Train Moving Load-Induced Vertical Superimposed Stress at Ballasted Railway Tracks

Wang, Heng; Zeng, Ling-Ling; Bian, Xia; Hong, Zhen-Shun

doi:https://doi.org/10.1155/2020/3428395

Advances in Civil Engineering

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Abstract Introduction Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 3428395 | https://doi.org/10.1155/2020/3428395

Train Moving Load-Induced Vertical Superimposed Stress at Ballasted Railway Tracks

Heng Wang,¹Ling-Ling Zeng,²Xia Bian,³and Zhen-Shun Hong⁴

Academic Editor: Jian Ji

Received28 Aug 2019

Revised17 Nov 2019

Accepted14 Jan 2020

Published21 Feb 2020

Abstract

Ballasted railway track is an important factor that forms railway transportation over the world, which may face severe damage during operation due to the deterioration of the track ballast geometry. A practical method of evaluating train moving load-induced vertical superimposed stress in substructure by incorporating the effects of ballast characteristics and multilayered substructure is presented. The proposed method is validated by comparing with the field measurements compiled from the literature with the calculated value. It is found that the prediction accuracy of the proposed method is within ±10%, in comparison with field measurements. Meanwhile, it should be emphasized that the predicted value by traditional methods was 1.4–5.0 times field measurements. Also, key factors affecting the predicted accuracy are identified through parameter analysis by using the proposed and the traditional methods.

1. Introduction

Nowadays, a ballasted railway support system typically consists of the superstructure (track and tie) and substructure (ballast and subgrade). Train load sequentially is transmitted through superstructure, towards ballast and subgrade [1]. With the substantial increase in train speeds and axle load, the subgrade is often confronted with settlement problems at various degrees due to the rapid increase in superimposed stress transferred into subgrade [2]. Hence, it is essential to calculate the vertical superimposed stress in subgrade (σ_z) for determining train load-induced settlement. Traditionally, σ_z was empirically determined by the trapezoidal method or the Boussinesq method in practice [3–5].

Note that the trapezoidal method was established based on the assumption of the perfectly flexible loaded area between ballast and subgrade layers, and homogeneous substructure without considering the ballast characteristics [6, 7]. In practice, the load area cannot be considered as perfectly flexible. As a result, the trapezoidal method significantly overestimated the vertical stress in comparison with the field measurement [5, 8]. In addition, the ballast aggregates subjected to the cycle train loading, resulting in considerable ballast fouling, such as particle degradation, infiltration of fines from surface, subballast and subgrade infiltration, and weathering factors [2, 3, 9–21]. Therefore, the properties of ballast (the internal friction angle of ballast (φ_B) and maximum particle size (D_max)) and stress distribution in substructure significantly varied with the real situations in practice [22–24]. Hence, the ballast characteristics should be taken into account for the determination of superimposed stress in the subgrade.

On the other hand, the Boussinesq method was based on the assumption of a homogeneous half-space for ballast and subgrade layers without considering the effect of multilayered structures [5, 7]. This assumption was far from the real condition of the railway substructure [25, 26]. This resulted in oversimplifying actual railway conditions, eventually leading to significant overestimation of train load-induced settlements [5, 8].

This study aims at proposing a practical approach for estimating σ_z by incorporating the effects of granular characteristics and multilayered substructure. Simple equations were proposed for estimating σ_z using the particulate-probabilistic theory [27] and the Kandaurov solution [28]. The validity of the proposed equations was investigated by the comparison between computed results and field measurements. Finally, the effects of key factors of φ_B and D_max on σ_z were studied.

2. Proposed Empirical Equation

2.1. A Brief Review of Traditional Stress Distribution Equations

Figure 1 shows typical load distribution from wheel to the railway track, tie, ballast, and subgrade layers for standard gauge track in China and France with railway gauge of 1.435 m and tie spacing of 0.6 m. Note that P_d represents the design load; q_r is the maximum rail seat load; σ_max is tie-ballast contact pressure; L is length of the tie; L is the effective length of tie supporting q_r; B is the tie width; 2a is the average contact width between tie and ballast; h_B is the ballast layer thickness; h_n is the nth layer thickness; H_n−1 is the equivalent depth; and Z_n is the distance from the top of nth layer. x, y, z represent the distance from load position. In the railway loading system, the rails transferred the wheel load to the ties, ballast layer, and subgrade sequentially. Thereby, tie-ballast contact stress (σ_max) was essential for determining σ_z.

Generally, P_d represents the design wheel load incorporating the dynamic effects, which can be determined by the empirical relation between static load P_s and dynamic amplification factor φ, expressed as follows [2–5, 29]:

Note that the calculation of φ can be practically related to the train speed by a power function based on kinematic theory [3]. As proposed by the Office of Research and Experiments of the International Union of Railways, the relation can be expressed as follows [2, 3, 5, 8]:where k was a constant depending on track and vehicle condition. Usually, k = 0.03 was used for common levelling defects and depressions [29]. Doyle [3] pointed out equation (2) was appropriate for the cases with < 200 km/h and the maximum value of φ was 1.9. Due to the fact that there was no adequate theoretical basis for this equation, a trial analysis was required to implement equation (2) for the case < 200 km/h. For the Qin-Shen railway and Orleans-Montauban railway, the measured stresses with a train speed of 5 km/h and 60 km/h are selected as the reference values, respectively. Figure 2 shows the amplification of measured stress with train speed up to 200 km/h. It can be seen that the cases investigated in this study are well evaluated by equation (2).

Based on the experiment data with various types of ties (tie spacing of 0.6 m), several researchers illustrated that only 32–76% of the wheel load was carried by the tie beneath the wheel and other parts of load were transmitted laterally to adjacent ties [2, 3]. For simplification in engineering practice, the design wheel load P_d on a rail seat was most commonly assumed distributed between three adjacent ties, and q_r = 0.5P_d was suggested for the case with tie spacing of 0.6 m [2–5].

Accordingly, the uniform contact stress σ_max distribution between the effective area (BL) of the tie and ballast was determined as follows:where safety factor F₂ = 2 was recommended by AREA [4] with considering possible excessive contact pressures due to nonuniform tie support. Field measurements indicated that the maximum contact stress was exerted by the tie to the underlying ballast [5]. The trial analysis and the results from the literature [2, 5, 8] shown in Figure 2 indicated that equation (2) was suitable for the case of train speed under 200 km/h. Hence, equation (3) could be only used for calculating the train-induced dynamic stress within this range.

The effects of train speed, track and vehicle condition, and railway geometric on train loading transfer were incorporated into the calculation of σ_max in equation (3). Finally, the vertical stress σ_z can be calculated based on σ_max using the trapezoidal method or Boussinesq method without considering the effects of granular characteristics and multilayered substructure. However, there was a lack of accuracy using the traditional methods compared to the field measurements due to the oversimplification of real situations.

2.2. Proposed Equation

In practice, the track substructure layers are composed of a complex conglomeration of discrete particles, in arrays of shape, size, and varying orientations [3, 5]. Note that the average size of the railway ballast was approximately 40 mm, and their present state differs from continuum mechanics [30–33]. On the other hand, the original shape of angular particles gradually became rounded grains during the ballast fouling induced by continuous cyclic train loading. The ballast size (D_max) and friction angle (φ_B) decreased with the increase in fouling content, resulting in a dramatic stress redistribution [22–24]. Hence, the random nature of the particles and the properties of ballast were important for evaluating the stress distribution induced by train loading.

Besides, the track substructure was a multilayered system. If the thickness of the top layer was large enough with respect to the radius of the loaded area, the multilayered system can be treated as a homogeneous layer [7]. However, the upper strata of the substructure are relatively thin in the field. For instance, the substructure of the Orleans-Montauban railway in France was composed of ballast, subballast, and subgrade, with thicknesses of 0.5 m and 0.4 m for the ballast layer and subballast [8, 25, 26]. Hence, the effect of layering must be taken into consideration on train loading-induced vertical stress distribution in the railway substructure.

In summary, the following key factors need to be incorporated into the empirical equation: (1) the random nature of the discrete particles; (2) the properties of the ballast (φ_B and D_max); and (3) multilayered substructures. Hence, the empirical equation incorporating these factors can be proposed as follows.

2.2.1. The Random Nature of Particles

Due to the random nature of granular material, the particulate-probabilistic theory proposed by Harr [27] was adopted in this study, which was a method of estimating the distribution of expected stresses in particulate media based on the central limit theorem of probability. The calculated vertical stress acting at a point in the medium was the total accumulated effect of many random variables: the shape, size, and distribution of the particles [34]; the spatial distribution of the voids; and the transmission of vertical forces proceed from a particle to its neighbours with depth [27]. Hence, this stochastic stress diffusion method was ascendant in incorporating particulate and inherently random nature of granular material [8, 31, 35, 36]. Note that the theory described by Harr was also introduced by Wang et al. [37] for proposing a quantitative method of determining embankment load-induced vertical superimposed stress in the subsoil. Harr provided the solution of σ_z under a uniform normal load P acting over strip of the width 2a as follows:where ν is the coefficient of lateral stress and ψ is the normal cumulative Gaussian distribution function [8, 27, 37]. Under the plane-strain conditions, assume that the uniform normal load P equals to the tie-ballast contact stress σ_max, and 2a equals to the average contact width between tie and ballast.

2.2.2. The Friction Angle F_n

Parameter ν was the coefficient of lateral stress in equation (4), which was related to the coefficient of the lateral earth pressure at rest (K₀) and obtained from the angle of internal friction (φ_n) of granular using Jaky’s formula [38]:

For the ballast layer, φ_n marked as φ_B, related to the particle shape, grain size, and stress level. It can be obtained from laboratory tests or calculated using the following empirical equation [39]:where φ_b represented the true interparticle friction angle determined from the tilt table test and c and d were dimensionless coefficients. Indraratna [39] suggested the average values of φ_b = 35°, c = 31.9, and d = −0.002 for the case with σ_max < 500 kPa, and C_u = 1.5–13, D_max = 38–80 mm, and ϕ_B = 45°–67°.

2.2.3. The Maximum Ballast Size D_max

Several researchers studied the distribution of tie-ballast contact stress in real track and illustrated that the typical maximum ballast size (D_max) ranged from 48 to 70 mm and the typical width of a tie (B) ranged from 200 to 290 mm [4, 5, 39]. In other words, the typical D_max/B was in the range of 16.6% to 35.0%. This implies that the number of ballast particles involved indirectly supporting the tie was relatively small, which had been shown as dotted lines in Figure 3.

The contact width 2a was a key parameter in equation (4), and the determination of the effective support width (2a) was very difficult. McHenry [40] studied the effective support area between tie and ballast via laboratory tests. They estimated that the effective support area 2a/B varied from 21.9% to 31.2% for new ballast, and that of fouled ballast was between 28.4% and 39.7% for the case with D_max = 40–76 mm, C_u = 1.5–6.5, and average axle load of 18 t. It is important to note that the values of 2a/B (21.9–39.7%) are approximate to the values of D_max/B (16.6–35.0%). Since the value of 2a/B and D_max/B was almost the same, 2a = D_max was suggested in this study for the sake of simplicity. Therefore, the value of 2a represented by the typical maximum ballast size D_max can be obtained from the gradation of ballast material for the investigated case.

2.2.4. The Multilayered Substructures

Equation (4) can be used for determining vertical superimposed stress for a single-layer structure. However, the track substructure is a multilayered system. Based on the linear elastic theory, Odemark [41] firstly developed an empirical method to convert the multilayered system to a single-layer system, and the equivalency was calculated by the elastic moduli (E_n) and Poisson’s ratio. Ullidtz [31] emphasized that this method only approximated for the case when elastic moduli decreased with depth (E_n/E_n+1 > 2) and the top layer was larger compared to the radius of the load area. However, these assumptions were far from the real condition of the railway substructure. For the Orleans-Montauban railway in France, the mean moduli estimated from the penetrometer test are 133 MPa, 103 MPa, and 75 MPa for ballast, subballast, and subgrade, and the top ballast layer of 0.5 m was less than the radius of 0.52 m of the load area (BL) [8, 25, 26].

For engineering applications, Kandaurov [28] proposed a comprehensive method of multilayered equivalency by a coefficient of lateral stress ν:where H_n−1 represents the equivalent depth, h_n represents the layer thickness, and ν_n represents the coefficient of lateral stress of the nth layer. This method can be applied to the multilayered system for the cases of any layer thickness and variation of ν values, and no assumptions regarding stress conditions were required [7, 31]. The values of ν can be obtained from equation (5).

Combining equations (3), (5), and (7) into equation (4), the proposed equation (8) presents the general way of determining σ_z in railway multilayered substructures. The detailed derivation of the empirical solution of the proposed method is presented in the Appendix:

3. The Validity of the Proposed Method

3.1. Vertical Superimposed Stress in Substructure

A total of 63 field measurements of ten well-documented railways in China and France were used to validate the proposed method of determining σ_z [8, 42–46]. The substructure granular material properties (D_max and φ_n) and railway geometric parameters (P_s, V, B, L, h_B, and h_n) obtained from the technical specifications of the investigated cases are presented in Table 1. In the analysis, the values of φ_B are obtained by laboratory triaxial tests reported in the literature or estimated by equation (6).

The field measurements of σ_z for the ten cases discussed in this study are listed in Table 2. The different σ_z values correspond to different V or z values. For comparison, the predicted value of σ_z determined by the trapezoidal method can be calculated using the following equations [5]:

A trapezoid with 2 : 1 inclined sides was generally adopted in this method.

Besides, σ_z with the Boussinesq method can be determined bywhere dε and dη are the length and width of a tie, respectively.

Figure 4 shows the typical comparisons between the predicted value of the trapezoidal method and the measured vertical stresses. The calculated results significantly deviated from the field measurements, varying within a wide range from 1.4 to 4.0 times the measured values. Figure 5 indicates that the Boussinesq method also yields higher stresses than the field measurements, varying within a wide range from 1.9 to 5.0 times the field measurements. These behaviours indicated that the effect of granular material characteristics and multilayered substructures should be taken into account to determine σ_z.

Figure 6 shows the typical distributions of σ_z along with the depth, along with the predicted value using the proposed method (equation (8)), the trapezoidal method, and the Boussinesq method. It can be seen that the calculated results using equation (8) are in agreement with the measured ones. The comparisons between field measurements and the predicted value using different methods for the ten cases are shown in Figure 7. It is encouraged that the results predicted by the proposed method possessed a high accuracy of ±10% in comparison with the field observations. The calculated results by the trapezoidal method and Boussinesq method are also shown in the same figure. The proposed method can significantly improve the accuracy compared with the traditional methods for the cases in China and France with railway track gauge of 1.435 m, tie spacing of 0.6 m, D_max = 50–80 mm, and ϕ_B = 45°–55°.

It should be emphasized that the standard of track spacing and tie spacing significantly varied around the world. For example, a COAL Link Line [47] with the track spacing of 1.065 m and tie spacing of 0.65 m in South Africa is presented in Tables 3 and 4. The maximum ballast size D_max was not reported in the literature and was assumed to be equal to 60 mm or 80 mm for trial analysis in this study. Figure 8 shows that the calculated results using equation (8) also have an acceptable accuracy compared with the measured ones. Due to the limited database, the application of the proposed method for cases with another standard of track spacing and tie spacing is still need to be further verified.

3.2. Parametric Analysis

With the increase in train speed and axle load in China and other countries, the ballast aggregates exhibited considerable ballast fouling due to cycle train loading. Indraratna et al. [22] introduced the void contaminant index (VCI) for railway ballast to quantify the extent of fouling. VCI = 0% represent fresh ballast. Laboratory test results indicated that the ballast size (D_max) and friction angle (φ_B) decreased with the increase in VCI. When ballast was fouled, the ballast breakage of the sharp corners and attrition of asperities occurred. The pore matrix of the ballast assembly changed substantially as the crushed fine particles clogging the voids and the number of particle contacts increased, resulting in vertical stress redistribution in the ballast layer [5]. As a result, an increasing percentage of horizontal diffuseness of train load may occur through the fine particle networks and the maximum vertical stress σ_z may reduce in the railway substructure [22–25].

To validate the application of the proposed method for simulating the above real ballast response, a parameter analysis was conducted. Section 1 of Qin-Shen passenger rail line in Table 1 was chosen for parameter analysis. Note that the calculated points located beneath the tie with a depth of z = 0.35 m, 0.50 m, 0.75 m, and 0.95 m were selected for analysis. Figure 9 presents the influences of φ_B on stress distributions. D_max = 50 mm and φ_sub = 25° were kept and φ_B = 50°, 37.5°, 25°, 12.5° were taken into consideration. It can be seen that the maximum vertical stress σ_z decreased nonlinearly with the decrease in φ_B. In Figure 10, φ_B = 50° and φ_sub = 25° were kept and D_max varied from 80, 70, 60, and 50 to 40 mm. As expected, the maximum vertical stress σ_z decreased nonlinearly with the decreases in D_max. These behaviours imply that the effects of ballast characteristics on train load transfer can be quantitatively evaluated using the φ_B and D_max by the proposed method. The decrease in maximum vertical superimposed stress for “the proposed equation with ballast characteristics” is consistent with the real ballast response under train loading.

It should be emphasized that the ballast fouling is a very complex problem, due to lack of quantitative equations describing the relationship between the extent of fouling (VCI) and D_max or φ_B; quantitatively assessing the fouling effect on train loading transfer is still need to be further studied. In conclusion, under the assumption that the aggregates are all connected during train loading and ignore the role of moisture on ballast fouling, the proposed method incorporating the effects of ballast characteristics (size and friction angle) and multilayered substructure on train loading transfer is recommended to empirically calculate σ_z, which only depends on the simple geometric parameters of track, the friction angle of granular material (ϕ), and the typical maximum ballast size (D_max).

4. Conclusions

A practical method of calculating train load-induced vertical superimposed stresses in subgrade is presented by incorporating the effects of ballast characteristics and multilayered substructure on load transfer.

The proposed approach is validated based on field measurements of train load-induced vertical superimposed stresses in subgrade compiled from the literature. It is found that the calculated results with the proposed method have a good agreement with the measurements within an accuracy of ±10%, with the railway track spacing of 1.435 m, tie spacing of 0.6 m, D_max = 50–80 mm, and ϕ_B = 45°–55°.

The proposed method incorporates the effects of ballast characteristics and multilayered substructure on vertical superimposed stresses in subgrade, substantially improving their calculating accuracy. The key influential factors responsible for ballast characteristics and multilayered substructure are found to be the maximum ballast size and the friction angle of granular material.

Appendix

Derivation of the Proposed Equation.

The solution of σ_z under a uniform normal load acting over strip of width 2a can be expressed as follows:

The coefficient of lateral stress ν was taken as the coefficient of the lateral earth pressure at rest (K₀) and obtained from the angle of internal friction (ϕ_n) of granular using Jaky’s formula:

Note that ψ () was the normal cumulative Gaussian distribution function expressed as follows:

Assuming that the uniform normal load equals to the tie-ballast contact stress σ_max, and 2a equals to the average contact width between tie and ballast, then

The effective support width equal to the ballast maximum size was suggested in this study as follows:

Hence, the σ_z at certain point for single-layer structure can be calculated as follows:

The Kandaurov method was introduced for determining the vertical stress in railway multilayered substructure:

Note that the equivalent distance from the calculated point to the tie (z) can be rewritten as follows:

Substituting equations (A.7) and (A.8) into equation (A.6) gives

Notations

a:	Half of the contact width between tie and ballast
B:	Width of tie
c, d:	Empirical coefficients
C_u:	Uniformity coefficients
D_max:	Maximum particle size
F₂:	Safety factor
h_B:	Thickness of the ballast layer
h_n:	Thickness of the nth layer
H_n−1:	Equivalent depth
K₀:	The coefficient of the lateral earth pressure at rest
k:	The constant depending on track condition
L:	Length of tie
L:	Effective length of tie supporting q_r
P_d:	Design load
P_s:	Static load
r:	The radius of a circle whose area equals to
t:	Gaussian function parameters
V:	Train speed
VCI:	Void contaminant index
x, y, z:	The distance from the load position
Z_n:	The distance from the top of nth layer
q_r:	Maximum rail seat load
φ:	Dynamic amplification factor
φ:	The internal friction angle of grain material
φ_B:	The internal friction angle of ballast
φ_b:	The true interparticle friction angle determined from tile table test
φ_n:	The internal friction angle of nth grain layer
φ_sub:	The internal friction angle of subgrade layer
σ_z:	Vertical superimposed stress in substructure
σ_zmax:	The maximum vertical stress applied on the subgrade
σ_max:	Tie-ballast contact pressure
dε:	The length of a tie
dη:	The width of a tie
ψ:	Cumulative Gaussian distribution function
ν:	Coefficient of lateral stress.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported by the National Natural Science Foundation of China (Grant nos. 51678157 and 41977243) and Fok Ying Tung Education Foundation (Grant no. 161070).

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Copyright © 2020 Heng Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Advances in Civil Engineering

Train Moving Load-Induced Vertical Superimposed Stress at Ballasted Railway Tracks

Abstract

1. Introduction

2. Proposed Empirical Equation

2.1. A Brief Review of Traditional Stress Distribution Equations

2.2. Proposed Equation

2.2.1. The Random Nature of Particles

2.2.2. The Friction Angle Fn

2.2.3. The Maximum Ballast Size Dmax

2.2.4. The Multilayered Substructures

3. The Validity of the Proposed Method

3.1. Vertical Superimposed Stress in Substructure

3.2. Parametric Analysis

4. Conclusions

Appendix

Notations

Data Availability

Conflicts of Interest

Acknowledgments

References

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2.2.2. The Friction Angle F_n

2.2.3. The Maximum Ballast Size D_max