1. Introduction
In recent years, high speed railways have been widely used around the world due to their outstanding advantages, such as their large capacity, comfort, punctuality, safety, and environmental friendliness. However, due to the dynamic impacts of high-speed trains, maintenance work for traditional ballasted tracks is difficult, and accompanied by a large consumption of labor and economic resources. To reduce maintenance work, ballastless track structures are widely used in high speed railway lines. Apart from the conventional ballastless track widely used in high speed railway, vibration-reduction ballastless tracks [
1] are applied to places where a reduction in vibrations transmitted from high speed railways into the surrounding soil and nearby buildings is highly demanded.
With this background, the China railway track system (CRTS) III slab track was researched and developed by China to overcome the shortcomings of the CRTS I slab track [
2], CRTS II slab track [
3], and the double-block ballastless track [
4]. The CRTS III slab track replaces the easily damaged CA mortar in the CRTS I and II slab tracks [
5,
6] with self-compacting concrete as a filling layer [
7]. The CRTS III slab track has been widely used in high-speed railway lines in China since 2010 due to its high performance.
With the development of high-speed railways and ballastless tracks, the issue of the dynamics of train–ballastless track subgrades become prominent, and a great deal of research work has been done on these aspects. Lei et al. [
8] developed a vehicle–CRTS II slab track-subgrade coupling dynamic model, in which a new type of slab track element was presented, and several application examples were illustrated. Zhu et al. [
9] established a 3D coupled dynamic model of vehicle–CRTS II slab tracks on a subgrade to calculate the vertical and lateral rail-supporting forces, and then those forces were inputted into a 3D nonlinear finite element model to investigate the evolution of interface damage and its influence on the dynamic response of the slab track. Yang et al. [
10] studied the effects of random track irregularity and vehicle velocity on the dynamic responses of slab tracks using a vehicle-CRTS I slab track in a subgrade interaction model, in which composite track elements were used to rapidly model the slab track. Later, Yang et al. [
11] developed a vehicle-slab track with a subgrade coupled dynamic model in the frequency domain to investigate the response and transfer characteristics of a ballastless track. Feng et al. [
12] studied the influence of the seam between a slab and CA mortar of a CRTS II slab track in terms of the vibration characteristics of the vehicle-track system using a vehicle-CRTS II slab track with a subgrade coupling dynamic model developed using ABAQUS
® software. Xu et al. [
13] proposed a probabilistic model for simulating random track irregularities in a vehicle-slab track with a subgrade coupled dynamic model to clarify the random vibration characteristics and probabilistic relationships between random track irregularities and dynamic behaviors of vehicle-track systems. Recently, using 8-node solid elements to model the subgrade, Xu et al. [
14] developed a matrix-coupled model for vehicle-slab track-subgrade interactions in a 3D space, and the accurateness, efficiency, and stability of the model were elaborated using numerical examples. Considering the influence of random track irregularity, Sun et al. [
15] proposed a numerical method, in which the vehicle-CRTS I slab track with a subgrade coupled dynamic model and the Kalker’s variational method and the material wear model were combined to predict non-uniform rail wear evolution. Chen et al. [
16] established a vertical model for vehicle-CRTS II slab track-subgrade dynamic interactions using the Green function method to study wheel polygonal wear. Aggestam et al. [
17] developed a 3D slab track model in ABAQUS
® using Python scripts, then the system matrices of the model were exported to MATLAB
® where the simulation of vehicle–track dynamic interactions was performed. Li et al. [
18] developed a nonlinear 3D-coupled vehicle-slab track model using LS-DYNA
® to investigate the influences of dynamic material properties of slab track components on train–track vibration interactions. Wang et al. [
19] investigated train-induced dynamic stress statistics in a subgrade surface using a vertical vehicle-track-subgrade coupled dynamic model. Xin et al. [
20] established a vehicle-slab track at the transition zone of a coupled dynamic model to study reasonable transition lengths, as well as the number and stiffness coefficient of the rubber mat at the transition zone between a fixed slab track and a floating slab track. Guo et al. [
21] put forward an iterative approach on the basis of vehicle–track coupled dynamics theory and an empirical model for cumulative plastic deformation of the subgrade to predict long-term track degradation of a ballastless track due to the evolution of differential subgrade settlement in a high-speed railway. Cai et al. [
22] established a rigid–flexible coupling vehicle-CRTS III slab track-subgrade dynamic model to research the influence of subgrade frost heave on the dynamic behavior of a high-speed railway vehicle.
A train running on realistic rail lines consists of several vehicles. However, in References [
8,
9,
10,
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22], only one vehicle was considered in the dynamic model. To be more consistent with actual operational situations, some scholars used train–ballastless track-subgrade-coupled dynamic models to analyze railway dynamics. Galvin et al. [
23] established a general and fully 3D multibody-finite element-boundary element model to research vibrations, where a train with 10 vehicle passing on non-ballast and ballast tracks for different train speeds was considered. Kece et al. [
24] developed a 2D slab track on a subgrade dynamic model to study slab track performance using the measured wheel load time history of an Amtrak Acela train with eight vehicles as the exciting force. Xu et al. [
25] established a 3D train-slab track on a subgrade dynamic model, which can consider the wheel–rail separation and the vehicle formation of a train, as well as the advantages and engineering applicability of the model, were illustrated using three numerical examples. Deng et al. [
26] established a wind–train-CRTS I slab track coupling dynamic model with MATLAB
®, in which wind loads were first obtained via a computational fluid dynamics simulation, and then the wind loads were inputted into the dynamic model. The traffic safety when a CRH3 high-speed train with three vehicles passed through two types of windproof facilities under crosswind conditions were studied and compared.
As can be seen from the research above, a great deal of work has been conducted on the vehicle–ballastless track–subgrade and train–ballastless track–subgrade coupled dynamics. However, these research works still have some limitations. Firstly, the dynamic response differences for the vehicle–ballastless track–subgrade coupled dynamics and train–ballastless track–subgrade coupled dynamics are lacking in-depth, comparative study. Secondly, the applicability of the vehicle–ballastless track–subgrade coupled dynamics is not distinct for different ballastless-track structures and track irregularity conditions. Thirdly, it is not clear how many vehicles should be used to achieve a balance between the calculation precision and efficiency for a train–ballastless track–subgrade coupled model. Fourthly, most of the previous research work focused on dynamic vibration responses and dynamic wheel–rail and fastener forces of the coupled system; however, the dynamic bending stresses of the ballastless track, which are key factors to design a ballastless track, especially for the vibration-reduction ballastless tracks, were not examined in detail.
To overcome these limitations, in this paper, a high-speed train-CRTS III slab track–subgrade coupling dynamic model, which can consider both the conventional CRTS III slab track, as well as the vibration-reduction CRTS III slab track, is established. The model is verified using results calculated by the ANSYS® program. With the model, the influence of the vehicle number on the dynamic characteristics of the train–CRTS III slab track–subgrade coupled system, for the conventional and vibration-reduction CRTS III slab tracks, with smooth and random track irregularity conditions, is studied and analyzed.
The novelties of this paper are as follows: Firstly, both the short and middle long wavelength random track irregularities were considered, so that the high frequency vibration of the track structure could be realistically simulated. Secondly, the influence of the vehicle number on the dynamic characteristics of the train–CRTS III slab track–subgrade coupled system for different ballastless track structures and track irregularity conditions were studied. Thirdly, in addition to the traditional dynamic vibration responses and the dynamic wheel–rail and fastener forces of the coupled system, the dynamic bending stresses of the ballastless track were also studied.
The research work can provide a theoretical basis to reasonably choose the vehicle number in modeling train–ballastless track–subgrade coupled systems with different ballastless track structures, track irregularity conditions, and research items, so that a balance between the calculation precision and efficiency could be achieved. Because of the required model length, the moving distance of the train and the simulation time can be greatly reduced with a low vehicle number in modeling the train–ballastless track–subgrade coupled system.
5. Verification of the Coupled System
The computer programs for the high speed train–CRTS III slab track–subgrade coupled system are, respectively, developed in MATLAB (MATLAB R2018a, MathWorks, Natick, Massachusetts, USA)® and ANSYS® (Canonsburg, PA, USA) platform.
In Reference [
29], the calculation results calculated using the program developed in MATLAB
® were verified by measured in situ data from the Shiziyang tunnel of the Guangzhou–Hong Kong high-speed railway line in China. The program was further verified by the calculation results using the program developed in the ANSYS
® platform.
The high-speed train sub-model, the wheel–rail interaction sub-model, the CRTS III slab track–subgrade sub-model, and the track irregularity sub-model in the ANSYS
® platform are the same as those in MATLAB
® and are described in detail in
Section 2.
In ANSYS
®, the rail, slab, and concrete base are modeled as BEAM3 linear beam elements. The fasteners, the connections between the slab and concrete base, the connections between the concrete base and the subgrade are modeled as linear COMBIN14 spring-damper elements. The detailed procedures for the establishment of the train sub-model and wheel–rail interaction sub-model in ANSYS can be found in Reference [
37].
With the developed programs, the dynamic characteristics of the CRH3 train–CRTS III slab track–subgrade coupled system for the conventional CRTS III slab track without track irregularity (Case 1), the vibration-reduction CRTS III slab track without track irregularity (Case 2), the conventional CRTS III slab track with random track irregularity (Case 3), and the vibration-reduction CRTS III slab track with random track irregularity (Case 4) are simulated, respectively. The largest dynamic responses of the accelerations of the car body, rail, slab, and concrete base, the positive and negative bending moments of rail, the compressive and tension forces of fastener, and the positive and negative bending stresses of slab and concrete base, which are calculated, respectively, using ANSYS
® and MATLAB
®, are listed in
Table 3.
It can be seen from
Table 3 that there are almost no differences between the largest dynamic responses of the coupled system calculated using ANSYS
® and MATLAB
®, and the maximum difference is less than 0.6% for all items and cases, except for the rail acceleration under random track irregularity (Cases 18 and 24 in
Table 4). Because the frequency of rail acceleration is very high under random track irregularity, and the influence factors for high-frequency vibrations are very complex, it is reasonable and acceptable that the difference for the largest rail acceleration calculated using ANSYS
® and MATLAB
® under random track irregularity be less than 5%. Thus, the calculated results of the coupled system for different cases are reliable and correct.