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

Authors: Heng Wang; Ling-Ling Zeng; Xia Bian; Zhen-Shun Hong
Journal: Advances in Civil Engineering
 (2020)

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## 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.


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## Body

## 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 (σz) for determining train load-induced settlement. Traditionally, σz 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 (φB) and maximum particle size (Dmax)) 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 estimatingσz by incorporating the effects of granular characteristics and multilayered substructure. Simple equations were proposed for estimating σz 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 φB and Dmax on σz were studied.


## 2. Proposed Empirical Equation

### 2.1. A Brief Review of Traditional Stress Distribution Equations

Figure1 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 Pd represents the design load; qr is the maximum rail seat load; σmax is tie-ballast contact pressure; L is length of the tie; L∗ is the effective length of tie supporting qr; B is the tie width; 2a is the average contact width between tie and ballast; hB is the ballast layer thickness; hn is the nth layer thickness; Hn−1 is the equivalent depth; and Zn 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 (σmax) was essential for determining σz.Figure 1
Schematic layout of a typical railway track system: (a) the trapezoidal method; (b) The Boussinesq method.Generally,Pd represents the design wheel load incorporating the dynamic effects, which can be determined by the empirical relation between static load Ps and dynamic amplification factor φ, expressed as follows [2–5, 29]:(1)Pd=φPs.Note that the calculation ofφ 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]:(2)φ=1+kV1003,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 V < 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 V < 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).Figure 2
Measured vertical stress amplification caused by train loading.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 Pd on a rail seat was most commonly assumed distributed between three adjacent ties, and qr = 0.5Pd was suggested for the case with tie spacing of 0.6 m [2–5].Accordingly, the uniform contact stressσmax distribution between the effective area (BL∗) of the tie and ballast was determined as follows:(3)σmax=qrBL∗F2=0.5PdBL∗F2=0.5ϕPsBL∗F2=0.5PsBL∗1+kV1003F2,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 ofσmax in equation (3). Finally, the vertical stress σz can be calculated based on σmax 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 (Dmax) and friction angle (φB) 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 (φB and Dmax); 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 σz under a uniform normal load P∗ acting over strip of the width 2a as follows:(4)σzx,z=P∗×ψx+azν−ψx−azν,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 σmax, and 2a equals to the average contact width between tie and ballast.


#### 2.2.2. The Friction AngleFn

Parameterν 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 (φn) of granular using Jaky’s formula [38]:(5)νn=K0=1−sinϕn.For the ballast layer,φn marked as φB, 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]:(6)ϕB=ϕb+expdσmax,where φb 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 φb = 35°, c = 31.9, and d = −0.002 for the case with σmax < 500 kPa, and Cu = 1.5–13, Dmax = 38–80 mm, and ϕB = 45°–67°.


#### 2.2.3. The Maximum Ballast SizeDmax

Several researchers studied the distribution of tie-ballast contact stress in real track and illustrated that the typical maximum ballast size (Dmax) 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 Dmax/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.Figure 3
The average pressure distribution and actual pressure distribution at the tie-ballast interface.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 Dmax = 40–76 mm, Cu = 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 Dmax/B (16.6–35.0%). Since the value of 2a/B and Dmax/B was almost the same, 2a = Dmax was suggested in this study for the sake of simplicity. Therefore, the value of 2a represented by the typical maximum ballast size Dmax 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 (En) and Poisson’s ratio. Ullidtz [31] emphasized that this method only approximated for the case when elastic moduli decreased with depth (En/En+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:(7)H¯n−1=h1ν1νn+h2ν2νn+∼+hn−1νn−1νn,where Hn−1 represents the equivalent depth, hn represents the layer thickness, and νn 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 σz in railway multilayered substructures. The detailed derivation of the empirical solution of the proposed method is presented in the Appendix:(8)σzx,z=σmaxψx+0.5DmaxH¯n−1+Zn1−sinϕn−ψx−0.5DmaxH¯n−1+Zn1−sinϕn.


## 2.1. A Brief Review of Traditional Stress Distribution Equations

Figure1 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 Pd represents the design load; qr is the maximum rail seat load; σmax is tie-ballast contact pressure; L is length of the tie; L∗ is the effective length of tie supporting qr; B is the tie width; 2a is the average contact width between tie and ballast; hB is the ballast layer thickness; hn is the nth layer thickness; Hn−1 is the equivalent depth; and Zn 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 (σmax) was essential for determining σz.Figure 1
Schematic layout of a typical railway track system: (a) the trapezoidal method; (b) The Boussinesq method.Generally,Pd represents the design wheel load incorporating the dynamic effects, which can be determined by the empirical relation between static load Ps and dynamic amplification factor φ, expressed as follows [2–5, 29]:(1)Pd=φPs.Note that the calculation ofφ 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]:(2)φ=1+kV1003,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 V < 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 V < 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).Figure 2
Measured vertical stress amplification caused by train loading.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 Pd on a rail seat was most commonly assumed distributed between three adjacent ties, and qr = 0.5Pd was suggested for the case with tie spacing of 0.6 m [2–5].Accordingly, the uniform contact stressσmax distribution between the effective area (BL∗) of the tie and ballast was determined as follows:(3)σmax=qrBL∗F2=0.5PdBL∗F2=0.5ϕPsBL∗F2=0.5PsBL∗1+kV1003F2,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 ofσmax in equation (3). Finally, the vertical stress σz can be calculated based on σmax 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 (Dmax) and friction angle (φB) 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 (φB and Dmax); 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 σz under a uniform normal load P∗ acting over strip of the width 2a as follows:(4)σzx,z=P∗×ψx+azν−ψx−azν,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 σmax, and 2a equals to the average contact width between tie and ballast.


### 2.2.2. The Friction AngleFn

Parameterν 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 (φn) of granular using Jaky’s formula [38]:(5)νn=K0=1−sinϕn.For the ballast layer,φn marked as φB, 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]:(6)ϕB=ϕb+expdσmax,where φb 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 φb = 35°, c = 31.9, and d = −0.002 for the case with σmax < 500 kPa, and Cu = 1.5–13, Dmax = 38–80 mm, and ϕB = 45°–67°.


### 2.2.3. The Maximum Ballast SizeDmax

Several researchers studied the distribution of tie-ballast contact stress in real track and illustrated that the typical maximum ballast size (Dmax) 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 Dmax/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.Figure 3
The average pressure distribution and actual pressure distribution at the tie-ballast interface.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 Dmax = 40–76 mm, Cu = 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 Dmax/B (16.6–35.0%). Since the value of 2a/B and Dmax/B was almost the same, 2a = Dmax was suggested in this study for the sake of simplicity. Therefore, the value of 2a represented by the typical maximum ballast size Dmax 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 (En) and Poisson’s ratio. Ullidtz [31] emphasized that this method only approximated for the case when elastic moduli decreased with depth (En/En+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:(7)H¯n−1=h1ν1νn+h2ν2νn+∼+hn−1νn−1νn,where Hn−1 represents the equivalent depth, hn represents the layer thickness, and νn 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 σz in railway multilayered substructures. The detailed derivation of the empirical solution of the proposed method is presented in the Appendix:(8)σzx,z=σmaxψx+0.5DmaxH¯n−1+Zn1−sinϕn−ψx−0.5DmaxH¯n−1+Zn1−sinϕn.


## 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 σz under a uniform normal load P∗ acting over strip of the width 2a as follows:(4)σzx,z=P∗×ψx+azν−ψx−azν,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 σmax, and 2a equals to the average contact width between tie and ballast.


## 2.2.2. The Friction AngleFn

Parameterν 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 (φn) of granular using Jaky’s formula [38]:(5)νn=K0=1−sinϕn.For the ballast layer,φn marked as φB, 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]:(6)ϕB=ϕb+expdσmax,where φb 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 φb = 35°, c = 31.9, and d = −0.002 for the case with σmax < 500 kPa, and Cu = 1.5–13, Dmax = 38–80 mm, and ϕB = 45°–67°.


## 2.2.3. The Maximum Ballast SizeDmax

Several researchers studied the distribution of tie-ballast contact stress in real track and illustrated that the typical maximum ballast size (Dmax) 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 Dmax/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.Figure 3
The average pressure distribution and actual pressure distribution at the tie-ballast interface.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 Dmax = 40–76 mm, Cu = 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 Dmax/B (16.6–35.0%). Since the value of 2a/B and Dmax/B was almost the same, 2a = Dmax was suggested in this study for the sake of simplicity. Therefore, the value of 2a represented by the typical maximum ballast size Dmax 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 (En) and Poisson’s ratio. Ullidtz [31] emphasized that this method only approximated for the case when elastic moduli decreased with depth (En/En+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:(7)H¯n−1=h1ν1νn+h2ν2νn+∼+hn−1νn−1νn,where Hn−1 represents the equivalent depth, hn represents the layer thickness, and νn 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 σz in railway multilayered substructures. The detailed derivation of the empirical solution of the proposed method is presented in the Appendix:(8)σzx,z=σmaxψx+0.5DmaxH¯n−1+Zn1−sinϕn−ψx−0.5DmaxH¯n−1+Zn1−sinϕn.


## 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 determiningσz [8, 42–46]. The substructure granular material properties (Dmax and φn) and railway geometric parameters (Ps, V, B, L∗, hB, and hn) obtained from the technical specifications of the investigated cases are presented in Table 1. In the analysis, the values of φB are obtained by laboratory triaxial tests reported in the literature or estimated by equation (6).Table 1
Database of railway geometric parameters in China and France.
Site
Year (y)
Axle load (t)
Track type (kg/m)
Type of ties
L
∗ (m)
B (m)
Ballast type
D
max (m)
h
B (m)
φB (°)
h
2 (m)
φ
2 (°)
Reference
Orleans-Montauban railway
2014
25
Concrete
1.3
0.3
Gravel
0.06
0.5
541
0.4
371
Lamas-lopez et al. [26]
10.5
Zhang et al. [8]
Qin-Shen passenger rail line
Section 1
0.4
Section 4
2002
14.5
60
Concrete
1.1
0.28
Gravel
0.06
0.35
521
0.4
251
Nie et al. [45]
Section 2
0.6
Da-Qin railway
2004
25
75
Concrete
1.0
0.26
Gravel
0.075
0.45
501
3
301
Zhao [46]
Guangzhou-Shenzhen
1994
22.5
60
Concrete
1.3
0.27
Gravel
0.048
0.45
482
Li [43]
Beijing circuit railway
1993
22.5
60
Concrete
1.2
0.26
Gravel
0.06
0.35
542
Li [43]
Han and Zhang [42]
Xiaoshan-Ningbo
2000
22.5
60
Concrete
1.2
0.27
Gravel
0.06
0.45
492
Han and Zhang [42]
Jinan-Qingdao
23
Concrete
1.1
0.29
Gravel
0.05
0.45
492
Liu and Xiao [44]
Tianjin-Pukou
14
Concrete
1.3
0.29
Gravel
0.05
0.6
532
Liu and Xiao [44]
Haerbin-Beian
19.6
43
Concrete
1.3
0.26
Gravel
0.06
0.4
452
Li [43]
Hangzhou-Nanchang
20.1
43
Concrete
1.3
0.26
Gravel
0.06
0.4
532
Li [43]
20
43
442
Li [43]
1Compiled from the literatures. 2Estimated by equation (6).The field measurements ofσz for the ten cases discussed in this study are listed in Table 2. The different σz values correspond to different V or z values. For comparison, the predicted value of σz determined by the trapezoidal method can be calculated using the following equations [5]:(9)σz=B⋅L∗⋅σmaxB+z2L∗+z.Table 2
Field measurements of fourteen cases used.
Site
V (km/h)
z (m)
σ
z (kPa)
Reference
Orleans-Montauban railway
60, 100, 140, 160, 180, 200
0.96, 2.3
12.2, 12.2, 12.4, 13.2, 13.2, 13.83
Lamas-Lopez et al. [26]
7.4, 7.7, 7.7, 8.0, 8.3, 8.9
0.96, 2.3
5.1, 5.0, 5.3, 5.4, 5.2, 5.4
Zhang et al. [8]
1.8, 1.9, 2.1, 2.2, 2.2, 2.3
Qin-Shen passenger rail line
Section 1
0.35, 0.75
31.4, 37.8, 38.5, 38.7
16.0, 16.6, 17.7, 17.2
Section 4
5, 160, 180, 200
0.35, 0.75
30.3, 32.8, 36.7, 38.5
Nie et al. [45]
19.8, 20.5, 21.6, 21.7
Section 2
0.35, 0.95
32.5, 38.2, 39.4, 39.0
16.5, 18.1, 17.5, 18.2
Da-Qin railway
75
0.45, 0.65, 0.95, 1.15, 1.45, 2.45, 3.45
32, 28, 23, 14, 10, 8, 5
Zhao [46]
Guangzhou-Shenzhen
160
0.45
35
Li [43]
Beijing circuit railway
160
0.35
120
Li [43]
Han and Zhang [42]
Xiaoshan-Ningbo
120
0.45
67
Han and Zhang [42]
Jinan-Qingdao
120
0.45
33
Liu and Xiao [44]
Tianjin-Pukou
200
0.6
16
Liu and Xiao [44]
Haerbin-Beian
65
0.4
66
Li [43]
Hangzhou-Nanchang
65
0.4
87
Li [43]
70
68
Li [43]A trapezoid with 2 : 1 inclined sides was generally adopted in this method.Besides,σz with the Boussinesq method can be determined by(10)σz=∑3σmax2π∫ε=−L∗/2ε=+L∗/2∫η=−B/2η=+B/2×z3x−η2+y−ε2+z22.5dεdη,where dε and dη are the length and width of a tie, respectively.Figure4 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 σz.Figure 4
Comparisons between predicted results in the trapezoidal method and measured vertical stresses.Figure 5
Comparisons between predicted results in the Boussinesq method and measured vertical stresses.Figure6 shows the typical distributions of σz 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, Dmax = 50–80 mm, and ϕB = 45°–55°.Figure 6
Stress distribution over depth calculations versus measurement at speed 200 km/h for the Orleans-Montauban railway.Figure 7
Comparisons between predicted results in the proposed method and measured vertical stresses.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 Dmax 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.Table 3
Database of railway geometric parameters in South Africa.
Site
Year (y)
Axle load (t)
Track type (kg/m)
Type of ties
L
∗ (m)
B (m)
Ballast type
D
max (m)
h
B (m)
φ
B (°)
h
n (m)
φ
n (°)
Reference
COAL Link Line1
2004
26
∗
2
60
Concrete
1.1
0.27
Gravel
0.3
542
0.2
303
Yang et al. [47]
0.2
253
0.2
203
0.2
153
1Track gauge = 1.065 m; tie spacing = 0.65 m. 2Estimated by equation (6). 3Estimated from Figures 5 and 8 in Yang et al. [47].Table 4
Database of railway field measurements in South Africa.
Site
V (km/h)
z (m)
σ
z (kPa)
Reference
COAL Link Line
47.5
0.5
95
Yang et al. [47]
0.7
85
0.9
77
1.1
60Figure 8
Comparisons between predicted results in the proposed method and measured vertical stresses for the case in South Africa.


### 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 (Dmax) and friction angle (φB) 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 σz 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. Section1 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 φB on stress distributions. Dmax = 50 mm and φsub = 25° were kept and φB = 50°, 37.5°, 25°, 12.5° were taken into consideration. It can be seen that the maximum vertical stress σz decreased nonlinearly with the decrease in φB. In Figure 10, φB = 50° and φsub = 25° were kept and Dmax varied from 80, 70, 60, and 50 to 40 mm. As expected, the maximum vertical stress σz decreased nonlinearly with the decreases in Dmax. These behaviours imply that the effects of ballast characteristics on train load transfer can be quantitatively evaluated using the φB and Dmax 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.Figure 9
Maximum vertical stress in subgrade with different friction angles of ballast.Figure 10
Maximum vertical stress in subgrade with different particle sizes.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 Dmax or φB; 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 σz, which only depends on the simple geometric parameters of track, the friction angle of granular material (ϕ), and the typical maximum ballast size (Dmax).


## 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 determiningσz [8, 42–46]. The substructure granular material properties (Dmax and φn) and railway geometric parameters (Ps, V, B, L∗, hB, and hn) obtained from the technical specifications of the investigated cases are presented in Table 1. In the analysis, the values of φB are obtained by laboratory triaxial tests reported in the literature or estimated by equation (6).Table 1
Database of railway geometric parameters in China and France.
Site
Year (y)
Axle load (t)
Track type (kg/m)
Type of ties
L
∗ (m)
B (m)
Ballast type
D
max (m)
h
B (m)
φB (°)
h
2 (m)
φ
2 (°)
Reference
Orleans-Montauban railway
2014
25
Concrete
1.3
0.3
Gravel
0.06
0.5
541
0.4
371
Lamas-lopez et al. [26]
10.5
Zhang et al. [8]
Qin-Shen passenger rail line
Section 1
0.4
Section 4
2002
14.5
60
Concrete
1.1
0.28
Gravel
0.06
0.35
521
0.4
251
Nie et al. [45]
Section 2
0.6
Da-Qin railway
2004
25
75
Concrete
1.0
0.26
Gravel
0.075
0.45
501
3
301
Zhao [46]
Guangzhou-Shenzhen
1994
22.5
60
Concrete
1.3
0.27
Gravel
0.048
0.45
482
Li [43]
Beijing circuit railway
1993
22.5
60
Concrete
1.2
0.26
Gravel
0.06
0.35
542
Li [43]
Han and Zhang [42]
Xiaoshan-Ningbo
2000
22.5
60
Concrete
1.2
0.27
Gravel
0.06
0.45
492
Han and Zhang [42]
Jinan-Qingdao
23
Concrete
1.1
0.29
Gravel
0.05
0.45
492
Liu and Xiao [44]
Tianjin-Pukou
14
Concrete
1.3
0.29
Gravel
0.05
0.6
532
Liu and Xiao [44]
Haerbin-Beian
19.6
43
Concrete
1.3
0.26
Gravel
0.06
0.4
452
Li [43]
Hangzhou-Nanchang
20.1
43
Concrete
1.3
0.26
Gravel
0.06
0.4
532
Li [43]
20
43
442
Li [43]
1Compiled from the literatures. 2Estimated by equation (6).The field measurements ofσz for the ten cases discussed in this study are listed in Table 2. The different σz values correspond to different V or z values. For comparison, the predicted value of σz determined by the trapezoidal method can be calculated using the following equations [5]:(9)σz=B⋅L∗⋅σmaxB+z2L∗+z.Table 2
Field measurements of fourteen cases used.
Site
V (km/h)
z (m)
σ
z (kPa)
Reference
Orleans-Montauban railway
60, 100, 140, 160, 180, 200
0.96, 2.3
12.2, 12.2, 12.4, 13.2, 13.2, 13.83
Lamas-Lopez et al. [26]
7.4, 7.7, 7.7, 8.0, 8.3, 8.9
0.96, 2.3
5.1, 5.0, 5.3, 5.4, 5.2, 5.4
Zhang et al. [8]
1.8, 1.9, 2.1, 2.2, 2.2, 2.3
Qin-Shen passenger rail line
Section 1
0.35, 0.75
31.4, 37.8, 38.5, 38.7
16.0, 16.6, 17.7, 17.2
Section 4
5, 160, 180, 200
0.35, 0.75
30.3, 32.8, 36.7, 38.5
Nie et al. [45]
19.8, 20.5, 21.6, 21.7
Section 2
0.35, 0.95
32.5, 38.2, 39.4, 39.0
16.5, 18.1, 17.5, 18.2
Da-Qin railway
75
0.45, 0.65, 0.95, 1.15, 1.45, 2.45, 3.45
32, 28, 23, 14, 10, 8, 5
Zhao [46]
Guangzhou-Shenzhen
160
0.45
35
Li [43]
Beijing circuit railway
160
0.35
120
Li [43]
Han and Zhang [42]
Xiaoshan-Ningbo
120
0.45
67
Han and Zhang [42]
Jinan-Qingdao
120
0.45
33
Liu and Xiao [44]
Tianjin-Pukou
200
0.6
16
Liu and Xiao [44]
Haerbin-Beian
65
0.4
66
Li [43]
Hangzhou-Nanchang
65
0.4
87
Li [43]
70
68
Li [43]A trapezoid with 2 : 1 inclined sides was generally adopted in this method.Besides,σz with the Boussinesq method can be determined by(10)σz=∑3σmax2π∫ε=−L∗/2ε=+L∗/2∫η=−B/2η=+B/2×z3x−η2+y−ε2+z22.5dεdη,where dε and dη are the length and width of a tie, respectively.Figure4 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 σz.Figure 4
Comparisons between predicted results in the trapezoidal method and measured vertical stresses.Figure 5
Comparisons between predicted results in the Boussinesq method and measured vertical stresses.Figure6 shows the typical distributions of σz 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, Dmax = 50–80 mm, and ϕB = 45°–55°.Figure 6
Stress distribution over depth calculations versus measurement at speed 200 km/h for the Orleans-Montauban railway.Figure 7
Comparisons between predicted results in the proposed method and measured vertical stresses.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 Dmax 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.Table 3
Database of railway geometric parameters in South Africa.
Site
Year (y)
Axle load (t)
Track type (kg/m)
Type of ties
L
∗ (m)
B (m)
Ballast type
D
max (m)
h
B (m)
φ
B (°)
h
n (m)
φ
n (°)
Reference
COAL Link Line1
2004
26
∗
2
60
Concrete
1.1
0.27
Gravel
0.3
542
0.2
303
Yang et al. [47]
0.2
253
0.2
203
0.2
153
1Track gauge = 1.065 m; tie spacing = 0.65 m. 2Estimated by equation (6). 3Estimated from Figures 5 and 8 in Yang et al. [47].Table 4
Database of railway field measurements in South Africa.
Site
V (km/h)
z (m)
σ
z (kPa)
Reference
COAL Link Line
47.5
0.5
95
Yang et al. [47]
0.7
85
0.9
77
1.1
60Figure 8
Comparisons between predicted results in the proposed method and measured vertical stresses for the case in South Africa.


## 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 (Dmax) and friction angle (φB) 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 σz 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. Section1 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 φB on stress distributions. Dmax = 50 mm and φsub = 25° were kept and φB = 50°, 37.5°, 25°, 12.5° were taken into consideration. It can be seen that the maximum vertical stress σz decreased nonlinearly with the decrease in φB. In Figure 10, φB = 50° and φsub = 25° were kept and Dmax varied from 80, 70, 60, and 50 to 40 mm. As expected, the maximum vertical stress σz decreased nonlinearly with the decreases in Dmax. These behaviours imply that the effects of ballast characteristics on train load transfer can be quantitatively evaluated using the φB and Dmax 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.Figure 9
Maximum vertical stress in subgrade with different friction angles of ballast.Figure 10
Maximum vertical stress in subgrade with different particle sizes.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 Dmax or φB; 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 σz, which only depends on the simple geometric parameters of track, the friction angle of granular material (ϕ), and the typical maximum ballast size (Dmax).


## 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,Dmax = 50–80 mm, and ϕB = 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.



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*Source: 3428395-2020-02-21.xml*