Train-Induced Vibration Analysis and Isolation Trench Measures in Metro Depot Structures

Abstract

Many cities around the world are developing over-track buildings above metro depots to achieve efficient and economical land use. However, the vibrations generated by frequent train operations have a significant impact on the over-track buildings. Therefore, the analysis and control of vibrations at metro depots are of great importance. This paper focuses on the train-induced vibration propagation law and the application of vibration isolation trench measures of the over-track building in the metro depot. To this end, a typical metro depot is taken as the research object. The train-track model, used for simulating wheel-rail force, and the track-soil-building model, used for predicting structural response, are established, respectively. Then, the vibration response of the over-track building of the metro depot is explored, and the effects of vibration isolation measures of the open trench and infilled trench in the metro depot are studied. The results show that the train-induced vibration excitation of the metro is mainly concentrated in the range of 1 to 80 Hz, and the predominant frequency range of the floor vibration is 25 to 50 Hz. The vibration response of the floor is mainly affected by the stiffness. The larger the floor area, the smaller the vertical natural frequency, and the wider the range between the train vibration excitation areas, the more prone to resonance. In addition, the vibration isolation effect of the open trench is better than that of the infilled trench. The primary factor affecting the vibration isolation performance of open trenches is their depth; the influence of trench position and width on the vibration isolation performance is weaker compared to the depth. In the predominant frequency range of floor vibration, the overall vibration isolation effect of the flexible infilled trenches is better than that of the rigid infilled trenches. The main factor affecting the vibration isolation effect of the infilled trenches is the impedance ratio of the material. Among the six kinds of filling materials selected in this paper, the barrier effect of gravel is the worst, and the barrier effect of foam is the best. Using the measure of a foam infilled trench, Z-vibration levels of floors can be reduced by 8.6–13.9 dB.

1. Introduction

In recent years, urban rail transit has developed rapidly in major cities around the world due to its high transportation efficiency, convenience, safety, and environmental friendliness. The metro depot, as an important component of the urban rail transit system, primarily provides necessary logistical services for the daily operation of metro trains. However, the large area of metro depots occupies significant land resources. Therefore, many cities both domestically and internationally have begun to implement Transit-Oriented Development (TOD) models. These models involve constructing large platforms on planned rail transit parking lots (stations) and developing residential real estate on these platforms [1,2]. While metro depots are extensively developed, vibrations caused by the frequent operation of trains are transmitted through the tracks, soil, and structural columns to the buildings above. This transmission leads to floor vibrations and radiated noise, as shown in Figure 1, which severely impacts people’s normal lives [3,4,5]. Therefore, it is of great significance to effectively reduce the train-induced vibrations of the floors.

Based on the vibration transmission chain, the problem of train-induced vibrations in metro systems can be divided into three subsystems: “vibration source, transmission path, and sensitive target”. Accordingly, train vibration control can be categorized into three parts: active vibration reduction at the source, vibration isolation along the transmission path, and passive vibration isolation of building structures. Vibration control at the source is typically divided into train-based measures and track-based measures. For train-based measures, researchers [6,7,8] have noted that train characteristics, such as train weight, operating speed, and the conditions of wheels and rails, are key factors influencing vibrations. Reducing train weight, lowering operating speed, and maintaining the smoothness of wheels and rails can effectively reduce train-induced vibrations. For track-based measures, methods such as vibration reduction at fasteners [9,10], sleepers [11], and ballast beds [12,13,14] have been proposed. These measures have demonstrated good vibration reduction effects when applied in engineering practices. For passive vibration isolation of building structures, many researchers have suggested laying flexible vibration isolation pads at building foundations [15] or installing steel spring supports and rubber bearings [16,17,18] to isolate vibrations. Others have proposed using floating floor systems [19,20] to reduce floor vibrations. Although these two methods can effectively reduce vibrations in building structures, their application to existing rail lines and nearby buildings often requires secondary modifications to tracks and structures, which increases construction difficulty. In contrast, vibration isolation along the transmission path can avoid these challenges. Furthermore, selecting appropriate vibration isolation measures along the transmission path can simultaneously reduce vibrations for multiple sensitive targets, providing higher efficiency. Currently, many studies focus on vibration isolation measures along the transmission path. In terms of isolation trenches, Ahmad [21] and Conte [22] investigated the vibration isolation performance of open trenches using the boundary element method. Alzawi [23] and Ulgen [24] conducted full-scale experiments on foam infilled trenches and explored their vibration isolation effects. Sun et al. [25] conducted large-scale experiments to study the isolation effects of open trenches and gravel infilled trenches under single-frequency load excitation. Numerous practical applications of infilled trenches have been reported worldwide, achieving favorable results. For example, in 1984, air-cushion-filled trench was first used in Stockholm, Sweden [26], reducing floor vibration in buildings by more than 70%. Subsequently, an air-cushion-filled trench was also implemented in Düsseldorf, Germany [26] to isolate vibration from high-speed railways. Additionally, the Brussels Public Transport Company [27] employed composite barriers made of polystyrene cores and concrete side panels near a test track. Apart from isolation trenches, other methods such as wave impeding blocks (WIBs) and pile rows have also been studied. Takemiya [28] and Hung et al. [29] analyzed the effectiveness of WIBs for vibration isolation. Kattis et al. [30], Tsai et al. [31], and Gao et al. [32] investigated the vibration isolation performance of pile rows.

Currently, research by scholars in various countries on vibration isolation in the propagation pathways of rail transit primarily focuses on the sections of subways or high-speed rail. However, studies on vibration isolation measures for the propagation pathways in metro depots are relatively scarce. In conventional railway sections, the above-ground or underground tunnel lines are generally located far from building structures. High-frequency vibrations become very weak after attenuation through the soil, resulting in building floor vibrations that are predominantly low-frequency. In contrast, metro depots typically have track lines that are located close to the over-track buildings. The high-frequency vibrations induced by trains do not completely attenuate and are transmitted through structural columns to the floors of the over-track buildings. The dominant frequency range of floor vibrations in this case is higher compared to conventional lines. Therefore, the vibration characteristics of the floors differ in these two scenarios. This paper focuses on the metro depot as the research target and investigates the impact of open trenches and infilled trenches on the vibration isolation effects for the over-track building of the metro depot.

2. Environmental Vibration Model of Metro Depot

Environmental vibrations caused by metro train operations are complex and involve the generation of train-induced vibrations and their propagation through the soil and buildings. In this study, numerical simulations are conducted to investigate this issue. The numerical model is divided into two substructure models: (a) the train-track model and (b) the track-soil-building model. First, a train-track dynamic model is established using SIMPACK software (https://www.3ds.com/zh-hans/products/simulia/simpack) to simulate the wheel-rail forces generated by the train. Then, a three-dimensional finite element model of the metro depot, including the track, soil, and building, is developed using the finite element software GFE (https://gfe.com.tr/en/downloads.html). The simulated wheel-rail forces are applied to the finite element model. Through calculations, the vibration response characteristics and propagation laws of the over-track building in the metro depot are analyzed. Based on these results, the effectiveness of vibration isolation measures, including open trenches and infilled trenches, is further studied.

2.1. Over-Track Platform and Building of the Metro Depot

In this study, a metro depot is used as an example for simulation analysis. The metro depot adopts an integrated structure combining the over-track platform and the building. As shown in Figure 2, the model focuses on the section near the depot gate in the operation depot. The over-track platform is a reinforced concrete frame structure. The first floor is used as the operation depot, with a floor height of 11.3 m. The second floor is used as a vehicle garage, with a floor height of 5.1 m. The platform spans four bays longitudinally along the track direction, with each bay measuring 12 m, resulting in a total length of 48 m. Transversely, the platform spans five bays, with each bay measuring 6.8 m, resulting in a total width of 34 m. A nine-story residential building is constructed above the platform. The building adopts a reinforced concrete frame structure. The first floor of the building has a floor height of 5.1 m, while the remaining standard floors have a uniform floor height of 3.1 m. The structural layout of a standard floor is shown in Figure 3, and the numbering of the standard floor slabs is presented in Figure 4.

The floor slabs and filling walls in the over-track platform and building structure are simulated using shell elements. Structural beams and columns are simulated using beam elements. The element size ranges from 0.5 to 1 m. The shell elements consist of two types: S3R and S4R. A total of 360 S3R elements and 11,030 S4R elements are used. The beam elements are of type B31, with a total of 5635 elements. The material parameters for the structural components of the building are listed in

Table 1. Material parameters of building structure.

Component	Material	Density (kg/m3)	Elastic Modulus (MPa)	Poisson’s Ratio
Filling wall	C30	2500	30,000	0.2
Structural beam	C40	2500	32,500	0.2
Floor slab
Structural column	C50	2500	34,500	0.2
	C60	2500	36,000	0.2
Reinforcement	HRB400	7800	200,000	0.3
I-beam	Q345	7800	206,000	0.3

.

The Rayleigh damping model is used to simulate the attenuation of train-induced vibrations during propagation. The damping ratio of structural component materials is set to 0.05. The lower and upper limits of the vibration frequency range of interest for structural components are set to 1 Hz and 80 Hz, respectively.

2.2. Track Structure

The material parameters of the track structure in the metro depot are shown in

Table 2. Material properties of track structure.

Component	Material	Density (kg/m3)	Elastic Modulus (MPa)	Poisson’s Ratio
Rail 60	Q235	7800	206,000	0.3
Sleeper	C30	2500	30,000	0.2
Ballast

.

The rails are connected to the sleepers through fasteners. In this study, spring-damper connectors are used to simulate the stiffness and damping of the fasteners. The specific parameters are shown in

Table 3. Fastener parameters.

Fastener Parameters	Numerical Value	Unit
Vertical stiffness	3 × 107	N/m
Lateral stiffness	3 × 107	N/m
Vertical damping	4 × 104	N·s/m
Lateral damping	4 × 104	N·s/m
Fastener spacing	0.6	m

.

2.3. Soil Model

The soil depth is set to 40 m. The soil dimensions are set to 90 m in the longitudinal direction of the track and 100 m in the transverse direction. Based on the geological survey report, the soil is simplified into five layers and is considered a linear elastic material. The parameters of the soil layers are shown in

Table 4. Soil parameters.

Soil Layer	Thickness (m)	Density (kg/m3)	Elastic Modulus (MPa)	Poisson’s Ratio
Artificial fill	2	1950	210	0.31
Clayey silt	5.6	2040	219	0.25
Fine sand	8.5	1980	364	0.3
Silty clay	7.2	2120	402	0.35
Sandy silt	16.7	2150	401	0.25

.

To prevent the reflection of train-induced vibration waves at the boundary of the soil medium from affecting the calculation results, a viscoelastic artificial boundary is applied at the soil boundary. Springs and dampers are added to the boundary to absorb the energy radiated outward from the vibration source during the vibration process. This eliminates the reflected waves caused by the truncation of the soil medium. As a result, the wave propagation from the near field to the far field can be accurately simulated. The parameters of the normal and tangential springs and dampers [33] at the soil model boundary in this study are shown as follows:

$$\left\{\begin{array}{l}{K}_N={\displaystyle \frac{\lambda +2G}{\left(1+A\right)r}}\\ C_N=B\rho c_P\end{array}\right.$$

(1)

$$\left\{\begin{array}{l}{K}_T={\displaystyle \frac{G}{\left(1+A\right)r}}\\ C_T=B\rho c_S\end{array}\right.$$

(2)

where $\lambda$ represents the first Lamé constant, and $G$ denotes the shear modulus; $r$ is the distance from the geometric center of the near-field structure to the boundary line or surface where the artificial boundary point is located; $\rho$ represents the density of the medium; $C_P$ and $C_S$ are the wave velocities of P-waves and S-waves in the medium, respectively; and $A$ and $B$ are correction coefficients, taken as 0.8 and 1.1, respectively.

The environmental vibration of the metro depot involves high-frequency vibration calculations, which are directly influenced by the size of the finite element mesh. If the mesh size is too large, high-frequency vibrations may be truncated, leading to distorted calculation results. Theoretically, the denser the finite element mesh, the more accurate the calculation results. However, the three-dimensional model of the depot structure covers a large area. If a uniformly dense mesh is applied to the entire soil region, the total number of elements in the model will become excessively large, resulting in prolonged computation time and higher hardware requirements. Therefore, the selection of mesh size should balance the frequency range of interest and the computation time. According to [34], when $\Delta x<{\lambda }_s/6$ ($\Delta x$ is the mesh size and ${\lambda }_s$ is the wavelength of the shear wave), satisfactory results can be obtained at most locations except near the wave source $0.5{\lambda }_s$. Due to the rapid attenuation of high-frequency vibrations in the soil and the gradual increase in shear wave velocity with soil depth, a locally refined mesh strategy is adopted for the soil model. The mesh size near the track vibration source is set to 0.4 m. Moving away from the vibration source toward deeper soil and the sides of the track, the mesh size is gradually increased, reaching 1.5 m at the edges of the soil region. The soil elements are modeled using C3D8 elements, with a total of 384,652 elements. Figure 5 shows the finite element model of the track-soil-building established in this study.

2.4. Train Model

The train model is a Chinese Type A train with an eight-car formation. A schematic diagram of the Type A metro train is shown in Figure 6.

The speed limit inside the operation depot is set at 5 km/h. The speed in the throat area is approximately 10 to 20 km/h. The section of the metro depot simulated in this study is located near the depot entrance. It is connected to the straight track of the throat area at a level crossing, where the speed is higher. To consider the worst-case scenario of vibrations from the over-track building, the train speed is set at 20 km/h. An eight-car train returning to the depot is simulated.

Table 5. Parameters of Type A metro train.

Parameters	Numerical Value	Unit
Vehicle body mass	23,825	kg
Bogie frame mass	3970	kg
Wheelset mass	1654	kg
Primary suspension vertical stiffness	1.26 × 106	N/m
Primary suspension longitudinal stiffness	1.0 × 107	N/m
Primary suspension lateral stiffness	6.5 × 106	N/m
Secondary suspension vertical stiffness	4.9 × 105	N/m
Secondary suspension longitudinal stiffness	2.31 × 105	N/m
Secondary suspension lateral stiffness	2.31 × 105	N/m
Primary suspension vertical damping	10,626	N·s/m
Primary suspension longitudinal damping	0	N·s/m
Primary suspension lateral damping	0	N·s/m
Secondary suspension vertical damping	20,590	N·s/m
Secondary suspension lateral damping	40,000	N·s/m

presents the parameters of the Type A metro train.

Train-induced vibrations are generated by moving loads caused by the weight of the train and track irregularities. In this study, the train load is simulated using the SIMPACK multibody dynamics software. Due to the use of seamless rails and high construction quality in the operation depot area, the overall track smoothness is relatively good. Therefore, the American Class 6 track spectrum is adopted in the model to impose irregularities on the rails [35]. Figure 7 shows the simulated wheel-rail forces of the train wheelsets in this study.

In the track-soil-building finite element model, the train load is applied under the assumption that the wheel-rail forces on both rails are identical. The wheel-rail forces are sequentially applied as moving loads to the rail nodes located above the fasteners.

3. Propagation Law of Train-Induced Vibration in Metro Depot

3.1. Verification of Model Calculation Accuracy

To better reflect the accuracy of the model calculation results and the reliability of the vibration propagation patterns, a comparison is made between the calculated results and the measured results for the metro depot model. The verification points are located at the footing of the track area structural column and the top of the cover structure column, both situated near the depot entrance, as shown in Figure 8. In this study, the background interference vibration elimination method is applied to ensure the accuracy of train-induced vibration measurement data in the depot area. The specific steps are as follows: First, the acceleration at the verification point is measured when no trains are operating, and this is defined as the background interference vibration baseline. Next, the acceleration at the same verification point is measured again during train operation, and both datasets are subjected to Fourier Transform (FFT). The frequency spectrum components corresponding to the background interference vibration are then subtracted in the frequency domain to eliminate the influence of background noise. Finally, the Inverse Fourier Transform (IFFT) is applied to reconstruct the time-domain signal, enabling the acquisition of accurate train-induced vibration measurement data.

Figure 9 and Figure 10 show the acceleration time history comparison and the vibration level of 1/3 octave band comparison at the two verification points, respectively. For verification point 1 (footing of the track area structural column), the measured peak acceleration is 0.0154 m/s², while the calculated peak acceleration is 0.0143 m/s², with an error of 7.7%. For verification point 2 (top of the cover structure column), the measured peak acceleration is 0.0225 m/s², while the calculated peak acceleration is 0.0245 m/s², with an error of 8.9%. The errors between the measured and calculated peak acceleration values at both points are within 10%. From the vibration level of 1/3 octave band comparison, it can be observed that the maximum vibration levels in the 1/3 octave band at both verification point 1 (footing of the track area structural column) and verification point 2 (top of the cover structure column) occur at 40 Hz. Within the frequency range of 2–80 Hz, the simulated vibration levels closely match the measured values. In summary, the metro depot model can effectively simulate the vibration issues caused by trains returning to the depot.

3.2. Vibration Response Analysis of the Floor in the Over-Track Building

Figure 11 shows the peak accelerations of all floor slabs from the 2nd to the 9th floors of the over-track building. It can be observed from Figure 11 that the overall peak accelerations of slabs 3 and 13 are relatively large, followed by slabs 1, 2, 4, 12, 14, and 15. The peak accelerations of slabs 5 to 11 are comparatively smaller. The maximum accelerations of slabs 1, 2, 3, 13, 14, and 15 are 0.0182 m/s², 0.0144 m/s², 0.0312 m/s², 0.0395 m/s², 0.0175 m/s², and 0.0180 m/s², respectively.

To further investigate the differences in vibration responses between different floor slabs, modal analysis was conducted on floor slabs 1 and 3 as examples.

Table 6. Floor slab modes.

Modal Order	Natural Frequency (Hz)
	Floor Slab 1	Floor Slab 3
1	45.7	29.4
2	49.4	44.3
3	49.7	44.4
4	54.7	51.5
5	56.1	54.2

presents the first five vertical natural frequencies obtained from the modal analysis of floor slabs 1 and 3.

The vibration in different frequency bands exhibits distinct variation trends during its transmission from the structural column to the center of the floor slab. In this study, the acceleration amplification factor is used to reflect these differences. The specific formula is shown as follows:

$$A_a={\displaystyle \frac{A_m}{A_c}}$$

(3)

where $A_a$ represents the acceleration amplification factor; $A_m$ denotes the 1/3 octave band acceleration root mean square (RMS) value at the midpoint of the floor slab; and $A_c$ refers to the 1/3 octave band acceleration RMS value at the corner point of the floor slab. Figure 12 illustrates the locations of different vibration measurement points on the floor slab.

Figure 13 shows the acceleration amplification factors in the 1/3 octave band ranging from 2 Hz to 80 Hz for floor slabs 1 and 3 of the over-track building. It can be observed that the vibration of floor slab 1 is amplified in the frequency range of 31.5 Hz to 80 Hz. The amplification is most significant at the frequencies of 50 Hz and 63 Hz. For floor slab 3, vibration amplification occurs across the frequency range of 5 Hz to 80 Hz. The amplification is particularly pronounced at the frequencies of 20 Hz, 25 Hz, 31.5 Hz, and 80 Hz. Considering the vibration frequency range of 2 Hz to 80 Hz, the overall amplification trend of floor slab 3 is stronger than that of floor slab 1.

The area of floor slab 1 is 21.76 m², while the area of floor slab 3 is 36.72 m². As shown in Figure 11, the peak acceleration of floor slab 3 is generally greater than that of floor slab 1. This indicates that the acceleration response of the floor slab is related to its area. As demonstrated by the modal analysis of the floor slabs in Table 6, under the condition of equal slab thickness, larger floor spans result in lower vertical stiffness. Consequently, the lower-order vertical natural frequency of the slab decreases. This increases the overlap between the natural frequency range of the slab and the dominant frequency range of vibration induced by train operations, making resonances more likely to occur.

In the vibration response analysis of the floor slab of the over-track building, the analysis can be conducted not only from the perspective of acceleration response but also by considering Z-vibration levels and vibration levels in the 1/3 octave band. The Z-vibration level reflects the overall vertical vibration level of the floor slab, while the vibration level in the 1/3 octave band represents the vibration levels within different frequency ranges of the floor slab. Different weighting correction methods were applied in this study for the analysis of Z-vibration levels and vibration levels in the 1/3 octave band of the floor slab. The Z-vibration levels were determined using the W_k weighting curve provided in ISO 2631-1:1997 [36], while the vibration levels in the 1/3 octave band were calculated using the weighting method specified in the Chinese standard “Limits and Measurement Methods for Building Vibration and Secondary Noise Induced by Urban Rail Transit”. Figure 14 shows the variation trends of Z-vibration levels for slabs 1 and 3 across the 1st to 9th floors. For slab 1, the minimum Z-vibration level is 62.8 dB, occurring on the 5th floor, while the maximum Z-vibration level is 68.6 dB, occurring on the top floor. The overall Z-vibration level of slab 1 decreases initially, fluctuates slightly in the middle floors, and then gradually increases toward the top floors. For slab 3, the minimum Z-vibration level is 65.6 dB, occurring on the 3rd floor, while the maximum Z-vibration level is 74.6 dB, also occurring on the top floor. The overall Z-vibration level of slab 3 decreases initially and then gradually increases toward the top floors. Both slabs 1 and 3 exhibit a significant amplification of Z-vibration levels near the top floors. This is similar to the conclusions drawn in reference [37]. After the vibrations propagate from the ground floor to the top floor, the upper part no longer contains structures that can absorb energy. Therefore, the floors near the top experience a significant amplification of vibrations due to the superposition of incident and reflected waves.

Figure 15 shows the vibration levels in the 1/3 octave band for slabs 1 and 3. As shown in Figure 15, the dominant vibration frequencies of slabs 1 and 3 are mainly concentrated in the range of 25 to 50 Hz. Vibrations above 100 Hz become weaker due to attenuation through the soil. The maximum vibration level of slab 1 is 67.1 dB, occurring at 40 Hz on the 9th floor of the building. For slab 3, the maximum vibration level is 73.7 dB, occurring at 31.5 Hz on the ground floor of the building.

The reason for the maximum vibration levels in the 1/3 octave band of floor slabs 1 and 3 occurring at different frequency points is mainly related to the first-order vertical natural frequency of the slabs. Figure 16 shows the first-order mode shape diagram of the slabs. As shown in Figure 16, the first-order frequency of floor slab 1 is 45.7 Hz, while that of floor slab 3 is 29.4 Hz. The frequency points corresponding to the maximum vibration levels in the 1/3 octave band are close to the fundamental frequency of each slab.

3.3. Vibration Assessment of the Over-Track Building

The analyzed over-track building is a nine-story residential structure. According to the “Urban Regional Environmental Vibration Standard” in China, the Z-vibration level limits for residential and educational areas are 70 dB during the daytime and 67 dB at night. The standard floor slabs numbered 1, 2, 14, and 15 in this building correspond to bedroom slabs.

Table 7. Floor Z-vibration level (dB).

Floor	Floor Slab Number
	1	2	14	15
1	65.9	65.8	65.4	65.4
2	64.1	65.3	63.9	64.5
3	65.9	62.5	61.8	64.0
4	63.6	60.4	61.2	63.2
5	62.8	60.4	61.1	63.3
6	63.8	61.1	62.3	64.9
7	65.3	62.5	64.2	66.2
8	67.1	64.5	65.7	67.2
9	68.6	65.2	66.7	67.7

presents the Z-vibration levels of bedroom slabs from the 1st to the 9th floor. It can be observed that, except for slabs 1 and 15 on the 8th and 9th floors, where the Z-vibration levels exceed the nighttime limit, the Z-vibration levels of bedroom slabs on all other floors are below both the daytime and nighttime limits.

Figure 17 shows the vibration levels in the 1/3 octave band for slabs 1, 2, 14, and 15 on the 1st, 5th, and 9th floors of the over-track building. From these diagrams, it can be observed that high-frequency vibrations in the range of 63–80 Hz gradually decrease with increasing floor height, while vibrations at certain frequency points in the range of 5 to 31.5 Hz exhibit amplification as the floor height increases. The vibration levels of 1/3 octave band in this study are calculated using the weighting method and control standards specified in the Chinese standard “Limits and Measurement Methods for Building Vibration and Secondary Noise Induced by Urban Rail Transit”. This standard sets the daytime vibration level limit in the 1/3 octave band at 65 dB and the nighttime limit at 62 dB for residential and educational areas. As shown in Figure 17, the maximum vibration levels in the 1/3 octave band for slabs 1, 2, 14, and 15 occur at 31.5 Hz and 40 Hz. At these frequencies, the vibration levels of bedroom slabs on certain floors exceed the daytime and nighttime limits.

4. Study on Vibration Isolation Trench Measures for Depot

Based on the vibration assessment of the building floor slabs, it is found that the vibration levels of some slabs in the over-track building exceed the standard limits. Therefore, appropriate vibration isolation measures should be implemented to reduce the vibration levels of the slabs. In this study, the method of vibration isolation through propagation paths is adopted to mitigate the floor vibrations of the over-track building. The effects of open trenches and infilled trenches on the vibration isolation performance of the over-track building are investigated in detail.

To compare the vibration isolation effects before and after the excavation of vibration isolation trenches, the acceleration attenuation coefficients are used for presentation. The specific calculation formula for the acceleration attenuation coefficients is shown as follows.

$$A_r={\displaystyle \frac{A_0}{A_1}}$$

(4)

where $A_r$ represents the attenuation coefficient of the acceleration root mean square (RMS) in each frequency band. $A_0$ is the 1/3 octave band acceleration RMS value of the floor slab after vibration isolation, and $A_1$ is the 1/3 octave band acceleration RMS value of the floor slab before vibration isolation. When $A_r=1$, it indicates no vibration isolation effect. When $A_r>1$, it indicates that the vibration in this frequency band is amplified under the influence of the isolation trench. When $A_r<1$, it indicates that the vibration in this frequency band is attenuated.

4.1. The Effect of Open Trench Depth on Vibration Isolation

Open isolation trenches are typically placed on both sides of the track. By excavating soil to a certain depth and width, a barrier is formed, which alters the propagation path of train-induced vibrations. This helps to reduce the vibrations caused by the train. Most of the vibration energy in the soil propagates outward in the form of Rayleigh waves [38]. The calculation formula for the Rayleigh wave velocity is shown as follows.

$$V_R={\displaystyle \frac{0.87+1.12\upsilon }{1+\upsilon }}{V}_S$$

(5)

where $V_R$ represents the Rayleigh wave velocity, $\upsilon$ is the Poisson’s ratio, and $V_S$ is the shear wave velocity.

In this study, the shear wave velocity of the first-layer soil in the metro depot is 203 m/s, and the Poisson’s ratio is 0.31. Through calculation, the Rayleigh wave velocity of the first-layer soil is determined to be 188 m/s. As shown in Figure 15, the frequency range corresponding to the maximum vibration levels in the 1/3 octave band of the floor slabs in the over-track building is 31.5–40 Hz. Therefore, the Rayleigh wavelength of the first-layer soil at 31.5 Hz is denoted as ${\lambda }_R=5.98\mathrm{m}\approx 6\mathrm{m}$. In the section investigating the effect of trench depth on vibration isolation performance, four working conditions are set: 1.5 m, 3 m, 4.5 m, and 6 m. The maximum trench depth corresponds to the Rayleigh wavelength of the first-layer soil at 31.5 Hz. For all four conditions, the trench width is set to 0.5 m, and the trench is located 2.4 m from the track centerline. The specific layout of the open trench is shown in Figure 18.

Figure 19 presents the acceleration attenuation coefficients of slab 1 on the 1st, 4th, 6th, and 9th floors under different trench depths. It can be observed that, for all four conditions, the acceleration attenuation coefficients of the slabs are concentrated in the range of 0.1 to 1, showing an overall attenuation trend. Additionally, as the trench depth increases, the acceleration attenuation coefficients of the slabs decrease, and the vibration isolation effect of the trench is gradually enhanced. From the acceleration attenuation curves of slab 1 on the 1st, 4th, 6th, and 9th floors, it can be seen that the isolation effect of the trench on low-frequency vibrations below 8 Hz is weaker than its effect on mid- to high-frequency vibrations in the range of 8–200 Hz. For trench depths of 4.5 m and 6.0 m, the minimum acceleration attenuation coefficients within the main vibration frequency range of 1–80 Hz are both less than 0.3, and the attenuation coefficients for high-frequency vibrations in the range of 160–200 Hz are even below 0.1. This indicates that the slab vibrations are effectively reduced. However, for the trench depth of 1.5 m, the acceleration attenuation coefficients for vibrations in the range of 1–200 Hz are generally higher than those of the other three conditions, with more frequency points showing vibration amplification. This indicates that the vibration isolation effect is relatively weaker.

Figure 20 shows the Z-vibration levels of slab 1 on each floor under different trench depths. When the trench depth is 1.5 m, the Z-vibration levels of the slabs on floors 1–9 are reduced by 3.3–5.9 dB. When the trench depth is 3 m, the Z-vibration levels are reduced by 4.8–10.2 dB. For a trench depth of 4.5 m, the Z-vibration levels are reduced by 8.9–13.2 dB. When the trench depth reaches 6 m, the Z-vibration levels are reduced by 11.0–17.2 dB. It can be observed that as the trench depth increases, the vibration isolation effect becomes stronger, and the Z-vibration levels of the slabs decrease continuously. When the trench depth reaches one Rayleigh wavelength of the surface soil, the Z-vibration levels of the over-track building slabs can be reduced by more than 10 dB.

4.2. The Effect of Open Trench Width on Vibration Isolation

The width of open trenches in the depot is restricted by the distance between the track and adjacent structural columns. Therefore, without altering the normal structural layout of the depot, the trench widths in this study were set to 0.5 m, 1 m, and 1.5 m. The trench depth and distance were uniformly set to 6 m and 2.4 m, respectively. The effect of different trench widths on the vibration isolation performance was investigated. Figure 21 shows the vibration acceleration attenuation coefficients of slab 1 on the 1st, 4th, 6th, and 9th floors under the three trench widths. It can be seen from Figure 21 that, as the trench width increases, the acceleration attenuation coefficients in the 1–10 Hz vibration frequency range remain nearly the same. In the 10–200 Hz frequency range, the acceleration attenuation coefficients show slight fluctuations but do not exhibit a clear trend of enhancement or reduction. The vibration isolation effects of the trenches under the three conditions show little difference.

Figure 22 shows the Z-vibration levels of slab 1 on floors 1–9 under three different trench widths. When the trench width is 0.5 m, the Z-vibration levels of the slabs decrease by 11.0–17.2 dB. When the trench width is 1.0 m, the Z-vibration levels decrease by 13.9–18.4 dB. When the trench width is 1.5 m, the Z-vibration levels decrease by 12.3–16.7 dB. The available space within the depot area is limited, making it impossible to set excessively large trench widths. Based on the attenuation of floor acceleration and the reduction in Z-vibration levels under three working conditions, it can be observed that changes in trench width within the range of 0.5–1.5 m have a relatively minor effect on vibration isolation performance. Therefore, in practical engineering trench excavation, the trench width should be determined by balancing economic excavation costs and construction difficulty.

4.3. The Effect of Open Trench Position on Vibration Isolation

Similar to the constraints on trench width, the position of open trenches is also restricted by the distance between the track and adjacent structural columns. Therefore, in this study, to investigate the effect of trench position on vibration isolation performance, the trench distances were set to 1.9 m, 2.4 m, and 2.9 m. Under all three scenarios, the trench width and depth were uniformly set to 0.5 m and 6 m, respectively. Figure 23 shows the vibration acceleration attenuation coefficients of slab 1 on the 1st, 4th, 6th, and 9th floors under the three distances. It can be observed that, in the 1–5 Hz frequency range, the acceleration attenuation coefficients of the slabs are nearly identical under all three conditions. In the 5–31.5 Hz frequency range, the acceleration attenuation effect shows an overall increasing trend as the distance of the trench increases. In the 40–200 Hz frequency range, the acceleration attenuation coefficients exhibit fluctuations under all three conditions, but the overall attenuation trend is approximately similar.

Figure 24 presents the Z-vibration levels of slab 1 on each floor under three different distances of the trench. When the distance is 1.9 m, the Z-vibration levels of the slabs decrease by 8.2–12.2 dB. When the distance is 2.4 m, the Z-vibration levels decrease by 11.0–17.2 dB. When the distance is 2.9 m, the Z-vibration levels decrease by 10.2–15.6 dB. Based on the Z-vibration levels attenuation values under the three conditions, it can be concluded that the attenuation effect is relatively better when the distance is 2.4 m, while the attenuation is relatively weaker when the distance is 1.9 m. Due to the close proximity between the tracks and the building structures within the depot, the active and passive vibration isolation characteristics of the trench are not clearly defined. However, considering the attenuation of both acceleration and Z-vibration levels under the three conditions, it can be observed that the vibration isolation effect is weakened to some extent when the trench is located too close to the track structure. It is recommended that the trench distance should be greater than 0.4 times the Rayleigh wavelength of the surface soil.

4.4. The Effect of Trench Filling Materials on Vibration Isolation

The effects of trench depth, width, and position on vibration isolation performance were discussed in detail above. It was found that trenches with a certain excavation depth can achieve good vibration isolation performance. However, in practical engineering, the excavation of vibration isolation trenches should also consider factors such as construction convenience, long-term maintenance, and the impact of the trench on the structural stability and safety of the depot. Therefore, compared to open trenches, infilled trenches have greater advantages. The vibration isolation mechanism of infilled trenches involves filling the trench with specific materials (e.g., gravel, concrete, rubber, fly ash, foam, etc.) to create differences in stiffness and shear wave velocity between the trench and the surrounding soil. This causes vibration waves to undergo reflection and refraction at the trench walls, which increases the propagation path and reduces the propagation speed of the waves, thereby achieving the effect of vibration isolation.

The concept of impedance was first introduced in the field of acoustics. It is defined as the product of the medium density $\rho$ and the wave velocity $v$ within the medium. The impedance ratio refers to the ratio of the impedance ${\rho }_1{v}_1$ of the first medium to the impedance ${\rho }_2{v}_2$ of the second medium [39]. The specific formula is shown as follows:

$$\alpha ={\displaystyle \frac{{\rho }_1{v}_1}{{\rho }_2{v}_2}}$$

(6)

For vibration isolation infilled trenches, the impedance ratio is an important factor for evaluating the vibration isolation performance of the medium. In practical engineering, the first medium is typically the filling material, while the second medium is the soil surrounding the infilled trench.

Table 8. Filling material parameters.

Filling Material	Density (kg/m3)	Elastic Modulus (MPa)	Poisson’s Ratio	Impedance Ratio
Gravel	2100	500	0.24	1.511
LAC	1800	14,000	0.2	8.712
C30 concrete	2500	30,000	0.2	13.935
Fly ash	500	25	0.35	0.174
Rubber	1480	4.5	0.48	0.123
Foam	50	5	0.4	0.024

presents the physical properties of commonly used filling materials and the impedance ratios between the filling materials and the topsoil. Based on the impedance ratio, filling materials can be classified into rigid materials and flexible materials. Materials with an impedance ratio greater than 1 are classified as rigid materials, while those with an impedance ratio between 0 and 1 are classified as flexible materials.

In this study, six materials from Table 8 were selected as filling materials. Among them, gravel, lightweight aggregate concrete (LAC), and C30 concrete have impedance ratios greater than 1 compared to the topsoil and are classified as rigid materials. Fly ash, rubber, and foam have impedance ratios less than 1 and are classified as flexible materials. The depth of the infilled trenches was set to 6 m, the width to 0.5 m, and the relative distance to 2.4 m from the track centerline. The specific layout of the infilled trench is shown in Figure 25.

Figure 26 shows the acceleration attenuation of slab 1 on the 1st, 4th, 6th, and 9th floors under the filling of three rigid materials. Figure 27 shows the acceleration attenuation of the same slabs under the filling of three flexible materials. It can be observed that different filling materials exhibit varying vibration isolation effects. For rigid infilled trenches, the overall vibration isolation effect of gravel infilled trench is the weakest. The acceleration attenuation coefficients fluctuate slightly around 1 in the 1–80 Hz frequency range, indicating almost no vibration isolation effect. The vibration isolation effects of the LAC infilled trench and C30 concrete infilled trench are better than that of the gravel infilled trench. Certain vibration isolation effects are observed in both types of trenches within the frequency ranges of 8–16 Hz, 31.5–63 Hz, and 160–200 Hz. For flexible infilled trenches, the foam infilled trench exhibits the best overall vibration isolation effect in the 5–125 Hz frequency range, followed by the rubber infilled trench, while the fly ash infilled trench performs relatively weaker. In the high-frequency range of 160–200 Hz, the rubber infilled trench demonstrates superior vibration isolation performance. According to the analysis in Section 3, the dominant vibration frequency range of the slabs in the over-track building is 25–50 Hz. A comparison of Figure 26 and Figure 27 shows that the effective vibration isolation frequency range of flexible infilled trenches is broader than that of rigid infilled trenches. Additionally, in the 25–50 Hz frequency range, flexible infilled trenches achieve better overall vibration isolation performance than rigid infilled trenches. Therefore, it can be concluded that flexible infilled trenches are more suitable for vibration isolation in metro depots.

Figure 28 presents the Z-vibration levels of slab 1 under six different filling materials. It can be observed that when flexible materials are used to fill the trench, the Z-vibration levels of the slabs on floors 1–9 are generally lower than those with rigid infilled trenches. For rigid infilled trenches, the vibration isolation performance of the C30 concrete infilled trench is the best, with Z-vibration levels of the slabs reduced by 1.5–6.4 dB. The LAC infilled trench performs slightly worse than C30 concrete infilled trench, with Z-vibration levels reduced by 1.5–5.1 dB. The gravel infilled trench amplifies the vibration in the dominant frequency range of the slabs, resulting in the worst vibration isolation performance, with Z-vibration levels increased by 0.3–1.6 dB for floors 1–9. For flexible infilled trenches, the foam infilled trench achieves the best vibration isolation performance, with Z-vibration levels of the slabs reduced by 8.6–13.9 dB. The vibration isolation effect of the foam infilled trench is closest to that of the open trench with the same dimensions. The fly ash infilled trench performs relatively weaker, with Z-vibration levels reduced by 2.2–7.0 dB. The rubber infilled trench achieves intermediate performance between the foam infilled and fly ash infilled trenches, with Z-vibration levels reduced by 4.5–9.1 dB.

Based on the comparative analysis of the impedance ratio of filling materials and the attenuation effects of slab acceleration and Z-vibration levels, it can be concluded that the impedance ratio is a key factor influencing the vibration isolation performance of infilled trench. The farther the impedance ratio deviates from 1, the better the vibration isolation performance of the infilled trench. Conversely, when the impedance ratio approaches 1, the vibration isolation effect decreases and may even disappear. This indicates that the stiffer the rigid material and the softer the flexible material, the better the vibration isolation performance of the infilled trench. For flexible infilled trenches, while they can provide better vibration isolation performance for the over-track building, flexible materials may face long-term stability issues such as compression creep, water absorption-induced swelling, and decomposition. In addition, flexible materials typically exhibit lower overall mechanical strength, and therefore often need to be used in conjunction with structural elements such as secant piles or diaphragm walls. This increases construction complexity and maintenance difficulty. For rigid infilled trenches, although the overall vibration isolation performance is weaker compared to flexible infilled trenches, rigid materials have higher overall mechanical strength and exhibit good resistance to compression creep. As a result, rigid infilled trenches are superior to flexible infilled trenches in terms of construction convenience and maintainability. However, as shown in Table 8 and Figure 28, when the impedance ratio between the rigid material and the surface soil reaches 8.7, further increasing the stiffness of the rigid material does not lead to significant improvement in vibration isolation performance.

The influence of open trenches and infilled trenches on the vibration isolation effects of the over-track building was discussed in detail in Section 4.1, Section 4.2, Section 4.3 and Section 4.4. However, with regard to the arrangement of trenches, this study considered the isolation trenches to be symmetrically arranged on both sides of the track. The effects of single-sided trenches or trenches asymmetrically arranged on both sides of the track were not analyzed. Moreover, there are multiple tracks within the depot area. The variation in distances between different tracks and the trenches leads to changes in the incidence angle of vibration waves when they propagate to the interior wall of the trenches. This study did not investigate the vibration isolation performance of the trenches under the operating conditions of trains running on different tracks. Therefore, further research on the above aspects is required in future studies.

5. Conclusions

In this study, numerical simulation analysis was conducted for a metro depot by establishing a train-track and track-soil-building substructure model. The vibration response and propagation characteristics of the slabs in the over-track building of the depot were investigated. Based on this analysis, vibration isolation trench measures for the metro depot were studied. The main conclusions are as follows:

(1)

The vibration response of slabs is significantly influenced by stiffness. When the slab thickness is the same, larger slab areas result in lower vertical natural frequencies. This leads to a wider frequency range under train load excitation, making resonance more likely to occur in the slabs. The dominant frequency range of slab vibration in the over-track building is between 25 and 50 Hz. The maximum vibration level in the 1/3 octave band occurs at 31.5–40 Hz. High-frequency vibration in the range of 63–80 Hz shows a decreasing trend with increasing floor height. In contrast, vibration at certain frequency points within the range of 5–31.5 Hz exhibits amplification as the floor height increases.

(2)

Within the main frequency range of slab vibration, open trenches exhibit the best vibration isolation performance. Numerical analysis indicates that depth is the primary factor affecting the isolation performance of open trenches. As the depth of the trench increases, the vibration isolation effect improves. When the trench depth reaches one Rayleigh wavelength of the surface soil, the Z-vibration levels of the slabs can be reduced by more than 10 dB. The influence of the position and width of open trenches on their vibration isolation performance is weaker than that of their depth. For the position of open trenches, the vibration isolation performance decreases when the trench is located closer to the track structure. It is recommended that the distance between the trench and the track structure should be greater than 0.4 times the Rayleigh wavelength of the surface soil. Regarding the width of open trenches, due to the limited space in the depot area, the trench width cannot be set too large. Therefore, within the width range of 0.5–1.5 m considered in this study, changes in trench width have a relatively minor influence on the vibration isolation performance.

(3)

Within the dominant frequency range of slab vibration, flexible infilled trenches exhibit better overall vibration isolation performance than rigid infilled trenches. The gravel infilled trench shows almost no vibration isolation effect within the 1–80 Hz frequency range of slab vibration, and the Z-vibration levels of the slabs are amplified by 0.3–1.6 dB. The foam infilled trench achieves the best vibration isolation performance. When a foam infilled trench is used, the Z-vibration levels of the slabs can be reduced by 8.6–13.9 dB. The impedance ratio between the filling material and the surrounding soil is an important factor affecting the vibration isolation performance of infilled trenches. When the impedance ratio deviates further from 1, the vibration isolation performance improves. However, when the impedance ratio approaches 1, the vibration isolation effect decreases and may even disappear. For rigid infilled trenches, when the impedance ratio between the filling material and the surface soil reaches 8.7, further increasing the stiffness of the filling material does not significantly enhance the vibration isolation performance.