ,x, oÞj do Lv ¼ 10 log10 RMS þ 10 log10 La (18) La Do k¼1

where the term

Eq. (18) has the same form as Eq. (1). The first term characterizes the source and corresponds to the force density LF. The second term characterizes the wave propagation through the soil and corresponds to the line source transfer mobility TML.

2.4. Assumption of equidistant impact points In Eq. (18), the line source transfer mobility can be rewritten as a superposition of point source transfer mobilities TMPk : " # na X TMPk =10 (19) TML ¼ 10 log10 La 10 k¼1

where TMPk is defined as "R o2 TMPk ¼ 10 log10

o1

jh^ zz ðxk ,x, oÞj2 do

#

Do

(20)

Eq. (19) corresponds to Eq. (2) for the determination of the line source transfer mobility TML according to the measurement setups in Fig. 1. In Eq. (19), na impact points are considered equal to the number of axles in the train and each impact point is located at the position of the corresponding axle (Fig. 3a). In the FRA procedure, however, na equidistant impact points are used (Fig. 3b). When the line source transfer mobility TML is determined according to Eq. (2), an arbitrary number n of impact points at interval h can be chosen, as long as the product nh corresponds to the total length L of the train. In the FRA manual, a third setup is suggested with impact points at the start and the end of the train (Fig. 3c). Using the trapezoidal rule, the edge points only account for half the length h/2 of the impact interval h, resulting in the following expression [8]: TML ¼ 10 log10 ½hð12 10TMP1 =10 þ 10TMP2 =10 þ    þ 10TMPn1 =10 þ 1210TMPn =10 Þ

(21)

In a typical application of the FRA procedure, the force density LF and the line source transfer mobility TML are determined experimentally. Eq. (18) now provides expressions that can be used to compute both terms numerically. In situations where no experimental force density is available, Eq. (18) can be used to compute the force density. When the computed force density is combined with an experimentally determined line source transfer mobility TML, a hybrid experimental–numerical prediction of the vibration velocity level is obtained.

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

1697

La

La

h/2

h

Fig. 3. Location of the impact points for the determination of the line source transfer mobility TML with (a) na impact points at the axle locations, (b) na equidistant impact points and (c) n impact points with two edge points.

3. Verification of the analytical expressions for the detailed vibration assessment of the FRA In Section 2, several assumptions have been introduced to obtain the analytical expressions for the force density and the line source transfer mobility in Eq. (18). First, a moving load is replaced by a load at a fixed position; second, incoherent point loads are considered and a mean value of the PSD is used; third, an average value of the transfer function in each frequency band is used and finally, the line source transfer mobility is calculated with equidistant point loads. These assumptions are now verified for the case of a train running in a tunnel. A coupled finite element–boundary element (FE–BE) method has recently been presented [6,14] to predict vibrations due to underground railway traffic. In this method, the tunnel geometry is assumed to be periodic in the longitudinal direction of the tunnel. The tunnel structure is modeled with the finite element method, while the soil is modeled with the boundary element method. A track is included in the tunnel model using the Craig–Bampton substructuring method. The coupled FE–BE method has been mutually verified with the pipe-in-pipe method [17–19] for the case of deep bored tunnels. Furthermore, the coupled FE–BE method has been experimentally validated by means of in situ measurements near the RER B line in Paris and near the Bakerloo line in London [13]. In the following, the case of the Bakerloo line tunnel is considered for verification of the expressions derived in Section 2. The Bakerloo tunnel is a deep bored tunnel with a cast iron lining and a single track, embedded in London clay at a depth of 28 m. The soil stratification has been derived from tests in situ and in the laboratory, revealing the presence of a shallow layer of 5 m on top of a homogeneous half-space consisting of London clay. The vibration velocity due to the passage of a train is computed. The only excitation mechanism considered is random track unevenness. In the present case, an unevenness according to FRA class 3 is taken. The train consists of seven cars of two different types—motor and trailer cars—with two bogies and four axles each. The train has a total of 28 axles with a length L of 112.30 m. The axles are characterized by the unsprung mass Mu, that corresponds to the mass of the wheelset. This model is accurate above a few Hertz, as the unsprung mass is dynamically uncoupled from the body and the bogie in this frequency range [12]. The mass Mu is equal to 1210 kg for motor cars and 950 kg for trailer cars. The vibration velocity in the free field is calculated due to a train passage at 30 and 48 km/h. A complete description of the train and the coupled FE–BE model of the track–tunnel–soil system is given by Gupta et al. [13]. In the following, the assumptions introduced in the derivation of the analytical expressions of the force density and the line source transfer mobility are verified by means of the coupled FE–BE model. A step-wise procedure is followed where the accuracy of each assumption is verified, before vibration velocities are compared to assess the global accuracy obtained with Eq. (18). The transfer functions and the time history of the axle loads are obtained with the coupled FE–BE method. Two points A and B are considered that are located at the free surface at 5.5 and 60 m from the tunnel centerline (Fig. 4). 3.1. Assumption of fixed point loads In Eq. (3), the vibration velocity in the free field is computed for moving axle loads, while in Section 2.1 the assumption of fixed point loads is introduced. This assumption is first verified by comparing the running RMS value of the vibration velocity due to moving and fixed sinusoidal loads, corresponding to axles running over a track with a sinusoidal unevenness. Fig. 5a shows the vertical vibration velocity in point A due to a single sinusoidal axle load of 80 Hz, moving at a speed of 30 km/h and at a fixed position. It is observed how the moving load generates a transient response while the fixed load generates a stationary response. A good agreement between both results is found when the position xk(t) of the moving load is equal to the position xk of the fixed load. Fig. 5b shows the vertical vibration velocity in point A due to a series of sinusoidal axle loads of 80 Hz, moving at a speed of 30 km/h and at a fixed position. The time history of the vibration velocity due to the moving train can be subdivided into three parts: an increasing level when the train approaches the measurement line, an approximately stationary level when the train passes the measurement line and a decreasing level when the train moves away from the

1698

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

60 m 5.5 m z B

28 m

A x

Fig. 4. Location of the tunnel and the measurement points A and B at the surface in the free field.

x 10−5

x 10−4 80 Hz

80 Hz

RMS velocity [m/s]

RMS velocity [m/s]

5 4 3 2 1

0 −20 −15 −10 −5

0

5

10

15

2

1

0 −20 −15 −10 −5

20

Time [s]

0

5

10

15

20

Time [s]

50 40 30 20 10 0 2 4 8 16 31.5 63 125 1/3 octave band center frequency [Hz]

RMS velocity [dB ref 10−8 m/s]

RMS velocity [dB ref 10−8 m/s]

Fig. 5. Running RMS value of the vertical velocity in point A due to (a) a single sinusoidal load and (b) a series of sinusoidal loads moving at a speed of 30 km/h (grey line) and at a fixed position (black line).

50 40 30 20 10 0

2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

Fig. 6. One-third octave band RMS value of the vertical velocity (a) in point A and (b) in point B due to a train moving at 30 km/h (grey line) and a train at a fixed position (black line).

measurement line. It is observed in Fig. 5b that the vibration level during the stationary part of the response is well approximated by the fixed load response. It can therefore be anticipated that the vibration velocity level Lv in the free field due to a train running on a track with random track unevenness can be well approximated by a prediction with fixed point loads. The one-third octave band RMS value of the vertical velocity in points A and B due to a train passage and due to a train at a fixed position with the load history of the moving axles is shown in Fig. 6 for a speed of 30 km/h and in Fig. 7 for a speed of 48 km/h. For the selection of the stationary part of the transient response, the procedure for the determination of the period T2 described in the DIN 45672-2 standard [20] is followed. In point A (Figs. 6a and 7a), a peak around 50 Hz is observed due to the resonance frequency of the coupled vehicle–track system. This peak is not present in the result for point B (Figs. 6b and 7b) as higher frequencies are more strongly attenuated with increasing distance from the track.

50 40 30 20 10 0 2

4

8

16 31.5 63 125

RMS velocity [dB ref 10−8 m/s]

RMS velocity [dB ref 10−8 m/s]

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

1699

50 40 30 20 10 0 2

1/3 octave band center frequency [Hz]

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

50 40 30 20 10 0 2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

RMS velocity [dB ref 10−8 m/s]

RMS velocity [dB ref 10−8 m/s]

Fig. 7. One-third octave band RMS value of the vertical velocity (a) in point A and (b) in point B due to a train moving at 48 km/h (grey line) and a train at a fixed position (black line).

50 40 30 20 10 0 2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

Fig. 8. One-third octave band RMS value of the vertical velocity (a) in point A and (b) in point B due to a fixed train with (grey line) and without (black line) the contribution of the cross-PSD matrices to the response.

In points A and B, drop-outs are observed at low frequencies in the frequency spectrum of the response to a moving train. These drop-outs appear when almost no transmission occurs in a particular frequency band due to interference between contributions to the free field response from different axles. As each axle is running over the same track, these contributions are similar except for a time delay which is determined by the axle position yk and the train speed v. The interference between the delayed contributions of different axles to the response results in characteristic lobes that are observed in typical response spectra [21–23]. When the response is determined in one-third octave bands, these lobes are averaged out when the onethird octave bands are sufficiently wide. At low frequencies, however, the bandwidth is small and single peaks and drop-outs still appear in the one-third octave band spectrum. In the frequency spectrum of the response to fixed point loads, no dropouts are observed as the effect of interference between contributions from different axles is considerably reduced in this case. This is due to the fact that the axles are fixed at different locations and, consequently, have different contributions to the response. As a result, the interference is therefore considerably reduced and drop-outs are no longer observed. Even when the response to moving loads and fixed point loads are considerably different in a few one-third octave bands at low frequencies, it can be concluded that a relatively good agreement is observed between both results. 3.2. Assumption of incoherent and equal point loads When fixed point loads are assumed, the interference between the contributions from different axles is considerably reduced. In the previous subsection, it was still assumed, however, that the axles are running over the same unevenness and therefore yield coherent axle loads. The coherence between the axle loads still leads to some interference between the contributions from different axles. In Eq. (13), the assumption of incoherent point loads is introduced by neglecting the crossPSDs S^ gkl ðoÞ. It has been shown by Wu and Thompson [16], however, that assuming incoherent point loads leads to a good approximation when the average response is studied in sufficiently wide frequency bands. The assumption of incoherent point loads is now verified by comparing the response obtained from Eq. (13) that includes all cross-PSDs S^ gkl ðoÞ with the response obtained from Eq. (14) that only accounts for the PSDs S^ gkk ðoÞ. Fig. 8 shows the onethird octave band RMS value of the vertical velocity in points A and B due to a series of fixed point loads with and without the contribution of the cross-PSDs S^ gkl ðoÞ to the response. A good agreement is found between both results, especially in the

1700

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

higher frequency range where the bandwidth is large and lobes in the response are averaged out. At frequencies below 2 Hz, the bandwidth is small and the agreement is less good. It should be noted, however, that the relatively good agreement between the results for coherent and incoherent loads is due to the fact that fixed point loads are considered here. As has been discussed in the previous subsection, this largely reduces the effect of interference between the contributions from different axles. In the case where a moving train is considered, assuming incoherent point loads will have greater impact and the agreement between the response to coherent and incoherent point loads will be less good. Excitation due to rail unevenness is the only excitation mechanism considered in this paper. The interference effect will occur as well for other excitation mechanisms that are perceived by each axle, such as parametric excitation due to the spatial variation of the track support system and transient excitation due to rail joints. Excitation due to wheel unevenness or wheel flats, however, is different for all axles and therefore leads to incoherent axle loads. Including excitation due to wheel defects therefore reduces the coherence between different axle loads and, consequently, reduces the effect of assuming fixed and incoherent point loads. In Eq. (14), the PSDs S^ gkk ðoÞ are also replaced by a mean value S^ g ðoÞ for all axles. In the present case where only random track unevenness is considered and each axle is modeled by its unsprung mass, differences in the axle loads are mainly due to a different value of the unsprung mass. When the axles have similar dynamic characteristics, it can therefore be assumed that the PSDs S^ gkk ðoÞ in Eq. (13) are similar and can be replaced by a mean value S^ g ðoÞ. Fig. 9 shows the one-third octave band RMS value of the vibration velocity in points A and B, obtained from Eq. (14) with the original PSDs S^ gkk ðoÞ of the axle loads and with a mean PSD S^ g ðoÞ for all axles. It can be concluded that replacing the original PSDs S^ gkk ðoÞ by a mean value S^ g ðoÞ allows for an accurate estimation of the one-third octave band RMS value of the vibration velocity level. 3.3. Assumption of a frequency-averaged value of the transfer function In Section 2.3, Eq. (16) has been derived for the calculation of the vibration velocity level Lv. In order to obtain Eq. (17), P where the force density and the line source transfer mobility are calculated separately, the term nk a¼ 1 jh^ zz ðxk ,x, oÞj2 is approximated by its average value over a one-third octave band. This assumption is verified by comparing the one-third octave band RMS value of the vertical velocity in points A and B computed with Eq. (16) based on the narrow band transfer function and computed with Eq. (17) using the frequency-averaged transfer function (Fig. 10). A good agreement is found between the two approaches. 3.4. Assumption of equidistant impact points In Eq. (19), the line source transfer mobility TML is determined with impact points corresponding to the axle positions. In Section 2.4, it is stated that the line source transfer mobility TML can also be calculated from point source transfer mobilities obtained from equidistant impact points. This assumption is verified by calculating the line source transfer mobility TML for three different setups of the impact points (Fig. 3) representing a train with length L: (1) na impact points corresponding to the axle positions, (2) na equidistant impact points with a spacing L/na and (3) n equidistant impact points with a spacing L/n including two edge points according to Eq. (21) given in the FRA procedure. Fig. 11 shows the line source transfer mobility TML in points A and B obtained from the three different setups. These results show that only very small differences are found for the line source transfer mobility TML. 3.5. Final comparison

50 40 30 20 10 0 2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

RMS velocity [dB ref 10−8 m/s]

RMS velocity [dB ref 10−8 m/s]

Finally, the overall accuracy of the expressions derived for the FRA method is verified. First, the vertical velocity due to a train running in the tunnel is calculated with the coupled FE–BE method. The RMS value of the vertical velocity in one-third

50 40 30 20 10 0 2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

Fig. 9. One-third octave band RMS value of the vertical velocity (a) in point A and (b) in point B due to a fixed train calculated with different axle loads (grey line) and with the mean value of axle loads (black line).

50 40 30 20 10 0 2

4

8

16 31.5 63 125

RMS velocity [dB ref 10−8 m/s]

RMS velocity [dB ref 10−8 m/s]

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

1701

50 40 30 20 10 0 2

1/3 octave band center frequency [Hz]

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

Fig. 10. One-third octave band RMS value of the vertical velocity (a) in point A and (b) in point B due to a fixed train calculated with the narrow band transfer function (grey line) and the frequency-averaged transfer function (black line).

(b) 0 −10 −20 −30 −40 −50 2 4 8 16 31.5 63 125 1/3 octave band center frequency [Hz]

RMS velocity [dB ref 10−8 m/s]

RMS velocity [dB ref 10−8 m/s]

(a)

0 −10 −20 −30 −40 −50 2 4 8 16 31.5 63 125 1/3 octave band center frequency [Hz]

50 40 30 20 10 0 2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

RMS velocity [dB ref 10−8 m/s]

RMS velocity [dB ref 10−8 m/s]

Fig. 11. One-third octave band value of the line source transfer mobility (a) in point A and (b) in point B for na impact points corresponding to the axle locations (light grey line), na equidistant impact points with a spacing L/na (dark grey line) and n equidistant impact points with a spacing L/n including two edge points (black line).

50 40 30 20 10 0

2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

Fig. 12. One-third octave band RMS value of the vertical velocity (a) in point A and (b) in point B due to a train moving at 30 km/h computed with the coupled FE–BE method (grey line) and with the FRA procedure (black line).

octave bands is calculated from this result. Second, the vibration velocity level is calculated with Eq. (18). The one-third octave band RMS value of the vertical velocity in points A and B is shown in Fig. 12 for a train moving at 30 km/h and in Fig. 13 for a train moving at 48 km/h. A good agreement is found between both results, especially at higher frequencies. The differences are mainly due to the assumption of fixed point loads, which significantly reduces the effect of interference between the contributions from different axles.

50 40 30 20 10 0 2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

RMS velocity [dB ref 10−8 m/s]

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

RMS velocity [dB ref 10−8 m/s]

1702

50 40 30 20 10 0 2

4

8

16 31.5 63 125

1/3 octave band center frequency [Hz]

Fig. 13. One-third octave band RMS value of the vertical velocity (a) in point A and (b) in point B due to a train moving at 48 km/h computed with the coupled FE–BE method (grey line) and with the FRA procedure (black line).

4. Conclusions The force density and the line source transfer mobility are concepts introduced by the detailed vibration assessment of the FRA. In this paper, the equations governing railway induced vibrations are used to derive analytical expressions for the force density and the line source transfer mobility. For this purpose, four assumptions are introduced: first, the assumption of fixed point loads is introduced; second, incoherent and equal axle loads are considered; third, the narrow band transfer function is replaced by its averaged value in one-third octave bands and finally, the line source transfer mobility is calculated with equidistant point loads. These assumptions are verified by means of a coupled FE–BE method in a case study where vibrations due to underground railway traffic are considered. This verification shows that a reliable prediction of the one-third octave band RMS value of the vertical velocity is obtained with the derived analytical expression and, consequently, the assumptions are justified under the conditions presented in this paper. The main difference between the numerical prediction and the prediction with Eq. (18) is due to the assumption of fixed point loads. It should be noted as well that only the case of underground railway traffic has been considered. A more elaborate study is needed to extend the generality of the conclusions. The two-step approach for the prediction of the vibration velocity level in case of new railway infrastructure that has been outlined in Section 1, is not assessed in the present study. The reliability of this approach can be investigated by comparing results obtained with a simulation of the two-step approach of the FRA procedure with results obtained with numerical predictions. The link that has been established in this paper between numerical and empirical predictions as the FRA procedure allows for so-called hybrid predictions based on empirical data and numerical results. Numerical models allow for a great flexibility in dealing with different train/track models, while empirical models allow for an accurate assessment of the vibration propagation through the soil. In hybrid predictions, the advantages of both approaches are combined.

Acknowledgments The research in this paper has been performed within the frame of the project OT-05-41 ‘‘A generic methodology for inverse modelling of dynamic problems in civil and environmental engineering’’. This project is funded by the Research Council of K.U.Leuven. Their financial support is gratefully acknowledged. The first author is a Research Assistant of the Research Foundation-Flanders (FWO). The support of FWO is gratefully acknowledged.

References [1] P. Galvı´n, J. Domı´nguez, Analysis of ground motion due to moving surface loads induced by high-speed trains, Engineering Analysis with Boundary Elements 31 (2007) 931–941. [2] H. Grundmann, M. Lieb, E. Trommer, The response of a layered half-space to traffic loads moving along its surface, Archive of Applied Mechanics 69 (1999) 55–67. [3] G. Lombaert, G. Degrande, J. Kogut, S. Franc-ois, The experimental validation of a numerical model for the prediction of railway induced vibrations, Journal of Sound and Vibration 297 (3–5) (2006) 512–535. [4] X. Sheng, C.J.C. Jones, M. Petyt, Ground vibration generated by a load moving along a railway track, Journal of Sound and Vibration 228 (1) (1999) 129–156. [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 (3–5) (2006) 611–625. [6] G. Degrande, D. Clouteau, R. Othman, M. Arnst, H. Chebli, R. Klein, P. Chatterjee, B. Janssens, 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 (3–5) (2006) 645–666.

H. Verbraken et al. / Journal of Sound and Vibration 330 (2011) 1692–1703

1703

[7] A.V. Metrikine, T. Vrouwenvelder, Surface ground vibration due to a moving train in a tunnel: two-dimensional model, Journal of Sound and Vibration 234 (1) (2000) 43–66. [8] C.E. Hanson, D.A. Towers, L.D. Meister, High-speed Ground Transportation Noise and Vibration Impact Assessment, HMMH Report 293630-4, U.S. Department of Transportation, Federal Railroad Administration, Office of Railroad Development, October 2005. [9] C.E. Hanson, D.A. Towers, L.D. Meister, Transit Noise and Vibration Impact Assessment, Report FTA-VA-90-1003-06, U.S. Department of Transportation, Federal Transit Administration, Office of Planning and Environment, May 2006. [10] E.C. Bovey, Development of an impact method to determine the vibration transfer characteristics of railway installations, Journal of Sound and Vibration 87 (2) (1983) 357–370. [11] J.T. Nelson, H.J. Saurenman, A prediction procedure for rail transportation groundborne noise and vibration, Transportation Research Record 1143 (1987) 26–35. [12] 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 (3–5) (2009) 1036–1066. [13] S. Gupta, G. Degrande, G. Lombaert, Experimental validation of a numerical model for subway induced vibrations, Journal of Sound and Vibration 321 (3–5) (2009) 786–812. ¨ [14] G. Degrande, M. Schevenels, P. Chatterjee, W. Van de Velde, P. Holscher, V. Hopman, A. Wang, N. Dadkah, Vibrations due to a test train at variable speeds in a deep bored tunnel embedded in London clay, Journal of Sound and Vibration 293 (3–5) (2006) 626–644. [15] M. Heckl, G. Hauck, R. Wettschureck, Structure-borne sound and vibration from rail traffic, Journal of Sound and Vibration 193 (1) (1996) 175–184. [16] T.X. Wu, D.J. Thompson, Vibration analysis of railway track with multiple wheels on the rail, Journal of Sound and Vibration 239 (1) (2001) 69–97. [17] S. Gupta, M.F.M. Hussein, G. Degrande, H.E.M. Hunt, D. Clouteau, A comparison of two numerical models for the prediction of vibrations from underground railway traffic, Soil Dynamics and Earthquake Engineering 27 (7) (2007) 608–624. [18] 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 (2006) 678–705. [19] J.A. Forrest, H.E.M. Hunt, Ground vibration generated by trains in underground tunnels, Journal of Sound and Vibration 294 (2006) 706–736. ¨ Normung. DIN 45672 Teil 2: Schwingungsmessungen in der Umgebung von Schienenverkehrswegen: Auswerteverfahren, 1995. [20] Deutsches Institut fur [21] L. Auersch, The excitation of ground vibration by rail traffic: theory of vehicle–track–soil interaction and measurements on high-speed lines, Journal of Sound and Vibration 284 (1–2) (2005) 103–132. [22] H.E.M. Hunt, Modelling of road vehicles for calculation of traffic-induced ground vibrations, Journal of Sound and Vibration 144 (1) (1991) 41–51. [23] H.E.M. Hunt, Modelling of rail vehicles and track for calculation of ground vibration transmission into buildings, Journal of Sound and Vibration 193 (1) (1996) 185–194.