J. Cent. South Univ. Technol. (2010) 17: 1109-1118

DOI: 10.1007/s11771-010-0604-3

Prediction of vibrations from underground trains on Beijing metro line 15

DING De-yun(丁德云)^{1, 2, 3}, LIU Wei-ning(刘维宁)¹, GUPTA S², LOMBAERT G², DEGRANDE G²

1. School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, China;

2. Department of Civil Engineering, K.U.Leuven, Leuven B3001, Belgium;

3. Beijing Urban Engineering Design and Research Institute Co., Ltd, Beijing 100037, China

? Central  South University Press and Springer-Verlag Berlin Heidelberg 2010

Abstract: The impact of vibrations due to underground trains on Beijing metro line 15 on sensitive equipment in the Institute of Microelectronics of Tsinghua University was discussed to propose a viable solution to mitigate the vibrations. Using the state-of-the-art three-dimensional coupled periodic finite element–boundary element (FE-BE) method, the dynamic track-tunnel-soil interaction model for metro line 15 was used to predict vibrations in the free field at a train speed of 80 km/h. Three types of tracks (direct fixation fasteners, floating slab track and floating ladder track) on the Beijing metro network were considered in the model. For each track, the acceleration response in the free field was obtained. The numerical results show that the influence of vibrations from underground trains on sensitive equipment depends on the track types. At frequencies above 10 Hz, the floating slab track with a natural frequency of 7 Hz can be effective to attenuate the vibrations.

Key words: vibration prediction; underground trains; coupled periodic FE-BE method; track types

1 Introduction

There are more and more environmental concerns which are focused on the vibrations induced by underground railway traffic, with the rapid development of urban mass transit. Vibrations can arise from the passage of trains in the tunnel and propagate through the tunnel and surrounding soil into nearby buildings. Vibrations may cause disturbance or annoyance and even affect people’s ability to work. They can also cause malfunction of vibration-sensitive equipment or machinery, as used in scientific research organizations, universities, hospitals and high-tech industries.

Especially in the past 15 years, there has been an increasing focus on the impact of vibrations induced by underground trains on sensitive equipment in buildings. For example, a low vibration environment in a medical building that was built over the subway structure was maintained by means of low resonance frequency floating slabs in Atlanta [1]. The potential vibration impact of trains operating along the underground section of the Link Light Rail alignment on sensitive equipment and experiments at the Physics Astronomy Building and the proposed Life Sciences III Building was concerned [2]. In Beijing, metro lines 4, 8, 10 and 15 respectively passed at close proximity of the Physics Laboratory of Beijing University, China Academy of Space Technology, the National Measurement Laboratory and Tsinghua University, where a lot of sensitive equipment could be affected by the operation of metro trains [3-6]. It is therefore very important to study the impact of  vibrations induced by metro trains on sensitive equipment, if necessary, to propose feasible vibration countermeasures.

A great amount of efforts were made to study the problem of vibrations. The methods mainly consisted of in situ measurements, theoretical analysis, numerical modeling and combination of above-mentioned methods. As far as numerical modeling was concerned, two-dimensional [7-10], two-and-a-half dimensional [11-12] and three-dimensional [13-17] models were developed in the past, based on the finite element (FE) formulation, the boundary element (BE) formulation or the coupled FE-BE method. However, two-dimensional or two-and-a-half dimensional models necessitated important simplifications to translate the three- dimensional (moving) load into an equivalent two-dimensional load and also underestimated geometric damping into the soil. Furthermore, the wave transmission in the direction of the tunnel axis was not accounted for. From the computational point of view, three-dimensional models became very expensive in the vibration analysis at higher frequencies. In this work, astate-of-the-art three-dimensional coupled periodic FE-BE method was introduced, by which the Floquet transform was used to exploit the longitudinal periodicity of the track-tunnel-soil system. The three-dimensional dynamic track-tunnel-soil interaction problem was solved with a subdomain formulation where a finite element method was used for the track and the tunnel, while a boundary element method was used for the soil, modelled as a horizontally layered half-space.

Based on the coupled periodic FE-BE method, the problem of  underground trains induced vibrations on Beijing metro line 15 at close proximity of the Institute of Microelectronics of Tsinghua University was mainly examined. The vibrations in the free field due to underground trains operating at a train speed of 80 km/h were predicted, considering three types of tracks used on the Beijing metro network: (1) direct fixation fasteners, (2) floating slab track and (3) floating ladder track.

2 Coupled periodic FE-BE method

The coupled periodic FE-BE method is based on the following hypotheses: (1) the tunnel is straight and periodic with period L in the longitudinal direction y along the tunnel axis, (2) the tunnel is embedded in a horizontal layered half space soil medium and (3) all displacements and strains remain sufficiently small so that the track-tunnel-soil interaction problem is linear and the superposition principle in the frequency domain is valid [18-20]. According to these hypotheses, the coupled periodic FE-BE method uses the Floquet transform to exploit periodicity L of geometry and restrict the problem domain Ω to a single bounded reference cell  (Fig.1), and the track-tunnel-soil interaction problem can be efficiently formulated and solved in the frequency-wavenumber domain.

Fig.1 Notations for periodic domains and moving loads in tunnel

2.1 Response to moving loads

The coupled track-tunnel-soil system is subjected to vertical loads moving at a speed v along the longitudinal direction y (Fig.1). In the fixed frame of reference, the distribution of n_a vertical axle loads on the coupled track-tunnel-soil is written as the summation of the product of Dirac functions that determine the time-dependent position x_k={x_k0, y_k0+vt, z_k0}^T and time history g_k(t) of the kth axle load:

                                            (1)

where F(x, t) is the axle load vector; y_k0 is the initial position of the kth axle that moves with the train speed v along the y-axle; δ is Dirac function and e_z denotes the vertical unit vector.

A Fourier transformation is applied to Eq.(1) to obtain the representation in the frequency-spatial domain:

                    (2)

where  is the frequency domain representation of the axle load and ω is the circular frequency.

The response in frequency domain at receiver x due to n_a axle loads is written as the superposition of the load distribution along the source line:

           (3)

where is the transfer function, expressing the displacement at x in the e_i direction, due to a unit load applied toin the e_j direction.

Substituting Eq.(2) into Eq.(3) and elaborating the dependency of the transfer function on the source coordinates gives:

(4)

Introducing a charge of variable according to , Eq.(4) becomes:

         (5)

If the spatial period of an infinite periodic structure is L, then the position x of any point in the problem domain is decomposed as , where  is the position in the reference cell; n is the cell number and e_y denotes the unit vector in the longitudinal directin y. The response due to moving loads in case of periodic domains is given by [21]:

     (6)

whereandThe transfer function in frequency-wavenumber domain   is the Floquet transform of the transfer function in frequency-spatial domain

It can be seen from Eq.(6) that the frequency content of the axle load  and transfer function  are needed to compute the response to moving loads.

2.2 Axle loads

There are various excitation mechanisms g_k(t) that are responsible for generating vibrations due to moving trains. In general, they mainly consist of two kinds of excitation: (1) quasi-static excitation g_s(t) and (2) dynamic excitation g_d(t). The former occurs when successive axles of the train pass over the track. The latter mainly includes the random excitation due to rail and wheel unevenness, the impact excitation due to rail joints and wheel flats and the parametric excitation due to sleeper periodicity [20]. In this work, the quasi-static excitation and the random excitation due to rail unevenness were just considered.

The quasi-static excitation can be modeled as constant forces moving along the track with a train speed v. Time history g_s(t) is equal to a single axle weight w_k. The Fourier transform ?_s(ω) equals 2pw_kδ(ω).

For the unevenness excitation, the contact force ?_d(ω) (dynamic wheel-track interaction force) in the frequency domain can be calculated by [22-23]:

                    (7)

where ?_v(ω) is the compliance of the vehicle; ?_t(ω) is the compliance of the track;   is a 3n_a×1 vector that collects the phase shift for each axle; v is the train speed and  is the single-sided power spectral density (PSD) of the rail unevenness.

No PSD for metro track has been obtained so far. A lot of PSDs for railway tracks were applied to predicting the contact forces on metro track [5, 20, 24]. On the basis of extensive measurements on the US railway network, a single-sided PSDas a function of the cyclic wavenumber (n_y=ω/(2pv)) was devised by the Federal Railroad Administration (FRA) [25]. Depending on the rail quality, the track was divided into six classes according to the FRA, class 1 being the poorest and class 6 the best. Based on good effect of FRA track classes applied in metro [5, 24], class 1 was adopted in this work.

The single-sided PSD as a function of wavenumber |ω_i|/v is computed as   varies approximately with n_y^-4 for FRA track classes, and therefore increases with v⁴. When the track compliance ?_t(ω) is assumed to be unaffected by train speed v, modulus |?_d(ω)| of the contact force due to unevenness excitation approximately increases with v^1.5 from Eq.(7).

2.3 Transfer functions

As the reference cell of the tunnel is bounded, the displacement fieldin the tunnel is decomposed on the basis of mode functions:

      (8)

where M is the number of modes; and the modal coordinates  are collected in vector  The kinematical basis for the tunnel is determined such that the modes are periodic of the second kind [18-19]:

                    (9)

where  is the reference cell of the tunnel.

Soil displacement can be written as the superposition of waves that are radiated by the tunnel into the soil:

            (10)

The weak or variational formulation of the problem results in a dynamic tunnel-soil interaction equation in the frequency-wavenumber domain [18-19, 26]:

                                      (11)

where K_t(κ_y) and M_t(κ_y) are the dynamic stiffness and mass matrixes of the tunnel, respectively; K_s(κ_y, ω) is the dynamic stiffness matrix of the soil calculated with a periodic boundary element formulation with Green- Floquet functions defined on the periodic structure with period L along the tunnel; and F_t(κ_y, ω) is the force matrix.

The BAMPTON-CRAIG substructuring technique [27] was used to efficiently incorporate a track in the tunnel, describing the kinematics of the track-tunnel system as a superposition of the track modes on a rigid base and the quasi-static transmission of the free tunnel modes into the track [19-20].

When the track is incorporated in the tunnel by the BAMPTON-CRAIG substructuring technique, Eqs.(8), (10) and (11) are solved to obtain the displacement field in the reference cell of the track-tunnel-soil system in the frequency-wavenumber domain. As displacements and stresses at the tunnel-soil interface are known, the wave field radiated into the soil is computed using the dynamic representation theorem in the unbounded layered soil domain in the reference cell. These soil displacements correspond to the transfer function in the frequency-wavenumber domain which is used in Eq.(6) to compute the incident wave field due to moving loads. The displacement field of the track-tunnel-soil system in the frequency domain at any point x is obtained after evaluation of the inverse Floquet transform.

3 Problem outline

The planned metro line 15 identified as one of the key projects in Beijing is 43.30 km long and stretches from the Summer Palace in the Haidian District to the Shunyi County. The metro line 15 is currently at the preliminary design stage. The running tunnel between Yuanmingyuan station and Shuangqinglu station will pass under the campus of Tsinghua University, and it is very close to the Institute of Microelectronics of Tsinghua University (Fig.2), where a lot of sensitive equipment is used for some important scientific researches and may be affected by the future operation of trains.

3.1 Characteristics of tunnel and soil

One of the proposed running tunnel schemes is the double-vault tunnel with a complex shape, which is 50 m far away from the Institute of Microelectronics of

Fig.2 Location of Institute of Microelectronics of Tsinghua University and metro line 15

Tsinghua University and at the depth of 6.66 m below the free ground surface (Fig.3). The tunnel has two levels: top level and bottom level. The former hosts two roads, and the latter hosts two underground railway tracks. The maximum width and height of the tunnel are of the order of 29.28 and 20.38 m, respectively. The thickness range of the tunnel lining is 1.15-3.05 m. The tunnel lining is made up of cast-in-place reinforced concrete. The reinforced concrete lining has an elastic modulus E_t=  3.0 GPa, a Poisson ratio n_t=0.25, a mass density r_t=    2 500 kg/m³ and a hysteretic material damping ratio b_t= 0.02.

The geological section map of the area of Tsinghua University shows that there are mainly three layers of soil: fill material, silty clay, gravel and pebble, as shown in Fig.3. The dynamic soil characteristics are summarized in Table 1.

3.2 Track characteristics

At present, a lot of tracks are applied to underground railway engineering to mitigate vibrations induced by trains. In this work, three kinds of tracks used on the Beijing metro network are considered: (1) direct fixation fasteners, (2) a floating slab track and (3) a

Fig.3 Cross section of running tunnel under campus of Tsinghua University (Unit: m)

Table 1 Dynamic soil characteristics in area of Tsinghua University

floating ladder track.

The DTVI₂ fasteners (Fig.4) are typical of the high resilience direct fixation fasteners (DF), which are directly installed on short sleepers. The rail pads have a high stiffness of 5×10⁷ N/m and discretely support the T-60 rails at an interval of 0.60 m. The T-60 rails have a mass per unit length of 60.64 kg/m and a bending stiffness of 6.434×10⁶ N?m².

Fig.4 Schematic diagram of direct fixation fasteners: (a) Plan; (b) Cross section

Floating slab track (FST) called mass-spring system has been largely applied to underground railway to mitigate vibrations [1, 28-31]. The natural frequency of FST is usually less than 15 Hz, and it can isolate vibration level up to 40 dB [32-33]. The FST (Fig.5) is supported by two rows of steel springs with a stiffness of 6.9×10⁶ N/m and an interval of 1.80 m along the tunnel axis direction. The width and thickness of the slab are 3.50 and 0.45 m, respectively. The slab has a density of  2 500 kg/m³. The rails and rail pads are the same to those in the DF.

Floating ladder track (FLT) is also named lightweight-spring system with a shape of ladder. Its structural concept is small-mass soft-suspended system so that it can reduce structure-borne noise with a very lightweight construction [34-36]. It is mainly composed of two longitudinal prestressed sleepers, resilient bearings and steel pipes (Fig.6). The length, width and thickness of the sleeper are 6.15, 0.46 and 0.185 m, respectively. The sleepers with a density of 2 680 kg/m³ rest on the resilient bearings with a stiffness of 2×10⁷ N/m and an interval of 1.25 m along the tunnel axis direction. The steel pipes connect two sleepers and have an interval of 2.40 m along the tunnel axis direction. The rails and rail pads are also the same to those in the DF.

3.3 Train characteristics

The rolling stock on metro line 15 consists of 6

Fig.5 Schematic diagram of floating slab track: (a) Plan;     (b) Cross section

carriages with each length of 19 m. The bogie and axle distances on all carriages are L_b=12.60 m and L_a=2.30 m, respectively. The total mass of the train is 342 t, resulting in an axle load of 1.398×10⁵ N. The mass of the coach full of passengers is 43 t, while the mass of the bogie and axle are 3.6 and 1.7 t, respectively. The primary suspension has the stiffness of 1.4×10⁶ N/m and the damping of 3×10⁴ N?s/m. The second suspension has the stiffness of 5.8×10⁵ N/m and the damping of 1.6×10⁵ N?s/m.

Speed v of the train varies between 30 and 80 km/h. These are typical values of the train speed on Beijing metro network. With the increase of the train speed v, the contact force increases with v^1.5 from Eq.(7), resulting in the rise of the response in the track-tunnel- soil system. Hence, the train speed of 80 km/h was adopted in this work.

4 Numerical results

In this work, the double-vault tunnel and three tracks were regarded as periodic structures with the same spatial period L=0.60 m. The tunnel could host two tracks, but only one track was considered in the analysis. It was expected that the presence of the other track in the tunnel did not affect the dynamic track-tunnel-soil interaction.

4.1 Kinematics of track and tunnel

The kinematical basis for the free tunnel consisting of 85 modes was used at each wavenumber, and for three types of tracks, the kinematical basis included 8, 10 and 10 modes on a rigid base for the DF, FST and FLT, respectively. Fig.7 shows the first in-plane

Fig.6 Floating ladder track: (a) Plan; (b) Cross section

Fig.7 Free tunnel modes and track mode on rigid base for each track: (a) The first in-plane free tunnel mode at 12.3 Hz; (b) The first out- of-plane free tunnel mode at 37.7 Hz; (c) The first DF mode at 186.4 Hz; (d) The first FST mode at 7.0 Hz; (e) The first FLT mode at 35.5 Hz

free tunnel mode at 12.3 Hz, the first out-of-plane free tunnel mode at 37.7 Hz and the first track modes on a rigid base at zero wavenumber for the DF at 186.4 Hz, the FST at 7.0 Hz and the FLT at 35.5 Hz.

4.2 Axle loads

Quasi-static force g_s(t) for every axle was equal to 139.8 kN due to the weight of the train. Using Eq.(7), contact force ?_d(ω) was calculated for the passage of a train on two uneven rails at a speed of 80 km/h.

Fig.8 shows the frequency content of the contact force ?_d(ω) at the front axle of the train at a train speed of 80 km/h. For each track, the first peak appears at about 3 Hz, which is the anti-resonance frequency of the bogie mass on the primary and the secondary suspension. For the DF, the second peak appears at the wheel-track resonance frequency around 60 Hz, which corresponds to the resonance of the unsprung mass of the rolling stock (wheel and axle) on the stiffness of the DF. For the FST, the second peak exhibits at about 63 Hz, however, for the FLT, it appears at around 26 Hz. There exist obvious differences among the frequency content of the contact forces for three tracks, which are due to the variation of eigenfrequencies of three tracks. Below 7 Hz, the vibration of the train plays a leading role. However, in the frequency range between 7 and 100 Hz, the vibration due to the wheel-track interaction takes a primary role.

Fig.8 Frequency content of contact force at front axle of train at speed of 80 km/h

4.3 Transfer functions

Three observation points A(0 m, 0 m, 0 m), B(20 m, 0 m, 0 m) and C(50 m, 0 m, 0 m) in the free field where the response is predicted, are marked in Fig.3.

Fig.9 shows the vertical transfer functions at points A, B and C in the free field for different tracks. Note that the displacement (unit: dB) is expressed by the formulation 20lg(D/D_ref), where D is the calculated displacement (unit: m/N) and D_ref=1 is the unit reference

Fig.9 Vertical transfer functions in free field at different points: (a) A (0 m, 0 m, 0 m); (b) B (20 m, 0 m, 0 m); (c) C (50 m, 0 m, 0 m)

displacement (unit: m/N). The response in the free field is characterized by an oscillating behavior due to the interference effect of layered soil. At frequencies above 10 Hz, the response in the free field from the FST is significantly reduced, compared with those from other two tracks. However, below 10 Hz, it amplifies around the resonance of the FST. There is little difference between the DF and the FLT in the frequency range below 20 Hz.

4.4 Response during passage of train

After the computation of the transfer functions and the excitation forces in the frequency-wavenumber domain, the response due to moving loads could be computed from Eq.(6). The response was calculated by adding the contribution of the dynamic excitation and the quasi-static excitation in the frequency domain.

Fig.10 shows the frequency content of the vertical acceleration in the free field at point C due to the passage of a train at a speed of 80 km/h. It is very clear that the contribution of the quasi-static excitation is negligible and the dynamic excitation due to the rail unevenness dominates the response in the free field. The dominant frequency content is around the wheel-track resonance

Fig.10 Frequency content of vertical acceleration at point C (50 m, 0 m, 0 m) due to passage of train at speed of 80 km/h for different tracks: (a) DF; (b) FST; (c) FLT

frequency for the DF and FLT. However, for the FST, it is around its natural frequency.

Fig.11 illustrates the time history of the vertical acceleration in the free field at point C due to the passage of a train at a speed of 80 km/h. From the time history, the peak particle acceleration (PPA) in the DF is the biggest, then followed by that in the FLT, and it is the lowest in the FST.

Fig.12 gives the one-third octave band RMS spectra of the vertical acceleration in the free field due to the passage of a train at a speed of 80 km/h at points A, B and C. Superimposed on the graphs are the maximum

Fig.11 Time history of vertical acceleration at point C (50 m,  0 m, 0 m) due to passage of train at speed of 80 km/h for different tracks: (a) DF; (b) FST; (c) FLT

Fig.12 One-third octave band RMS spectra of vertical acceleration due to passage of train at speed of 80 km/h at different points: (a) A (0 m, 0 m, 0 m); (b) B (20 m, 0 m, 0 m); (c) C (50 m, 0 m, 0 m)

allowable RMS values of the vertical acceleration for sensitive equipment in the Institute of Microelectronics. There are two curves:Ⅰand Ⅱ. The former is from special equipment that has a very high requirement for environmental vibration, and the latter is from some equipment which has a relatively high vibration requirement.

At point A, the RMS values of the predicted acceleration in the DF and FLT are above curve Ⅱ at frequencies around the wheel-track resonance frequency, however, the RMS value in the FST lies above curveⅠ and below curve Ⅱ in the frequency range below   100 Hz. At point B, all of the predicted RMS values lie above curveⅠand below curve Ⅱ. At point C, the predicted RMS values in the DF and FLT also lie above curveⅠand below curve Ⅱ, but, the predicted RMS value in the FST is below curve I at frequencies above  35 Hz. At frequencies between 10 Hz and 100 Hz, the acceleration response can be significantly reduced by means of the FST, compared with other two tracks.

As above-mentioned, a lot of sensitive equipment in the Institute of Microelectronics is placed in the region of 50 m far away from the tunnel. Hence, some equipment could not be affected, but special equipment might be seriously interrupted.

5 Conclusions

(1) Using the rail unevenness from FRA track class 1, the comparison of the one-third octave band RMS spectrum of the predicted acceleration and the maximum allowable RMS values of the vertical acceleration for sensitive equipment in the Institute of Microelectronics of Tsinghua University suggests that some equipment which has a relatively high vibration requirement may not be interrupted by underground trains induced vibrations, but special equipment that has a very high vibration requirement would be seriously affected.

(2) Compared with the direct fixation fasteners and the floating ladder track, the floating slab track with a natural frequency of 7 Hz can be more effective to attenuate the vibrations at frequencies above 10 Hz.

(3) The applicability of the coupled periodic FE-BE method is demonstrated to study the efficiency of vibration isolation measures. It is necessary to validate the FE-BE model by in situ measurements in further work to improve the accuracy of predicted results.

(4) The isolation measures at the source (track) are just shown in this work. In order to more effectively control the impact of vibrations due to underground trains on sensitive equipment, it is necessary to further study the vibration transmission path between the source and the receiver and isolation measures at the receiver.

References

[1] NELSON J T. Recent developments in ground-borne noise and vibration control [J]. Journal of Sound and Vibration, 1996, 193(1): 367-376.

[2] WOLF S. Potential low frequency ground vibration (＜6.3 Hz) impacts from underground LRT operations [J]. Journal of Sound and Vibration, 2003, 267(3): 651-661.

[3] LIU Wei-ning, DING De-yun, SUN Xiao-jing, LIU Wei-feng. Study on special vibration reduction measures in Beijing metro line 4 [R]. Beijing: Beijing Jiaotong University, 2005: 1-10. (in Chinese)

[4] LIU W F, GUPTA S, DEGRANDE G, LIU W N. Numerical modelling of vibrations induced by underground railway traffic on metro line 4 in Beijing [R]. Leuven: K.U. Leuven and Beijing Jiaotong University, 2006: 12-14.

[5] GUPTA S, LIU W F, DEGRANDE G, LOMBAERT G, LIU W N. Prediction of vibrations induced by underground railway traffic in Beijing [J]. Journal of Sound and Vibration, 2008, 310(1): 608-630.

[6] DING D Y, GUPTA S, LOMBAERT G, LIU W N, DEGRANDE G. The prediction of vibrations induced by underground railway traffic on line 8 of Beijing metro [R]. Leuven: K.U.Leuven and Beijing Jiaotong University, 2008: 13-15.

[7] CHUA K H, BALENDRA T, LO K W. Groundborne vibrations due to trains in tunnels [J]. Earthquake Engineering and Structural Dynamics, 1992, 21(5): 445-460.

[8] JONES C J C, THOMPSON D J, PETYT M. A model for ground vibration from railway tunnels [J]. Proceedings of the ICE—Transport, 2002, 153(2): 121-129.

[9] LIN Q H, KRYLOV V V. Effect of tunnel diameter on ground vibrations generated by underground trains [J]. Low Frequency Noise, Vibration and Active Control, 2000, 19(11): 17-25.

[10] von ESTORFF O, STAMOS A A, BESKOS D E, ANTES H. Dynamic interaction effects in underground railway traffic systems [J]. Engineering Analysis with Boundary Elements, 1991, 8(4): 167- 175.

[11] SHENG X, JONES C J C, PETYT M. Ground vibration generated by a harmonic load acting on a railway track [J]. Journal of Sound and Vibration, 1999, 225(1): 3-28.

[12] LOMBAERT G, DEGRANDE G, CLOUTEAU D. Numerical modelling of free field traffic induced vibrations [J]. Soil Dynamics and Earthquake Engineering, 2000, 19(7): 473-488.

[13] HANAZATO T, UGAI K, MORI M, SAKAGUCHI R. Three-dimensional analysis of traffic induced ground vibrations [J]. Journal of Geotechnical Engineering, 1991, 117(8): 1133-1151.

[14] MOHAMMADI M, KARABALIS D L. Dynamic 3-D soil-railway track interaction by BEM-FEM [J]. Earthquake Engineering and Structural Dynamics, 1995, 24(9): 1177-1193.

[15] FORREST J A, HUNT H E M. A three-dimensional tunnel model for calculation of train-induced ground vibration [J]. Journal of Sound and Vibration, 2006, 294(4/5): 678-705.

[16] HUSSEIN M F M, HUNT H E M. A numerical model for calculating vibration from a railway tunnel embedded in a full-space [J]. Journal of Sound and Vibration, 2007, 305(3): 401-431.

[17] GARDIEN W, STUIT H G. Modelling of soil vibrations from railway tunnels [J]. Journal of Sound and Vibration, 2003, 267(3): 605-619.

[18] CLOUTEAU D, ARNST M, AL-HUSSAINI T M, DEGRANDE G. Free field vibrations due to dynamic loading on a tunnel embedded in a stratified medium [J]. Journal of Sound and Vibration, 2005, 283(1/2): 173-199.

[19] DEGRANDE G, CLOUTEAU D, OTHMAN R, ARNST M, CHEBLI H, KLEIN R, CHATTERJEE P, JANSSENS B. A numerical model for ground-borne vibrations from underground railway traffic based on a periodic finite element-boundary element formulation [J]. Journal of Sound and Vibration, 2006, 293(3/5): 645-666.

[20] GUPTA S. Numerical modelling of subway induced vibrations [D]. Leuven: Department of Civil Engineering, K.U.Leuven, 2008: 24- 76.

[21] CHEBLI H, OTHMAN R, CLOUTEAU D. Response of periodic structures due to moving loads [J]. Comptes Rendus Mécanique, 2006, 334(6): 347-352.

[22] LOMBAERT G, DEGRANDE G, KOGUT J, FRAN?OIS S. The experimental validation of a numerical model for the prediction of railway induced vibrations [J]. Journal of Sound and Vibration, 2006, 297(3/5): 512-535.

[23] LOMBAERT G, DEGRANDE G. Ground-borne vibration due to static and dynamic axle loads of intercity and high-speed trains [J]. Journal of Sound and Vibration, 2009, 319(3/5): 1036-1066.

[24] LIU Wei-feng. Study on numerical prediction model of metro train induced vibrations in tunnels and free field [D]. Beijing: Beijing Jiaotong University, 2010: 90-108. (in Chinese)

[25] HAMID A, YANG T L. Analytical description of track-geometry variations [J]. Transportation Research Record, 1982, 838: 19-26.

[26] CLOUTEAU D, ELHABRE M L, AUBRY D. Periodic BEM and FEM-BEM coupling: Application to seismic behaviour of very long structures [J]. Computational Mechanics, 2000, 25(6): 567-577.

[27] BAMPTON M C C, CRAIG R R Jr. Coupling of substructures for dynamic analyses [J]. Journal of Aircraft Industries Association of America, 1968, 6(7): 1313-1319.

[28] WILSON G P, SAURENMAN H J, NELSON J T. Control of ground-borne noise and vibration [J]. Journal of Sound and Vibration, 1983, 87(2): 339-350.

[29] LOMBAERT G, DEGRANDE G, VANHAUWERE B, VANDEBORGHT B, FRAN?OIS S. The control of ground-borne vibrations from railway traffic by means of continuous floating slabs [J]. Journal of Sound and Vibration, 2006, 297(3/5): 946-961.

[30] KUO C M, HUANG C H, CHEN Y Y. Vibration characteristics of floating slab track [J]. Journal of Sound and Vibration, 2008, 317(3/5): 1017-1034.

[31] XIANG Jun, WANG Yang, HE Dan, KONG Fan-bing, GUO Gao-jie. Model of vertical vibration of train and floating slab track system in urban rail transit system [J]. Journal of Central South University: Science and Technology, 2008, 39(3): 596-601. (in Chinese)

[32] DING De-yun, LIU Wei-ning, ZHANG Bao-cai, SUN Xiao-jing. Modal analysis on the floating slab track [J]. Journal of the China Railway Society, 2008, 30(3): 61-64. (in Chinese)

[33] DING De-yun, LIU Wei-ning, ZHANG Bao-cai, XIE Da-wen. 3-D numerical study on vibration isolation performance of special floating slab track in lab [J]. Journal of the China Railway Society, 2009, 31(6): 58-62. (in Chinese)

[34] OKUDA H, ASANUMA K, MATSUMOTO N, WAKUI H. Dynamic load, resistance and environmental performance of floating ladder track [J]. Quarterly Report of RTRI, 2004, 45(3): 149-155.

[35] HOSKING R J, MILINAZZO F. Floating ladder track response to a steadily moving load [J]. Mathematical Methods in the Applied Sciences, 2007, 30(14): 1823-1841.

[36] XIA H, DENG Y S, ZOU Y W, de ROECK G, DEGRANDE G. Dynamic analysis of rail transit elevated bridge with ladder track [J]. Frontiers of Architecture and Civil Engineering in China, 2009, 3(1): 2-8.

(Edited by YANG You-ping)

Foundation item: Projects(50538010, 50848046) supported by the National Natural Science Foundation of China; Project(BIL07/07) supported by the Research Council of K.U.Leuven and the National Natural Science Foundation of China

Received date: 2009-12-25; Accepted date: 2010-04-26

Corresponding author: DING De-yun, PhD; Tel: +86-13661363252; E-mail: dyding2301@163.com