r - The Hong Kong Polytechnic University

vertical stiffness of primary suspension system k s = 1.4 MN/m, vertical damping of primary suspension system
d p = 0.03MN-s/m, and characteristic length α = 0.745 m. FRA track class 1 is adopted in generating the
artificial irregularities profile u w/r (y) for the rails, which corresponds to the poorest rail quality. The lower and
upper bounds of the wavenumbers [k yl , k yu ] defined in the single sided PSD are chosen to be 0.1 and 15 rad/m,
respectively, with n = 40 intervals. The corresponding wavelengths of rail unevenness considered range from L i
= 0.42 to 62.8 m. By assuming the train speed to be c = 50 m/s, the major frequencies involved in the rail
irregularities range from 7 to 119 Hz. Thus the analysis frequency range selected is from 0 to 150 Hz.
Table 2. Material Properties.
Young’s modulus
E (MPa)
Poisson’s ratio
υ
Mass density
ρ (kg/m 3 )
Damping ratio
β
Material
Concrete tunnel lining 35,000 0.25 2,500 0.02
Concrete slab 28,500 0.2 2,500 0.02
Elastic Foundation 0.5 0.25 150 0.1
Fill material 116.6 0.341 1,900 0.05
Silty clay 289 0.313 2,023 0.04
Gravel and pebble 704 0.223 1,963 0.03
For train speed equal to c = 50 m/s, the responses computed for X = 0 m and X = 50 m on the ground with and
without an elastic foundation supporting the concrete slab track were plotted in Figure 8, with parts (a) to (c)
showing the time histories of displacement, velocity and acceleration, respectively. As indicated by Figure 8(a),
there exists a localized quasi-static displacement combined with distinct fluctuating vibrations for the case
without floating slab track. This is mainly caused by the moving tributary weight of the train, quasi-static in
nature, while the fluctuating vibrations are induced by rail irregularities. For the case with floating slab track,
although the fluctuating vibrations with high frequencies are reduced by the elastic foundation, the localized
displacements and the fluctuating vibration with low frequencies increase.
(a) Displacement
(b) Velocity
(c) Acceleration
Figure 8 Effect of elastic foundation on the ground response caused by a moving train over uneven rails: (a)
displacement, (b) velocity, (c) acceleration
The isolation effect of the floating slab track is clearer from the velocity and acceleration responses in Figures
8(b) and (c), as the high-frequency components have been suppressed. Such an observation is also confirmed by
the spectra shown in Figures 9(a) and (b) for the responses at X = 0 m and 50 m, respectively. The isolation
effect of elastic foundations is generally poor for vibrations at lower frequencies, but effective for higher
frequencies. Because the acceleration has a frequency content covering mostly higher frequencies, the isolation
effect of floating slab track is generally significant. In contrast, the frequency content of the displacement
response is mostly localized on the low-frequency region, which makes the effect of isolation not so effective.
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