Coupling Train-Track Models with the Discrete Element Method for a More Realistic Simulation of Ballasted Track Dynamic Behavior

Abstract

Train-track interaction models are widely used to simulate the dynamic responses of train and track components. In a conventional train-track analytical model, the ballast layer is often simplified as mass blocks, interacting with other components through a spring and dashpot system. Such an idealization ignores the particle-level information, that is interaction of different sized and shaped aggregates and related degradation characteristics linked to fouling behavior of the ballast layer. On the other hand, realistic ballast models based on the discrete element method (DEM) can capture the particle-level information but require predefined external loading patterns as inputs to mimic the train passages. To overcome such drawbacks of train-track and DEM models, this paper proposes to couple the two calibrated models together to build a more realistic ballasted track model for predicting dynamic responses of the train and track. The coupled model was first validated with detailed field data collected from Amtrak’s Northeast Corridor. The validated model was then used to study the effects of crosstie spacing realizing that a smaller crosstie spacing than regular often results in a higher construction cost. Increasing crosstie spacing, however, was found to result in larger track displacements, crosstie accelerations and reaction forces, particle accelerations, and local average normal contact forces. Therefore, vibration patterns with a smaller crosstie spacing were more stable. Such observations suggest that crosstie spacing plays an essential role in controlling track dynamic responses, and an optimum crosstie spacing could be determined by using the newly introduced coupled model.

Keywords

railroad ballast crossties track modeling

Railroad is one of the most economical and energy-efficient transportation modes for both freight and passenger traffic all over the world. In general, there are two types of railroad track structure: ballasted and ballastless. In the United States, ballasted track is dominant for both freight and passenger routes because of its lower construction cost and more readily accessible maintenance activities. According to the literature, most track defects and failures are caused by dynamic load-related issues such as broken rails in the US ( 1 ). Therefore, a better understanding of track dynamics with calibrated, realistic, and well-functioning simulation models requires mechanistic interactions among various track components to be properly identified to ensure a safe, sustainable, and reliable railroad infrastructure.

Plenty of interaction numerical models have been developed to predict the dynamic responses of ballasted track. Zhai and Cai integrated train, wheelset, and track into one model ( 2 ). Based on the hypothesis that load was transmitted within a cone region in the ballast layer, Zhai et al. proposed a ballast modeling method and validated it with field experiments ( 3 ). A “sandwich” model was used by Huang et al. to simulate the track behavior when the thickness of the asphalt track bed varied ( 4 ). Hou et al. developed a nonlinear support train-track model to simulate the hanging crosstie situations which are commonly caused by nonuniform support in transition zones such as bridge approaches ( 5 ). In addition to the interaction models, some models based on the finite element method (FEM) have emerged in recent decades. Boler et al. constructed an FEM model to simulate the transient displacement of a ballasted track ( 6 ). Long-term track deformations were estimated through a nonlinear FEM model by Varandas et al. ( 7 ). In the conventional train-track models, the ballast layer is typically idealized as mass blocks, which are assumed to interact with other components through springs and dashpots, while in the FEM-based numerical models, the ballast layer is modeled as a continuum medium that is split into several finite sized elements. Even though these numerical models may often predict accurately the macroscale dynamic responses of the track structure, such as wheel–rail interaction force, they commonly lack a realistic representation of the ballast layer, and therefore, the particle-level information is missing. In addition, previous studies have shown the importance of aggregate morphological (size and shape) properties and their effects on ballast performance ( 8 – 10 ). Therefore, improving the numerical models by introducing a realistic representation of ballast layer that can capture the particle-level characteristics is necessary.

In a ballast layer simulation based on the DEM, each ballast aggregate can be modeled as an individual particle, so the aggregate morphology and assembly properties are retained. Guo et al. and Mahmoud et al. generated 2-D particles in their DEM models based on real ballast aggregates ( 11 , 12 ). Moaveni et al. developed an enhanced aggregate image analyzer that could quantify the aggregate morphology properties, based on which some realistic 3-D particles were established in a DEM model to study the effects of ballast particle shape on track lateral stability ( 9 , 13 ). The gradations of ballast can also be represented in DEM models by adjusting the virtual particle size and proportion ( 10 , 14 , 15 ). Previous research efforts of DEM modeling on railroad ballast achieved a realistic ballast representation that overcomes the shortcomings in FEM and train-track models. On the other hand, significant research progress made on crosstie-ballast DEM models in the last decade guarantees that the DEM models can accurately simulate both the response and degradation behavior of ballast layer subjected to dynamic train loadings. A series of studies related to single-crosstie DEM models were carried out at the University of Illinois at Urbana-Champaign (UIUC) to simulate ballast dynamic responses under different conditions ( 9 , 10 , 16 –18). A large-scale DEM model with eight crossties and over 150,000 particles was validated with the full-scale experiments conducted at Zhejiang University’s innovative high-speed rail tester (ZJU-iHSRT) to successfully predict the long-term permanent deformation/settlement behavior in a subsequent study ( 19 ). Different from the train-track interaction model, in which no external loading is required because a virtual train is simulated in the model, most DEM models used for ballast modeling require predefined external loading patterns on a simulated crosstie to mimic the train passages. To obtain realistic track behavior trends caused by train loadings in the DEM models, the external loading patterns need to be identical to those applied in the field.

Model coupling means integrating two or more models together through interaction laws. There have been only a few research studies on coupled models in railroads. To study the load distribution in the track bed, Gao et al. proposed a train-track-soil model by coupling a soil FEM model below a train-track model, and Ngo et al. coupled a subgrade finite difference method (FDM) model below a ballast DEM model ( 20 , 21 ). Interestingly, Shi et al. coupled a crosstie-ballast DEM model with a tamping machine multi-body model to determine optimum tamping procedures in maintenance applications ( 22 ). Further, Guo et al. coupled a train multi-body model with a track DEM-FDM model to investigate the dynamic behavior when ballast was mixed with crumb rubber ( 23 ). Coupling a train-track model and a crosstie-ballast DEM model can combine their strengths and eliminate the drawbacks mentioned above. However, little literature can be found in this area. This paper attempts to fill the research gap by coupling a train-track model with a DEM model for a more realistic simulation of the ballasted track dynamic responses.

Objective and Scope

The objective of this paper is to establish a validated, realistic and well-functioning simulation model by coupling a train-track model with a DEM model to predict ballasted track dynamic responses. To achieve this goal, a 30-crosstie train-track analytical model and a single-crosstie DEM model were coupled through a proposed approach. The coupled model was validated with detailed field data collected from Amtrak’s Northeast Corridor related to certain load-displacement, acceleration, and crosstie reaction force aspects. The validated coupled model is then used to study the effects of crosstie spacing, and both macroscale and mesoscale dynamic responses are investigated.

Coupled Model

Train-Track Model

A train-track-bridge model was developed by Hou et al. to predict railway structure dynamic responses at both open track and near-bridge locations ( 5 ). The train was modeled as a multi-body system in accordance with the actual train configurations of Amtrak’s ACELA Express passenger train (eight cars with 32 axles), and a time step of 0.01 s was sufficient to predict the transient displacement for the operating speed (177 km/h or 110 mph) near the Upland Street bridge approach on the Northeast Corridor in Chester, Pennsylvania. A typical crosstie spacing of 0.61 m (2 ft.) was used. All other train and track parameters including mass, moment of area, stiffness, and damping ratio were calibrated and verified. Since the coupled model in this paper would be validated with the same field measurements as in Hou et al.’s study ( 5 ), the same parameters were used.

The original train-track-bridge model developed by Hou et al. ( 5 ) comprises 1,000 crossties (i.e., 500 crossties for the embankment side and 500 crossties for the bridge side). Because this coupled model only focuses on a ballasted track on the embankment, and a long track span would dramatically increase the computational cost with the DEM simulation, a smaller model with 30 crossties and a total length of 18.3 m was constructed on the embankment side for the coupled model. It was assumed that the ballasted track was fully supported with no hanging crossties in the open track far away from the bridge abutment. Therefore, the nonlinear support stiffness feature developed for hanging crossties in the original analytical model was not implemented here.

Figure 1 compares the vertical displacements predicted in the middle of the open track by both the original model (at the No. 250th crosstie or 152 m [500 ft] away from the bridge approach) and the smaller model (at the No. 15th crosstie or 9 m [30 ft] away from the bridge approach) when an ACELA Express passenger train passes. Note that negative values represent downward movements, while positive values represent upward movements. Figure 1 shows that the displacement trends and magnitudes predicted by the smaller model are identical to those of the original model predictions. This finding corroborates with the research findings of Shi et al. who used a similar 25.3 m train-track model and claimed that the boundary effects could be ignored ( 24 ). The shortened train-track model with 30 crossties could, therefore, behave with similar dynamic trends to the original model for simulating the track behavior on an open track location and thus function properly with much less intense computational time and memory requirements.

Single-Crosstie DEM Simulation Model

Single-crosstie models have been used successfully to simulate ballasted track mechanical and dynamic response behaviors when subjected to train loading ( 10 , 16 –18). A single-crosstie model is in fact ideal for the initial investigation because of the requirement for only limited computational resources. In this study, a polyhedral 3-D DEM algorithm, BLOKS3D, was used to construct a single-crosstie DEM model to provide a realistic representation of crosstie and ballast for the proposed coupled model investigation ( 25 ). Figure 2a visualizes the layout of the DEM simulation developed for the coupled model. A crosstie with dimensions of 2.6 m × 0.29 m × 0.2 m is placed on top of a ballast layer, which is composed of over 10,000 ballast particles. The ballast layer is 5.4 m long, 0.6 m wide, 0.4 m deep in the crib and 0.305 m deep below the crosstie, which is consistent with the field conditions observed at the Upland Street bridge approach on Amtrak’s Northeast Corridor. Ballast shoulders have a slope of 1:1.75 on both sides (see Figure 2a).

Capturing aggregate particle size and shape accurately is essential for realistic simulations of ballast behavior. With the assumption that ballast in the open track area is fresh and clean without degradation, the gradation and shape properties, that is, flat and elongated ratio, surface texture index, and angularity index (AI), of clean granite ballast samples measured from image analysis by Qian et al. ( 26 ), were applied in the DEM model. Figure 2a shows the three typical ballast particles used in the DEM model with AI of 570°, 448°, and 390°. They were mixed in certain percentages (see Figure 2a) to achieve an average AI of 550° as observed from the clean granite ballast samples ( 26 ). Figure 2b shows the grain size distribution used in the DEM model, and the ballast material met American Railway Engineering and Maintenance-of-Way Association (AREMA) No.24 gradation requirements. The ballast particles had a maximum aggregate size of 63.5 mm (2.5 in.), D₅₀, which denotes the particle size equivalent to 50% of materials passing, was 42 mm (1.8 in.), and all materials were greater than 25.4 mm (1 in.). The coefficient of uniformity, C_u, was 1.61, and the coefficient of curvature, C_c, was 1.10.

Table 1 summarizes the main parameters used in the DEM model. The surface friction angle and damping remained the same as those in the previous studies (16 –18, 26 , 27 ). Inter-particle normal and shear stiffness, which dominate the dynamic responses of unbound aggregate assemblies, typically require additional tuning in DEM models ( 28 , 29 ). When tuning the two parameters, a constant ratio of normal stiffness over shear stiffness is usually assumed ( 11 , 30 ). Therefore, the same stiffness ratio, 2:1, obtained from the previously validated crosstie-ballast DEM models using BLOKS3D was applied ( 16 , 17 , 19 ). Some previous studies calibrated the two parameters by matching target assembly stiffness: Coetzee and Els calibrated the inter-particle stiffness by matching the assembly stiffness obtained from laboratory experiments in confined dynamic compression tests ( 30 ); Zhang et al. calibrated the inter-particle stiffness by matching the ballast layer support stiffness ( 31 ). Similarly, normal and shear contact stiffness model parameters in this study were calibrated through a dynamic loading test as described in the following steps: (1) build single-crosstie DEM models with different normal and shear stiffness properties but the ratio of normal stiffness over shear stiffness remains at 2:1; (2) apply a half sinusoidal loading pattern on the crossties, which was calculated from Amtrak’s ACELA Express train configurations by Feng et al. ( 17 ); (3) calculate ballast support stiffness in each scenario by dividing peak force by peak crosstie displacement; (4) pick the normal contact stiffness that is corresponding to the target ballast stiffness. Figure 3 illustrates the calibration results and infers a linear relationship between ballast layer (assembly) support stiffness and particle contact (particle-to-particle interaction) stiffness, interestingly consistent with the observations in Coetzee and Els’s study. To match the ballast support stiffness used in the original train-track-bridge model (120 MN/m as illustrated in the horizontal dashed line in Figure 3), normal and shear contact stiffness properties of 2 MN/m and 1 MN/m, respectively, were used in DEM simulations as highlighted with vertical dashed line in Figure 3.

Table 1.

Discrete Element Method (DEM) Model Parameters Used for the Coupled Model

Model parameter	New ballast
Surface friction angle, θ	31°
Normal contact stiffness, Kn	2 MN/m
Shear contact stiffness, Ks	1 MN/m
Contact damping	0.4

Coupled Model Approach

As visualized in Figure 4, the coupled model consists of a multi-body train model, representing a combination of the smaller 30-crosstie track model and the single-crosstie DEM model, as introduced in the previous sections. The dynamic equations for rail from the Euler–Bernoulli beam assumption are formulated in Equations 1 and 2, and parameter descriptions are summarized in Table 2. The train loading used in the coupled model has the same configuration of the ACELA Express passenger train, which has eight four-axle cars in total (two locomotives at both ends and six passenger cars in the middle). To reduce any boundary effects, the DEM model was coupled at the location of the 15th crosstie, where the mass blocks for crosstie and ballast were replaced with the single-crosstie DEM model. As expressed in Equation 2, the two models are coupled by inputting the same magnitude of force (F_a) at bottom of rail in the train-track model and on top of crosstie in the DEM model, whereas the support forces exerted on the rail beam from uncoupled crossties are calculated based on the rail-crosstie vertical interaction (i.e., relative deflections and velocities of rail and crossties).

Table 2.

Parameter Description for Rail Dynamic Equations

Parameter	Description
E	Young’s modulus of rail
I	Second moment area of rail section
k_p	Rail pad stiffness
c_p	Rail pad damping coefficient
U_r	Vertical deflection of rail
V_r	Vertical vibration velocity of rail
U_ti	Vertical deflection of i-th crosstie
V_ti	Vertical vibration velocity of i-th crosstie
m_r	Rail mass per unit length
F_rsi	Support force acting on rail from i-th crosstie
p_j	Wheel force acting on rail
x	Longitudinal coordinate on rail
x_ti	Longitudinal coordinate of i-th crosstie
x_wj	Longitudinal coordinate of j-th wheel
t	Time
NT	Number of crossties
NW	Number of wheels
CP	Index of coupled crosstie

Figure 5 illustrates the proposed coupling approach. A train passage is divided into equal timesteps, and there could be some iterations in each timestep before achieving the equilibrium on input force (F_a) and simulated rail-crosstie reaction force (F_rt). At the i^th time step, the input force, (F_a)ⁱ, is first initialized with the same value as the last timestep’s, (F_a)^i-1. The corresponding rail displacement, (u_r)ⁱ, and velocity, (v_r)ⁱ, will be derived from the train-track model, while the crosstie displacement, (u_t)ⁱ, and velocity, (v_t)ⁱ, will be derived from the DEM model. Sequentially, the simulated reaction force, (F_rt)ⁱ, can be calculated using a relative displacement and a relative velocity (see the right side of Figure 5 for the equation). To end iterations, the difference between the input force and simulated reaction force needs to be less than a threshold denoted as A. Input force, (F_a)ⁱ, will be updated according to the equation on the left of Figure 5 until such condition is satisfied. The two parameters in the coupling algorithm, threshold and relaxation factor, dominate the convergence efficiency. The threshold determines the tolerated error and thus controls the accuracy of the coupled model; a smaller threshold leads to more iterations but higher accuracy. The relaxation factor controls the step size of error compensation. Theoretically, abundant iterations are required when an extremely small relaxation factor is used, whereas overshooting or overcompensating will also induce redundant iterations when the relaxation factor is too high. A stringent threshold of 100 N (small compared with the wheel load on a scale of 100 kN) was chosen for accurate predictions. To determine the optimum relaxation factor, a grid search from 0.20 to 0.50 was conducted, but there was no significant difference in the convergence efficiency for all tested relaxation factors (all relaxation factors lead to about five iterations on average for each timestep). Note that the coupling accuracy has been guaranteed with a small threshold, so the selection of relaxation factor is not vital and a relaxation factor of 0.35 was used in this study.

\begin{matrix} EI \frac{\partial^{4} U_{r} (x, t)}{\partial x^{4}} + m_{r} \frac{\partial^{2} U_{r} (x, t)}{\partial t^{2}} = \\ - \sum_{i = 1}^{NT} F_{rsi} (t) δ (x - x_{ti}) + \sum_{j = 1}^{NW} p_{j} δ (x - x_{wj}) \end{matrix}

(1)

\begin{matrix} F_{rsi} (t) = \\ {\begin{matrix} F_{a} (t) & i = CP \\ k_{p} [U_{r} (x, t) - U_{ti} (t)] + c_{p} [V_{r} (x, t) - V_{ti} (t)] & i \neq CP \end{matrix} \end{matrix}

(2)

Model Validation

Vertical Displacement

A multi-depth deflectometer (MDD), consisting of five linear variable differential transformers (LVDTs), was instrumented at an open track location, which was 18.3 m (60 ft) away from the Upland Street bridge approaches on Amtrak’s Northeast Corridor ( 32 ). It was assumed that 18.3 m (60 ft) was far enough to neglect the influences of complicated dynamic response trends near the bridge approach, so, the coupled model, which simulated the behavior of an open track with an ideal crosstie support condition, could be compared with the measured data by the MDD. Figure 6 shows the track substructure profile and installation depths of the five LVDTs at layer interfaces and an anchor. Each LVDT could measure individual layer displacement with respect to the LVDT or anchor immediately below it. The anchor at the bottom of the MDD was buried sufficiently deep below the surface, so it was assumed to be fixed under traffic loadings. Design and installation details of MDDs can be found in Mishra et al.’s study ( 32 ).

To validate the coupled model, the total displacement measured by the MDD when the ACELA Express passenger train passed at a speed of 177 km/h (110 mph) was compared with the model predictions at the coupling location (i.e., at 15th crosstie), as shown in Figure 7. Positive values indicate upward movements, while negative values indicate downward movements. A total of 32 peaks, which are identical to the number of axles in an ACELA Express passenger train, are registered in both the field measurements and model predictions in Figure 7. The predicted displacements by the coupled model match well with the field data for trends and magnitudes. Thus, the coupled model can predict track displacements accurately.

Crosstie Acceleration

Figure 8 illustrates an accelerometer installed on a crosstie near the Upland Street bridge approach after an under-tie pad (UTP) was installed to remediate a hanging crosstie problem at near-bridge locations in 2014 ( 33 ). With the assumption that the gap between the crosstie and ballast layer was eliminated and the crosstie was uniformly supported after the UTP was applied, it is reasonable to compare the measured crosstie accelerations with the model prediction, which are only applied to the ideal situation without a hanging crosstie.

The crosstie accelerations collected from the field and simulated by the coupled model during the passage of an ACELA Express passenger train are compared in Figure 9. Figure 9a shows the crosstie accelerations in the time domain below one standard gravitational acceleration, g = 9.8 m/s². Positive values indicate upward accelerations, while negative values indicate downward accelerations. Both the field measured and predicted crosstie accelerations have magnitudes around 0.3g, and the accelerations induced by locomotives at two ends are slightly larger than those caused by the passenger cars since locomotives apply heavier axle loads. Interestingly, the coupled model can also capture the behavior in the field data that upward accelerations are greater than downward accelerations for most cars. One can also note that there are recorded low acceleration values between two axles of the same car from the field data, but they are almost 0 in the model predictions (e.g., 1.7 s to 1.9 s in Figure 9a). One possible reason is that the crosstie in the field continued to vibrate between two axles or this was some signal noise from the accelerometer.

After performing Fourier transform analysis, the crosstie accelerations are decomposed in the frequency domain as shown in Figure 9b. Frequencies related to ACELA Express passenger train axle spacings are highlighted. Detailed calculations about these frequencies can be found in Boler et al.’s study ( 6 ). It can be observed that crosstie accelerations from the field and the model predictions are both dominant at 6 Hz (bogie), 11 Hz, 13 Hz, 16 Hz (passenger car axle), and 18 Hz (locomotive axle). So, the coupled model can accurately simulate crosstie accelerations in both time and frequency domains.

Crosstie Reaction Force

Figure 10 shows the two dual-element strain gauge circuits instrumented on the rail web at an open track location near Upland Street bridge approaches on Amtrak’s Northeast Corridor ( 29 ). The strain gauges were calibrated to correlate the applied vertical load levels with the voltages induced in the strain gauges. The left circuit was installed on top of the crosstie to measure the bending force carried by rail, whereas the right one was installed on the crib to measure the wheel loading. Mishra et al. proposed the crosstie reaction force (i.e., force between rail and tie) could be calculated by subtracting the load levels registered by the circuit on the crosstie from the circuit on the crib, and this method has been further verified with recent field experiments ( 32 , 34 ).

The crosstie reaction forces calculated from the strain gauge measurements and those predicted by the coupled model as a result of an ACELA Express passenger train passage are compared in Figure 11. In total, 32 peaks were registered in both field data and model predictions. One can conclude that the coupled model can accurately predict the trends and the magnitudes of crosstie reaction forces when the six passenger cars passed. However, the magnitudes of crosstie reaction forces for the two locomotives (in the beginning and the end in Figure 11) were underestimated. This could possibly be attributed to the incorrect axle loads, not identical to those subjected by the train loading in the field, which were used as inputs for the coupled model. On the other hand, the strain gauge-based measurement of crosstie reaction was validated with only static loading at the current stage ( 34 ), and dynamic train loading effects, especially caused by higher train speed (i.e., 177 km/h or 110 mph) passes, could introduce noise and possible dynamic amplifications to the strain gauge measurements. No doubt further research efforts are needed to improve the model predictions related to crosstie reaction forces.

Effects of Crosstie Spacing

Coupled Model Set-Ups

The structural functions of crossties are to distribute wheel loads and maintain the track gauge. Using large crosstie spacings can reduce overall construction costs, but could cause a reduction in track stiffness, which is detrimental to track serviceability ( 35 ). A better understanding of the effects of crosstie spacing on track components could provide more insights into determining an optimum crosstie spacing for controlling track dynamic response trends. The coupled model introduced in this paper was, therefore, used to study the effects of crosstie spacing on dynamic responses of track structure.

Concrete crosstie spacings between 0.51 m and 0.76 m (20 in. to 30 in.) are recommended by the AREMA specification. Accordingly, three different crosstie spacings, 0.51 m, 0.61 m, and 0.71 m (20 in., 24 in., and 28 in.) were used to construct three coupled models to study the related implications. All other model parameters were kept the same as described in the previous section. Note that the same single-crosstie DEM model (identical ballast particle assembly) was used in the three coupled models for a fair comparison. A passage of the ACELA Express passenger train traveling at 177 km/h (110 mph) was simulated in each scenario. The validated coupled model was used in this section for these analyses and both macroscale and mesoscale dynamic responses were predicted from simulations.

Macroscale Dynamic Response

Changing tie spacing could lead to changes in support conditions along the track and, therefore, the related displacements, force distributions, and vibrations of track components would vary. To study such effects, the macroscale dynamic responses including track displacement, crosstie reaction force, and crosstie acceleration were first investigated.

Figure 12, a and b , compare peak track displacements and peak crosstie reaction forces, respectively, as induced by the locomotives and passenger cars at different crosstie spacings. As expected, the locomotives generate higher displacements and crosstie reaction forces than passenger cars, and the track obtains greater displacements and crosstie reaction forces when the crosstie spacing increases. Note that the peak displacements and crosstie reaction forces induced by locomotives can be amplified by 28% and 42%, respectively, when the crosstie spacing increases from 0.51 m to 0.71 m (20 in. to 28 in.). Such an observation indicates that excessive crosstie spacings can increase the risks of jumps in the wheel–rail rolling contact and track degradation.

The effects of crosstie spacing on crosstie accelerations during a train passage are demonstrated in Figure 12c. The accelerations are below a standard gravitational acceleration. Positive values represent upward accelerations, while negative values represent downward accelerations. The magnitudes of crosstie accelerations become larger when tie spacing increases. Nevertheless, the crosstie accelerations at the smallest tie spacing (i.e., 0.51 m or 24 in.) acquire the most stable and identical vibration pattern for each car. Irregular and excessive accelerations induced by large crosstie spacings can expedite wear of track components (e.g., fasteners between crosstie and rail) and influence passenger comfort ( 35 ). Therefore, selecting an optimum crosstie spacing is essential for ensuring railway safety and serviceability, and the coupled model could be used to provide guidance on this.

Mesoscale Dynamic Response

The coupled model can also simulate mesoscale track response behavior. Different from the macroscale responses, mesoscale concentrates on individual particle responses and particle-to-particle interactions. In this section, individual ballast particle accelerations and inter-particle forces under different crosstie spacings were investigated.

Over 10,000 ballast particles were simulated in the coupled model, so it is not feasible to inspect every particle. Figure 13a demonstrates the two representative particles traced at mid-depth of the ballast layer: Particle 1 and Particle 2 are below the tie edge and the crosstie center, respectively. The three coupled models with different crosstie spacings share the same ballast particle assembly, so the two representative particles are the exact same among the three coupled models. Figure 13, b and c , illustrates the accelerations of Particle 1 and Particle 2, respectively, under different crosstie spacings. Positive values represent upward accelerations, while negative values represent downward accelerations. Like crosstie accelerations, the magnitudes of Particle 1 and Particle 2 accelerations both increase with increasing crosstie spacings for most train cars, and the magnitudes of accelerations subjected to locomotives are greater than those of the passenger cars. Particle 1 experienced relatively greater accelerations than Particle 2. One can note that Particle 1 obtained more unstable vibration patterns (see Figure 13b), namely irregular excessive peaks, especially when the locomotives passed. This can be explained by Particle 2 at the center of the ballast layer being constrained symmetrically, while Particle 1 right below the crosstie edge seems not to have such an ideal constraint. These observations infer that increasing crosstie spacing could accelerate the ballast degradation, and the particles around crosstie edges might be relatively more severely influenced.

Other than individual particle dynamic responses, the investigations on particle-to-particle interactions help researchers understand how forces are distributed in a discrete medium ( 10 , 17 , 36 , 37 ). Figure 14 shows the local average normal contact force distributions under different crosstie spacings when the axle of the first locomotive was right at top of the observation point (i.e., 15th crosstie). The dashed line at the top serves as a reference. The three 3-D histograms correspond to the distributions with crosstie spacings of 0.71 m, 0.61 m and 0.51 m (28 in., 24 in., and 20 in.) from left to right. The 3-D histograms are viewed along the y-axis, which is parallel to the train’s moving direction. Each bar indicates the average of normal contact force falling within that angle interval. The length and color of a bar are proportional to the magnitude of the local average normal contact force. Detailed explanations of the local average normal contact force can be found in Liu et al.’s study ( 10 ). All three 3-D histograms show anisotropic distributions in which loads primarily concentrate along the vertical direction. The distributions for the three crosstie spacings are similar, because the same ballast layer assembly was used to construct the model and the formed aggregate skeleton, which primarily sustains the external loads, did not vary among the three scenarios. The maximum local average normal contact forces are listed below the 3-D histograms, and they become smaller when crosstie spacings decrease. These findings also address the concern that larger crosstie spacing can expedite ballast degradation and reduce its service life.

Summary and Conclusions

The objective of this study was to develop a realistic coupled model that could be validated to predict track dynamic responses in both macroscale (e.g., for capturing crosstie responses) and mesoscale (ballast particle interactions) caused by the passage of a higher speed passenger train. The coupled model consists of a 30-crosstie train-track analytical model and a single crosstie supported by ballast modeled based on the discrete element method (DEM). The train-track model was constructed based on a previously proposed and fully validated train-track-bridge model, also verified for accuracy for a shorter 30-crosstie domain in this study. The single-crosstie DEM model was also established based on the previous significant research efforts focused on the development of the BLOKS3D DEM model, and additional calibrations were also performed through a dynamic loading test. These well-established train-track and DEM models were successfully coupled in this study via a proposed coupling approach based on force equilibrium.

The coupled model was validated with three types of field data collected from open track near Upland Street bridge approaches on Amtrak’s Northeastern Corridor in Chester, Pennsylvania. Crosstie and ballast displacements were collected from an MDD instrumentation at the open track location. An accelerometer installed on the crosstie registered the crosstie accelerations after an UTP installation on the site. Crosstie reaction forces were calculated using data from the strain gauges welded on the rail web. The predictions by the coupled model successfully matched the field collected data of tie and ballast displacements and accelerations in the time and frequency domain. The coupled model could predict the crosstie reaction forces accurately when subjected to the ACELA Express passenger cars, but it underestimated the reaction forces caused by the locomotive passage.

To reveal the potential of the coupled model, a study on the effects of crosstie spacings on track dynamic responses was conducted both in macroscale and mesoscale. Three coupled models were developed to study crosstie spacings of 0.51 m, 0.61 m, and 0.71 m (20 in., 24 in., and 28 in.). Main study findings and conclusions are summarized below:

The crosstie vertical displacements and reaction forces induced by the locomotives decreased by 28% and 48%, respectively, on decreasing crosstie spacing from 0.71 m to 0.51 m (28 in. to 20 in.). These findings reveal that the crosstie spacing plays a significant role in maintaining a stable wheel–rail interaction and extending the railway service life.

Higher crosstie accelerations and more irregular vibration patterns were observed in the models with larger crosstie spacings. Thus, the wear of track components (e.g., fasteners between crosstie and rail) could be expedited, and passenger comfort would be influenced when the crosstie spacing increased.

Accelerations of two individual particles at mid-depth of the ballast layer (i.e., Particle 1 below the crosstie edge and Particle 2 below the crosstie center) were investigated. Particle 2 obtained smaller accelerations and more stable patterns than Particle 1, which might be a result of the symmetric constraints below the center of crosstie. The highest accelerations were found in the models with the largest crosstie spacing.

Local average normal contact forces were compared among the three coupled models with varying crosstie spacings, that is, 0.51 m, 0.61 m, and 0.71 m (20 in., 24 in., and 28 in.). A similar distribution pattern was observed because the same ballast layer assembly was used. The model with the highest crosstie spacing acquired the highest maximum local average normal contact force, and it brought concerns about potentially much faster ballast degradation.

The coupled model introduces a realistic representation of the ballast layer into the framework of analytical train-track models and brings the possibility of real-time train loadings into the DEM model simulations. It can be used widely to study the dynamic responses of trains and tracks in the future. Note that there are still several limitations in the current model: (1) using a single-crosstie DEM might have boundary effects; and (2) the coupled model is only designed for an ideal support condition without hanging crosstie issues which would require nonlinear support and stiffness conditions. Future research efforts are needed to push the boundary and make further progress on this innovative coupled modeling approach.

Footnotes

Acknowledgements

Field data used in this paper for model validation were collected from a U.S. Federal Railroad Administration (FRA) BAA funding project, Mitigation of Differential Movement at Railway Transitions for High Speed Passenger Rail and Joint Passenger/Freight Corridors. The authors would like to thank Professor Jianfeng Mao from Central South University, and Dr. Wenting Hou and Wenjing Li from the University of Illinois at Urbana-Champaign for their help with the train-track-bridge model development. In addition, the authors would like to acknowledge the China Scholarship Council (CSC) scholarship, which provided funding for Zhongyi Liu’s time and research effort in this paper.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: Zhongyi Liu, Bin Feng, and Erol Tutumluer; data collection: Zhongyi Liu; analysis and interpretation of results: Zhongyi Liu, Bin Feng, and Erol Tutumluer; draft manuscript preparation: Zhongyi Liu, Bin Feng, and Erol Tutumluer. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Zhongyi Liu

Bin Feng

Erol Tutumluer

Data Accessibility Statement:

The data that support the findings of this study are available from the corresponding author, Erol Tutumluer, on reasonable request.

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