Dynamic analysis of rail transit elevated bridge with ladder track

Abstract

In this paper, a dynamic analysis model of an elevated bridge with ladder tracks under moving train load is established. The whole process of a train running through an elevated bridge at different speeds is simulated. The dynamic responses of the elevated bridge with ladder track and the running safety and comfort index of train vehicles are evaluated. Compared with the dynamic responses of an elevated bridge with ordinary non-ballasted slab track, the ladder track’s effect on reducing the vibration of an elevated bridge is analyzed. The analysis results show that the ladder track has good vibration reduction characteristics as compared to ordinary non-ballasted track.

Keywords

rail transit / elevated bridge / ladder track / dynamic response

Introduction

The emergence and development of elevated rail transit has brought us a rapid and convenient traffic medium, but at the same time some new problems have arisen. Since the elevated rail transit systems run in urban regions, and often through city downtowns and residential areas, the train-induced vibrations and noise are serious, which has become a big obstacle to the planning and design of elevated rail transit systems. There have been some lessons in the construction of the Beijing metro line, Shanghai metro line, Japan Shinkansen and some others, where the vibration and noise problem was neglected during their construction, and thus brought complex technology and higher cost for reforming engineering during the operation stage.

As one of the main environmental pollutants, vibration in the environment that seemed to have been tolerated in the past is increasingly being considered a nuisance today. The influences of vibrations on the living and working environments of people have attracted the close attention of many metropolitan governments as well as engineers and researchers in China and abroad. The generation mechanism of vibrations, their propagation properties in the ground, pollution to the environment, harm to man’s health, and countermeasures against this problem, such as the reduction of vibration sources, the attenuation and cutting of vibration transmission in ground soils and the isolation of buildings, are all under study [1-10].

Many kinds of reduction measures have been studied by designers and researchers in China and abroad. The floating ladder track, a new vibration reduction track which has been well used in Japan and the U.S., is one of them. Several theoretical analyses and engineering applications performed in Japan proved that the ladder track system, with light weight, sufficient and effective elasticity, low maintenance and low cost, is an ideal track system that can effectively reduce the vibration and noise of the track while ensuring good train running safety and stability [11-14].

Ladder track system

The elastically supported ladder track system is composed of ladder sleepers, ductile bearings and a concrete base. The ladder sleeper is made up of prestressed concrete (PC) longitudinal beams, rubber bearings, buffer pads, isolators and steel pipe connectors, just like a ladder structure supported on L-shaped reinforced concrete bases at regular intervals (see Fig. 1).

The ladder sleeper is supported by polyurethane damping materials between two fasteners to restrict track vibration, reduce vibration intensity spreading on the supporting structure and reduce structure noise. Using buffer pads, the longitudinal and transverse horizontal forces of the track are resisted. The PC longitudinal beam can be regarded as the secondary longitudinal beam except for the rail; the rail and PC longitudinal beam bear the train load together. Thus, a composite track with high stiffness was formed, increasing the performance of load dispersion.

The elastically-supported ladder track system has been successfully used in several subways and elevated bridges in Japan and installed as trial sections in the U.S. In China, a ladder track trial section with a full length of 177 m has been installed on the elevated bridge of Beijing Metro Line 5. The trial section crosses two continuous beams with continuous spans of 3 m×30 m and 3 m×29 m respectively. It was designed by the Beijing Urban Engineering Design & Research Institute Corporation, Ltd, and constructed by the 17th Railway Engineering Bureau (see Fig. 2).

Experimental

Track irregularity measurement

The track irregularity plays an important role in the dynamic analysis of the train-track interaction system, which is the main excitation to train-bridge system vibrations, the factor controlling the running safety and stability of vehicles, and the main cause to structural damage and faults of track components. Therefore, measurements were taken on the track irregularity of ladder tracks at the trial section, see Fig. 3.

Several geometric parameters can be used to quantitatively describe the rail irregularities, i.e., the vertical profile z_v, the cross-level z_c, the lateral (alignment) y_a, and the gauge y_g. In Fig. 4, the rotational irregularity can also be expressed in terms of the angle formed by the level difference between the left rail and the right rail, namely, θ_c=z_c/2b.

The lateral and vertical track irregularities of the inner rail and outer rail of the ladder track were measured by means of a surveyor’s level and transit instrument (see Fig. 2), from which the track central line orientation, rotational and gauge irregularities were obtained. The statistical standard variation of track irregularities and the power spectrum were calculated using the following formulas:

(1){\sigma }_{\eta }^{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{X}}{\displaystyle {\int }_{\mathrm{0}}^x{[\eta (x)-{\mu }_{\eta }]}^{\mathrm{2}}\mathrm{d}x}={\displaystyle \frac{\mathrm{1}}{X}}{\displaystyle {\int }_{\mathrm{0}}^X{\eta }^{\mathrm{2}}(x)\mathrm{d}x}-{\mu }_{\eta }^{\mathrm{2}}={\phi }_{\eta }^{\mathrm{2}}-{\mu }_{\eta }^{\mathrm{2}},

(2)S_{\eta }(f)=\underset{\mathrm{\Delta }f\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{\mathrm{\Delta }f}}\left[\underset{X\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{X}}{\displaystyle {\int }_{\mathrm{0}}^X{\eta }^{\mathrm{2}}(x,f,\mathrm{\Delta }f)\mathrm{d}x}\right],

where \eta (x) is the profile, {\mu }_{\eta } is the mean, X is the total sampling length, f is the space frequency, and Δf is the frequency interval of track irregularity. Detailed explanation can be found in Ref. [6].

Table 1 shows the statistical parameters of the measured track irregularities, and Fig. 5 is the power spectra of the lateral and vertical rail irregularities.

From the power spectra, it can be seen that the irregularity wavelength is mainly between 0.5 m and 80 m, in which the amplitude of the long wave is large and that of middle wave is small, and the shorter the wave length is, the smaller the amplitude is. Besides, the peaks in the power spectra show that there are many periodic wave components in the random track irregularity in this section. This is in line with the irregularity state of a new rail where wave wearing was formed in the rail rolling process. Table 1 shows that the maximum and standard deviation of the measured track irregularities are reasonable compared with the railway track code in China [15].

Laboratory test of ladder track system

A dynamic loading test was carried out in the Structural Laboratory of Beijing Jiaotong University to study the vibration reduction performance of the ladder track system, as shown in Fig. 6.

The test sample, a 6.25 m long ladder sleeper assemble element, was loaded with harmonic forces generated by the Pseudo-dynamic-loading device, and with impact load by a hammer. Mounted at the rail, sleeper and bearing are KISLER8305A accelerometers. The loading frequencies are in the range of 2-20 Hz with an increment of 1 Hz, the sampling frequency is 500 Hz, and loading duration is 2 min in each loading stage.

The vibration reducing ability of the track is described by vibration reduction level as

(3)\mathrm{V}\mathrm{L}=\mathrm{20}\lg (a_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}}/a_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}}),

where a_{\mathrm{r}\mathrm{a}\mathrm{i}\mathrm{l}} and a_{\mathrm{b}\mathrm{a}\mathrm{s}\mathrm{e}} are the accelerations measured at the rail and base, respectively.

Figure 7(a) is the vibration reduction level of the track under harmonic load, which shows that in the range of 2-20 Hz, the reduced vibration level is about 20 dB at 2 Hz and 25 dB over 2 Hz. Figure 7(b) is the vibration reduction level of the track under impact load, which shows that in the frequency bond of 0-80 Hz, the vibration reduction levels are greater than 20 Hz, except for those around 30 Hz which are about 15 dB. The conclusion is drawn by taking the rail vibration as 0 dB. The result shows good vibration reducing ability of the ladder track system.

Dynamic response measurement

A field measurement was carried out on the ladder track trial section of the Beijing Metro Line 5. The measurement points were selected at the mid-span of the 3 m×30 m continuous elevated bridge, as shown in Fig. 8.

The accelerometers LC0123ICP were mounted on the rail bottoms and sleepers at the ladder track, the common slab track and the concrete beam deck, as shown in Fig. 9. An Intelligence Data Acquisition System Wavebook-512 with 32 CHs was adopted for signal processing, and the sampling frequency was 5000 Hz.

The acceleration responses of tracks and the beam deck were measured at each train passage, and there were 30 groups of data obtained. The typical acceleration histories of the rail and beam deck measured at the common slab track and the ladder track are shown in Figs.10 and 11, respectively. The maximum accelerations and their statistical data are listed in Table 2.

It can be seen that the average maximum acceleration of rail on the common slab track is 244.45 m/s², while that on the ladder track is 149.40 m/s², which is lower by 39%. The average maximum accelerations of the bridge deck, when the train runs on the common slab track and the ladder track, are 3.08 m/s² and 1.4 m/s², respectively. Another factor that should be pointed out is that the average speed of the train running on the ladder track is 72 km/h, while that on the common slab track is 49 km/h.

In order to compare the vibration reduction properties of the two types of tracks, the average frequency spectra are calculated, as shown in Fig. 12.

In Fig.12, the vibration levels are calculated with the following equation:

(4)L_{\mathrm{V}}=\mathrm{20}\lg (a_{\mathrm{r}\mathrm{m}\mathrm{s}}/a_{\mathrm{0}}),

where a_{\mathrm{0}}=\mathrm{1}\times {\mathrm{10}}^{-\mathrm{6}}\mathrm{m}/{\mathrm{s}}^{\mathrm{2}} is the base acceleration; a_{\mathrm{r}\mathrm{m}\mathrm{s}} is the effective value of acceleration in m/s².

Figure 13 shows the transfer relations of the acceleration levels from rail to bridge deck of the two types of tracks, which are obtained from the spectrum analysis of the measured data.

It can be seen from the figures that the vibration level of the ladder track is a little bit higher than the common slab track in the frequency range lower than 160 Hz, while it is obviously lower in the frequency range higher than 160 Hz, with the maximum reduction being 20.3 dB.

Numerical study

The numerical study is based on the dynamic analysis model of train-ladder track-elevated bridge system, which is used to study the dynamic responses and serviceability of a bridge with ladder track system. The model is composed of three parts: the train model, the ladder track model and the bridge model, as shown in Fig.14.

The train model consists of several passenger vehicles and each is considered as a dynamic system with 27 degrees of freedom. The motion equations of the train-ladder track-elevated bridge system can be expressed as

(5)\left[\begin{array}{lll}{\mathit{M}}_{\mathrm{v}\mathrm{v}}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& {\mathit{M}}_{\mathrm{t}\mathrm{t}}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& {\mathit{M}}_{\mathrm{b}\mathrm{b}}\end{array}\right]\left\{\begin{array}{l}{\ddot{\mathit{X}}}_{\mathrm{v}}\\ {\ddot{\mathit{X}}}_{\mathrm{t}}\\ {\ddot{\mathit{X}}}_{\mathrm{b}}\end{array}\right\}+\left[\begin{array}{lll}{\mathit{C}}_{\mathrm{v}\mathrm{v}}& {\mathit{C}}_{\mathrm{v}\mathrm{t}}& \mathbf{0}\\ {\mathit{C}}_{\mathrm{t}\mathrm{v}}& {\mathit{C}}_{\mathrm{t}\mathrm{t}}& {\mathit{C}}_{\mathrm{t}\mathrm{v}}\\ \mathbf{0}& {\mathit{C}}_{\mathrm{b}\mathrm{t}}& {\mathit{C}}_{\mathrm{b}\mathrm{b}}\end{array}\right]\left\{\begin{array}{l}{\dot{\mathit{X}}}_{\mathrm{v}}\\ {\dot{\mathit{X}}}_{\mathrm{t}}\\ {\dot{\mathit{X}}}_{\mathrm{b}}\end{array}\right\}+\left[\begin{array}{lll}{\mathit{K}}_{\mathrm{v}\mathrm{v}}& {\mathit{K}}_{\mathrm{v}\mathrm{t}}& \mathbf{0}\\ {\mathit{K}}_{\mathrm{t}\mathrm{v}}& {\mathit{K}}_{\mathrm{t}\mathrm{t}}& {\mathit{K}}_{\mathrm{t}\mathrm{v}}\\ \mathbf{0}& {\mathit{K}}_{\mathrm{b}\mathrm{t}}& {\mathit{K}}_{\mathrm{b}\mathrm{b}}\end{array}\right]\left\{\begin{array}{l}{\mathit{X}}_{\mathrm{v}}\\ {\mathit{X}}_{\mathrm{t}}\\ {\mathit{X}}_{\mathrm{b}}\end{array}\right\}=\left\{\begin{array}{l}{\mathit{F}}_{\mathrm{v}}\\ {\mathit{F}}_{\mathrm{t}}\\ {\mathit{F}}_{\mathrm{b}}\end{array}\right\},

where the subscripts v, t and b are the vehicle, track and bridge, respectively, which can be found in Refs. [6,16].

The train adopted in the calculation is a light-rail train composed of 8 vehicles, with average static axle loads of 101 kN (tare) and 132 kN (crush), respectively. The main parameters of the coach used in the case study can be found in Ref. [16]. The principal vibration mode frequency of the coach is about 1.04 Hz in the vertical direction and 0.68 Hz in the lateral direction. The track vertical, lateral and rotational irregularities are taken into consideration by using the measured data from the trial section of the elevated bridge of the Beijing Metro Line 5. The damping ratio of bridge vibration is 2.0 %, and the integral time step 0.002 s.

Figure 15 shows the acceleration and displacement histories of the bridge with ladder track at the mid-span, and Fig. 16 shows the car-body acceleration histories of the first vehicle of the train running on the elevated bridge. The calculated train speed is 80 km/h.

The dynamic responses of the train-bridge system for the elevated bridge with ladder track and ordinary track are shown in Tables 3 and 4. The vertical acceleration at the mid-span of the bridge versus the train speed is shown in Fig.17.

The results show that the elasticity of the ladder track has little influence on the operating properties of a vehicle. Compared with common slab tracks, the vibration reduction effects are mainly reflected in the decrease of the mid-span vertical acceleration of the bridge, and it varies with train speed. At the train speed of 80 km/h, the vibration of the bridge is reduced by 42.8%, about 4.8 dB, showing an obvious reduction effect.

Conclusions

Theoretical analysis and experimental study prove that the ladder track has good vibration reduction characteristic. Compared with the common non-ballasted slab track, the vibration reduction effect of ladder tracks is mainly embodied in the vertical acceleration of the bridge, while the effect on the vertical displacement of the bridge is very small. The ladder track has little effect on the lateral dynamic response of the bridge and the dynamic response of the train. The higher the train speed, the more obvious the ladder track’s vibration reduction effect is.

References

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