Experimental validation of a numerical model for subway induced vibrations

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1 Experimental validation of a numerical model for subway induced vibrations S. Gupta, G. Degrande, G. Lombaert Department of Civil Engineering, K.U.Leuven, Kasteelpark Arenberg, B-3001, Leuven, Belgium Tel: , Fax: , shashank.gupta@bwk.kuleuven.be Summary This paper presents the experimental validation of a numerical model for the prediction of subway induced vibrations. The model fully accounts for the dynamic interaction between the train, the track, the tunnel and the soil. The periodicity or invariance of the tunnel and the soil in the longitudinal direction is exploited using the Floquet transformation, which allows for an efficient formulation in the frequency-wavenumber domain. A general analytical formulation is used to compute the response of three-dimensional invariant or periodic media that are excited by moving loads. The numerical model is validated by means of several experiments that have been performed at a site in Regent s Park on the Bakerloo line of London Underground. Vibration measurements have been performed on the axle boxes of the train, on the rail, the tunnel invert and the tunnel wall, and in the free field, both at the surface and at a depth of 15 m. Prior to these vibration measurements, the dynamic soil characteristics and the track characteristics have been determined. The Bakerloo line tunnel of London Underground has been modelled using the coupled periodic FE-BE approach and free field vibrations due to the passage of a train have been predicted and compared to the measurements. The correspondence between the predicted and measured response in the tunnel and in the free field is reasonably good, given the large amount of uncertainties involved. 1. Introduction 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 buildings, causing annoyance to people. Vibrations are perceived directly or they are sensed indirectly as re-radiated noise. The frequency range of interest for subway induced vibrations is 1- Hz and for the re-radiated noise it is 30-0 Hz. To quantify these vibrations, great efforts have been made in recent years to develop the prediction models [1,2,3,4,5,6] that account for the three-dimensional dynamic track-tunnel-soil interaction. These advanced models take advantage of the invariance (or the periodicity) of the geometry along the tunnel axis using a Fourier or Floquet transformation. This paper concentrates on the coupled periodic finite element-boundary element (FE-BE) model [1,2] that was developed within the frame of the CONVURT project [7]. Elaborate in situ vibration measurements have been performed at a site in Regent s Park situated above the north- and south-bound Bakerloo line tunnels of London Underground [8]. These tunnels are deep-bored segmented tunnels with a cast iron lining and a single track, embedded in London clay at a depth of 28 m. The reference section is situated at kilometer post , which is 581 m west of Regent s Park station or approximately 0 m east of Baker Street station. S. Gupta, G. Degrande, G. Lombaert 1

2 Vibration measurements have been performed during engineering hours at night for 35 passages of a test train in the north-bound Bakerloo line tunnel at a speed between and 50 km/h. In addition, rail and wheel roughness have been measured, while the track characteristics have been determined by rail receptance measurements [9]. The dynamic soil characteristics have been determined by in situ tests (SCPT, SASW) and by laboratory testing [10]. The results of the vibration measurements are presently used to validate the coupled periodic FE-BE model. 2. The numerical method Within the frame of the CONVURT project [7], a coupled periodic FE-BE model has been developed that exploits the longitudinal invariance or periodicity of the track-tunnel-soil system [1,2]. The response to the moving loads in the periodic domains is given by Chebli et al. [11]. The response to moving loads is deduced from the transfer function in the frequency-wavenumber domain and the frequency content of the axle loads Modelling of the track-tunnel-soil system The transfer functions are computed using the classical domain decomposition approach based on the finite element method for the tunnel and the boundary element method for the soil [1,2]. The Floquet transform is used to exploit the periodicity of geometry and to restrict the problem domain to a single bounded reference cell. The track-tunnel-soil interaction problem is solved in the frequencywavenumber domain and the wave field radiated into the soil is computed. Reader is referred to complimentary literature [1,2] for more details on the coupled periodic FE-BE model. The Bakerloo line tunnel of London Underground 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 tunnel has an internal radius of 1.83 m and a wall thickness of m. There are six longitudinal stiffeners and one circumferential stiffener at an interval of m, resulting in a periodic structure. The tunnel s reference cell is modelled using finite element method, where shell elements have been used for the cast iron lining, while the longitudinal and circumferential stiffeners are modelled using beam elements. The concrete on the tunnel invert has been modelled using 8-node brick elements with incompatible bending modes. Dynamic soil characteristics have been determined by in situ and laboratory testing [10,12]. The testing revealed that the tunnel is embedded in a layered soil consisting of a shallow layer with a thickness of 5 m on top of a homogeneous half space consisting of London clay. The top layer has a shear wave velocity C s = 275 m/s, a longitudinal wave velocity C p = 1964 m/s, a density ρ s = 19 kg/m 3 and a material damping ratio β s = The underlying half space has a shear wave velocity C s = 2 m/s, a longitudinal wave velocity C p = 1571 m/s, a density ρ s = 19 kg/m 3 and a material damping ratio β s = [8]. The track is a non-ballasted concrete slab track with Bullhead rail supported on hard wooden sleepers nominally spaced at d = 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. The rails have a mass per unit length ρ r A r = 47 kg/m and a bending stiffness E r I r = Nm 2. The rails are not supported by rail pads and the resilience is mainly provided by the timber sleepers, which have a varying stiffness depending on its moisture content. The track model consists of two infinite Euler beams representing the rails and the mass elements representing the sleepers. The mass of the sleepers is distributed in the longitudinal direction with a mass per unit length m s = M s /d = 70 kg/m. As there are no rail pads, a stiff connection is assumed between the rails and the sleepers, while the sleepers are continuously supported on the tunnel invert with springs of vertical stiffness k s = k s /d = MN/m 2. The S. Gupta, G. Degrande, G. Lombaert 2

3 continuous elastic support below the sleepers accounts for the resilience of the track. A high value of the damping c s = Ns/m 2 is assumed in the elastic support to suppress the resonance, as no resonance has been observed in the rail receptance measurements within 0 Hz The train-track interaction forces Ground vibrations generated by moving trains arise from the combination of various excitation mechanisms. For the experimental validation of the numerical model, two main excitation mechanisms are considered: the quasi-static excitation and the unevenness excitation. The test train employed for vibration measurements on the Bakerloo line consisted 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 m, while the length of the trailer car is m. The bogie and axle distances on all cars are m and 1.91 m, respectively. The distance between the first and the last axle of the train is m. The tare mass of a motor car is kg, while the bogie mass is 6690 kg and the mass of wheelset is 1210 kg. The tare mass of a trailer car is kg, while the bogie mass is 4170 kg and the mass of a wheelset is 950 kg. The quasi-static excitation occurs when the axles of the train pass over the track and can be modelled as constant forces moving along the track with the train speed v. The constant load is equal to the total weight of the train distributed on the axles. Roughness is the main generation source of vibrations from moving trains. Rail roughness has been measured on the site using Müller BBM rail roughness measurement equipment (RM10E) [8]. Unevenness of the rail with wavelength longer than 0.10 m could not be measured with the available equipment, thus restricting the analysis to frequencies above 130 Hz for a train speed of 47.6 km/h [8]. Due to this limitation of the roughness measurements, axle box vibrations are used to estimate the wheel-rail interface forces ĝ d (ω) [13]: Ĉ v (ω)ĝ d (ω) = û a (ω) (1) where Ĉv (ω) is the vehicle s compliance and û a (ω) is the response of the axle of the train. For the computation of the dynamic forces, the train can be well represented with the vehicle s unsprung mass [13]. In this case, the vehicle compliance matrix is equal to the diagonal matrix Ĉv (ω) = diag{ 1/(M u ω 2 )} of order 28. It should be mentioned that the vehicle compliance is significantly influenced by the vehicle s suspensions at low frequencies and therefore, the estimation of the forces at frequencies below 10 Hz will be inaccurate. Moreover, the effect of wheel-axle bending resonance, which occurs at higher frequencies has not been accounted for. Axle box vibrations have been measured on six axle boxes during the whole journey of the test train on the section between Regent s Park and Baker Street stations [8]. Figure 1a shows the time history of the acceleration of axle box 1 for a period of time corresponding to the passage of a train over the test section at a speed of 47.6 km/h. The advantage of using equation (1) to estimate the interaction force is that the various excitation mechanisms such as the unevenness excitation, parametric excitation and excitation due to rail joints and wheel flats are intrinsically accounted for. Visual inspection of the track revealed that there were a number of rail joints in the vicinity of the reference section, which could generate significant impact forces at the wheel-rail interface. The peaks in the axle box response shown in figure 1a are clearly due to the passage of the axle over joints in the rail. Figure 1b shows the one-third octave band spectra of the axle box displacements. The high response of the axle boxes indicate that the rails are of very poor quality. This has also been confirmed by the observation on the site that the rails were indeed heavily corrugated. S. Gupta, G. Degrande, G. Lombaert 3

4 (a) Acceleration [m/s 2 ] (b) Displacement [db ref µ m] Frequency [Hz] (c) RMS force [db ref 1 N] Frequency [Hz] Fig. 1. (a) Time history and (b) one-third octave band spectrum of the axle box response for a period of time corresponding to the passage of a train length over the test section at a train speed of 47.6 km/h. (c) The one-third octave band spectrum of the contact force at the front axle of the train for a speed of 47.6 km/h. Figure 1c shows the contact force at the front axle of the train for a train speed of 47.6 km/h. The frequency content of these forces exhibits a clear maximum near the train-track resonance frequency between 50 and Hz. 3. Response during the passage of a train in the Bakerloo line tunnel The response can be calculated by adding the contribution of the dynamic forces and the quasistatic forces in the frequency domain. In the following, the running RMS and the one-third octave band RMS spectra of the measured and predicted vertical vibration velocity are compared. The running RMS has been calculated with an averaging time of one second according to ISO (a) (b) 1 1 A A A1 1 A6 Fig. 2. The experimental (gray line) and computed (black line) running RMS and one-third octave band RMS spectra of the vertical velocity on (a) the rail (A1) and (b) the tunnel invert (A6) during the passage of a test train at a speeds 47.6 km/h. Firstly, the response in the tunnel is compared to the experimental results on the rail and the tunnel invert [9]. Figure 2 compares the predicted and measured vertical velocity on the rail (A1) and tunnel invert (A6) during the passage of the test train at a speed of 47.6 km/h. On the rail, the contribution of the quasi-static forces and the dynamic forces can be distinguished. The quasi-static forces are important at low frequencies below 10 Hz. A peak corresponding to the axle passage frequency of f a = v/l a = 6.92 Hz (L a = 1.91 m) is observed in the predicted as well as the measured spectra. S. Gupta, G. Degrande, G. Lombaert 4

5 (a) (b) 90 FF02z FF03z FF02z FF03z Fig. 3. The experimental (gray line) and computed (black line) running RMS and one-third octave band RMS spectra of the vertical velocity on the surface (FF02) and (b) at the depth (FF03) during the passage of a test train at a speed of 47.6 km/h. The vibration levels are maximum on the rail and decrease on the tunnel invert. The contribution of the quasi-static forces has diminished on the tunnel invert, and the dominant frequency content is situated in the frequency range above 30 Hz, where the dynamic forces are significant. The response at low frequencies below 12 Hz is underestimated by the numerical model. This could be due to the underestimation of the dynamic forces at these low frequencies. Also the influence of the train suspensions is significant at low frequencies, which has been disregarded in the model. The running RMS values show that the prediction accuracy on the rail and the tunnel invert is within 10 db. Free field vibration measurements have been performed in Regent s Park above the Bakerloo line tunnels. The vibrations measurements have been performed on the surface as well as at a depth of 15 m, where tri-axial accelerometers have been installed in a seismic cone [10]. In this paper, the computed vertical response is compared to the measurements on the surface (FF02z) and at a depth of 15 m (FF03z), at a distance of 23.5 m from the tunnel. Figure 3 compares the experimental and computed running RMS and the one-third octave band spectra of the vertical free field vibration at points FF02 and FF03 during the passage of a test train at a speed of 47.6 km/h. Both the experimental and numerical results show that the dominant frequency content is around the wheel-track resonance frequency of about 50 Hz. A reasonably good agreement between the experimental and numerical results is observed for the response in the free field. The quasi-static contribution at low frequencies is not important in the free field and only the dynamic forces prevail. The vertical response at the surface (FF02z) has approximately the same magnitude as the vertical component at depth at the same location (FF03z). Furthermore, the amplitude decreases for increasing distance from the tunnel due to geometrical damping, while higher frequency components at increasing distance are significantly attenuated by material damping in the soil. 4. Conclusions In this paper, the experimental validation of a numerical model for the prediction of subway induced vibrations has been presented. An elaborate measurement campaign has been conducted at a site in Regent s Park situated above the Bakerloo line tunnels of London Underground. In situ vibra- S. Gupta, G. Degrande, G. Lombaert 5

6 tion measurements have been performed on the axle boxes of the test train, in the tunnel and in the free field. Apart from these measurements, other tests and measurements have also been performed to determine the soil properties and track characteristics. The coupled periodic FE-BE model fully accounts for the dynamic interaction between the train, the track, the tunnel and the soil. The response to moving loads (trains) is computed by first estimating the excitation forces and then solving the track-tunnel-soil interaction problem to compute the vibrations in the free field. The interaction force determined from the axle box vibrations accounts for various excitation mechanisms such as unevenness excitation, excitation due to rail joints and wheel flats as well as parametric excitation. The free field vibrations for the passage of a test train in the Bakerloo line tunnel have been predicted and validated. The correspondence between the predicted and experimental results is reasonably good, given the large number of modelling uncertainties. This paper demonstrates the applicability of the state-of-the-art 3D model in accurate prediction of vibrations from underground railways. The advanced model enables to investigate the inherent physics of ground vibrations, which is not possible with empirical methods or simplified deterministic models. References [1] D. Clouteau, M. Arnst, T.M. Al-Hussaini, and G. Degrande. Freefield vibrations due to dynamic loading on a tunnel embedded in a stratified medium. Journal of Sound and Vibration, 283(1 2): , 05. [2] G. Degrande, D. Clouteau, R. Othman, M. Arnst, H. Chebli, R. Klein, P. Chatterjee, and 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): , 06. [3] J.A. Forrest and H.E.M. Hunt. A three-dimensional tunnel model for calculation of train-induced ground vibration. Journal of Sound and Vibration, 294(4-5): , 06. [4] M.F.M. Hussein. Vibration from underground railways. PhD thesis, Department of Engineering, University of Cambridge, 04. [5] X. Sheng, C.J.C. Jones, and D.J. Thompson. Prediction of ground vibration from trains using the wavenumber finite and boundary element methods. Journal of Sound and Vibration, 293: , 06. [6] L. Andersen and 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: , 06. [7] [8] G. Degrande, M. Schevenels, P. Chatterjee, W. Van de Velde, P. Hölscher, V. Hopman, A. Wang, and 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): , 06. [9] A. Wang. Track measurements on London Underground Bakerloo Line. Report , Pandrol, May 03. CONVURT EC-Growth Project G3RD-CT [10] P. Hölscher and V. Hopman. Test site Regent s Park London. Soil description. Report , Version 2, GeoDelft, December 03. CONVURT EC-Growth Project G3RD-CT [11] H. Chebli, O. Ramzi, and D. Clouteau. Response of periodic structures due to moving loads. Comptes Rendus Mécanique, 334: , 06. [12] L. Pyl and G. Degrande. Determination of the dynamic soil characteristics with the SASW method at Regent s Park in London. Report BWM-03-17, Department of Civil Engineering, K.U.Leuven, December 03. CONVURT EC-Growth Project G3RD-CT [13] G. Lombaert, G. Degrande, J. Kogut, and S. François. The experimental validation of a numerical model for the prediction of railway induced vibrations. Journal of Sound and Vibration, 297(3-5): , 06. S. Gupta, G. Degrande, G. Lombaert 6

A numerical model for ground-borne vibrations from underground railway traffic based on a periodic FE-BE formulation

A numerical model for ground-borne vibrations from underground railway traffic based on a periodic FE-BE formulation D. Clouteau, R. Othman, M. Arnst, H. Chebli Ecole Centrale de Paris, LMSSMat, F-99 Chˆatenay-Malabry,