1 Introduction

Ballast track is a common type of track structure adopted in heavy-haul railways [1]. As an essential structure of the ballast track, the granular ballast bed is significantly important for ensuring the safe operation of railways [2]. In recent years, as the axle load and transportation volume of heavy-duty trains continued to increase, this led to a severe increase in problems like crushing and pulverization of the gravel ballast in the ballast bed, resulting in a continuous decrease in the maintenance cycle [3, 4]. At the same time, the maintenance time allowed for heavy-haul freight lines is reducing continuously due to a steady increase in transportation volume pressure, leading to increased challenges for railway maintenance. Studies suggest that the ballast breakage of railway tracks mainly occurs under the sleepers [5,6,7]. This is due to the high concrete sleeper stiffness, in which the ballast and the bottom surface of the sleeper exhibit point contact relation. Under the dynamic train load, the ballast is subjected to significant force and is thus prone to crushing. In response, studies have been carried out by many scholars in different countries, and they opine that using the under-sleeper pad (USP) is an important measure to address this critical issue, which severely affects the life of the ballast bed [8,16] investigated the vibration-dam** effect of the USP on track structures through vibration tests. The results show that it can reduce the vibration of track structures by about 30%. Kaewunruen et al. [17] analyzed the impact of laying USP on the vibration response at rail joints. The results show that it increases the vibration of both the rail and the sleepers. The amplitude of the rail vibration increased by approximately 66.7%. Paixão et al. [18] conducted in situ tests to explore the influence of the application of USP in transition sections on the dynamic response of the ballast track. They found that the USP increased the vertical compliance of the track, which, to a certain extent, enlarged the acceleration of the sleeper. Abadi et al. [19] observed in their field studies that the USP leads to an increase in the contact area between the ballast and the sleeper and a decrease in the contact pressure, thus reducing ballast wear. Safari et al. [20] conducted indoor tests, and they opined that the USP could reduce the sleeper settlement and improve their lateral stability and load distribution. Mottahed et al. [21] investigated the vibration response of the sleeper when the train passed through the bridge deck fitted with the USP through field tests. The results show that when the train passes through the bridge deck fitted with the USP, the vibration of the sleeper can be reduced to 58% of its maximum value. Ngamkhanong et al. [22] carried out field tests to study the vibration behavior of the USP under impact loads. The results show that the USP can reduce the wheel–rail force by nearly 10% under extreme impact loads.

Regarding numerical simulation, Chen et al. [23] established a simplified model for the transition section of a multi-track sleeper road and bridge by changing the stiffness of the subgrade wall unit. They found that increasing train speed and axle load can accelerate and deteriorate the process at the road and bridge transition section. Krishnamoorthy et al. [24] established a finite element method (FEM) model of the track structure. They studied the effect of the USP with varying thicknesses on the mechanical characteristics of the sleeper and the ballast bed. Qu et al. [25] established a FEM structural model to study the vibration-dam** effect of the ballast track laying USP. The results show that the vibration is amplified between 5 and 30 Hz and exhibited a better dam** effect above 30 Hz. Paixão et al. [26] established a 3D road and bridge transition section model of a vehicle–rail–ballast bed and concluded that laying USP can effectively control the vertical stiffness of the track and reduce the load transferred to the ballast layer by the sleepers, and the reasonable laying of the USP can improve the stiffness balance in the transition section. Li et al. [27] established a ballast box model for the USP based on discrete element method (DEM), which reveals that the number of ballast contacts with the sleeper can be increased, thereby making the load more uniform in the distribution on the ballast bed. Wan et al. [28] studied the impact of laying USP in the switches area on track performance using numerical optimization methods. The results indicate that using softer track pads combined with USP can significantly reduce dynamic forces. Cao et al. [29] adopted a biphasic porous elastic soil model to establish a vehicle–track–ground model and analyze the vibration reduction effect of USP. The results show that USP can effectively reduce free field ground vibration at lower train speeds, but the vibration isolation effect is relatively minor at higher speeds. Insa et al. [30] explored the feasibility of laying USP in transition zones. The results show that laying USP can achieve a more uniform level of vertical stiffness in the track line, thereby improving the dynamic behavior of the vehicle–track system. Navaratnarajah et al. [31] carried out experimental research and three-dimensional FEM analysis to evaluate the performance of USP. They found that the USP reduces the ballasts’ vertical and lateral strain when axle load increases. However, with the increase in speed, the reduction proportion is not particularly significant.

It is observable that the use of USP plays a commendable dam** role in areas such as transition zones, switches, and rail joints, with the performed function being even more effective under low-speed, heavy-load conditions. Moreover, the current study focuses mainly on the impact of laying USP on the macro-mechanical characteristics of the track structure and the smoothness of the train's ride, primarily analyzing the dampening effects from a macro perspective. Many analyses are based on simulation models developed using the FEM and vehicle–track coupled dynamics theory, overlooking the mesoscopic mechanism of the interaction between the ballast and the sleeper, as well as between the ballast particles themselves under the function of USP. Consequently, these failures to address the critical perspective do not effectively reveal the evolution of the different lateral resistance distribution at various positions, the level of ballast bed stress, and the corresponding impacts on the ballast bed's deformation characteristics in both the macro- and mesoscopic mechanical behaviors.

With this taken into consideration, a 3D coupling analysis model of the interaction between the USP and granular ballast was established using the discrete-continuous coupling analysis method. Further, field tests of ballast bed support stiffness and static and dynamic mechanical testing under train loads were conducted. From the experimental studies and numerical simulation perspectives, the present study meticulously analyzes the influence mechanism of laying USP on the dynamic and static mechanical behavior of ballast tracks under heavy-load operating conditions. It aims to reveal the working mechanism of the USP on the ballast bed, thereby providing theoretical guidance for improving the performance of the heavy-load railway ballast bed, delaying ballast degradation, and reducing maintenance workloads.

2 In situ test

2.1 Site overview

In order to study the mechanical quality state characteristics of the ballast bed after laying the USP, the selected site for the study is a heavy-haul section of the Shijiazhuang–Taiyuan Railway (Shitai Railway) in China, with areas both containing and lacking USP, respectively. The site overview is shown in Fig. 1. It is worth mentioning that the section where USPs are laid is part of an actively operated railway line. The Shitai Railway plays a crucial role in transporting coal and other essential materials from Shanxi to Hebei, making its stability and efficiency vital to the transportation system in North China.

Fig. 1
figure 1

Test site: a with USP; b without USP

Full size image

The on-site track structure is as follows: U75 V, U71 Mn, and other 60 kg/m on-line heat treatment rails are used. Type III concrete sleepers are laid and equipped with Type II elastic fasteners. The ballast bed is constructed using a mixture of first-grade limestone and basalt crushed stone. The width of the ballast bed top surface is 3500 mm, and the thickness is 350 mm. The side slope is 1:1.75. The width and height of the ballast shoulder are 450 and 150 mm, respectively. The gradation curve of the ballast particles is shown in Fig. 2. The material of the USP is rubber, with a stiffness of 80 kN/mm and a thickness of 15 mm. It is adhered to the bottom of the type III concrete sleeper, and the schematic diagram of the on-site track structure section is shown in Fig. 3.

Fig. 2
figure 2

Gradation curve of ballast particles

Full size image
Fig. 3
figure 3

Schematic diagram of the cross section of the track structure (unit: mm)

Full size image

2.2 Test methods and steps

2.2.1 Ballast bed support stiffness test

The ballast bed support stiffness reflects the elasticity and load-bearing capacity of the ballast track, primarily stemming from the support provided by the ballast particles at the bottom of the sleeper [32]. Reasonable ballast bed support stiffness is necessary to ensure the smoothness, comfort, and safety of train operation.

When conducting ballast bed support stiffness tests, it is necessary to remove the fastenings and pads under the rails at the test sleeper location to separate the rail from the sleeper. Further, to ensure the accuracy of the test data, a displacement reference base is installed in between two sleepers by replacing the central sleeper. The displacement sensor is fixed at the center of the displacement reference bracket to synchronously collect the settlement deformation of the test sleeper under vertical load. A reaction frame is installed between the rail and the test sleeper, and vertical loading is generated by jacks. The field testing setup for the ballast bed support stiffness is shown in Fig. 4.

Fig. 4
figure 4

Ballast bed support stiffness test: a loading diagram; b test site

Full size image

To avoid the impact of excessive force applied by the jack on the geometric position of the track, in the present study, the maximum force applied by the jack is set to 40 kN by referring to the field test load limit in [33]. After loading, gradually unload slowly and continue to record sleeper displacement changes. Considering the variability in the support conditions of each sleeper, seven sleepers are tested for each condition, and the average values are taken for subsequent analysis and research.

2.2.2 Dynamic test of track structure

After laying the USP, the contact relationship between the sleeper and the ballast has changed, thereby affecting the dynamic response and overall stability of the track structure. To investigate the impact of the USP on track structure under train operation conditions and to evaluate their effectiveness, tests for rail vertical displacement, sleeper vertical displacement, and dynamic pressure on the sleeper are conducted separately in sections with and without USP. The arrangement of the measurement points is shown in Fig. 5a.

Fig. 5
figure 5

Track structure dynamic test: a schematic diagram of measuring point layout; b test site

Full size image

The field testing setup is shown in Fig. 5b. Cantilever beam-type displacement sensors made of elastic elements are typically used for displacement testing of the track structure. The width of the beam gradually decreases from the fixed end to the free end, ensuring equal strength. One end of the elastic element is fixed on the mounting frame with bolts, the free end is provided with a bolt hole, and the connecting rod is fixed on the elastic element with a nut. The trapezoidal section between the fixed and free ends is called the working section of the displacement sensor. On an appropriate section within this range, a strain gauge is attached symmetrically on the rail top and bottom. To ensure the accuracy of the absolute displacement test of the track structure, it is usually necessary to drive a long steel pin into the ballast to obtain an absolutely fixed reference point. Additionally, the pressure on the sleeper is tested using the shear stress method.

2.3 Test results

2.3.1 Test analysis of ballast bed support stiffness

To analyze the impact of the USP on the static mechanical properties of the ballast bed, the curves of vertical loading and unloading test results for both the ballast bed (with USP) and the ballast bed (without USP) are drawn, as shown in Fig. 6.

Fig. 6
figure 6

Test results of the ballast bed support stiffness: a with USP; b without USP

Full size image

From Fig. 6, it can be observed that the vertical load stiffness of the ballast bed (with USP) has decreased compared to the ballast bed (without USP). According to “Code for Design of Railway Track” [34], the ballast bed support stiffness is defined as the secant stiffness on the load–displacement curve during the vertical loading process under unrestricted conditions. The specific analytical formula is

$${{K}}=\frac{{P}_{35}-{P}_{7.5}}{{S}_{35}-{S}_{7.5}},$$
(1)

where K is ballast bed support stiffness, \({P}_{35}\) and \({P}_{7.5}\) are the vertical load on the sleeper, and \({S}_{35}\) and \({S}_{7.5}\) are the vertical displacement of the sleeper under the vertical load of \({P}_{35}\) and \({P}_{7.5}\), respectively.

Based on the calculation method described in Eq. (1), the calculated value of the support stiffness of the ballast bed (with USP) is 85.08 kN/mm. In comparison, the support stiffness of the ballast bed (without USP) is higher, at 149.28 kN/mm. This indicates that using USP leads to a reduction of about 43.01% in the ballast bed support stiffness. This significant reduction implies that the USP provides greater flexibility.

The whole process of vertical loading and unloading of the sleeper is further analyzed, and the total energy, elastic deformation energy, and dissipated energy of the ballast bed (with USP) and the ballast bed in the whole process are calculated. The dissipated energy is represented by the area enclosed by the loading and unloading curves, which reflects the extent of energy dissipation by the ballast bed [2, 35]. Extensive area results in increased energy dissipation by the ballast bed. The ratio of dissipated energy to total energy is defined as the energy consumption ratio [36]. The energy variation pattern of the ballast bed is shown in Fig. 7.

Fig. 7
figure 7

The law of ballast bed energy change

Full size image

From Fig. 7, it can be observed that the values of total energy and elastic deformation energy of the ballast bed (with USP) are higher than those of the ballast bed (without USP), indicating that the USP is effective in absorbing energy. Notably, the increase in elastic deformation energy suggests that the USP is good at maintaining the ballast bed shape and function when subjected to load, thus providing more stable support for the railway system. Moreover, the energy consumption ratio of the ballast bed (with USP) is 43.98%, compared to 36.30% of the ballast bed (without USP). The 7.68% increase indicates it is more efficient in dissipating the energy transferred from external loads.

2.3.2 Test analysis of track structure dynamics

Testing and controlling the displacement of track structures under the action of trains is an important aspect of ensuring driving safety. Similarly, the pressure on the sleepers is also a significant factor, as its magnitude reflects the dynamic pressure exerted on them during the train passage [37, 38]. The selection, design, and arrangement of sleeper materials must consider this dynamic pressure to ensure the long-term safety and reliable operation of railways. To analyze the impact of the USP on critical components of the track structure during the operation of heavy-haul railways, the curves of vertical displacement of the rail, vertical displacement of the sleepers, and variations in pressure on the sleepers are illustrated in Fig. 8.

Fig. 8
figure 8

Dynamic test results of track structure: a rail displacement; b sleeper displacement; c dynamic pressure on sleeper

Full size image

From Figs. 8a, b it could be observed that the vertical displacement of the rail in sections with USP increased by 13.70% when compared with the sections without USP. Simultaneously, the vertical displacement of the sleepers shows a significant increase of 21.35%. This change is primarily attributed to the USP. Due to its unique material and design, the USP provides higher elasticity and shock absorption when the train passes over it, consequently affecting the vertical displacement of both the rail and the sleepers.

Further observation of Fig. 8c reveals that the dynamic pressure on sleepers in sections with the USP is 73.04 kN. For sections without USP, the dynamic pressure exhibited is 79.30 kN, showing a net decrease of 7.89% compared to the USP. This indicates that laying the USP positively affects load sharing, effectively reducing the pressure on the sleepers and, consequently, the stress on the ballast bed. This could be attributed to the fact that the USP increases the range of flexural changes in the rail, leading to a more uniform distribution of rail load on the sleepers. These changes are significant in reducing ballast bed stress, extending the lifespan of the track, and improving the stability of train travel.

3 Numerical simulation model establishment and verification

3.1 Discrete–continuous coupling analysis method

For the granular ballast bed, the discrete element method (DEM) is often used for analysis and research due to the discrete characteristics of ballast particles so that the movement patterns and contact characteristics of ballast particles can be analyzed from a mesoscopic perspective [39, 40]. However, the ballast track is a multi-element heterogeneous structure containing continuous medium (rails, sleepers) and discrete medium (ballast bed). The connections and contact forms between the structural layers are complex. Therefore, there are difficulties and deficiencies in numerical simulation using only the DEM, which cannot accurately reflect the granular characteristics of the ballast bed and the mesoscopic stress characteristics of the continuous medium simultaneously. To overcome these difficulties, the discrete–continuous coupling analysis method must be considered for the present study.

For the continuous medium, solutions could be solved by the FEM and multi-flexible body dynamic (MFBD), both of which can help identify the mechanical changes of the continuum medium. However, most of the coupling methods between the DEM and the FEM are single coupling, which can only transfer the mechanical information solved by the DEM to the FEM model for mechanical response analysis of the continuum medium. However, it is difficult to consider the impact of the deformation of the continuum medium on the particles. Therefore, it is difficult to accurately analyze the mechanical characteristics of the ballast bed with USP for heavy-haul railways. Since these characteristics are seriously affected by boundary effects, the size of the numerical model needs to be relatively large, which necessitates the need for higher computational speeds for effective analysis.

Coupling the DEM with MFBD involves using wall elements as coupling boundaries. These wall elements transfer mechanical information from the contact surfaces of the DEM to MFBD. The basic principle of the coupling between DEM and MFBD is as follows: within each time step, the motion information of the discrete media obtained from the DEM calculations, including spatial position and the forces exerted by the discrete media on the continuous medium, can be transmitted in real-time to the MFBD calculations. The MFBD then solves for the continuous medium based on the received information and updates its spatial position. The effects of the continuous medium, after its spatial position has changed, on the discrete media are transmitted back to the DEM, and the calculation proceeds to the next time step. This process is repeated in a loop until the sum of the calculated time steps reaches the preset simulation duration, at which point the simulation calculation is stopped, and the results are outputted. The basic principle of this coupling is illustrated in Fig. 9.

Fig. 9
figure 9

Basic principle of DEM–MFBD coupling

Full size image

This coupling method is a real-time two-way coupling analysis method that can be applied to structural deformation analysis and can accurately simulate the contact between flexible bodies and rigid bodies in the system, thereby effectively reflecting the contact between discrete medium and continuous medium and their respective physical and mechanical properties. Therefore, the present study chooses the discrete–continuous coupling analysis method coupled with DEM–MFBD.

When solving the DEM-MFBD coupling problem, the force–displacement parameters between the DEM and MFBD models are transferred through the coupling transition boundary-wall element. However, the contact force on the wall unit grid surface is often not presented on the grid nodes, and there may be multiple contact points on each grid. It is necessary to use the shape function interpolation method combined with the principle of virtual work to act on the contact force of the particles on the coupling surface. When converted into equivalent nodal force, the transformation formula between its global coordinates and local coordinates is shown as Eq. (2).

$${\left[{{x}},{{y}},{{z}}\right]}^{{\text{T}}}={{\varvec{T}}}_{{\text{a}}}{\left[X,Y,Z\right]}^{{\text{T}}},$$
(2)

where X, Y, and Z are the coordinates in the global coordinate system; x, y and z are the coordinates in the local coordinate system; and \({{\varvec{T}}}_{{\text{a}}}\) is the coordinate transformation matrix.

The analytical formula for equivalent nodal force is shown as Eqs. (3)–(5);

$${\varvec{F}}={\sum }_{m=1}^{M}{{\varvec{T}}}_{{\text{b}}}{\varvec{N}}{{\varvec{T}}}_{{\text{c}}}{{\varvec{F}}}_{{\text{a}}},$$
(3)
(4)
(5)

where \({\varvec{F}}\) is the equivalent node force vector, \({{\varvec{T}}}_{{\text{b}}}\) is the equivalent node force conversion matrix, N is the node shape function matrix, \({{\varvec{T}}}_{{\text{c}}}\) is the contact force conversion matrix, \({{\varvec{F}}}_{{\text{a}}}\) is the contact force vector of the contact point, and M is the number of contact points on the unit.

After conversion, the equivalent node force is loaded into the MFBD model for stress and strain calculations. Later, the node coordinates are updated on the coupling boundary of the DEM wall unit based on the deformed node information to complete the data transmission by the two-way coupling process. It is worth mentioning that the time step required for DEM analysis is usually tiny. To ensure real-time interaction at fixed time nodes, the time step of MFBD analysis needs to be set to an integer multiple of the time step of the DEM computation.

3.2 The establishment of the ballast bed (with USP) 3D coupling analysis model

The railway ballast stones vary in shape and size, and their irregular forms significantly impact the mechanical properties of the granular ballast bed. Accurately simulating the angularity and shape characteristics of ballast particles is a crucial factor affecting the precision of computational results [41, 42 ]. In this study, a 3D laser scanner is utilized to precisely acquire the 3D coordinate data of ballast particles and their 3D profiles. These are then imported into EDEM, where the spherical particle stacking method is used to generate clump models of the ballast particles, as shown in Fig. 10. It is also observed that as the number of filled spherical particles increases, the geometric description of the ballast particles becomes more realistic and detailed, but this also significantly reduces the model computational efficiency. Existing research [43] indicates that for the granular ballast bed, the computational results do not vary significantly when the number of simulated ballast shapes exceeds 10. Considering both the simulation accuracy of ballast particles and computational efficiency, this study adopts 22 spheres to simulate a single ballast particle in the actual modeling process.

Fig. 10
figure 10

Ballast particle modeling

Full size image

Modeling is done by first creating an ordinary sleeper ballast track model and, later, by generating a coupling model of the ballast bed (with USP). As shown in Fig. 2, ballast particles that meet the gradation requirements are generated for the first level ballast particle gradation curve. The falling rain method is used to accumulate the ballast particles, and combined with layered compaction, an ordinary sleeper ballast bed model of three-sleeper length is created. An ordinary sleeper ballast track model is obtained by assembling the ballast bed model with 60 kg/m rails and type III concrete sleepers. The built model simplifies the subgrade structure of the lower part of the track, uses wall units to simulate the subgrade, and simulates the elastic characteristics of the subgrade by reasonably setting the stiffness of the wall units. The dimensions of the ballast track structure section in the model show consistency with the actual field condition, as shown in Fig. 3.

The USP improves the contact between the bottom surface of the concrete sleeper and the ballast particles, thereby enhancing the service performance of the ballast track. During modeling, the under-sleeper pad is first meshed, and the discretization is performed using 8-node hexahedral Solid8 elements, with the smallest mesh cell size being approximately 0.015 m. The total number of mesh cells obtained is 3287 as shown in Fig. 11a. Subsequently, a force distributing rigid (FDR) is established across the entire upper surface of the USP, and fixed constraints are applied to the lower surface of the sleeper structure, resulting in a bonding effect. Import the rail model, the rail-sleeper–USP model is obtained (as shown in Fig. 11b). An upward displacement of 15 mm is applied to the rail and sleeper in the ordinary sleeper ballast track model, the rail-sleeper–USP model generated in the previous step is imported in the “wall” format, and then the model under the action of gravity is rebalanced. Ultimately, a stable and compact ballast bed (with USP) 3D coupling model is obtained, as shown in Fig. 11d, along with its detailed contact model, as shown in Fig. 11c.

Fig. 11
figure 11

Coupling model establishment: a grid division of the USP; b rail-sleeper–USP; c USP–ballast bed contact surface; d ballast bed (with USP) 3D coupling analysis model

Full size image

Since the ballast bed (with USP) 3D coupling model is formed by translation and replacement of the rail and sleeper structure based on the ordinary sleeper ballast bed model, it does not cause any noticeable disturbance to the ballast bed structure below the sleeper. Moreover, statistical analysis shows that the bulk density of the ballast bed with USP and without USP is both 1750 kg/m3 and the compactness is both 70%. Therefore, it can be concluded that the differences in the simulation results between the two models are due to the presence of the USP.

The intrinsic parameters of granular materials, track structure, and the USP, as well as their contact parameters, play a significant role in the accuracy of the coupling model's computational results. In this paper, the Hertz–Mindlin (no-slip) contact model [2, 44] commonly used in ballast track simulations, is adopted to simulate the contact conditions between ballast particles as well as between ballast particles and sleepers, USPs, and boundaries. The model parameters mainly involve the basic physical properties of the materials and the contact parameters. In combination with the actual conditions on site and referring to [4, 47,48] the computation parameters for the coupling model are established, as shown in Table 1.

Table 1 Coupling model calculation parameters
Full size table

3.3 Model verification

For verifying the accuracy of the ballast bed (with USP) and ballast bed (without USP) models established earlier, the numerical simulation results of the ballast bed support stiffness are compared with the field test results from Sect. 2.3.1, as shown in Fig. 12.

Fig. 12
figure 12

Comparison of test and simulation values of ballast bed support stiffness

Full size image

Various parameters that could be observed from Fig. 12 are as follows: a) test value of the ballast bed (with USP) support stiffness 85.08 kN/mm, b) simulation value is 86.44 kN/mm, c) relative error of 1.60% (considerably small error), and d) relative error between the test and simulation value of the ballast bed (without USP) is 1.38%. The relative errors are both minor. The load–displacement change curve lines’ shape and key points exhibit relative closeness.

To validate the effectiveness of the established models from a dynamic perspective even further, the time history curve (Fig. 8c) of the dynamic pressure on the sleeper obtained from the experiments in Sect. 2.3.2 is applied as an input at the centroid of the sleeper in the model. The operating vehicle type is C80B, with an axle load of 25 t, a speed of 60 km/h, and a fixed axle spacing of 1830 mm. This analysis compares the simulation and experimental values of the dynamic displacement of the sleeper, as shown in Fig. 13.

Fig. 13
figure 13

Comparison of test and simulation values of dynamic sleeper displacement: a ballast bed (with USP); b ballast bed (without USP)

Full size image

It can be seen from Fig. 13a that the average dynamic displacement test of the sleeper (with USP) is 0.895 mm, while the average numerical simulation value is 0.911 mm. The difference between the simulation and test results is 1.79%, and both exhibit relative consistency with one another. It can be seen from Fig. 13b that the average dynamic displacement test of the sleeper (without USP) is 1.043 mm, while the average value of numerical simulation is 1.063 mm. The difference between the simulation and test results is 1.92%, which again shows consistency with one another.

From the previous sections, it could be inferred that the accuracy of the model has been verified from both static and dynamic perspectives, considering the ballast bed support stiffness and the dynamic displacement of the sleeper. This indicates that the established model can be used for subsequent computational analysis.

4 Analysis and discussion on calculation results

4.1 Evolution law of lateral resistance at different positions

Ballast bed lateral resistance is a crucial factor in preventing track buckling and maintaining track stability [49, 50]. The installation of the USP alters the friction between the sleepers and the ballast particles, affecting the lateral resistance of the ballast bed. To comparatively analyze the impact of installing USP on the lateral stability of the ballast bed, a mesoscopic contact force distribution cloud map of two types of ballast beds under a 2 mm lateral displacement of the sleeper is illustrated, as shown in Fig. 14. To ensure the authenticity and reliability of the simulation and to reduce the impact of boundary conditions, only the middle single sleeper is analyzed and studied.

Fig. 14
figure 14

Microscopic contact characteristics of ballast bed: a with USP; b without USP

Full size image

As can be observed from Fig. 14, there are specific differences in the distribution and characteristics of contact forces between the two types of ballast beds. For the ballast bed (with USP), the areas of greater contact force are primarily distributed at the ballast shoulder, and due to the lateral movement of the sleeper to the left, the ballast particles near the sleeper position exhibit significant resistance. It can also be noticed that the strong contact force gathers along the direction of sleeper movement at an angle of 30 ° along the horizontal line of movement. In contrast, the contact force in other places, such as the sleeper bottom, is always less.

For the ballast bed (without USP), the average contact force between ballast particles is higher. High contact force extends its influence across a broad area. It could also be observed that the sleeper bottom exhibits a concentration of more significant contact force to a certain extent. This is due to the existence of the USP, which causes the rigid–rigid contact between the sleeper and the ballast in some places to be converted into a flexible–rigid contact between the USP and the ballast when the ballast bed moves laterally. The deformation of the USP improves the contact relationship and releases greater contact force.

A study conducted by **g et al. [51] has shown that the lateral resistance of the ballast bed is provided by the friction between the ballast particles and the sleeper, and its resistance is closely related to the frictional resistance provided by the ballast shoulders, sleeper sides, and sleeper bottoms. To further study and compare the influence of the USP on the evolution of lateral resistance, the lateral resistance values at different positions under different sleeper displacements are calculated, as shown in Fig. 15.

Fig. 15
figure 15

Variation of lateral resistance with sleeper displacement at different positions: a ballast bed (with USP); b ballast bed (without USP)

Full size image

It can be seen from Fig. 15 that under different sleeper displacements, the ballast shoulder resistance of the ballast bed (with USP) is greater than that of the sleeper side and sleeper bottom. It can also be observed that the ballast shoulder resistance is dominant. The resistance of the ballast bed (without USP) is greater than that of the ballast shoulder and the sleeper side, and the resistance of the sleeper bottom is the main component of the total resistance. For the sleeper displacement of 2 mm, the ballast shoulder resistance, sleeper side resistance, and sleeper side resistance are 5.40, 2.48, and 2.06 kN, respectively. Compared with the ballast bed (without USP), the change rates are 57.43%, −9.16%, and −67.91%, respectively. This is because the ballast bed contains USP, turning the rigid–rigid contact at the sleeper bottom into a flexible–rigid contact. This reduces the force directly resisting lateral movement. However, the resistance at the sleeper side remains essentially unchanged compared to the ballast bed (without USP). Consequently, while the resistance at the sleeper bottom decreases and the side resistance remains relatively stable, the shoulder resistance becomes the dominant force.

Figure 16 further shows the sharing ratio of the resistance of each part under different displacements.

Fig. 16
figure 16

Variation in the sharing ratio of lateral resistance at different positions: a ballast bed (with USP); b ballast bed (without USP)

Full size image

As can be seen from Fig. 16, with the increase in sleeper displacement, there is a slight increase in the sharing ratio of resistance exhibited by the sleeper bottom in the ballast bed. While the sharing ratio of the ballast shoulders decreases slightly, the sharing ratio of the sleeper sides shows minimal change. The order of resistance sharing ratio is consistently high at the ballast shoulder, followed by that at the sleeper side, and it is lowest at the sleeper bottom. When the sleeper displacement reaches 2 mm in the ballast bed (without USP), the order of resistance sharing ratio changes, with the sleeper bottom’s bearing exhibiting maximum change. It is followed by the ballast shoulders showing moderate change and the sleeper side showing the lowest change. In the ballast bed (with USP), the sharing ratio of resistance at the bottom decreases by 29.24% compared to the ballast bed (without USP), the sharing ratio at the sleeper sides increases by 2.24%, and at the ballast shoulders it increases by 27.00%. This indicates that the presence of the USP affects the distribution of lateral resistance in the ballast bed, primarily altering the sharing ratio of resistance at the sleeper bottom and the ballast shoulders.

4.2 Analysis of ballast bed stress

The internal stress level of the ballast bed is a crucial indicator for measuring the service condition of the ballast bed and evaluating the smoothness of train operation [52, 53]. In order to further analyze the influence of the USP on the mechanical properties and service state of the ballast bed under train load from the dynamic analysis, the area under the sleeper is selected (0.3 m × 0.35 m), as shown in Fig. 17. A planar Cartesian coordinate system is established, and stress level contour maps under the sleepers under the influence of train load for different types of railway ballast beds are shown in Fig. 18.

Fig. 17
figure 17

Schematic diagram of the area under the sleeper with train load

Full size image
Fig. 18
figure 18

Stress analysis of ballast bed: a with USP; b without USP

Full size image

It can be seen from Fig. 18 that from the vertical direction, under the action of train load, the stress under different types of ballast beds both show a decreasing trend along the depth direction, and the ballast bed exhibits an increased attenuation rate for the induced stress. In the horizontal direction, the ballast bed (with USP) is distributed almost symmetrically around the center of the sleeper. The ballast bed stress in the outer area of the sleeper is found to be high. However, the center position of the sleeper on the ordinary ballast bed shows greater stress. Also, the presence of the USP reduces the red area in the ballast bed stress. The results show significant differences during quantitative analysis of the stress distribution characteristics of the ballast bed (with USP) versus the ballast bed (without USP). Specifically, the average stress of the ballast bed (with USP) is 116.88 kPa, while the corresponding value for the ballast bed (without USP) is 159.07 kPa. These data indicate that the average stress of the ballast bed (with USP) is reduced in comparison with the ballast bed (without USP) by 26.52%. This could be attributed to the presence of the USP, which optimizes the distribution of the train load transfer forces, thereby reducing the overall stress on the ballast bed. This has a positive impact on extending the lifespan of the ballast bed, reducing ballast breakage, reducing maintenance costs, and enhancing the safety of heavy-haul railway operations.

4.3 Analysis of deformation characteristics of ballast bed

In order to ensure the smooth and safe operation of trains, the ballast bed needs to be in good service condition. The mutual dislocation and rearrangement of gravel ballast particles is one of the crucial reasons for the cumulative settlement deformation of ballast tracks [53]. A comparative analysis of the deformation behavior of the two types of ballast beds under the train’s dynamic load is conducted to analyze the impact of the USP on the overall deformation of the ballast bed. Simulation calculations are performed after completing a single section of train dynamic load for determining the residual vertical plastic displacement values of all ballast particles in the area beneath the sleeper (2.6 m × 0.3 m × 0.35 m). The results are then subjected to grouped statistical analysis, as shown in Fig. 19.

Fig. 19
figure 19

Statistics of plastic displacement values of different ballast particles: a with USP; b without USP

Full size image

According to Fig. 19, in both types of ballast beds, most of the ballast particles exhibit relatively small plastic displacement, remaining below 0.15 mm, with only a few particles experiencing larger plastic displacement. The plastic displacement of ballast particles in the ballast bed (with USP) is predominantly less than 0.03 mm. In contrast, in the ballast bed (without USP), few ballast particles have plastic displacement less than 0.03 mm. However, they are mainly concentrated between 0.045 and 0.15 mm displacement values. Considering the maximum plastic displacement, the maximum plastic displacement of ballast particles in the ballast bed (with USP) is found to be 0.255 mm, which is 35.11% lower than the displacement of 0.393 mm found in the ballast bed (without USP). This indicates that the rearrangement behavior of ballast particles in the ballast bed (with USP) is weaker than that in the ballast bed (without USP), which is more conducive to maintaining the stability of the ballast bed.

A study conducted by Zhao et al. [54] shows that the rotation of ballast particles from a mesoscopic perspective is one of the causes of the macroscopic settlement of the ballast bed. Therefore, the present study also focuses on statistical analysis of the ballast particle rotation. The peak rotational angular velocity of the ballast particles at 50, 150, 250, and 350 mm is extracted below the sleeper when the train load is applied, as shown in Table 2.

Table 2 Angular velocity of ballast particles’ rotation
Full size table

It can be observed from Table 2 that the load from the passage of trains causes ballast particles to rotate. Maximum rotational angular velocity decreases with an increase in the ballast bed depth, though the attenuation rate varies. For the ballast bed (with USP), the attenuation rates are 44.97%, 25.00%, and 5.13%, respectively, while the rates are 4.53%, 19.27%, and 16.26% for the ballast bed (without USP). The USP exhibits a higher attenuation rate at shallow depths of the ballast bed, reflecting the excellent vibration-dam** characteristics of the pad beneath the sleeper. At the same depth, the peak rotational acceleration of the ballast bed (with USP) is found to be about 50% to 70% lower than that of the ballast bed (without USP). Thus, the USP can reduce the ballast particles’ rotation angle, thereby contributing to the stability of the ballast bed structure and reducing the macroscopic settlement deformation of the ballast bed.

5 Conclusions

This paper conducts the field test of ballast bed support stiffness and dynamic sleeper displacement on the actual operation of the heavy-haul railway line and obtains the mechanical characteristic data of ballast beds (with and without USP). Based on the DEM–MFBD coupling analysis method, models of ballast beds (with and without USP) are established. The differences in the mechanical behavior of the track structure between the two are analyzed and compared. The conclusions are as follows:

  1. (1)

    Laying the USP enhances the flexibility and vibration-dam** performance of the ballast bed significantly, resulting in approximately a 43.01% reduction in the support stiffness while increasing the energy consumption ratio by 7.68%. This makes the ballast bed more efficient at dissipating energy transmitted from external loads, which is of great significance in extending the service life of the ballast bed.

  2. (2)

    Laying the USP results in a 13.70% increase in vertical displacement of the rail and a 21.35% increase in vertical displacement of the sleeper. In comparison, the dynamic pressure on the sleeper decreases by 7.89%. This is beneficial for achieving a more uniform load distribution, effectively reducing the load borne by the sleeper directly under the wheel load, which has a significant engineering value for enhancing the stability and safety of railway lines.

  3. (3)

    Laying the USP optimizes the lateral resistance distribution of the ballast bed, with the sharing ratio of the sleeper bottom decreasing by 29.24%, while the sharing ratio of the sleeper side and the ballast shoulder increased by 2.24% and 27.00%, respectively. The order of sharing ratio always remains as follows: ballast shoulder sharing the most, followed by the sleeper side, and the sleeper bottom sharing the least. This optimized sharing ratio is suitable for heavy-haul railway sections. It is recommended that the compactness of the ballast at the shoulders be improved through tam** operations to ensure good lateral stability of the ballast bed.

  4. (4)

    Laying the USP reduces the stress levels of the ballast bed significantly. The average ballast bed stress is reduced from 159.07 to 116.88 kPa, a reduction of 26.52%. This is beneficial for reducing ballast breakage and maintaining the ballast bed in good service condition.

  5. (5)

    Laying the USP can reduce the number of ballast particles with large plastic displacements under train loads, with the maximum plastic displacement reduced by 35.11%. Additionally, the peak angular velocity of particle rotation at different depths decreased by about 50% to 70%, which is beneficial for slowing down the settlement deformation of the ballast bed and extending the maintenance and repair cycle.