# Dynamic Stress of Subgrade Bed Layers Subjected to Train Vehicles with Large Axle Loads

Authors: Fang Xu; Qi Yang; Wenjie Liu; Wuming Leng; Rusong Nie; Huihao Mei
Journal: Shock and Vibration
 (2018)

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## Abstract

The dynamic responses of subgrade bed layers are the key factors affecting the service performance of a heavy-haul railway. A 3D train-track-subgrade interaction finite element (FE) model was constructed using the ABAQUS code, where different vertical irregular track spectra were simulated by modifying the vertical node coordinates of the FE mesh of the rail. Then, the dynamic stresses in the subgrade bed layers subjected to heavy-haul trains were studied in detail. The results showed the following: (1) the transverse distribution of the dynamic stress transformed from a bimodal pattern to a unimodal pattern with increasing depth; (2) the pass of adjacent bogies of adjacent carriages can be simplified once loaded on the subgrade since the dynamic stresses are maintained around the peak value during the pass of the adjacent bogies; (3) the dynamic stress at the bottom of the subgrade bed surface layer was more sensitive to the train axle load compared with that at the subgrade surface because the dynamic stresses induced by the two rails were gradually overlaid with increasing depth; (4) the maximum dynamic stress at the subgrade bed bottom was reduced by approximately 70% compared with that at the subgrade surface; (5) the vertical track irregularities intensified the vertical excitation between the train vehicle wheels and rails, and the maximum dynamic stress at the subgrade surface under the action of the irregular heavy-haul track spectrum increased by 23% compared with the smooth rail condition; and (6) the possible maximum dynamic stress (σdm) at the subgrade surface under the action of irregular track spectra can be predicted using the triple standard deviation principle of a normally distributed random variable, i.e., σdm = μ + 3σ (where μ and σ are the expectation and standard deviation of σdm, respectively).


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## Body

## 1. Introduction

Heavy-haul railways have significant economic and social benefits because of their large freight transport capacity. More than 70% of China’s heavy-haul railway lines are composed of embankments that are compacted from different types of soils. The subgrade bed layers at the top of an embankment are directly subjected to the static and dynamic loads transferred from the track structures and running trains, and their dynamic performances are the key factors determining the service performance of a heavy-haul railway [1, 2].Recently, the increase in axle loads and the formation lengths of heavy-haul trains has intensified the dynamic response (e.g., dynamic stress and displacement) of subgrade bed layers, thus bringing new challenges to the serviceability of existing heavy-haul railways. Han and Zhang [3] showed that the dynamic stress of subgrade bed layers is markedly increased when the train axle load is increased to 25 t. Wei et al. [4] suggested that the upper limit of a train’s axle load was approximately 27 t for the existing railway lines in China. Di et al. [5] assessed the subgrade strength of the Shuo-Huang heavy-haul railway and demonstrated that the train axle load should not exceed 27 t, even if the existing subgrade is reinforced using oblique jet grouting piles. Hence, the existing heavy-haul railways (especially the subgrade beds) in China cannot handle the operation of heavy-haul trains with large axle loads, and the fundamental design criteria may need to be improved, whereas the premise for optimizing the design of subgrade beds is to master their dynamic performances subjected to heavy-haul trains with large axle loads.In the past few decades, the dynamic response of railway subgrades was mainly studied based on field tests and/or laboratory model tests [6–11], which are uneconomical and time-consuming. Recently, with the development of computer technology, numerical simulations have gradually become an important research technique. Numerical simulations on the dynamic responses of railway subgrades can be divided into two main types, i.e., the track-subgrade model and the train-track-subgrade interaction model. A track-subgrade model simplifies the train-induced dynamic load and directly applies it on the rail track. Xiao [12] simplified the train load as the summation of a static load and a series of sine waves, whereas Bian et al. [13], Shan et al. [14], Ang and Dai [15], El Kacimi et al. [16] and Hall and Lars [17] simulated the train loads as moving loads applied directly on the rail. These methods are widely used since they are simple and can significantly reduce the computation time. However, the interactions between the train and track system were not properly considered. Hence, a number of train-track-subgrade interaction models have also been established during the last decade. Ma et al. [18], Giner-Navarro et al. [19], and Cai et al. [20] constructed full train-track-subgrade interaction models to study the wheel-track interaction behavior; and Kouroussis et al. [21], Chiang and Tsai [22], Kouroussis and Verlinden [23], Alves Costa et al. [24] and Zhai et al. [25] investigated train-induced ground vibrations using train-track-subgrade interaction simulations. Galvin et al. (2013) [26] applied a two-series suspension structure to simulate a train and constructed a 3D train-track-subgrade model to study the contact force between the train wheels and the rail of a high-speed railway, and their model obtained an acceptable subgrade dynamic stress compared with the field measurements.Currently, numerical simulations on train-track-subgrade systems mainly focus on high-speed railways and are aimed at the train-track interaction and ground vibrations; few train-track-subgrade system simulations are aimed at the dynamic response of the subgrade beneath the track structure, especially for subgrade bed layers of heavy-haul railways. In this study, a 3D finite element (FE) model of a train-track-subgrade interaction system was constructed to study the dynamic stress in the subgrade bed layers of a heavy-haul railway while considering the rail track irregularities.


## 2. Train-Track-Subgrade Interaction Model

The finite element simulation was performed using the ABAQUS code (version 6.14-5). The prototype of the FE model is a soil embankment of the Shuo-Huang heavy-haul railway that is located in the Hebei province of China. The embankment has a top width of 7.0 m, a height of 6.0 m, a slope ratio of 1:1.5, and a length of 72 m. The cross section of the embankment is presented in Figure1. The ballast layer has a top width of 3.0 m, a thickness of 0.5 m, and a slope ratio of 1:1.75. The subgrade bed is 2.5 m thick and consists of a 0.6 m surface layer (SBSL) and a 1.9 m bottom layer (SBBL). The embankment foundation is composed of Quaternary clay with a thickness of 5.0 m. The groundwater level is relatively deep. The rail has a unit length weight of 70 kg/m, and the sleeper spacing is 0.543 m.Figure 1
Embankment cross section.


### 2.1. Train Vehicle Model

The train was simplified as two connected carriages to simulate the superposition effects of the adjacent bogies of adjacent carriages. The train bogies were modeled as two-series suspension systems consisting of spring-damping units. This study is mainly aimed at the vertical dynamic stress response in the subgrade bed layers; hence, the considered displacement freedoms were the nodding and bouncing movements of the train carriages and train bogies and the bouncing movements of the train wheels [27]. A schematic diagram of the train model is illustrated in Figure 2(a). The train carriage, train bogies, and train wheelset were simulated as rigid bodies. The perspective view of the constructed train model is presented in Figure 2(b). The main dimensions of the simulated trains are listed in Table 1, and the parameters of the spring-damping units that are used to simulate the two-series suspension system are listed in Table 2 (where K1 and C1 are the stiffness and damping of the first suspension, respectively; and K2 and C2 represent the values of the second suspension).Figure 2
Train vehicle model: (a) carriage components; (b) perspective view of the train model.
(a)
(b)Table 1
Main parameters of simulated trains.
Train model
Axle load (t)
Bogie axle spacing (mm)
Bogie center spacing (mm)
Carriage length (mm)
C80
25
1830
8200
12000
C96
30, 32.5, 35
1860
9926
13726Table 2
Parameters of the spring-damping units.
K
1 (kN/m)
K
2 (kN/m)
C
1 (kN·s/m)
C
2 (kN·s/m)
17000
5900
90
100


### 2.2. Interaction between the Train Vehicle Wheel and Rail

The interaction between the train wheel and rail was simulated using Hertz nonlinear contact theory [28]. The surfaces of the wheel and rail were selected as the master surface and slave surface of the contact pair, respectively. The tangential contact behavior was simulated using a penalty friction model with a friction coefficient of 0.2. The vertical contact behavior was modeled using the Hertz nonlinear contact theory, where the vertical contact force is formulated as(1)Pj=1GZwj−δrj−ηj2/3,j=1to4,0,when  separation,where Pj is the vertical contact force between the jth train wheel and rail, Zwj is the vertical displacement of the jth train wheel, δrj and ηrj are the vertical displacement and irregularity of the rail directly below the jth train wheel, and G is a contact constant.A soft contact mode was applied to model the vertical Hertz nonlinear contact between the train wheels and rail [28]. The schematic diagram of the soft contact mode is presented in Figure 3, where overclosure was defined to simulate the vertical contact behavior.Figure 3
Soft contact mode between the train vehicle wheel and rail in the vertical direction.


### 2.3. Track and Subgrade

The fasteners and cushion blocks used to connect the rail and sleepers were simplified as spring-damping units as illustrated in Figure4. The sleepers, ballast layer, subgrade bed layers, soil layers below the subgrade bed (LBSB), and the embankment foundation were treated as continuum media but with different mechanical parameters. The rail, sleepers, and ballast layer were modeled as elastic media, while the substructures were simulated as elastoplastic media using the Mohr–Coulomb constitutive model. The detailed mechanical properties of the FE model are listed in Table 3, where E, ν, ρ, φ, c, and D represent the elastic modulus, Poisson’s ratio, density, frictional angle, cohesion, and damping ratio of the corresponding component, respectively.Figure 4
Connection between the rail and sleeper.Table 3
Mechanical parameters of different parts of the FE model.
Part
E (MPa)
ν
ρ (kg/m3)
φ (°)
c (kPa)
D
Train
Rigid body
—
—
—
—
—
Rail
2.1 × 105
0.25
7850
—
—
—
Sleeper
30000
0.2
2500
—
—
—
Ballast
200
0.25
2400
—
—
0.05
SBSL
180
0.3
2300
33
58
0.1
SBBL
150
0.3
2200
31
45
0.1
SBSB
70
0.35
1800
17
30
0.15
Foundation
50
0.35
1700
15
25
0.15The bottom width of the FE model was 55 m and the damping ratios of the FE model boundaries were set as 1.0 to mitigate the reflection effects of the dynamic stress waves. The horizontal displacements of the left, right, front, and back boundaries were fixed but vertical movement was allowed, while at the bottom both the vertical and horizontal displacements were fixed. The FE model was meshed using a hexahedron element with eight nodes (C3D8). The elements used to model the subgrade bed layers had a side length of 0.1 to 0.2 m. The FE mesh of the entire train-track-subgrade model is presented in Figure5.Figure 5
Train-track-subgrade interaction model.


## 2.1. Train Vehicle Model

The train was simplified as two connected carriages to simulate the superposition effects of the adjacent bogies of adjacent carriages. The train bogies were modeled as two-series suspension systems consisting of spring-damping units. This study is mainly aimed at the vertical dynamic stress response in the subgrade bed layers; hence, the considered displacement freedoms were the nodding and bouncing movements of the train carriages and train bogies and the bouncing movements of the train wheels [27]. A schematic diagram of the train model is illustrated in Figure 2(a). The train carriage, train bogies, and train wheelset were simulated as rigid bodies. The perspective view of the constructed train model is presented in Figure 2(b). The main dimensions of the simulated trains are listed in Table 1, and the parameters of the spring-damping units that are used to simulate the two-series suspension system are listed in Table 2 (where K1 and C1 are the stiffness and damping of the first suspension, respectively; and K2 and C2 represent the values of the second suspension).Figure 2
Train vehicle model: (a) carriage components; (b) perspective view of the train model.
(a)
(b)Table 1
Main parameters of simulated trains.
Train model
Axle load (t)
Bogie axle spacing (mm)
Bogie center spacing (mm)
Carriage length (mm)
C80
25
1830
8200
12000
C96
30, 32.5, 35
1860
9926
13726Table 2
Parameters of the spring-damping units.
K
1 (kN/m)
K
2 (kN/m)
C
1 (kN·s/m)
C
2 (kN·s/m)
17000
5900
90
100


## 2.2. Interaction between the Train Vehicle Wheel and Rail

The interaction between the train wheel and rail was simulated using Hertz nonlinear contact theory [28]. The surfaces of the wheel and rail were selected as the master surface and slave surface of the contact pair, respectively. The tangential contact behavior was simulated using a penalty friction model with a friction coefficient of 0.2. The vertical contact behavior was modeled using the Hertz nonlinear contact theory, where the vertical contact force is formulated as(1)Pj=1GZwj−δrj−ηj2/3,j=1to4,0,when  separation,where Pj is the vertical contact force between the jth train wheel and rail, Zwj is the vertical displacement of the jth train wheel, δrj and ηrj are the vertical displacement and irregularity of the rail directly below the jth train wheel, and G is a contact constant.A soft contact mode was applied to model the vertical Hertz nonlinear contact between the train wheels and rail [28]. The schematic diagram of the soft contact mode is presented in Figure 3, where overclosure was defined to simulate the vertical contact behavior.Figure 3
Soft contact mode between the train vehicle wheel and rail in the vertical direction.


## 2.3. Track and Subgrade

The fasteners and cushion blocks used to connect the rail and sleepers were simplified as spring-damping units as illustrated in Figure4. The sleepers, ballast layer, subgrade bed layers, soil layers below the subgrade bed (LBSB), and the embankment foundation were treated as continuum media but with different mechanical parameters. The rail, sleepers, and ballast layer were modeled as elastic media, while the substructures were simulated as elastoplastic media using the Mohr–Coulomb constitutive model. The detailed mechanical properties of the FE model are listed in Table 3, where E, ν, ρ, φ, c, and D represent the elastic modulus, Poisson’s ratio, density, frictional angle, cohesion, and damping ratio of the corresponding component, respectively.Figure 4
Connection between the rail and sleeper.Table 3
Mechanical parameters of different parts of the FE model.
Part
E (MPa)
ν
ρ (kg/m3)
φ (°)
c (kPa)
D
Train
Rigid body
—
—
—
—
—
Rail
2.1 × 105
0.25
7850
—
—
—
Sleeper
30000
0.2
2500
—
—
—
Ballast
200
0.25
2400
—
—
0.05
SBSL
180
0.3
2300
33
58
0.1
SBBL
150
0.3
2200
31
45
0.1
SBSB
70
0.35
1800
17
30
0.15
Foundation
50
0.35
1700
15
25
0.15The bottom width of the FE model was 55 m and the damping ratios of the FE model boundaries were set as 1.0 to mitigate the reflection effects of the dynamic stress waves. The horizontal displacements of the left, right, front, and back boundaries were fixed but vertical movement was allowed, while at the bottom both the vertical and horizontal displacements were fixed. The FE model was meshed using a hexahedron element with eight nodes (C3D8). The elements used to model the subgrade bed layers had a side length of 0.1 to 0.2 m. The FE mesh of the entire train-track-subgrade model is presented in Figure5.Figure 5
Train-track-subgrade interaction model.


## 3. Track Irregularity

Track irregularity is a key factor that intensifies the dynamic response of a train-track-subgrade interaction system. Track irregularity consists of vertical irregularity, horizontal irregularity, direction irregularity, and rail spacing irregularity. The vertical and horizontal irregularities may significantly affect the vertical interaction of the system [29]. This study mainly considers vertical irregularity.There has not been an authoritative track spectrum available for describing the track irregularities of heavy-haul railways in China. Wei [30] described that the vertical track irregularity of America’s fifth-grade spectrum is similar to that of the China’s heavy-haul railways; hence, it was applied as the heavy-haul track irregular spectrum in the established FE model. The vertical track irregularity of America’s fifth-grade spectrum is expressed as(2)SvΩ=kFAvΩc2Ω2Ω2+Ωc2,where SvΩ is the power spectrum density of irregularity, Ω is the spatial frequency of irregularity, Ωc is the cutoff frequency, Av is the roughness constant (=0.2095), and kF is the safety factor with a value of 0.25 generally used.The basic principle for modeling the random track irregularity is converting the irregular power spectrum density to a time domain sample using the inverse fast Fourier transform (IFFT). The relationship between the discrete power spectral density (Sxk) and time series xs in the IFFT is formulated as(3)Sxk=1N2Dxs2=1N2X∗kXk,where N is the number of discrete sampling points, Dxs means conducting a discrete Fourier transform on xs (s = 0 to N − 1), Xk is the Fourier frequency spectrum of the time series xs, and X∗k is the adjoint of Xk (k = 0 to N − 1).According to equation (3), Xk can be back-analyzed using the available power spectral density of irregularity. Then, the serialized time domain model for the track irregularity can be obtained by conducting an inverse discrete Fourier transform on Xk:(4)xn=1N∑k=1NXkexp2iπknN,Xk=NξkSxkΔωΔω,Δω=t2π,ξk=expiϕk,where xn is the discrete time domain irregularity, t is the time, i is the imaginary unit, ξk is an independent phase series, and ϕk is a random variable obeying a uniform distribution.Specifically, the discrete time domain track irregularity can be obtained as follows: (1) discretely sampling on the power spectrum density of irregularity, (2) bank-deducingXk using equation (3), and (3) conducting an inverse discrete Fourier transform on Xk to obtain xn. The MATLAB software was adopted to improve the analysis efficiency, and the obtained vertical time domain irregularity of America’s fifth-grade spectrum is presented in Figure 6. By comparison, the vertical time domain irregularity of the left rail of the China’s three main railway lines (Beijing-Shanghai railway, Beijing-Guangzhou railway, and Beijing-Jiulong railway) is also shown in Figure 6. It is found that the vertical irregularity of China’s three main railway lines is less than that of America’s fifth-grade spectrum; hence, it may not be available for heavy-haul railways in China.Figure 6
Vertical track irregularity in the time domain.It is assumed that the vertical track irregularity is fully attributed to the surface undulation of the rail. The node coordinates matrix of the smooth rail surface was first exported from the ABAQUS FE model (INP file); then, the vertical node coordinates were modified according to the vertical track irregularity illustrated in Figure6 using the MATLAB software. Finally, the modified node coordinate matrix was imported to the smooth rail surface. Figure 7 presents the rail models before and after modification. It shows that the vertical track irregularity was properly simulated.Figure 7
Rail model: (a) before modification; (b) after modification.
(a)
(b)


## 4. FEA Results and Analysis

### 4.1. Verification of the FE Model

Ding et al. [31] reported the measured maximum vertical displacements of the rail and sleeper and the maximum acceleration of the ballast at a test site of the Shuo-Huang heavy-haul railway under a train with an axle load of 25 t; the FE analyzed (FEA) results are compared with the field measurements as listed in Table 4. It shows that the analyzed values agreed well with the field measurements. Miao et al. [32] reported the dynamic stress-time curve measured at the subgrade surface under a trial-train with an axle load of 30 t; the FE analyzed curve is compared with the measured curve, as illustrated in Figure 8. It is observed that the calculated curve well simulated the measurements in both wave shape and peak values. The good agreements between the FEA results and field measurements demonstrated the effectiveness and reliability of the FE model.Table 4
Measured and simulated vertical displacement of the rail and sleeper.
Parameter
Measured value
Calculated value
Vertical displacement of rail (mm)
2.01
1.92–2.21
Vertical displacement of sleeper (mm)
1.55
1.48–1.73
Acceleration of ballast (m/s2)
1.31
0.81–1.13Figure 8
Measured and calculated subgrade surface dynamic stress.


### 4.2. Dynamic Stress of the Subgrade Bed Layers

The train speed of the Shuo-Huang heavy-haul railway is approximately 80 km/h; hence, the default speed in the FEA was 80 km/h. The analyzed dynamic stress contour at the subgrade surface induced by a heavy-haul train with an axle load of 30 t is presented in Figure9. In the specified case, the analyzed results all corresponded to a train axle load of 30 t in the following parts of this article. It shows that the dynamic stress is symmetrically distributed along track centerline; the maximum dynamic stress appeared directly under the rail and was attenuated toward the track centerline and the embankment shoulder; the transverse distribution of the dynamic stress exhibited a bimodal pattern with two peak points and a tensile stress appeared under the middle part of the train carriage. In addition, the train ran smoothly on a smooth track; the difference of the dynamic stress at the subgrade surface under the four bogies was less than 10 kPa, and the effects of a bogie were similar to a nonuniform rectangular pressure, as shown in Figure 9(a). Conversely, the train exhibited bouncing movements on the vertically irregular track, thus the maximum dynamic stress was greater than that under a smooth track condition; nodding motion appeared on the train carriages and bogies, hence the dynamic stresses at the subgrade surface under the four bogies appeared to be obviously different with a greater deviation of approximately 20 kPa (Figure 9(b)).Figure 9
Dynamic stress contour at the subgrade surface: (a) smooth track; (b) vertically irregular track.
(a)
(b)Figures10 and 11 present the dynamic stress contour at the bottom of the SBSL and SBBL under irregular track conditions, respectively. It is determined that the transverse distribution of the dynamic stress varied from a bimodal pattern to a unimodal pattern with an increase in the depth from the subgrade surface (Figures 9(b) and 10). The dynamic stress directly beneath the rail was close to that under the track centerline at the bottom of the subgrade bed surface layer, as illustrated in Figure 10. The distribution of the dynamic stress induced by adjacent bodies of adjacent carriages exhibited an elliptical shape and the superimposed dynamic stress was apparently greater than that induced by a single bogie (Figure 11). Therefore, the superposition effects of the adjacent bogies could enlarge the influencing depth of the dynamic stress because of the small bogie spacing of the heavy-haul train.Figure 10
Dynamic stress contour at the bottom of the subgrade bed surface layer.Figure 11
Dynamic stress contour at the bottom of the subgrade bed bottom layer.Figures12(a) and 12(b) present the subgrade surface dynamic stress-time curves directly below the rail and track centerline under different train axle loads, respectively. It shows that the maximum dynamic stresses (σdm) at the subgrade surface directly below the rail and track centerline had increments of 33.8 kPa and 32 kPa (corresponding to percent increases of 39.8% and 47.3%) when the train axle load increased from 25 t to 35 t, respectively. An increase in the train axle load intensified the subgrade dynamic stress and the interaction between the train and track system. In addition, the subgrade surface dynamic stresses induced by the adjacent bogies exhibited an obvious superposition effect, and the dynamic stresses were maintained around the peak value during the pass of the adjacent bogies; hence, the pass of adjacent bogies of adjacent carriages may be simplified once loaded on the railway subgrade.Figure 12
Dynamic stress at the subgrade surface: (a) below the rail; (b) below track centerline.
(a)
(b)The relationships between the maximum dynamic stress and train axle load (P) at different locations below a sleeper are illustrated in Figure 13. It shows that the maximum dynamic stresses increased approximately linearly as the train axle load increased. The average increase rate of the dynamic stress directly below the rail was 3.17 kPa/t, 3.73 kPa/t, and 1.52 kPa/t at the subgrade surface and the bottoms of SBSL and SBBL, respectively. For the locations directly below the track centerline, the average increase rates were 3.19 kPa/t, 3.59 kPa/t, and 1.78 kPa/t. Generally, the dynamic stress at the bottom of the SBSL was more sensitive to the train axle load compared to that at the subgrade surface because the dynamic stresses induced by the two rails were gradually overlaid with an increase in the depth from the subgrade surface.Figure 13
Relationship between the maximum dynamic stress and train axle load.Figure14 presents the relationship between the maximum dynamic stress and the depth from the subgrade surface. σdm at the subgrade surface directly below the rail as well as its attenuation rate was greater than that below the track centerline. σdm below the rail became less than that below the track centerline at a depth of approximately 0.3 m. σdm at the subgrade bed bottom was reduced by approximately 70% compared with that at the subgrade surface. The ratio of σdm to the self-weight stress (σs) at the subgrade bed bottom (2.5 m below the subgrade surface) under a train axle load of 25 t was 0.21 and 0.22 for the locations directly below the rail and the track centerline, respectively; the σdm/σs values increased to 0.34 and 0.36 under a train axle load of 30 t, respectively. Zhou [33] suggested that the σdm/σs value should be less than 0.20 to maintain the long-term stability of a railway subgrade. Consequently, the thickness of the subgrade bed (2.5 m, a subgrade bed surface layer of 0.6 m and a subgrade bed bottom layer of 1.9 m) of existing railways in China may meet the requirements of running a heavy-haul train with an axle load of 25 t but cannot enable the operation of heavy-haul trains with axle loads greater than 25 t unless the subgrades have been reinforced.Figure 14
Relationship between the maximum dynamic stress and depth from the subgrade surface.


### 4.3. Effects of Irregular Track Spectrum

Figure15 illustrates the subgrade surface dynamic stress-time curves under different track conditions. The vertical track irregularity intensified the vertical excitation between the train wheels and rails and induced a greater dynamic stress compared with the smooth rail condition. The maximum dynamic stress at the subgrade surface increased by 23% and 15.8% under the action of the heavy-haul irregular track spectrum and China’s three main railway lines irregular track spectrum, respectively.Figure 15
Dynamic stress at the subgrade surface under different irregular track spectra: (a) below rail; (b) below centerline.
(a)
(b)In China, the irregular track spectra of the left and right rails are different. The dynamic stress contour at the subgrade surface taking this difference into account is shown in Figure16. It is determined that the FE model effectively reflected the irregularity difference of the left and right rails, and the dynamic stress was not symmetrically distributed along the track centerline.Figure 16
Dynamic stress contour at the subgrade surface considering the irregularity difference of the left and right rails.Figure17 presents the transverse distributions of the dynamic stress at different depths. The subgrade surface dynamic stress exhibited a bimodal pattern (or an “M” shape) and the value under the left rail was approximately 19% greater than that under the right rail, whereas the asymmetry of the dynamic stress gradually decreased as the depth increased and was negligible at the subgrade bed bottom; hence, the irregularity difference between the left and right rails had a slight impact on the embankment soils below the subgrade bed bottom.Figure 17
Transverse distributions of dynamic stresses at different depths.


### 4.4. Statistical Analysis of the Maximum Dynamic Stress at the Subgrade Surface

The maximum dynamic stress at the subgrade surface was randomly distributed under different sleepers because of the vertical track irregularity. Figure18 illustrates the statistical results of σdm at the subgrade surface directly below the intersection points of the rail and sleepers. It is observed that the track conditions significantly affected the distribution of σdm; σdm was relatively small and the distribution was relatively concentrated when the rail track was smooth; and σdm increased and the distribution range became wider when the track condition worsened, indicating that the subgrade bed layers were more likely to withstand a greater dynamic stress, which is harmful for maintaining the long-term service performance of the subgrade.Figure 18
Statistical graph of the maximum dynamic stress at the subgrade surface: (a) smooth rail; (b) irregularity in China’s three main lines; (c) heavy-haul track irregularity.
(a)
(b)
(c)The statistical distributions ofσdm under different irregular track conditions can be well-fitted by normal distribution curves as illustrated in Figure 18, and the expectation (μ) and standard deviation (σ) increased as the track condition worsened.The possible maximum dynamic stress at the subgrade surface was predicted using the normal distribution theory. Ifσdm∼(μ, σ2), then(5)Pσdm−μ≥3σ=1−Pσdm−μ<3σ=1−0.9973=0.0027<0.003.Therefore, the probability ofσdm outside the interval of (μ − 3σ, μ + 3σ) was less than 0.003. Statistically, the interval of (μ-3σ, μ+3σ) is generally taken as the actual possible range of a normal distribution random variable, namely, the triple standard deviation principle. Consequently, the possible maximum value of the dynamic stress at the subgrade surface can be estimated as μ + 3σ. The analyzed possible maximum dynamic stresses at the subgrade surface under different irregular track spectra are listed in Table 5. The maximum dynamic stress at the subgrade surface of the Shuo-Huang heavy-haul railway induced by a train with an axle load of 30 t could reach 123 kPa [34], which was favorably simulated with a predicted value of 116.9 kPa (with a deviation less than 5%).Table 5
Statistical results of the maximum dynamic stress at the subgrade surface.
Statistical parameter
Axle load (t) for heavy-haul track irregularity
Axle load (t) for China’s three main lines track irregularity
30
32.5
35
30
32.5
35
μ (kPa)
86.6
97.3
110.3
74.4
82.1
91.0
σ (kPa)
10.1
9.82
10.23
8.1
8.34
8.91
μ + 3σ (kPa)
116.9
126.8
141.0
98.7
107.1
117.7


## 4.1. Verification of the FE Model

Ding et al. [31] reported the measured maximum vertical displacements of the rail and sleeper and the maximum acceleration of the ballast at a test site of the Shuo-Huang heavy-haul railway under a train with an axle load of 25 t; the FE analyzed (FEA) results are compared with the field measurements as listed in Table 4. It shows that the analyzed values agreed well with the field measurements. Miao et al. [32] reported the dynamic stress-time curve measured at the subgrade surface under a trial-train with an axle load of 30 t; the FE analyzed curve is compared with the measured curve, as illustrated in Figure 8. It is observed that the calculated curve well simulated the measurements in both wave shape and peak values. The good agreements between the FEA results and field measurements demonstrated the effectiveness and reliability of the FE model.Table 4
Measured and simulated vertical displacement of the rail and sleeper.
Parameter
Measured value
Calculated value
Vertical displacement of rail (mm)
2.01
1.92–2.21
Vertical displacement of sleeper (mm)
1.55
1.48–1.73
Acceleration of ballast (m/s2)
1.31
0.81–1.13Figure 8
Measured and calculated subgrade surface dynamic stress.


## 4.2. Dynamic Stress of the Subgrade Bed Layers

The train speed of the Shuo-Huang heavy-haul railway is approximately 80 km/h; hence, the default speed in the FEA was 80 km/h. The analyzed dynamic stress contour at the subgrade surface induced by a heavy-haul train with an axle load of 30 t is presented in Figure9. In the specified case, the analyzed results all corresponded to a train axle load of 30 t in the following parts of this article. It shows that the dynamic stress is symmetrically distributed along track centerline; the maximum dynamic stress appeared directly under the rail and was attenuated toward the track centerline and the embankment shoulder; the transverse distribution of the dynamic stress exhibited a bimodal pattern with two peak points and a tensile stress appeared under the middle part of the train carriage. In addition, the train ran smoothly on a smooth track; the difference of the dynamic stress at the subgrade surface under the four bogies was less than 10 kPa, and the effects of a bogie were similar to a nonuniform rectangular pressure, as shown in Figure 9(a). Conversely, the train exhibited bouncing movements on the vertically irregular track, thus the maximum dynamic stress was greater than that under a smooth track condition; nodding motion appeared on the train carriages and bogies, hence the dynamic stresses at the subgrade surface under the four bogies appeared to be obviously different with a greater deviation of approximately 20 kPa (Figure 9(b)).Figure 9
Dynamic stress contour at the subgrade surface: (a) smooth track; (b) vertically irregular track.
(a)
(b)Figures10 and 11 present the dynamic stress contour at the bottom of the SBSL and SBBL under irregular track conditions, respectively. It is determined that the transverse distribution of the dynamic stress varied from a bimodal pattern to a unimodal pattern with an increase in the depth from the subgrade surface (Figures 9(b) and 10). The dynamic stress directly beneath the rail was close to that under the track centerline at the bottom of the subgrade bed surface layer, as illustrated in Figure 10. The distribution of the dynamic stress induced by adjacent bodies of adjacent carriages exhibited an elliptical shape and the superimposed dynamic stress was apparently greater than that induced by a single bogie (Figure 11). Therefore, the superposition effects of the adjacent bogies could enlarge the influencing depth of the dynamic stress because of the small bogie spacing of the heavy-haul train.Figure 10
Dynamic stress contour at the bottom of the subgrade bed surface layer.Figure 11
Dynamic stress contour at the bottom of the subgrade bed bottom layer.Figures12(a) and 12(b) present the subgrade surface dynamic stress-time curves directly below the rail and track centerline under different train axle loads, respectively. It shows that the maximum dynamic stresses (σdm) at the subgrade surface directly below the rail and track centerline had increments of 33.8 kPa and 32 kPa (corresponding to percent increases of 39.8% and 47.3%) when the train axle load increased from 25 t to 35 t, respectively. An increase in the train axle load intensified the subgrade dynamic stress and the interaction between the train and track system. In addition, the subgrade surface dynamic stresses induced by the adjacent bogies exhibited an obvious superposition effect, and the dynamic stresses were maintained around the peak value during the pass of the adjacent bogies; hence, the pass of adjacent bogies of adjacent carriages may be simplified once loaded on the railway subgrade.Figure 12
Dynamic stress at the subgrade surface: (a) below the rail; (b) below track centerline.
(a)
(b)The relationships between the maximum dynamic stress and train axle load (P) at different locations below a sleeper are illustrated in Figure 13. It shows that the maximum dynamic stresses increased approximately linearly as the train axle load increased. The average increase rate of the dynamic stress directly below the rail was 3.17 kPa/t, 3.73 kPa/t, and 1.52 kPa/t at the subgrade surface and the bottoms of SBSL and SBBL, respectively. For the locations directly below the track centerline, the average increase rates were 3.19 kPa/t, 3.59 kPa/t, and 1.78 kPa/t. Generally, the dynamic stress at the bottom of the SBSL was more sensitive to the train axle load compared to that at the subgrade surface because the dynamic stresses induced by the two rails were gradually overlaid with an increase in the depth from the subgrade surface.Figure 13
Relationship between the maximum dynamic stress and train axle load.Figure14 presents the relationship between the maximum dynamic stress and the depth from the subgrade surface. σdm at the subgrade surface directly below the rail as well as its attenuation rate was greater than that below the track centerline. σdm below the rail became less than that below the track centerline at a depth of approximately 0.3 m. σdm at the subgrade bed bottom was reduced by approximately 70% compared with that at the subgrade surface. The ratio of σdm to the self-weight stress (σs) at the subgrade bed bottom (2.5 m below the subgrade surface) under a train axle load of 25 t was 0.21 and 0.22 for the locations directly below the rail and the track centerline, respectively; the σdm/σs values increased to 0.34 and 0.36 under a train axle load of 30 t, respectively. Zhou [33] suggested that the σdm/σs value should be less than 0.20 to maintain the long-term stability of a railway subgrade. Consequently, the thickness of the subgrade bed (2.5 m, a subgrade bed surface layer of 0.6 m and a subgrade bed bottom layer of 1.9 m) of existing railways in China may meet the requirements of running a heavy-haul train with an axle load of 25 t but cannot enable the operation of heavy-haul trains with axle loads greater than 25 t unless the subgrades have been reinforced.Figure 14
Relationship between the maximum dynamic stress and depth from the subgrade surface.


## 4.3. Effects of Irregular Track Spectrum

Figure15 illustrates the subgrade surface dynamic stress-time curves under different track conditions. The vertical track irregularity intensified the vertical excitation between the train wheels and rails and induced a greater dynamic stress compared with the smooth rail condition. The maximum dynamic stress at the subgrade surface increased by 23% and 15.8% under the action of the heavy-haul irregular track spectrum and China’s three main railway lines irregular track spectrum, respectively.Figure 15
Dynamic stress at the subgrade surface under different irregular track spectra: (a) below rail; (b) below centerline.
(a)
(b)In China, the irregular track spectra of the left and right rails are different. The dynamic stress contour at the subgrade surface taking this difference into account is shown in Figure16. It is determined that the FE model effectively reflected the irregularity difference of the left and right rails, and the dynamic stress was not symmetrically distributed along the track centerline.Figure 16
Dynamic stress contour at the subgrade surface considering the irregularity difference of the left and right rails.Figure17 presents the transverse distributions of the dynamic stress at different depths. The subgrade surface dynamic stress exhibited a bimodal pattern (or an “M” shape) and the value under the left rail was approximately 19% greater than that under the right rail, whereas the asymmetry of the dynamic stress gradually decreased as the depth increased and was negligible at the subgrade bed bottom; hence, the irregularity difference between the left and right rails had a slight impact on the embankment soils below the subgrade bed bottom.Figure 17
Transverse distributions of dynamic stresses at different depths.


## 4.4. Statistical Analysis of the Maximum Dynamic Stress at the Subgrade Surface

The maximum dynamic stress at the subgrade surface was randomly distributed under different sleepers because of the vertical track irregularity. Figure18 illustrates the statistical results of σdm at the subgrade surface directly below the intersection points of the rail and sleepers. It is observed that the track conditions significantly affected the distribution of σdm; σdm was relatively small and the distribution was relatively concentrated when the rail track was smooth; and σdm increased and the distribution range became wider when the track condition worsened, indicating that the subgrade bed layers were more likely to withstand a greater dynamic stress, which is harmful for maintaining the long-term service performance of the subgrade.Figure 18
Statistical graph of the maximum dynamic stress at the subgrade surface: (a) smooth rail; (b) irregularity in China’s three main lines; (c) heavy-haul track irregularity.
(a)
(b)
(c)The statistical distributions ofσdm under different irregular track conditions can be well-fitted by normal distribution curves as illustrated in Figure 18, and the expectation (μ) and standard deviation (σ) increased as the track condition worsened.The possible maximum dynamic stress at the subgrade surface was predicted using the normal distribution theory. Ifσdm∼(μ, σ2), then(5)Pσdm−μ≥3σ=1−Pσdm−μ<3σ=1−0.9973=0.0027<0.003.Therefore, the probability ofσdm outside the interval of (μ − 3σ, μ + 3σ) was less than 0.003. Statistically, the interval of (μ-3σ, μ+3σ) is generally taken as the actual possible range of a normal distribution random variable, namely, the triple standard deviation principle. Consequently, the possible maximum value of the dynamic stress at the subgrade surface can be estimated as μ + 3σ. The analyzed possible maximum dynamic stresses at the subgrade surface under different irregular track spectra are listed in Table 5. The maximum dynamic stress at the subgrade surface of the Shuo-Huang heavy-haul railway induced by a train with an axle load of 30 t could reach 123 kPa [34], which was favorably simulated with a predicted value of 116.9 kPa (with a deviation less than 5%).Table 5
Statistical results of the maximum dynamic stress at the subgrade surface.
Statistical parameter
Axle load (t) for heavy-haul track irregularity
Axle load (t) for China’s three main lines track irregularity
30
32.5
35
30
32.5
35
μ (kPa)
86.6
97.3
110.3
74.4
82.1
91.0
σ (kPa)
10.1
9.82
10.23
8.1
8.34
8.91
μ + 3σ (kPa)
116.9
126.8
141.0
98.7
107.1
117.7


## 5. Conclusions

A 3D train-track-subgrade interaction finite element (FE) model was constructed using the ABAQUS code (version 6.14-5), where different vertical irregular track spectra were simulated by modifying the rail node coordinates in the ABAQUS INP file. Then, the dynamic stress of the subgrade bed layers of a heavy-haul railway embankment was analyzed in detail using the 3D FE model. The main conclusions drawn from this study are as follows:(1)
The transverse distribution of the dynamic stress in the subgrade bed layers gradually transformed from a bimodal pattern to a unimodal pattern with an increase in the depth from the subgrade surface.(2)
The dynamic stress at the subgrade surface induced by the adjacent bogies of adjacent carriages exhibited obvious superposition effects because of the small adjacent bogie spacing as well as the dynamic stresses maintained around the peak value during the pass of the adjacent bogies; hence, the pass of adjacent bogies can be simplified once loaded on the subgrade.(3)
The maximum dynamic stress at different locations increased approximately linearly as the train axle load increased. The dynamic stress at the bottom of the subgrade bed surface layer was more sensitive to the train axle load compared with that at the subgrade surface because the dynamic stresses induced by the two rails were gradually overlaid with an increase in the depth from the subgrade surface.(4)
The maximum dynamic stress at the subgrade bed bottom was reduced by approximately 70% compared with that at the subgrade surface. The ratio of maximum dynamic stress to the self-weight stress at the subgrade bed bottom under a train axle load of 25 t exceeded the recommended value of existing design standards (0.2); consequently, the existing railways in China cannot enable the operation of heavy-haul trains with axle loads greater than 25 t unless the railway subgrades have been reinforced.(5)
The vertical track irregularity intensified the vertical excitation between the train wheels and rails, and the maximum dynamic stress at the subgrade surface under the action of the irregular heavy-haul track spectrum increased by 23%. The maximum dynamic stress and its distribution range increased with the track irregularity increasing, which was harmful for maintaining the long-term service performance of the subgrade.(6)
Based on the normal distribution theory, the possible maximum dynamic stress (σdm) at the subgrade surface under the action of an irregular track spectrum can be predicted using the triple standard deviation principle, i.e., σdm = μ + 3σ (where μ and σ are the expectation and standard deviation of σdm, respectively).



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*Source: 2916096-2018-12-23.xml*