Keywords

1 Introduction

Urban rail transit has great advantages in reducing the highway traffic in metropolitan city. For urban rail transit, subways have been widely developed due to its characteristics of efficient operation and less occupation of the ground areas. By the end of December 2017, in-service subway lines have spread over 34 cities with a total of 3884 km in mainland China [1]. The train depots are the subway or railway facilities for the regular maintenance, examination, and storage for the trains and usually occupy the large ground areas [2]. Traditional metro depots with low density of buildings hold larger ground spaces which make significant influences on the space and the environment, and it’s not conducive to efficient utilization of the urban land and space due to the low density of the buildings [3]. Comprehensive utilization of the spaces above the depots can not only improve the utilization efficiency of the urban land, but also have great economic advantages. So, many cities in mainland China like Beijing and Wuhan have begun developing the over-track building complexes, which are built with a new structural system with large platforms, for residents and workers above the metro depot. The platforms above the metro depot are used for entertaining and parking, supported by the columns which are set in between train tracks.

However, the subway operation into and out of depots can generate excessive vibrations due to interaction between rails and wheels, which propagate from the tracks directly into the over-track buildings through ballast, columns, and platforms, radiating noises within the buildings that adversely affect the living qualities and even the health of the building occupants. It is essential to predict the train-induced vibration level prior to construction within the over-track buildings and take appropriate measures to mitigate the vibration for human comfort.

Massive researches have been conducted to characterize the vibration at the sources, in the propagation paths, and in the receivers inside the buildings induced by subway trains and operation lines. The vibration characteristics at the sources and in the propagation paths induced by railways have been summarized in detail [4,5,6,7,8]. Anderson [9] conducted measurements to collect the vibration data from two buildings caused by the railway operation and got that the perceptible vibrations, which are mainly between 5 and 50 Hz, can transmit into the building through the ground. Xia et al. [10] took in situ experiment to examine the vibration levels induced by running trains near Shenyang–Harbin railway line within two buildings and found the acceleration levels in the buildings exceeded the allowance by the Chinese Code. Sanayei et al. [11, 12] performed measurements of train-induced and subway-induced vibrations in different areas in Boston. It provided the basis that the vibration levels in sensitive locations inside the buildings induced by train operations should be estimated precisely. Numerous prediction modeling methods have been developed to predict the subway-induced vibration from the underground lines [13,14,15,16].

The vibration transmission in metro depots entirely differs from that around the operation line and subways, for the trains work on the ground right under the platform and the over-track buildings which is not the case with the operation lines and subways. Some studies have been conducted on the transmission characteristics and levels of the vibration on the grounds, platforms, and over-track buildings in the metro depots. Field measurements were conducted to study the law of the vibration attenuation in the ground, platforms, and the over-track buildings subject to the vibration input in throat area [17], in testing line area [18], and in both throat and testing line areas [2, 3]. The measured maximum vertical vibration levels were lower than the limitation by Chinese standard [19] according to these researches. Numerical models were established to analyze the vibration response of the over-track buildings [20,21,22]. Vibration data of the sources were measured in the time domain on site and applied into the established numerical model. The results from the numerical model were well consistent with the measured vibration in the over-track buildings in both time and frequency domain, and the models were used to predict the vibration levels prior to the construction of the over-track buildings. He et al. [23] studied the impact of the train speed on the vibration within the over-track buildings and concluded that limiting the train speed into and out of the metro depot is an efficient way to attenuate the vibration within the over-track buildings.

This passage is to study the characteristics of the vibration transmission inside the over-track building above the main body in a typical metro depot under the influence of subway operation. A three-dimensional dynamic finite element model of soil-platform-building was built according to a metro depot to characterize the vibration transmission, and the train-induced vibrations were compared with the limitation by Chinese standard. It is efficient and reliable to apply this model to predict the vibration levels in the over-track building before constructing.

2 Description of Metro Depot and Over-Track Building

The metro depot related to this study is situated in Beijing, China. Its total occupying land use is approximately 314,200 m2, including 266,000 m2 for the functions of metro depot and 48,200 m2 for the exploitation, which makes the metro depot the biggest one in China. Two avenues are situated at the north and west of the metro depot, respectively, as shown in Fig. 1. The soil layers consist of artificial accumulation layer, recent sedimentary layer, new Quaternary alluvium, and late Quaternary Pleistocene flood alluvium below the ground in this area. There is a main body, a throat area, and so on in the metro depot, and the tracks in the main body are straight. The area above the metro depot preserved for the construction of the over-track buildings with 12 floors is about 150,000 m2, which will serve for the occupants and workers.

Fig. 1
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Areas distribution in Beijing metro depot

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All the buildings are constructed using reinforced concrete columns with large open spans from the ground to the platform, whose height is 16.3 m to guarantee enough space for the trains. The columns supporting the platform have the sectional dimensions of 1.2 m by 1.2 m and are located on a grid of 6.0 m × 6.0 m, which is the same case with the columns supporting the over-track buildings according to the plan. The support structures for over-track building from the first floor to the top consist of frame structures with 12 floors with the height of 5.6 m in the first floor and the height of 4 m in standard floors, and load-bearing walls with the thick of 0.2 m, as shown in Fig. 2. The beams inside over-track building have the sectional size of 0.4 m × 0.6 m.

Fig. 2
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A sketch map of the metro depot in vertical direction

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This study focused on vibration transmissions within an over-track building with 12 floors above the main body of the depot induced by the train operation into and out of the metro depot.

3 Establishment of the Finite Element Model

3.1 The Finite Element Model

As described above, the studied metro depot captures a large land use area. The vibration induced by the subway operation attenuates quickly with the propagation of the vibration wave, which makes it unnecessary to take all areas of metro depot into account in the study. A part of metro depot is applied in this model that must take a reasonable size for the reasons that a model which is too large can increase the computational cost and which is too small can affect the accuracy and reliability of the results [24]. The lengths of this model which are parallel to the track lines are 96 m and the widths which are perpendicular to the track lines are 51 m to make sure it is accurate and reliable to predict the vibrations. The height of the platform is 16.3 m to provide enough space for the trains.

The structure modeling and dynamic analysis are accomplished in the finite element software, ANSYS, which is widely used for modeling and analysis in statics and dynamics. The finite element model is shown in Fig. 3, and Fig. 2 shows the positions of train-induced forces which is a function of time acting at the point where fasteners and sleepers contact with each other. The points where the vertical and horizontal vibration data is gathered inside the over-track buildings, including the points on the floors around the column and the points in the middle of floors, are also illustrated in Fig. 2. In this model, all the beams and columns with different sectional size are simulated by the beam elements, BEAM4, which do not allow for shear deformations of the units. All the floors and walls are modeled using the surface elements, PLANE42, and soil and ballasts are regarded as the physical elements, SOLID45. The elements should be divided into small elements with the right sizes, which should be small enough to accurately determine the full waveform when vibration waves are propagating in the units and big enough to reduce the calculation time [24]. For the purpose of impeding wave reflection in the cut-off boundaries which are employed to take constraints to models from the infinite stratum into consideration, the viscoelastic boundaries are applied with the advantages of good behaviors in numerical stability and accuracy and its convenience of application in the finite element models [25].

Fig. 3
figure 3

Finite element model of metro depot

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Impacts of ballast, formation, building foundations, frame structures, and wall panels on the wave propagation in the depot are taken into account, and the effects of decorative components such as wall materials are ignored in this three-dimensional finite element dynamic model.

3.2 Damping

Damping is a so complicated mechanism of energy dissipation that many damping models have been put forward to make it easy to consider it in the researches and the Rayleigh damping model with a linear combination of stiffness and mass matrix is widely used. In order to determine the damping coefficients, the modal analysis is required to obtain the natural frequency of the building, and the first five levels of modal frequencies obtained from the modal analysis of the soil-platform-structure model are 1.00, 1.36, 2.40, 3.47, and 5.48 Hz, respectively. The mass damping coefficient and stiffness damping coefficient are acquired by taking the first two natural frequencies and damping ratios into Eq. (1).

$$\left\{ \begin{aligned} \alpha = & \frac{{4\pi \,f_{i} f_{j} \left( {\varepsilon_{j} f_{i} - \varepsilon_{i} f_{j} } \right)}}{{f_{i}^{2} - f_{j}^{2} }} \\ \beta = & \frac{{\left( {\varepsilon_{i} f_{i} - \varepsilon_{j} f_{j} } \right)}}{{\pi \left( {f_{i}^{2} - f_{j}^{2} } \right)}} \\ \end{aligned} \right.$$
(1)

where α and β are mass and stiffness damping coefficient, fi, fj, \(\varepsilon_{i} ,\varepsilon_{j}\) are the ith and jth natural frequencies and damping ratios, which are 1.0 Hz, 1.36 Hz, 0.03, and 0.03 when i equals to 1 and j equals to 2, respectively. The value of α and β on the basis of Eq. (1) are 0.217 and 4.04 × 10−3.

4 Vibration Sources

Simulation of train-induced excitation at the sources is one of the key processes in this study on the train-induced vibration transmission within the over-track building, because whether the input excitation is well consistent with the measured excitation directly make impacts on correctness of the results. The method to calculate the excitation in frequency domain with coupled train–track dynamic model presented by Ma [26, 27], which has been confirmed by a number of field measurements, is studied and applied in this work.

4.1 The Coupled Train–Track Model

In the coupled train–track system, the trains which can move forward at the speed of v on the rails consist of several individual vehicles, and each vehicle is made up of three parts including a body, two bogies, and four wheelsets that each part connects to the others with parallel springs and dampers, as shown in Figs. 4 and 5. There are 10 degrees of freedom for a single vehicle, including ups and downs of the body, two bogies, and four wheelsets, as well as rotations about the axial direction of the body and two bogies. The suspended mass and the interactions between wheels and rails are also taken into account in this model. The interactions across the interfaces between rails and sleepers are nonlinear that are represented by a series of springs and dampers, which is the same case with sleepers and ballasts, as well as ballasts and foundations.

Fig. 4
figure 4

Coupled train–track model

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Fig. 5
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Basic cell in global coordinate system

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In vehicle model, there are 10 degrees of freedom for a certain vehicle, and the equation of motion in different degree of freedom should be formed to obtain the dynamic response. The equation of motion is shown as Eq. (2), on the basis of D’Alembert principle.

$${\mathbf{M}}_{m} {\ddot{\mathbf{A}}}_{m} (t) + {\mathbf{C}}_{m} {\dot{\mathbf{A}}}_{m} (t) + {\mathbf{K}}_{m} {\mathbf{A}}_{m} (t) = {\mathbf{Q}}_{m} (t)$$
(2)

where \({\ddot{\mathbf{A}}}_{m} (t),{\dot{\mathbf{A}}}_{m} (t),{\mathbf{A}}_{m} (t)\) are the acceleration, velocity, and displacement vector of the certain vehicle, Qm(t) are the excitation vector and Mm, Cm, and Km are the mass, damping, and stiffness matrix.

Rails in this model are simplified as the Euler–Bernoulli beams ignoring the shear deformations across the transverse sections, which are infinite in the space and periodically supported by the vertical springs and dampers. The dynamic displacement of the infinite beam can be written as Eq. (3).

$$E_{\text{r}}^{*} I_{\text{r}} \frac{{\partial^{4} u_{\text{r}} }}{{\partial x^{4} }} + m\frac{{\partial^{2} u_{\text{r}} }}{{\partial t^{2} }} = {\text{e}}^{{{\text{i}}\omega_{l} t}} \delta (x - x_{0}^{\text{F}} - vt) - \sum\limits_{n = - \infty }^{n = + \infty } {f_{n} (t)} \delta (x - x_{n} )$$
(3)

where ur represents the vertical displacement, \(E_{\text{r}}^{ *} = E_{\text{r}} (1 + {\text{i}}\eta_{\text{r}} )\) are the elastic modulus which allows for the real elastic modulus \(E_{\text{r}}\) and the damping loss factor \(\eta_{\text{r}}\). Ir and m are the sectional moment of inertia and mass per unit length, respectively. \(x_{0}^{\text{F} }\) is the location of the load at the initial time, and \(x_{n} = nL\) is the location of the nth fastener where L represents the spacing between the adjacent fasteners. fn(t) is the support force from the nth fastener.

Solution of dynamic responses of the infinite beams can be carried out in a basic cell with the length of L, and the operations are conducted in global coordinate system in frequency domain, as shown in Fig. 5 and expressed as Eq. (4).

$$E_{\text{r}}^{*} I_{\text{r}} \frac{{\partial^{4} u_{\text{r}} }}{{\partial x^{4} }} + m\frac{{\partial^{2} u_{\text{r}} }}{{\partial t^{2} }} = {\text{e}}^{{{\text{i}}\omega_{l} t}} \delta (x - x_{0}^{\text{F}} - vt)$$
(4)

Fourier transformation is used to transform Eq. (4) from the time domain to the frequency domain, expressed as Eq. (5)

$$E_{\text{r}}^{*} I_{\text{r}} \frac{{{\text{d}}^{4} \hat{u}_{\text{r}} }}{{{\text{d}}x^{4} }} - \omega^{2} m\hat{u}_{\text{r}} = \frac{1}{v}{\text{e}}^{{{\text{i}}\frac{{\omega_{l} - \omega }}{v}\left( {x - x_{0}^{\text{F}} } \right)}}$$
(5)

where the \(\hat{u}_{\text{r}}\) is the vertical displacement of the basic cell in frequency domain.

The displacements of a fixed point in the basic cell due to the unit harmonic moving load acting at the beams can be calculated using the method proposed by Ma [26, 27], and displacements of the points in other ranges are derived from solutions of the basic cell, which must comply with the periodic-infinite structure theory, expressed as Eq. (6).

$$\hat{u}_{\text{r}} (\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} ,\omega ,\omega_{l} ) = {\text{e}}^{{{\text{i}}(\omega_{l} - \omega )n_{L} L/v}} \cdot \hat{u}_{\text{r}} (x,\omega ,\omega_{l} )$$
(6)

where \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x}\) and x obey the relation of \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x} = x + n_{L} L\), L represents the length of basic cells, nL refers to the number of the basic cells between the point at \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{x}\) and x, and \(\hat{u}_{\text{r}} (x,\omega ,\omega_{l} )\) is the displacement of the basic cell in frequency domain.

Continuous discrete displacements of points in whole beams induced by the unit harmonic moving load \(F(t) = {\text{e}}^{{{\text{i}}\omega_{l} t}}\) at the speed of v are obtained through making the angular frequency \(\omega\) continuous discrete values at the focused range in frequency domain. Then, applying the inverse discrete Fourier transformation, the displacements in frequency domain can be transformed to the displacements in time domain.

Train’s passage would generate dozens of moving loads with certain magnitude, and all of them make contribution to the total displacements of the targeted point that are the linear superposition of influence under the individual one. The actual forces interacting across interfaces between rails and foundation are products of the actual displacements and composite stiffness allowing for the stiffness between the rails and sleepers, sleepers and ballasts, as well as the ballasts and foundation.

4.2 Excitation Due to Moving Trains

This work is based on operation of the Type-B train at the speed of 25 km/h which is integrated by six single vehicles. The vehicle parameters and track parameters are listed in Tables 1 and 2. The simulated excitation is illustrated as Fig. 6. The maximum of interacting forces between the rails and foundation induced by train operations are approximately 6 × 104 N, and the forces are mainly in low frequency within 10 Hz. The force magnitude in high frequency is low.

Table 1 Vehicle parameters
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Table 2 Track parameters
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Fig. 6
figure 6

Train-induced interacting force between rails and foundation

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5 Train-Induced Vibrations Inside the Over-Track Building

In order to characterize the vibration transmissions inside the over-track building subjected to the train operation, points for gathering vertical and horizontal vibration data are appointed to be located at each floor around the column and in the middle of each floor, as illustrated in Fig. 2. The analyses of vibration responses are conducted in three aspects including vibration acceleration levels, vibration acceleration spectra in frequency domain, and in 1/3 octave frequency. According to ISO 2631-1 [28], the main frequencies of vibration impacts on humans are within 1–80 Hz. As a result, frequencies in this study range from 1 to 100 Hz.

5.1 Vibration Acceleration Level

The vibration acceleration levels are used for evaluating impacts of vibrations inside buildings, which are induced by the vibration sources within the buildings, such as elevators, motors, and so on, formulated by GB/T 50355-2005 [29]. The method of calculating the vibration acceleration level has been made by it, expressed as Eq. (7).

$$L_{\text{a}} = 20\lg \frac{{a_{\text{rms}} }}{{a_{0} }}$$
(7)

where a0 is the reference acceleration with the value of 1 × 10−6 m/s2, arms is the root mean square of acceleration value (m/s2), and La represents the vibration acceleration level (dB).

Figure 7 shows vertical vibration acceleration levels around the column and in the middle of the floor inside the building. The vertical vibration acceleration levels are attenuated between the first to the ninth floor around the column and the first to tenth floor in the middle of floors and amplified greatly at the top floor, with the minimum of approximately 46.2 and 52.6 dB, respectively. What is more, the vertical vibration acceleration level around the column is less than that in the middle of the floor by 1.8–6.9 dB in the same floor. Vertical constraint from the column makes the floor vibration weaker around the column, and vibration waves from various propagation paths strengthen each other in amplitude in the middle of floors, which may contribute to that phenomenon.

Fig. 7
figure 7

Vertical acceleration level inside the building

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The horizontal vibration acceleration levels around the column and in the middle of the floor inside the building are shown in Fig. 8. The horizontal vibration acceleration levels around the column and in the middle of the floor in the same height are comparably close, with little differences in magnitude. The horizontal vibration levels show the tendency of slight amplification on the whole, with no attenuation, with the increase in height, which can be explained by the cause that the floors are more weakly restrained in the horizontal direction in higher floors.

Fig. 8
figure 8

Horizontal acceleration level inside the building

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Comparing Figs. 7 and 8, it is evident that the vibration acceleration levels in vertical orientation are greater than those in horizontal orientation by 7.8–14.6 dB in the same floor. This is because the moving loads induced by the train operation in the straight track are mainly vertical and stiffness in vertical direction are smaller than that in horizontal direction in the middle of the floor that vibrations are amplified with the stiffness degradation.

5.2 Spectrum in Frequency Domain

Figures 9, 10, and 11 typically show a portion of vibration acceleration spectra in frequency domain both around the column and in the middle of the floors. In both vertical and horizontal directions, the train-induced vibrations inside the building are mainly distributed in high frequency within 30–80 Hz, which could make impacts on the occupants, and the dominant frequency decreases with the rise in height. As shown in Figs. 9 and 10, the dominant frequencies of vertical vibrations in the middle of the floors are basically consistent with these on the same floor around the column, which range from 50 to 70 Hz. Moreover, the dominant frequencies of the floor vibrations in horizontal direction, varying between 30 and 55 Hz, are lower in comparison with these in vertical direction by about 15 Hz, as shown in Fig. 11. Also, the maximums of the horizontal vibration amplitudes are obviously below these in vertical direction.

Fig. 9
figure 9

Train-induced vertical vibrations in the middle of the floors

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Fig. 10
figure 10

Train-induced vertical vibrations around the column

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Fig. 11
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Train-induced horizontal vibrations in the middle of the floors

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There exist the regularities that the vibrations inside the over-track building are enhanced at certain frequencies, which are near 23, 35, 75 Hz. That is because the train-induced vibrations within the building have triggered the certain bending mode of the floor slabs, whose frequencies are exactly corresponding to these where the vibrations are largely amplified. Additionally, it is quite clear from the graphic area enclosed by the curves of the vibration acceleration spectra and the transverse axis that the horizontal vibration energy, which is mostly distributed at some specific frequencies, is more concentrated than the vertical vibrations in frequency domain.

5.3 1/3 Octave Spectrum

The vertical vibration acceleration spectra in 1/3 octave frequency on the floors around the column and in the middle of the floors are shown in Figs. 12 and 13. It can be obviously found that when the central frequency is below 10 Hz, the vertical acceleration levels of each floor are approximately equivalent, with the differences below 5 dB. The vertical vibrations attenuate slowly in the low frequency during the transmission between floors. When beyond 16 Hz, the acceleration levels change quickly with the rise in height. It also can be noticed that the vertical acceleration level reaches a peak value at 16 Hz and the maximum at 63 Hz. Then above 63 Hz, the acceleration levels fall off rapidly. Comparing to allowance by the standard of limit and measurement method of vibration in the room of residential buildings [29], the vertical acceleration levels inside the over-track building are far below the limitation.

Fig. 12
figure 12

1/3 octave spectra of the vertical vibrations in the middle of the floors

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Fig. 13
figure 13

1/3 octave spectra of the vertical vibrations around the column

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Figure 14 shows the 1/3 octave spectra of the horizontal vibration acceleration levels in the middle of the floors. The horizontal vibration levels increase monotonously and greatly with the rise in height, in which the vibration level in the first floor is lower than that in the top floor by about 20 dB, when the central frequency is smaller than 6.3 Hz. Inversely, when above that, the vibration levels change slowly and the differences are not that great between different floors. Meanwhile, the horizontal vibration levels are of little changes in the low frequency, and these are in sharp variations when the central frequency goes beyond 6.3 Hz. And the acceleration levels in horizontal direction reach the maximum at the 31.5 Hz.

Fig. 14
figure 14

1/3 octave spectra of the horizontal vibrations in the middle of the floors

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It can be inferred from the analyses above that the vibration transmission law across the floors in the horizontal orientation inside the over-track building absolutely differs from that in the vertical orientation. What is more, there are some diversities between the distribution of the horizontal vibration 1/3 octave spectra in the first floor of the building and that in standard floor in high frequency, whose heights are 5.6 and 4 m, respectively. This is because the horizontal constraints to the first floor are imposed by the platform, which could make it vibrate in a different way.

6 Conclusion

Three-dimensional dynamic finite element model of soil-platform-building was established to characterize the vibration transmissions inside the over-track building induced by the train operations into and out of the metro depot. The coupled train–track model was studied and applied as well. These findings have revealed the key insights into the vibration transmissions inside the over-track buildings, including:

  1. 1.

    Vibration waves from various propagation paths strengthen each other in amplitude in the middle of floors, which results in amplifying the vertical vibrations by more than 1.8 dB in comparison with the vertical vibration levels around the columns inside the building. The horizontal vibration levels are comparable in the same floor.

  2. 2.

    The weaker constraints to the higher floors in horizontal direction make main contributions to the vibration enhancement that makes horizontal vibration levels with the tendency of amplification on the whole with the rise in height. The vertical vibrations are attenuated by the floors and amplified at the top floor.

  3. 3.

    Train-induced vibrations inside the over-track building are mainly distributed in high frequency within 30–80 Hz, which have potential to make impacts on the occupants, and the dominant frequency decreases with the rise in height in both vertical and horizontal direction.

  4. 4.

    In the 1/3 octave frequency, the vertical vibration acceleration levels in high frequencies change more obviously with the increasing floor, and the horizontal vibration acceleration levels in low frequencies change more obviously with the increasing floor.