Abstract
The metro tunnel lines built in a soft soil area may suffer from tunnel differential settlement due to the high compressibility of soft soil, the engineering constructions nearby tunnel lines, and the cyclic load of metro trains. In this paper, a dynamic coupling model for a metro train-monolithic bed track system under tunnel differential settlement is established. A cosine function is introduced to simulate a real settlement curve measured from a metro tunnel in southern China, and the vibration performance of the train-track system under tunnel settlements is investigated in both the time domain and frequency domain. Based on the standards for the train safety and passengers’ comfort, the speed limit for the metro train traveling on a monolithic bed track with different settlement distributions are concluded. The present research could be useful for the operation and maintenance of metro tunnels in soft soil areas.
1. Introduction
In recent years, the urban rail transit systems have been developed rapidly, especially in cities with high population density. The metro tunnels are important parts of the urban rail transit system, which can greatly alleviate urban traffic pressure and promote public transportation [1]. However, it also brings new challenges to urban residents, such as environmental vibration or noise problems [2, 3]. In addition, for the metro tunnels constructed in thick soft deposit areas (such as the cities in southern China) [4–6], it may suffer from tunnel differential settlement due to the high compressibility of soft soils, the differences in soil properties, engineering constructions near the tunnels [7, 8], and the cyclic load of metro trains [9]. The differential settlement of tunnels may lead to the deformation of the metro track, which can aggravate the train-track vibration and cause the damage of tunnel structures.
The problems related to differential settlement of the railway system have attracted wide attention of researchers. Frohling et al. [10] provided a simulation procedure for the dynamic analysis of the train-track system, which can predict the current and future performance of the vehicle/track system, including the track settlement. Based on the model test, finite element method, and principal stress axis rotation test, Momoya et al. [11] studied the settlement development mechanism of subgrade under train load. Abadi et al. [12] introduced several empirical ballast settlement models of ballasted tracks and evaluated the models with the laboratory test data.
The aforementioned researches are mainly focused on the ground railway system. For the differential settlement of metro tunnels, Huang et al. [13] investigated the train-induced vibration and the long-term settlement of a metro tunnel in saturated soft clay by using the finite element method. The differences between the 2D and 3D soil-water coupling analyses were studied, and the tunnel settlement was predicted by the 2D model. Jiang et al. [14] established a theoretical model for the metro train-track system, in which a quarter of a train vehicle was modelled as a rigid mass-spring-damping system and the rail-track system was simulated as two Euler beams supported by distributed spring-dashpot elements. The tunnel differential settlement was simulated by a cosine function, and the reaction forces of fasteners and the vertical displacement of the soil were calculated under different cases.
In this paper, a dynamic coupling model is established for a metro train-monolithic bed track system. A differential settlement curve measured from a metro tunnel in southern China is considered, and the effect of tunnel differential settlement on the dynamic performance of train-monolithic bed track system is studied. According to the standards of the train safety and passengers’ comfort, the speed limits for a metro train traveling on monolithic bed track under tunnel differential settlement are also suggested.
2. Metro Train-Track Model and Solution
2.1. Theoretical Model
Consider a metro train-monolithic bed track system, as shown in Figure 1. The train is composed of several cars, each of which consists of one carriage, two bogies, and four wheelsets. The carriage, bogies, and wheelsets are connected with each other by primary suspensions k1 and c1 and secondary suspensions k2 and c2, respectively. A total of 2 degrees of freedom are assigned to the carriage and also to each bogie to consider the vertical motion and the rotational motion ψ. For each wheelset, we consider only the vertical motion . In this way, each train car is simulated by a rigid mass-spring-damping system with a total of 10 degrees of freedom [15–17].
Based on the Euler–Bernoulli beam theory, the rail is simplified as a simply supported beam with finite length lr, bending stiffness ErIr, and linear mass density . The fasteners are simulated as discrete spring-dashpots with distance , stiffness , and damping coefficient . Since the monolithic track bed is integrated with the shield tunnel, a simply supported Timoshenko beam is used to model the monolithic bed (together with the tunnel lining), with bending stiffness EhIh, shear stiffness κAhGh, and linear mass density ρhAh. The soil surrounding the metro tunnel is modelled as uniform distributed spring-dashpots with the stiffness coefficient and damping coefficient .
2.2. Governing Equations and Solutions
2.2.1. Equations for Metro Train and Rail
Based on the D’Alembert’s principle, the dynamic equilibrium equation of the metro train is established as follows:where M, C, and K are the mass matrix, damping matrix, and stiffness matrix of the train, respectively; , , and are the displacement vector, velocity vector, and acceleration vector of the train, respectively; and F is the external force vector of the train.
The equation of the rail can be written as follows:where is the bending stiffness of the rail; is the linear mass density of the rail; is the vertical displacement of the rail; is the external force applied on the rail; is the position of the ith wheelset at time t, and is the wheel-rail contact force for the ith wheelset of the ath car; is the position of the jth fastener, and is the corresponding fastener force of the jth fastener; and and are the numbers of cars and fasteners, respectively. The expressions for the wheel-rail contact force and the fastener reaction force are respectively given byin which is the wheel-rail contact stiffness; is the location of the ith wheelset of the ath car at arbitrary time t; is the corresponding vertical displacement of the ith wheelset of the ath car; is the vertical displacement of the monolithic bed; is the length of the rail; and and are the stiffness coefficient and damping coefficient of rail fasteners, respectively.
2.2.2. Equations for Monolithic Bed-Tunnel Lining System
The metro monolithic bed is usually connected integrally with the tunnel lining. In this paper, we use a simply supported Timoshenko beam to model the monolithic bed and tunnel lining. The governing equation of the Timoshenko beam can be given bywhere and are the bending stiffness and shear stiffness of the monolithic track bed-tunnel lining system, respectively; and are the linear mass density and rotational inertia of the monolithic track bed-tunnel lining system, respectively; and and are the vertical displacement and rotational angle, respectively.
The external force applied on the monolithic bed-tunnel lining system is written as follows:where and are the stiffness coefficient and damping coefficient of the soil foundation, respectively, and is the vertical velocity of the track bed-lining system.
2.2.3. Solutions of Equations
As mentioned above, the rail is modelled as a simply supported Euler–Bernoulli beam. According to the modal superposition method, the vertical displacement of the rail can be assumed asin which is the uth modal function of the rail displacement; is the uth generalized function of the rail displacement; and is the mode number adopted in the calculation.
The vertical displacement of the monolithic track bed-lining system can be similarly obtained bywhere and are the uth modal functions of vertical displacement and rotational angle of the track bed-lining system and is the generalized function and is the total number of the mode functions of the track bed-lining system.
By substituting equations (8) into (2), and then applying the orthogonality of the modal functions, we can obtain the ordinary differential equation of the rail as follows:
The ordinary equations for the monolithic track bed-lining system can be similarly obtained by substituting equation (9) into equation (6).
Combining equations (10) and (11) with equation (1), the global governing equations for the metro train-monolithic bed track system can be derived. By virtue of the Newmark-β method, the dynamic responses of the metro train and the monolithic bed track system can finally be calculated.
3. Metro Train-Track System under Tunnel Differential Settlement
The metro tunnels built in a soft clay area usually suffer from long-term settlement and deformation-related groundwater infiltration. Significant differential settlement of metro tunnels has been found in cities such as Shanghai and Nanjing in southeastern China [13, 14]. The tunnel differential settlement can increase the vibration of the metro train-track system and damage the tunnel structures. In order to investigate the influence of tunnel differential settlement on the vibration performance of the metro train-monolithic bed track system, a theoretical model for tunnel differential settlement is introduced, as shown in Figure 2. The following assumptions are adopted for the tunnel differential settlement model [14]: (1) the deformation of the rail is consistent with the tunnel differential settlement; (2) all the supporting elements/layers (including fasteners and soil) remain in good contact with structures; (3) the rail deformation caused by the tunnel settlement is regarded as an initial deformation of the rail, which is similar to the track irregularity.
According to the assumptions described above, the initial track deformation considering both the tunnel differential settlement and random track irregularity is written aswhere is the random track irregularity related to the track quality which is referenced from the US Class 6 railways [18] and is the track deformation caused by the tunnel differential settlement.
4. Numerical Results and Discussion
First, a simple numerical example for a metro train-monolithic bed track system without tunnel settlement is studied to validate the numerical methods proposed in this paper. The vibration responses of the metro train-track system are calculated and compared with those obtained by Wei et al. [19]. Here, a metro train grouped with two cars is considered, and the train speed is assumed to be 20 m/s. The random track irregularity of the US Class 5 railways is adopted, and the train-track parameters used for the numerical validation are the same as those in [19], as listed in Table 1.
The rail acceleration and the wheel-rail contact force obtained by the present method are shown in Figure 3. It is concluded that the present results are generally consistent with those in [19], but there are still some numerical differences between them, as shown in Table 2. The main reason for the numerical differences is that some of the calculation parameters were not mentioned in [19], e.g., the bending stiffness and shear stiffness of the tunnel, the distance between fasteners, and the equivalent stiffness of the soil. Therefore, numerical deviations may exist between these two results.
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4.1. Parameters of Train-Track System with Tunnel Settlement
Figure 4(a) shows the measured differential settlement curve for a metro tunnel with monolithic bed track in southern China, and Figure 4(b) shows the fitting curve of that in Figure 4(a) which can be represented by a polynomial function. This metro tunnel is constructed by the shield method and is buried in soft deposit with high compressibility, low permeability, and low shear strength. The settlement of the tunnel was observed during the construction of the tunnel, and it became severe due to the cyclic dynamic loading of the metro train. So far, the maximum differential settlement of the tunnel has exceeded 100 mm, and it is still developing according to the monitoring records.
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The material and geometric parameters of the metro train-monolithic bed track system used in the numerical examples are mainly referenced from this real monolithic bed track tunnel in southern China, and the detail values are listed in Table 3. The cross-section of the metro tunnel is circular, with an outer diameter of 6.2 m and the thickness of 0.35 m. The tunnel lining is precast with C50 concrete and its modulus of elasticity is taken as 34.4 GPa. The B-type metro train is grouped with two cars, and the train speed is assumed to be 72 km/h. The length of the rail is taken as , which is the same as the length of the monolithic bed. The type of fasteners is DTVI2-1 with a spacing of 0.625 m, i.e., the rail and monolithic track bed are connected to each other by 520 fasteners.
For the numerical analysis in this paper, the tunnel differential settlement from section 585.901 m to section 790.901 m in Figure 4(a) is considered and the corresponding fitting curve is shown in Figure 4(b). The initial time t = 0 is taken as the moment when the first wheelset of the train arrives at the position to reduce the boundary influence on the calculation results.
4.2. Time-Domain Analysis
The vibration responses of the metro train-monolithic bed track system under tunnel differential settlement are calculated. Here, the random track irregularity is considered based on the power spectral density of the US Class 6 railways. As shown in Figure 5, the vertical acceleration of the first carriage increases obviously from 0.109 m/s2 to 0.14 m/s2 at t = 1.5 s (the moment when the first wheelset enters the differential settlement section). The largest change occurs around t = 2.5 s due to the complex effects combined with the differential settlement and random irregularity of the track. After the train leaves the settlement section at t = 13.4 s, the effect of tunnel settlement on the carriage acceleration will gradually vanish, as shown in Figure 5.
It is also found that the carriage acceleration is sensitive to the local curvature of the tunnel settlement curve. The variation of the carriage acceleration is relatively small when the settlement curve changes smoothly. However, the carriage acceleration varies significantly even for a slight irregularity in the settlement curve, as shown in Figure 5, at t = 10.2 s (corresponding to the position of lr = 249 m in Figure 4(b)). In this case, the passengers’ comfort will be adversely affected.
Figure 6 shows the wheel-rail contact force of the first wheelset. Similar to the carriage acceleration, the tunnel settlement mainly affects the wheel-rail contact force during the period from 1.5 s to 13.4 s. When the first wheelset of the train enters the settlement section at t = 1.5 s, the contact force decreases significantly from 77 kN to 49 kN. The minimum wheel-track contact force is 41.94 kN at t = 2.5 s, which may lead to derailment and is dangerous for subway operation.
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The vertical displacement and acceleration at the midpoint of the rail are also calculated. As shown in Figures 7 and 8, the tunnel differential settlement has little effect on the vertical displacement of the rail but has significant influence on the rail acceleration. For three specific moments in Figure 8, i.e., point A (the moment when the first wheelset arrives at the midpoint of the rail), point B (the moment when the midpoint of the train arrives at the midpoint of the rail), and point C (the moment when the last wheelset arrives at the midpoint of the rail), the corresponding accelerations increase from 252 m/s2, 299 m/s2, and 357 m/s2 to 388 m/s2, 349 m/s2, and 456 m/s2, respectively. Among them, the maximum increase rate of the rail acceleration is 54%.
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4.3. Frequency-Domain Analysis
In order to further study the dynamic characteristics of the train-monolithic bed track system under tunnel differential settlement, the vibration of the train-track system in the frequency domain is also analyzed [20, 21]. Figure 9 shows the frequency spectrum of the wheel-rail contact force. It is found that the tunnel differential settlement has obvious effect on the high-frequency part of the wheel-rail contact force. In detail, the frequency amplitude between 300 Hz and 400 Hz increases from 210 N to 450 N, while the frequency amplitude between 600 Hz and 700 Hz increases from 405 N to 1180 N, with a maximum increase rate of 191%.
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A similar phenomenon can be observed in the 1/3 octave band of the wheel-rail contact force, as shown in Figure 10. It can be found that the amplitude increases by 38.8% between 250 Hz and 800 Hz. It is concluded that the tunnel differential settlement can amplify the dynamic reaction between the train wheel and the rail, thereby increasing the damage/loss of the wheel and the rail track and reducing their service life.
Figure 11 shows the 1/3 octave bands of the rail acceleration and lining acceleration under different conditions. It can be seen from Figure 11 that the tunnel differential settlement can also increase vibration levels of the rail acceleration and lining acceleration in the high-frequency region but has little effect on the low-frequency region. As shown in Figures 11(a) and 11(b), the rail acceleration increases by 2.569 dB at 500 Hz, and the vibration level of the lining acceleration increases by 10.6% at 630 Hz. It is noted that the increase of vibration level in the high-frequency band can intensify environmental noises and may have adverse effect on the lives of residents near the metro lines.
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For the dynamic analysis of the train-track system under different settlement conditions, a trigonometric function is usually adopted for the theoretical modelling of the tunnel settlement [12]. In order to verify the trigonometric model and extend the present research, this paper compares the numerical results obtained by the trigonometric model with those obtained by the real tunnel settlement curve. Here, we assume a cosine function expressed as follows:where is the depth of the tunnel settlement and is the distribution length of the settlement, as shown in Figure 2 and is the position where the tunnel settlement starts.
Figure 12 shows the comparison of the theoretical and practical tunnel settlement curves. The dynamic responses of the metro train-track system due to the theoretical and real tunnel settlement curves are studied both in the time and frequency domains. It is found that the vibration difference due to the two settlement curves is small. Figure 13 shows the 1/3 octave bands of wheel-rail contact force under the theoretical and practical tunnel settlement curves. It is found that the difference in the frequency bands of wheel-rail contact force is less than 5%. Therefore, it is confirmed that the cosine function can be used as an approximation of the practical tunnel settlement curve. This allows for further theoretical analysis to investigate the influence of different tunnel settlements on the dynamic characteristics of the train-track system.
4.4. Effects of Tunnel Settlement Waveform on Train-Track System
As mentioned in the previous section, the practical tunnel settlement curve used in this paper can be approximated by a cosine function. Therefore, we can use the cosine model to theoretically study the dynamic responses of the metro train-track system due to various tunnel differential settlements. In this section, five different cases are discussed with the tunnel settlement length assumed as 10 m, 50 m, 100 m, 150 m and 200 m, respectively. In each case, the maximum carriage acceleration and the minimum wheel-rail contact force are discussed due to different settlement depths, as shown in Figures 14 and 15. The speed of the metro train is still assumed as 72 km/h.
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As shown in Figure 14(a), when the settlement length is relatively small, the maximum carriage acceleration increases linearly with the increase of the tunnel settlement depth. However, for the settlement length larger than 150 m, no obvious changes can be observed. The minimum wheel-rail contact force decreases significantly with increase of settlement depth, as shown in Figure 14(b). For the settlement depth-to-length ratio () larger than , the contact force becomes zero and the train derailment may occur in this case. Therefore, the tunnel settlement with a large depth-to-length ratio has adverse influence on the vibration of the metro train-track system, which may threaten the safety of subway operation.
The effects of train speed on the carriage acceleration and wheel-rail contact force are also studied, as shown in Figure 15. Here, the wavelength of the tunnel settlement is taken as . It can be observed from Figure 15(a) that the maximum carriage acceleration increases obviously with the increase of train speed. It is also found that the maximum carriage acceleration is very sensitive to the settlement depth for the train traveling at a higher speed. However, the effect of the settlement depth on the maximum carriage acceleration can be negligible when the train speed is lower than 20 m/s. The minimum wheel-rail contact force due to different train speeds is shown in Figure 15(b). With the increase of train speed and settlement depth, the wheel-rail contact force decreases significantly, or even becomes zero, which is dangerous for the train operation.
According to the Code for Design of Metro in China (GB50157-2013) [22], the train safety and passengers’ comfort indicators should satisfy the following requirements:where is the carriage acceleration; is the gravity acceleration taken as 9.8 m/s2; and is the reduction of wheel load and is the average wheel load (usually taken as the static wheel weight).
The vibration responses of the train-monolithic bed track system under different settlement distributions and different train speeds are calculated here. Based on equation (14), we can finally conclude the speed limit for the metro train traveling on the monolithic bed track with different settlement distributions, as listed in Table 4.
It can be seen from Table 4 that the speed limit of the train decreases correspondingly with the increase depth of the tunnel settlement. It should be noted that the train speed limit shown in Table 4 is only suitable for the metro train-monolithic bed track system discussed in this paper. Actually, the speed limit of the metro train is decided by many factors, including the material and geometric parameters of the train-track system, the soil parameters, and the practical distribution curve of the tunnel differential settlement. However, the methods presented in this paper can be extended easily to other cases and the train speed limit calculated due to different cases can be a theoretical reference for the safety operation of the metro system.
5. Conclusions
In summary, the vibration responses of the metro train-monolithic bed track system under tunnel differential settlement are analyzed based on a two-dimensional dynamic model. The effects of a practical tunnel differential settlement on the dynamic responses of the train-track system are studied. According to the curvature of the practical settlement curve, a cosine function is introduced for modelling the practical tunnel settlement. Based on this cosine model, the vibration responses of the train-track system under different tunnel settlement distributions are further investigated, and the speed limit of a metro train is obtained based on the requirements for the train safety and passengers’ comfort.(1)The tunnel differential settlement has a great influence on the carriage acceleration, the wheel-rail contact force, and the rail acceleration but has little effect on the rail displacement.(2)The tunnel differential settlement can amplify the dynamic responses of accelerations and forces in the high-frequency range, which may enhance the environment noise.(3)The dynamic responses of the metro train-track system will increase significantly with the increase of the train speed and settlement depth but will decrease with the increasing settlement length.(4)According to the train safety and passengers’ comfort, the speed limit of a metro train can be obtained under different settlement distributions, which can provide a reference for subway operation.
It should be noted that the dynamic response of the metro train-track system is highly dependent on the material and geometric parameters of the train-track system. Therefore, the quantitative results obtained in this paper are mainly suitable within the range of parameters considered in this paper. However, the method used in this paper can be easily extended for vibration analysis of other train-track structures.
Data Availability
All the underlying data related to this article are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest with respect to the research, authorship, and/or publication of this article.
Acknowledgments
This work was supported by the Zhejiang Provincial Natural Science Foundation of China (Grant no. LY17E080005), the National Natural Science Foundation of China (Grant no. 51778576), and the Hangzhou Science and Technology Plan Project (Grant nos. 20191203B40 and 20172016A06).