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

School of Civil Engineering, Fujian University of Technology, Fuzhou 350108, China College of Civil Engineering and Architecture, Zhejiang University of Technology, Hanzhou 310014, China Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210098, China Institute of Geotechnical Engineering, School of Transportation, Southeast University, Nanjing 210096, China


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][4][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][10][11][12][13][14][15][16][17][18][19][20][21]. erefore, 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][23][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]. is assumption was far from the real condition of the railway substructure [25,26]. is resulted in oversimplifying actual railway conditions, eventually leading to significant overestimation of train load-induced settlements [5,8].
is 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]. e 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.

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 tieballast 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. ereby, 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][3][4][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 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).
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][3][4][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 F 2 � 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]. e 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. e 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.

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][31][32][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. e ballast size (D max ) and friction angle (φ B ) decreased with the increase in fouling content, resulting in a dramatic stress redistribution [22][23][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.

e 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. e 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 ] 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.

e Friction Angle F n .
Parameter ] was the coefficient of lateral stress in equation (4), which was related to the coefficient of the lateral earth pressure at rest (K 0 ) and  Advances in Civil Engineering 3 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°.

e 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%. is 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. e 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. ey 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. erefore, 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. (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].

e Multilayered Substructures. Equation
For engineering applications, Kandaurov [28] proposed a comprehensive method of multilayered equivalency by a coefficient of lateral stress ]: where H n− 1 represents the equivalent depth, h n represents the layer thickness, and ] n represents the coefficient of lateral stress of the nth layer. is method can be applied to the multilayered system for the cases of any layer thickness and variation of ] values, and no assumptions regarding stress conditions were required [7,31]. e values of ] 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. e detailed derivation of the empirical solution of the proposed method is presented in the Appendix:

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][43][44][45][46]. e 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). e field measurements of σ z for the ten cases discussed in this study are listed in Table 2. e 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 by where 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. e 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. ese 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. e 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. e calculated results by the trapezoidal method and Boussinesq method are also shown in the same figure. e 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°.

Advances in Civil Engineering
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  Advances in Civil Engineering 5 is presented in Tables 3 and 4. e 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.

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 Other case in Table 2 Da-Qin railway Other case in Table 2 Da-Qin railway Qin-Shen passenger railway Orleans-Montauban railway σ z (measured) = 0.20 σ z (predicted) σ z (measured) = 0.53 σ z (predicted) Advances in Civil Engineering corners and attrition of asperities occurred. e 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][23][24][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    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 . ese 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. e 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 ).

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. e 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°. e proposed method incorporates the effects of ballast characteristics and multilayered substructure on vertical superimposed stresses in subgrade, substantially improving their calculating accuracy.
e 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.

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 F 2 : Safety factor h B : ickness of the ballast layer h n : ickness of the nth layer H n− 1 : Equivalent depth K 0 : e coefficient of the lateral earth pressure at rest k: e 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: e radius of a circle whose area equals to BL * t: Gaussian function parameters V: Train speed VCI: Void contaminant index x, y, z: e distance from the load position Z n : e distance from the top of nth layer q r : Maximum rail seat load φ: Dynamic amplification factor φ: e internal friction angle of grain material φ B : e internal friction angle of ballast φ b : e true interparticle friction angle determined from tile table test φ n : e internal friction angle of nth grain layer φ sub : e internal friction angle of subgrade layer σ z : Vertical superimposed stress in substructure σ zmax : e maximum vertical stress applied on the subgrade σ max : Tie-ballast contact pressure dε: e length of a tie dη: e width of a tie ψ: Cumulative Gaussian distribution function ]: Coefficient of lateral stress.