Vibration Induced by Subway Trains: Open-Trench Mitigation Analysis in the Time and Frequency Domains

In this paper, we analyse the mitigation effects of open trenches on the vibrations induced by subway trains. The study is performed by using both physical model tests and numerical simulations. The effectiveness is evaluated by calculating the frequency response function (FRF) and the vibration acceleration peak (VAP) in both time and frequency domains. The experimental and numerical results demonstrate that the open trench has clear effects on the dynamic soil response. Both time and frequency domain results suggest that the dynamic response of the soils beyond the open trenches could be significantly affected, due to the existence of the open trench. According to the frequency domain analysis, the inclusion of open trenches could effectively reduce the soil response in a higher frequency range. Due to reflection effects at the boundaries of the trench, an amplification of the soil response in front of the open trench is observed. Parametric study by means of numerical simulations is also performed. The width of the open trench demonstrates negligible effects on the dynamic soil response, whilst the trench depth exhibits a large influence on the trench isolation performance. With an increase in the trench depth, the isolation performance is significantly improved. It is concluded that the open trenches perform well as an isolation barrier, in mitigating the vibration induced by subway trains.


Introduction
e rapid extension and extensive use of trains have enhanced the convenience of public transportation significantly. However, the vibrations that are generated by rail traffic can cause significant problems. ey can cause distress to people who live near the rails, and they can threaten nearby structures that house sensitive machinery. Vibrations could propagate from the tunnels to nearby buildings. is can cause a perceptible vibration as well as reradiated noise which may have a significant impact on the comfort of residents of buildings [1].
Many vibration countermeasures have been developed to reduce the vibration effects from railways. Various types of isolation are discussed in the literature, e.g., open and filled trenches, concrete walls or piles, and flexible gas cushions [2][3][4][5][6][7]. Among these types of isolation, the open trench has been demonstrated as an effective intervention, and it is the most common intervention in practical traffic applications, especially for mitigating vertical vibrations [6].
ere are mainly two reasons that open trenches can be an effective way to mitigate the vibration transmitted from the soil to buildings, i.e., (1) they are one of the lowest cost isolation measures [8], (2) they provide better vibration reduction capacity [9][10][11][12].
In the past, efforts were made to use open trenches to analytically and experimentally solve vibration reduction problems. Closed-form solutions [13,14] were obtained, and model tests for particular cases [15][16][17] were conducted, but they were restricted to simple geometries and idealized problems. To complement the analytical and experimental studies, numerical simulations have been used extensively to investigate the performance of open trenches because they can be used to analyse complicated geometries and conditions. Hence, it is possible to provide design guidelines for practical traffic applications. Twodimensional (2D) Finite Element (FE) models were established by May and Bolt to analyse the influence of open trenches on various types of incident waves [18]. Ahmad and Al-Hussaini conducted 2D boundary element (BE) simulations to study the performance of trench barriers with various geometrical and material parameters [12]. Coupled finite/infinite methods (e.g., [19]), finite difference methods (e.g., [20]), and coupled FE/BE methods (e.g., [8,[21][22][23]), varying from two-dimensional models to 2.5-dimensional cases, have been used by many authors to study the effectiveness of open trenches in reducing vibrations. Different types of the grounds, i.e., homogeneous grounds [24,25] and nonhomogeneous grounds [11,26], have been analysed. More recently, high-speed railways have attracted a lot of attention, and several studies have been conducted to investigate the use of trenches as vibrating attenuating barriers for the mitigation of vibrations [27,28].
Although the effectiveness of open trenches for mitigating train-induced vibrations generally has been recognized, as mentioned above, most published results have concentrated on the ground-borne vibrations induced by trains running on surface railways. However, subway trains have become much more prevalent in urban areas, especially in densely populated cities. Several empirical procedures have been proposed for estimating the level of ground-borne vibration due to subway trains [29][30][31], but their scope is rather limited. Sheng et al. pointed out that the oscillation frequencies induced and transmitted by subway trains via propagation through the ground are in the range of 15-200 Hz [32]. ompson et al. have agreed that subway trains induce higher frequency vibrations at considerably lower amplitudes than trains on surface railways, and this groundborne noise has a greater adverse effect on the sound inside buildings [33]. Field data have indicated that vertical ground vibration is more important than the horizontal ground vibration induced by subway trains [34,35]. Although open trenches are effective at attenuating vibrations that propagate in the vertical direction, few studies have been performed to analyse the effectiveness of open trenches in mitigating the vibrations caused by subway trains.
In our study, both physical model tests and numerical simulations are used to evaluate the effectiveness of open trenches for mitigating the vibrations induced by subway trains. We investigate cases with and without open trenches. We analyse the effectiveness of open trenches for mitigating the dynamic response of the soils surrounding the subway tunnels, in the time and frequency domains. e effects of vibration frequencies on the trench isolation performance are examined. Parametric studies are performed to determine the critical case with best vibration isolation effects, by means of numerical simulations.

Model Tests
Physical model tests have been conducted in this study. Due to the fact that the dynamic force from an underground tunnel can be relatively small, the behavior of soils would usually remain within the linear range [36][37][38][39]. us, an elastic scaling law was used. Elastic scaling laws were determined according to Iai, 1989 [40] and Iai et al., 2005 [41]. ree fundamental scaling factors, i.e., geometry, density, and Young's modulus, were 1/20, 1/1, and 1/30, respectively. Other relative parameters were altered based on the elastic scaling law, as detailed in Table 1.

Test Facilities.
e experiments were performed using a steel model box with effective length, width, and height dimensions of 1.5 m, 0.9 m, and 1.35 m, respectively [42]. Any undesirable reflections of the compressional wave and shear wave from the rigid boundaries of the steel box could reduce the accuracy of the test results. To eliminate boundary effects, an energy absorbing material, i.e., Duxseal (30 mm thick), was installed on the bottom and side walls of the box to absorb incident waves [43][44][45][46]. e test model was mainly composed of a subway tunnel and the surrounding soil layer (Figure 1). e typical subway tunnel was modelled as a prototype in the experimental tests, and the corresponding diameter and lining thickness were 5.7 m and 0.3 m, respectively. e model subway tunnel was supported by a segment lining that had an inner diameter of 270 mm. Each segment was 15 mm thick and 75 mm wide. For the construction of the lining of the segment, we believed that controlling its overall bending moment characteristics was one of the most significant problems, so this aspect was given careful consideration in the design of the lining. We used the staggered joint method in the assembling mode in order to control the design values of the positive and negative moments, the shearing force of the lining, and the shearing forces of the longitudinal bolts in the universal segment lining. Each ring of the segment lining contained three standard blocks, i.e., two abutment blocks and one block to seal the roof. A 9 mm groove was made in which the longitudinal joints were placed to reduce the bending stiffness; hence, the bending characteristics of the model joint were consistent with those of its corresponding prototype [47]. e longitudinal joint was represented by a series of steel sticks with diameters of 4 mm. e transverse shear stiffness was assumed to be infinite, because any sliding between the adjacent segments was ignored due to the few deformations of the structure of the tunnel [42]. Two series of experiments were conducted to examine the isolation performance of the open trench. Figure 2 e subway soil formation prototype consisted of homogeneous soft soils. In this study, a uniform soil layer was modelled. To satisfy the scaling laws of the test, a mixed material of quartz sand, coal ash, river sand, and oil was used 2 Shock and Vibration as tested soils (the corresponding mass ratio is 54 : 27 : 12 : 7, respectively) [42]. e lining segments of the subway tunnel were modelled by a mixture of diatomite, plaster stone, and water (the corresponding mass ratio is 0.4 : 1.0 : 1.8) [48]. Uniaxial compression tests were conducted to determine the mechanical parameters of the model lining and soil. By varying the amount of components of the mixed material, different elastic parameters can be obtained, as shown in Table 2. e material properties of the soils and lining segments for the prototype and the testing model have been presented in Tables 3 and 4, respectively. e soils were poured into the steel box from a constant height at a constant velocity in order to ensure the uniformity of the soil and to control its density [46,49].
To simulate the excitation caused by a subway train, we used an electromagnetic dynamic shaker (type JM-20) in the tests. Figure 3 shows that the shaker was placed vertically at the bottom of the tunnel lining to provide a vertical dynamic excitation at the center of the tunnel invert. In order to apply the dynamic load accurately, the shaker worked in conjunction with a JM-1230 wave generator and a corresponding JM5801 power amplifier. A JM0710-001 washershaped dynamic load cell was used to measure the force from the shaker. Eighteen JM0213 piezoelectric accelerometers were placed at the free surface, and accelerometers were also installed in the interior of the soil layer adjacent to the open bench, to measure the dynamic response of the model. e vertical vibration component is most commonly representative of the vibration field. In addition, the horizontal component could be hard to measure due to the limitations of the instruments and technologies. Due to the two reasons, the vertical vibration has been measured and recorded in the Shock and Vibration 3 present experimental tests. Figure 1 shows that testing points A10-A13 were located in front of the open trench, while the other testing points were located beyond the open trench.

Test
Performance. Two series of experiments were performed, one with the open trench and one without. ree kinds of dynamic forces were applied on the tunnel invert during each test, i.e., harmonic loading, sweep loading, and train-induced vibration loading. During the experiment, firstly, three types of vibration signals were generated in the JM-1230 type wave generator and then passed to the corresponding JM5801 power amplifier. en, the amplified vibration signal was sent to the JM-20 electromagnetic shaker to vibrate the model subway tunnel. In order to record the data accurately, a sampling frequency of 8000 Hz was chosen, approximately 10 times to the maximum frequency component of the measured signal.
At the first instance, the harmonic loading was applied so that the results of the test could be compared directly with the numerical results in order to validate the numerical formulation and solution procedures. Figure 4(a) shows an example of the harmonic signal in which the fixed frequency was 200 Hz (prototype scale). In order to study the isolation performance of the open trench at various vibration frequencies, we used a sweep sinusoidal frequency with a period of 5 seconds (Figure 4     results [50,51], we established a three-dimensional model that coupled the vehicle and the track. e vibration load caused by irregularities in the track emphatically was considered. e train load of two track spectra (level 5 and level 6) and three speeds (40,50, and 60 km/h) was obtained. In this paper, we used the train load of the level 5 track spectrum, the speed limit of which was 60 km/h.

Numerical Simulations
e finite difference software Fast Lagrangian Analysis of Continua (FLAC 3D ) was used to perform the numerical simulations and to compare with the model tests. e threedimensional FLAC 3D model was designed to replicate, as closely as possible, the scale of the dimensions of the prototype used in the model tests. A rectangle was used to model the soil medium, and a horizontal cylinder was buried in it to model the tunnel, as shown in Figure 5. e length, width, and height dimensions of the rectangle were 30 m, 18 m, and 27 m, respectively. e open trench was also modeled by a rectangle that was buried in the soil medium next to the subway tunnel.
ere was a distance of 9 m between the centerline of the trench and the vertical centerline of the tunnel. e tunnel in the numerical model was designed to coincide with the dimensions of the prototype experimental tunnel, with an outer diameter of 6.0 m, a lining thickness of 0.3 m, and a length of 18 m. e size of the elements in the mesh of the model was less than 1/10 to 1/8 of the corresponding wavelength of the vibration, with 2,000,000 meshed elements [52]. Quiet boundaries [53], which were used to absorb incident waves at the boundaries of the model, simulated an infinite medium. e quiet-boundary scheme proposed by Lysmer and Kuhlemeyer [54] involved dashpots attached independently to the boundary in the normal and shear directions. e dashpots provided viscous normal and shear tractions given by where v n and v s are the normal and shear components of the velocity at the boundary; ρ is the mass density; and C P and C s are the pressure and shear wave velocities. e fixed boundary (at the bottom) represented the restraint of the displacements in all three coordinate directions (x, y, and z). A model made of an elastic material was used because the deformation induced by the train was relatively small [55,56].
A harmonic load at a single frequency was applied at the tunnel invert. By varying the frequency of the harmonic load (from 0 to 200 Hz), we measured the vertical acceleration of the soil surrounding the tunnel with and without an open trench.
Rayleigh damping was taken as the damping model in this study, and the equation of motion is given as where M and K denote the mass matrix and the stiffness matrix, respectively, α and β denote the mass damping coefficient and the stiffness damping coefficient, respectively. Two parameters were used to define Rayleigh damping in FLAC 3D , i.e., the center frequency and the fraction of critical damping. e center frequency was set to be consistent with the vibration frequency that existed at the tunnel invert. e fractions of critical damping were set as 0.1 and 0.05 for the lining of the tunnel and soil medium, respectively.
A harmonic load at a single frequency with an amplitude of 1 N was applied at the centerline of the numerical tunnel invert (x � 0 m, y � 9 m, z � 0.3 m).
e amplitudes of the model's responses at the imposed loading frequency were recorded after the model reached the steady state. By varying the frequency of the harmonic load (i.e., from 0 to 200 Hz), the vertical dynamic responses of the surrounding soil, for cases with or without an open trench, were calculated at the same testing points that were used in the tests of the model. In the parametric studies, the train load was applied at the centerline of the tunnel invert. Vertical peak particle acceleration of soil at ground surface was calculated to study the effect of trench dimensions on vibration mitigation effect.

Results and Discussions
e comparison between the experimental and numerical results is performed, for the case of the first instance of harmonic loading, to validate the numerical formulation and the procedures used to obtain the solution. e FRFs and VAPs are calculated and evaluated in order to analyze the effectiveness of the open trench for different sweep frequencies.

Dynamic Response of Soils Undergoing Harmonic
Loading. Figure 6 shows the dynamic responses of the soil for A1, A4, A13, and A14 during the harmonic loading at fixed frequencies of 50 Hz, 100 Hz, and 200 Hz obtained from experimental testing and from numerical simulation. e red curve represents the case with an open trench, and the black curve represents the case without an open trench. Figure 6 shows that the numerical results (dashed lines) and the experimental results (solid lines) are almost identical. Only a small difference occurs at the peak acceleration, which could be because the material parameters that are used as inputs to the numerical model do not exactly represent the model tunnel and the soil.
By comparing the two cases, i.e., with and without an open trench, it is apparent that the open trench provides attenuation for the propagation of vibrations. is attenuation is also observed for the soils beyond the open trench, as shown in Figures 6(a), 6(c)-6(e), 6(g)-6(i), 6(k), and 6(l). For example, at the fixed frequency of 200 Hz, the amplitude of the attenuation of A14 is 68%. is result demonstrates that the open trench efficiently isolates the vibration isolation, and Woods [15] has concluded that a reduction of 0.25 should be considered as "effective." e reason for this observation is that the open trench can interrupt the path in the soil along which the vibration is propagating, so it can effectively reduce the dynamic response of the soil behind the trench. With respect to the soils in front of the open trench (measurement point A13), the dynamic responses of these soils are amplified, with an increase of 24% in at the fixed frequency of 200 Hz. e amplification of the vibration in the soil occurs because reflection waves are generated at the boundaries of the trench, and these waves can propagate back to the ground, resulting in increasing the response of the soil.

Dynamic Response of Soils to Sweep Loading.
e measured time domain data are transferred to the frequency 6 Shock and Vibration where ω is the frequency, S FF (ω) is the autospectrum of the force, S AA (ω) is the autospectrum of the response in acceleration, S FA (ω) is the cross spectrum of the force and acceleration response, and S AF (ω) is the cross spectrum of the acceleration response and force [41,57,58]. Figure 7 shows the experimental results of values of the coherence function for testing points A1, A4, A7, A10, A13, and A14 without the open trench, and it shows that the values of the coherence function are close to 1 when the excitation signal is larger than 50 Hz. A coherence value of 1 indicates that the measured response is 100% due to the measured excitation. However, when the excitation signal ranges from 0 Hz to 50 Hz, the values of the coherence function generally are less than 0.8. As mentioned earlier, this may due to the e ects of unwanted noise, which are proportionately higher when the magnitude of the measured signal is low. Hence, only numerical simulations are used to investigate the vertical dynamic response of soils when the measured signal is low (i.e., less than 50 Hz).  Figures 8 and 9 show that, generally, a reasonably good match of the experimental and numerical results can be obtained. e average di erence is within 5 dB, which may due to the e ect of unwanted noise (mainly due to the driving system, associated on-board electronics, and heavy machinery located in the laboratory). When the cases with and without the open trench are compared, the FRFs of the vertical dynamic soil responses have similar trends. In the rst stage, i.e., before 60 Hz, the FRFs of the three test points (i.e., A14, A16, and A18) increase rapidly as the frequency increases. As 60 Hz is approached, the FRFs decrease as the frequency increases, but there are a few uctuations. With respect to measuring points A10 and A13, the FRFs increase as the excitation signal increases before reaching the magnitude of 40 Hz. Measuring point A13 is closer to the open trench, and the FRFs obtained from this measuring point uctuates, i.e., they increase initially, and then they decrease. However, for testing point A10, which is somewhat far away from the open trench, the FRFs increase slowly at rst and then decrease. e results obtained at testing point A14 indicate that the average di erence between the cases with and without the open trench is only 1.6 dB when the frequency ranges from 5 Hz to 50 Hz. However, the average di erence increases to 7.8 dB when the period of the amplitude increases to the range of 150-200 Hz. is is because the velocities of the elastic waves (shear wave, compressional wave, and surface wave) in the soil are constant. e higher excitation frequency causes a shorter wavelength. It is more di cult for the short-wavelength waves to go over the isolation trench. erefore, the higher the frequency of the excitation load is, the better the isolation performance becomes. In addition, the e ectiveness of the open trench decreases gradually as its distance from the vibration source increases. An average di erence of 5.9 dB is obtained at testing  Figure 10 shows the vertical acceleration responses that are obtained from the surface testing points, and they are characterised by a cycle e ect that corresponds to the wheel-set of the subway trains. When the wheel-set passes by the testing points, the vertical acceleration responses increase rapidly. However, when the wheel-set moves away from the measuring points, the vertical acceleration responses exhibit a transient reduction.  Table 5 provides the details. e peak vertical acceleration of the particles of soil on the surface of the ground due to the loads of the trains is calculated. In this paper, the insertion loss is used to examine the results in order to gain better insight concerning the test data. e insertion loss is calculated by the following formula: where IL is the insertion loss, a is the peak vertical acceleration of the particles at the surface of the soil with an open trench, a 0 is the peak vertical acceleration of the particles at the surface of the soil without an open trench. Figure 11 shows the insertion loss of the response of the soil at the surface of the ground. Figure 11(a) shows that an open trench can e ectively reduce the response of the soil behind the trenches. However, the width of open trenches has a relatively small e ect on the isolation performance of an open trench. As the width of the trench is increased, the response of the soil is reduced slightly.
e average reductions in the amplitudes of the soil's dynamic response  behind the trenches with widths of 0.5, 1.0, and 2.0 m are 3.6, 3.7, and 4.2 dB, respectively. e phenomenon of amplifying the response of the soil in front of the trench is not a ected by increases in the width of the trench. Figure 11(b) shows that depth of the trench is a critical parameter for the trench isolation performance. e depth of the trench shows relatively small e ects on the dynamic response of the soil in front of an open trench. However, the depth of the trench has a signi cant impact on its isolation performance. e dynamic response of the soil behind the trench is a ected mostly by the depth of the trench. With an increase in the trench depth, a clear improvement of isolation performance is observed. At three di erent trench depths, i.e., 6, 12, and 19 m, the reductions of the soil response behind the trench are 1.3, 3.2, and 4.2 dB, respectively. ese results suggest that a deep trench should be used to obtain a better reduction of the vibration. Figure 11(c) shows the e ects of the location of the trench on the isolation performance. e ampli cation of the response of the soil in front of the trench is more obvious, when the trench is closer to the source of the vibration. As the distance from the trench to tunnel is decreased, the peak response in front of the trenches  Type of isolation trench  1  Trench width  18  0.5, 1, 2  9  Open trench  2  Trench depth  6, 12, 18  2  9  Open trench  3  Trench position  18  Shock and Vibration 13 increases slightly. A better reduction of vibration occurs when the trench is closer to the measurement points. For example, at the measurement point A17 (16 m away from the centre of the tunnel), the decreases of soil responses for three different locations of the trenches (i.e., 5, 9, and 13 m) are 2.7, 4.1, and 5.5 dB, respectively.

Concluding Remarks
Both physical modelling and numerical simulations were performed to analyse the isolation performance of open trenches on the subway train-induced vibrations. e conclusions can be drawn as below: (iii) A parametric study was performed in terms of numerical simulations. e effects of depths, widths, and locations of trenches on the isolation performance were determined. e results showed that the width of the trench demonstrated negligible effects on the dynamic response of the soil on the surface of the ground. However, increasing the depth of open trenches could significantly improve their isolation performances. In addition, a better reduction of the vibration was observed when the trench was closer to the measurement points.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.