Research ArticleDynamic Stress of Subgrade Bed Layers Subjected to TrainVehicles with Large Axle Loads

Fang Xu 12 Qi Yang 12 Wenjie Liu 3 Wuming Leng12 Rusong Nie 12

and Huihao Mei12

1School of Civil Engineering Central South University Changsha China2National Engineering Laboratory for High Speed Railway Construction Changsha China3Hunan Communications Research Institute Co Ltd Changsha China

Correspondence should be addressed to Qi Yang qiyang123csueducn and Wenjie Liu liuwje163com

Received 10 August 2018 Revised 25 November 2018 Accepted 2 December 2018 Published 23 December 2018

Academic Editor Nicola Nistico

Copyright copy 2018 Fang Xu et al0is is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

0e dynamic responses of subgrade bed layers are the key factors affecting the service performance of a heavy-haul railway A 3Dtrain-track-subgrade interaction finite element (FE) model was constructed using the ABAQUS code where different verticalirregular track spectra were simulated by modifying the vertical node coordinates of the FE mesh of the rail 0en the dynamicstresses in the subgrade bed layers subjected to heavy-haul trains were studied in detail 0e results showed the following (1) thetransverse distribution of the dynamic stress transformed from a bimodal pattern to a unimodal pattern with increasing depth (2)the pass of adjacent bogies of adjacent carriages can be simplified once loaded on the subgrade since the dynamic stresses aremaintained around the peak value during the pass of the adjacent bogies (3) the dynamic stress at the bottom of the subgrade bedsurface layer was more sensitive to the train axle load compared with that at the subgrade surface because the dynamic stressesinduced by the two rails were gradually overlaid with increasing depth (4) the maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 compared with that at the subgrade surface (5) the vertical track irregularitiesintensified the vertical excitation between the train vehicle wheels and rails and the maximum dynamic stress at the subgradesurface under the action of the irregular heavy-haul track spectrum increased by 23 compared with the smooth rail conditionand (6) the possible maximum dynamic stress (σdm) at the subgrade surface under the action of irregular track spectra can bepredicted using the triple standard deviation principle of a normally distributed random variable ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdm respectively)

1 Introduction

Heavy-haul railways have significant economic and socialbenefits because of their large freight transport capacity Morethan 70 of Chinarsquos heavy-haul railway lines are composed ofembankments that are compacted from different types ofsoils 0e subgrade bed layers at the top of an embankmentare directly subjected to the static and dynamic loadstransferred from the track structures and running trains andtheir dynamic performances are the key factors determiningthe service performance of a heavy-haul railway [1 2]

Recently the increase in axle loads and the formationlengths of heavy-haul trains has intensified the dynamicresponse (eg dynamic stress and displacement) of subgrade

bed layers thus bringing new challenges to the serviceabilityof existing heavy-haul railways Han and Zhang [3] showedthat the dynamic stress of subgrade bed layers is markedlyincreased when the train axle load is increased to 25 t Weiet al [4] suggested that the upper limit of a trainrsquos axle loadwas approximately 27 t for the existing railway lines inChina Di et al [5] assessed the subgrade strength of theShuo-Huang heavy-haul railway and demonstrated that thetrain axle load should not exceed 27 t even if the existingsubgrade is reinforced using oblique jet grouting pilesHence the existing heavy-haul railways (especially thesubgrade beds) in China cannot handle the operation ofheavy-haul trains with large axle loads and the fundamentaldesign criteria may need to be improved whereas the

HindawiShock and VibrationVolume 2018 Article ID 2916096 12 pageshttpsdoiorg10115520182916096

premise for optimizing the design of subgrade beds is tomaster their dynamic performances subjected to heavy-haultrains with large axle loads

In the past few decades the dynamic response of railwaysubgrades was mainly studied based on field tests andorlaboratory model tests [6ndash11] which are uneconomical andtime-consuming Recently with the development of com-puter technology numerical simulations have graduallybecome an important research technique Numerical sim-ulations on the dynamic responses of railway subgrades canbe divided into two main types ie the track-subgrademodel and the train-track-subgrade interaction model Atrack-subgrade model simplifies the train-induced dynamicload and directly applies it on the rail track Xiao [12]simplified the train load as the summation of a static loadand a series of sine waves whereas Bian et al [13] Shan et al[14] Ang and Dai [15] El Kacimi et al [16] and Hall andLars [17] simulated the train loads as moving loads applieddirectly on the rail0ese methods are widely used since theyare simple and can significantly reduce the computationtime However the interactions between the train and tracksystem were not properly considered Hence a number oftrain-track-subgrade interaction models have also beenestablished during the last decade Ma et al [18] Giner-Navarro et al [19] and Cai et al [20] constructed full train-track-subgrade interaction models to study the wheel-trackinteraction behavior and Kouroussis et al [21] Chiang andTsai [22] Kouroussis and Verlinden [23] Alves Costa et al[24] and Zhai et al [25] investigated train-induced groundvibrations using train-track-subgrade interaction simula-tions Galvin et al (2013) [26] applied a two-series sus-pension structure to simulate a train and constructed a 3Dtrain-track-subgrade model to study the contact force be-tween the train wheels and the rail of a high-speed railwayand their model obtained an acceptable subgrade dynamicstress compared with the field measurements

Currently numerical simulations on train-track-subgrade systems mainly focus on high-speed railwaysand are aimed at the train-track interaction and groundvibrations few train-track-subgrade system simulations areaimed at the dynamic response of the subgrade beneath thetrack structure especially for subgrade bed layers of heavy-haul railways In this study a 3D finite element (FE) modelof a train-track-subgrade interaction system was constructedto study the dynamic stress in the subgrade bed layers of aheavy-haul railway while considering the rail trackirregularities

2 Train-Track-Subgrade Interaction Model

0e finite element simulation was performed using theABAQUS code (version 614-5) 0e prototype of the FEmodel is a soil embankment of the Shuo-Huang heavy-haulrailway that is located in the Hebei province of China 0eembankment has a top width of 70m a height of 60m aslope ratio of 115 and a length of 72m0e cross section ofthe embankment is presented in Figure 1 0e ballast layerhas a top width of 30m a thickness of 05m and a sloperatio of 1175 0e subgrade bed is 25m thick and consists

of a 06m surface layer (SBSL) and a 19m bottom layer(SBBL) 0e embankment foundation is composed ofQuaternary clay with a thickness of 50m 0e groundwaterlevel is relatively deep 0e rail has a unit length weight of70 kgm and the sleeper spacing is 0543m

21 Train Vehicle Model 0e train was simplified as twoconnected carriages to simulate the superposition effects ofthe adjacent bogies of adjacent carriages 0e train bogieswere modeled as two-series suspension systems consisting ofspring-damping units 0is study is mainly aimed at thevertical dynamic stress response in the subgrade bed layershence the considered displacement freedoms were thenodding and bouncing movements of the train carriages andtrain bogies and the bouncingmovements of the train wheels[27] A schematic diagram of the train model is illustrated inFigure 2(a) 0e train carriage train bogies and trainwheelset were simulated as rigid bodies 0e perspectiveview of the constructed train model is presented inFigure 2(b)0emain dimensions of the simulated trains arelisted in Table 1 and the parameters of the spring-dampingunits that are used to simulate the two-series suspensionsystem are listed in Table 2 (where K1 and C1 are the stiffnessand damping of the first suspension respectively andK2 andC2 represent the values of the second suspension)

22 Interaction between the Train Vehicle Wheel and Rail0e interaction between the train wheel and rail was sim-ulated using Hertz nonlinear contact theory [28] 0e sur-faces of the wheel and rail were selected as the master surfaceand slave surface of the contact pair respectively 0etangential contact behavior was simulated using a penaltyfriction model with a friction coefficient of 02 0e verticalcontact behavior was modeled using the Hertz nonlinearcontact theory where the vertical contact force is formulatedas

Pj

1GZwj minus δrj minus ηj1113960 11139611113882 1113883

23 j 1 to 4

0 when separation

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(1)

where Pj is the vertical contact force between the jth trainwheel and rail Zwj is the vertical displacement of the jthtrain wheel δrj and ηrj are the vertical displacement andirregularity of the rail directly below the jth train wheel andG is a contact constant

A soft contact mode was applied to model the verticalHertz nonlinear contact between the train wheels and rail[28] 0e schematic diagram of the soft contact mode ispresented in Figure 3 where overclosure was defined tosimulate the vertical contact behavior

23 Track and Subgrade 0e fasteners and cushion blocksused to connect the rail and sleepers were simplified asspring-damping units as illustrated in Figure 4 0e sleepersballast layer subgrade bed layers soil layers below thesubgrade bed (LBSB) and the embankment foundation were

2 Shock and Vibration

treated as continuum media but with dierent mechanicalparameterse rail sleepers and ballast layer were modeledas elastic media while the substructures were simulated aselastoplastic media using the MohrndashCoulomb constitutivemodel e detailed mechanical properties of the FE modelare listed in Table 3 where E ] ρ φ c and D represent theelastic modulus Poissonrsquos ratio density frictional anglecohesion and damping ratio of the corresponding com-ponent respectively

e bottom width of the FE model was 55m and thedamping ratios of the FE model boundaries were set as 10 tomitigate the reshyection eects of the dynamic stress wavese horizontal displacements of the left right front andback boundaries were xed but vertical movement wasallowed while at the bottom both the vertical and horizontaldisplacements were xed e FE model was meshed using a

hexahedron element with eight nodes (C3D8) e elementsused tomodel the subgrade bed layers had a side length of 01to 02m e FE mesh of the entire train-track-subgrademodel is presented in Figure 5

3 Track Irregularity

Track irregularity is a key factor that intensies the dynamicresponse of a train-track-subgrade interaction system Track

Rail

05m06m

19m

35m

Embankment foundation

1175SBSL 06m

SBBL 19m

Layers below subgrade bed115

30m70m

SleeperBallast

25m

Figure 1 Embankment cross section

v

Bogie

Train carriage

RailWheel-rail contact

Spring

K1 D1

K2 D2 Damping

Bogie

First suspension

Second suspension

(a)

Bogie axle spacing

Bogie center spacing Adjacent bogies

(b)

Figure 2 Train vehicle model (a) carriage components (b) perspective view of the train model

Table 1 Main parameters of simulated trains

Trainmodel

Axleload (t)

Bogie axlespacing (mm)

Bogie centerspacing (mm)

Carriagelength (mm)

C80 25 1830 8200 12000C96 30 325 35 1860 9926 13726

Table 2 Parameters of the spring-damping units

K1 (kNm) K2 (kNm) C1 (kNmiddotsm) C2 (kNmiddotsm)17000 5900 90 100

(h4 p4)

(hn pn)

(h3 p3)(h2 p2)

Overclosure (h)

Cont

act p

ress

ure (

p)

(h1 0)

Figure 3 Soft contact mode between the train vehicle wheel andrail in the vertical direction

Shock and Vibration 3

irregularity consists of vertical irregularity horizontal ir-regularity direction irregularity and rail spacing irregu-larity e vertical and horizontal irregularities maysignicantly aect the vertical interaction of the system [29]is study mainly considers vertical irregularity

ere has not been an authoritative track spectrumavailable for describing the track irregularities of heavy-haulrailways in China Wei [30] described that the vertical trackirregularity of Americarsquos fth-grade spectrum is similar tothat of the Chinarsquos heavy-haul railways hence it was appliedas the heavy-haul track irregular spectrum in the establishedFE model e vertical track irregularity of Americarsquos fth-grade spectrum is expressed as

Sv(Ω) kFAvΩ2cΩ2 Ω2 +Ω2c( )

(2)

where Sv(Ω) is the power spectrum density of irregularityΩis the spatial frequency of irregularity Ωc is the cutofrequency Av is the roughness constant (02095) and kF isthe safety factor with a value of 025 generally used

e basic principle for modeling the random track ir-regularity is converting the irregular power spectrum densityto a time domain sample using the inverse fast Fouriertransform (IFFT) e relationship between the discretepower spectral density (Sx(k)) and time series (xs) in theIFFT is formulated as

Sx(k) 1N2 D xs( )∣∣∣∣

∣∣∣∣2 1N2 X

lowast(k)X(k)∣∣∣∣

∣∣∣∣ (3)

where N is the number of discrete sampling points D(xs)means conducting a discrete Fourier transform on xs (s 0to N minus 1) X(k) is the Fourier frequency spectrum ofthe time series xs and Xlowast(k) is the adjoint of X(k) (k 0to N minus 1)

According to equation (3) X(k) can be back-analyzedusing the available power spectral density of irregularityen the serialized time domain model for the track ir-regularity can be obtained by conducting an inverse discreteFourier transform on X(k)

x(n) 1NsumN

k1X(k)exp

2iπknN

( )

X(k) Nξ(k)Sx(kΔω)Δωradic

Δω t

(2π)

ξ(k) exp iϕk( )

(4)

where x(n) is the discrete time domain irregularity t is thetime i is the imaginary unit ξ(k) is an independent phaseseries and ϕk is a random variable obeying a uniformdistribution

Specically the discrete time domain track irregularitycan be obtained as follows (1) discretely sampling on thepower spectrum density of irregularity (2) bank-deducingX(k) using equation (3) and (3) conducting an inversediscrete Fourier transform on X(k) to obtain x(n) eMATLAB software was adopted to improve the analysiseiexclciency and the obtained vertical time domain irregularityof Americarsquos fth-grade spectrum is presented in Figure 6By comparison the vertical time domain irregularity of theleft rail of the Chinarsquos three main railway lines (Beijing-Shanghai railway Beijing-Guangzhou railway and Beijing-Jiulong railway) is also shown in Figure 6 It is found that thevertical irregularity of Chinarsquos three main railway lines is lessthan that of Americarsquos fth-grade spectrum hence it maynot be available for heavy-haul railways in China

It is assumed that the vertical track irregularity is fullyattributed to the surface undulation of the rail e nodecoordinates matrix of the smooth rail surface was rstexported from the ABAQUS FE model (INP le) thenthe vertical node coordinates were modied according tothe vertical track irregularity illustrated in Figure 6 usingthe MATLAB software Finally the modied node co-ordinate matrix was imported to the smooth rail surfaceFigure 7 presents the rail models before and after modi-cation It shows that the vertical track irregularity wasproperly simulated

Fasteners and cushion blocks

Rail

Rail

SleeperSleeper

Figure 4 Connection between the rail and sleeper

Table 3 Mechanical parameters of dierent parts of the FE model

Part E (MPa) ] ρ (kgm3) φ (deg) c (kPa) DTrain Rigid body mdash mdash mdash mdash mdashRail 21 times 105 025 7850 mdash mdash mdashSleeper 30000 02 2500 mdash mdash mdashBallast 200 025 2400 mdash mdash 005SBSL 180 03 2300 33 58 01SBBL 150 03 2200 31 45 01SBSB 70 035 1800 17 30 015Foundation 50 035 1700 15 25 015

4 Shock and Vibration

4 FEA Results and Analysis

41Verication of the FEModel Ding et al [31] reported themeasured maximum vertical displacements of the rail andsleeper and the maximum acceleration of the ballast at a testsite of the Shuo-Huang heavy-haul railway under a trainwith an axle load of 25 t the FE analyzed (FEA) results arecompared with the eld measurements as listed in Table 4 Itshows that the analyzed values agreed well with the eldmeasurements Miao et al [32] reported the dynamic stress-time curve measured at the subgrade surface under a trial-train with an axle load of 30 t the FE analyzed curve iscompared with themeasured curve as illustrated in Figure 8It is observed that the calculated curve well simulated themeasurements in both wave shape and peak valuese good

agreements between the FEA results and eld measurementsdemonstrated the eectiveness and reliability of the FEmodel

42 Dynamic Stress of the Subgrade Bed Layers e trainspeed of the Shuo-Huang heavy-haul railway is approxi-mately 80 kmh hence the default speed in the FEA was80 kmh e analyzed dynamic stress contour at the sub-grade surface induced by a heavy-haul train with an axle loadof 30 t is presented in Figure 9 In the specied case theanalyzed results all corresponded to a train axle load of 30 tin the following parts of this article It shows that the dy-namic stress is symmetrically distributed along track cen-terline the maximum dynamic stress appeared directlyunder the rail and was attenuated toward the track centerlineand the embankment shoulder the transverse distribution ofthe dynamic stress exhibited a bimodal pattern with twopeak points and a tensile stress appeared under the middlepart of the train carriage In addition the train ran smoothlyon a smooth track the dierence of the dynamic stress at thesubgrade surface under the four bogies was less than 10 kPaand the eects of a bogie were similar to a nonuniformrectangular pressure as shown in Figure 9(a) Converselythe train exhibited bouncing movements on the verticallyirregular track thus the maximum dynamic stress wasgreater than that under a smooth track condition noddingmotion appeared on the train carriages and bogies hence thedynamic stresses at the subgrade surface under the fourbogies appeared to be obviously dierent with a greaterdeviation of approximately 20 kPa (Figure 9(b))

Figures 10 and 11 present the dynamic stress contourat the bottom of the SBSL and SBBL under irregulartrack conditions respectively It is determined that thetransverse distribution of the dynamic stress varied from abimodal pattern to a unimodal pattern with an increase inthe depth from the subgrade surface (Figures 9(b) and 10)

0 50 100 150 200ndash10

ndash5

0

5

10

Ver

tical

irre

gula

rity

(mm

)

Distance (m)

Americarsquos fifth-grade track irregularityChinarsquos three mail lines track irregularity

Figure 6 Vertical track irregularity in the time domain

TrainRailBallast

72m

5m

55m

Subgrade bed layersEmbankment foundation

Figure 5 Train-track-subgrade interaction model

Shock and Vibration 5

0e dynamic stress directly beneath the rail was close to thatunder the track centerline at the bottom of the subgrade bedsurface layer as illustrated in Figure 10 0e distribution ofthe dynamic stress induced by adjacent bodies of adjacentcarriages exhibited an elliptical shape and the superimposeddynamic stress was apparently greater than that induced by asingle bogie (Figure 11) 0erefore the superposition effectsof the adjacent bogies could enlarge the influencing depth of

the dynamic stress because of the small bogie spacing of theheavy-haul train

Figures 12(a) and 12(b) present the subgrade surfacedynamic stress-time curves directly below the rail and trackcenterline under different train axle loads respectively Itshows that the maximum dynamic stresses (σdm) at thesubgrade surface directly below the rail and track centerlinehad increments of 338 kPa and 32 kPa (corresponding topercent increases of 398 and 473) when the train axleload increased from 25 t to 35 t respectively An increase inthe train axle load intensified the subgrade dynamic stressand the interaction between the train and track system Inaddition the subgrade surface dynamic stresses induced bythe adjacent bogies exhibited an obvious superposition ef-fect and the dynamic stresses were maintained around thepeak value during the pass of the adjacent bogies hence thepass of adjacent bogies of adjacent carriages may be sim-plified once loaded on the railway subgrade

0e relationships between the maximum dynamic stressand train axle load (P) at different locations below a sleeperare illustrated in Figure 13 It shows that the maximumdynamic stresses increased approximately linearly as thetrain axle load increased 0e average increase rate of thedynamic stress directly below the rail was 317 kPat373 kPat and 152 kPat at the subgrade surface and thebottoms of SBSL and SBBL respectively For the locationsdirectly below the track centerline the average increase rateswere 319 kPat 359 kPat and 178 kPat Generally thedynamic stress at the bottom of the SBSL was more sensitiveto the train axle load compared to that at the subgradesurface because the dynamic stresses induced by the two railswere gradually overlaid with an increase in the depth fromthe subgrade surface

Figure 14 presents the relationship between the maxi-mum dynamic stress and the depth from the subgradesurface σdm at the subgrade surface directly below the rail aswell as its attenuation rate was greater than that below the

(a) (b)

Figure 7 Rail model (a) before modification (b) after modification

Table 4 Measured and simulated vertical displacement of the railand sleeper

Parameter Measuredvalue

Calculatedvalue

Vertical displacement of rail (mm) 201 192ndash221Vertical displacement of sleeper (mm) 155 148ndash173Acceleration of ballast (ms2) 131 081ndash113

04 06 08 10 12

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

FE analyzedMeasured

Figure 8 Measured and calculated subgrade surface dynamicstress

6 Shock and Vibration

track centerline σdm below the rail became less than thatbelow the track centerline at a depth of approximately 03mσdm at the subgrade bed bottom was reduced by approxi-mately 70 compared with that at the subgrade surface eratio of σdm to the self-weight stress (σs) at the subgrade bedbottom (25m below the subgrade surface) under a train axle

load of 25 t was 021 and 022 for the locations directly belowthe rail and the track centerline respectively the σdmσsvalues increased to 034 and 036 under a train axle load of30 t respectively Zhou [33] suggested that the σdmσs valueshould be less than 020 to maintain the long-term stabilityof a railway subgrade Consequently the thickness of thesubgrade bed (25m a subgrade bed surface layer of 06mand a subgrade bed bottom layer of 19m) of existingrailways in China may meet the requirements of running aheavy-haul train with an axle load of 25 t but cannot enablethe operation of heavy-haul trains with axle loads greaterthan 25 t unless the subgrades have been reinforced

43 Eects of Irregular Track Spectrum Figure 15 illustratesthe subgrade surface dynamic stress-time curves underdierent track conditions e vertical track irregularityintensied the vertical excitation between the train wheelsand rails and induced a greater dynamic stress comparedwith the smooth rail condition e maximum dynamicstress at the subgrade surface increased by 23 and 158under the action of the heavy-haul irregular track spectrumand Chinarsquos three main railway lines irregular track spec-trum respectively

In China the irregular track spectra of the left and rightrails are dierent e dynamic stress contour at the sub-grade surface taking this dierence into account is shown inFigure 16 It is determined that the FE model eectivelyreshyected the irregularity dierence of the left and right railsand the dynamic stress was not symmetrically distributedalong the track centerline

Figure 17 presents the transverse distributions of thedynamic stress at dierent depths e subgrade surfacedynamic stress exhibited a bimodal pattern (or an ldquoMrdquoshape) and the value under the left rail was approximately19 greater than that under the right rail whereas theasymmetry of the dynamic stress gradually decreased as thedepth increased and was negligible at the subgrade bedbottom hence the irregularity dierence between the leftand right rails had a slight impact on the embankment soilsbelow the subgrade bed bottom

Adjacent b

ogies

Tension zone

Field 2 S22(avg 75)

+2359 times 104

+0000 times 100

ndash3793 times 103

ndash7586 times 103

ndash1138 times 104

ndash1517 times 104

ndash1896 times 104

ndash2276 times 104

ndash2655 times 104

ndash3034 times 104

ndash3414 times 104

ndash3793 times 104

ndash4172 times 104

ndash4552 times 104

ndash4931 times 104

ndash5310 times 104

ndash5689 times 104

ndash6069 times 104

ndash6448 times 104

ndash6827 times 104

ndash7207 times 104

ndash7586 times 104

(a)

Adjacent bogies

Tension zone

Field 3 S22(avg 75)

+2729 times 104

+0000 times 100

ndash5198 times 103

ndash1040 times 104

ndash1559 times 104

ndash2079 times 104

ndash2599 times 104

ndash3119 times 104

ndash3638 times 104

ndash4158 times 104

ndash4678 times 104

ndash5198 times 104

ndash5718 times 104

ndash6237 times 104

ndash6757 times 104

ndash7277 times 104

ndash7797 times 104

ndash8316 times 104

ndash8836 times 104

ndash9356 times 104

ndash9876 times 104

ndash1040 times 105

(b)

Figure 9 Dynamic stress contour at the subgrade surface (a) smooth track (b) vertically irregular track

Adjacent bogies

Field 3 S22(avg 75)

+1246 times 104

+0000 times 100

ndash3797 times 103

ndash7595 times 103

ndash1139 times 104

ndash1519 times 104

ndash1899 times 104

ndash2278 times 104

ndash2658 times 104

ndash3038 times 104

ndash3418 times 104

ndash3797 times 104

ndash4177 times 104

ndash4557 times 104

ndash4937 times 104

ndash5316 times 104

ndash5696 times 104

ndash6076 times 104

ndash6056 times 104

ndash6835 times 104

ndash7215 times 104

ndash7595 times 104

Figure 10 Dynamic stress contour at the bottom of the subgradebed surface layer

Adjacent bogies

Field 3 S22(avg 75)

+1487 times 100

+0000 times 100

ndash1945 times 103

ndash3891 times 103

ndash5836 times 103

ndash7781 times 103

ndash9727 times 103

ndash1167 times 104

ndash1362 times 104

ndash1556 times 104

ndash1945 times 104ndash1751 times 104

ndash2140 times 104

ndash2334 times 104

ndash2529 times 104

ndash2724 times 104

ndash2918 times 104

ndash3113 times 104

ndash3303 times 104

ndash3502 times 104

ndash3696 times 104

ndash3891 times 104

Figure 11 Dynamic stress contour at the bottom of the subgradebed bottom layer

Shock and Vibration 7

44 Statistical Analysis of theMaximumDynamic Stress at theSubgrade Surface e maximum dynamic stress at thesubgrade surface was randomly distributed under dierentsleepers because of the vertical track irregularity Figure 18illustrates the statistical results of σdm at the subgrade surfacedirectly below the intersection points of the rail and sleepersIt is observed that the track conditions signicantly aectedthe distribution of σdm σdm was relatively small and thedistribution was relatively concentrated when the rail trackwas smooth and σdm increased and the distribution rangebecame wider when the track condition worsened indicating

that the subgrade bed layers were more likely to withstand agreater dynamic stress which is harmful for maintaining thelong-term service performance of the subgrade

e statistical distributions of σdm under dierent ir-regular track conditions can be well-tted by normal dis-tribution curves as illustrated in Figure 18 and theexpectation (μ) and standard deviation (σ) increased as thetrack condition worsened

e possible maximum dynamic stress at the subgradesurface was predicted using the normal distribution theoryIf σdmsim(μ σ

2) then

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

140

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(a)

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(b)

Figure 12 Dynamic stress at the subgrade surface (a) below the rail (b) below track centerline

24 26 28 30 32 34 36

20

40

60

80

100

120

140

160

Bottom of SBSL rail

Subgrade surface railSubgrade surface centerline

Bottom of SBSL centerlineBottom of SBBL railBottom of SBBL centerline

Max

imum

dyn

amic

stre

ss (k

Pa)

Axle load (t)

Figure 13 Relationship between the maximum dynamic stress andtrain axle load

0 20 40 60 80 100

20

15

10

05

25σdmσs = 036

σdmσs = 034

σdmσs = 021

Maximum dynamic stress (kPa)

Dep

th (m

)

σdmσs = 022

30 t below rail

25 t below rail25 t below centerline

30 t below centerlineSelf-weight stress

Figure 14 Relationship between the maximum dynamic stress anddepth from the subgrade surface

8 Shock and Vibration

P σdm minus μ∣∣∣∣

∣∣∣∣ge 3σ( ) 1minusP σdm minus μ∣∣∣∣

∣∣∣∣lt 3σ( )

1minus 09973 00027lt 0003(5)

erefore the probability of σdm outside the interval of(μ minus 3σ μ + 3σ) was less than 0003 Statistically the intervalof (μ-3σ μ+3σ) is generally taken as the actual possible rangeof a normal distribution random variable namely the triplestandard deviation principle Consequently the possiblemaximum value of the dynamic stress at the subgradesurface can be estimated as μ + 3σ e analyzed possiblemaximum dynamic stresses at the subgrade surface underdierent irregular track spectra are listed in Table 5 emaximum dynamic stress at the subgrade surface of theShuo-Huang heavy-haul railway induced by a train with anaxle load of 30 t could reach 123 kPa [34] which was fa-vorably simulated with a predicted value of 1169 kPa (with adeviation less than 5)

5 Conclusions

A 3D train-track-subgrade interaction nite element (FE)model was constructed using the ABAQUS code (version614-5) where dierent vertical irregular track spectra weresimulated by modifying the rail node coordinates in theABAQUS INP le en the dynamic stress of the subgradebed layers of a heavy-haul railway embankment was ana-lyzed in detail using the 3D FE model e main conclusionsdrawn from this study are as follows

(1) e transverse distribution of the dynamic stress inthe subgrade bed layers gradually transformed from

00 02 04 06 08 10 12 14

20

40

60

80

100

120D

ynam

ic st

ress

(kPa

)

Time (s)

Heavy-haul track irregularityChinarsquos three main lines track irregularitySmooth rail

(a)

00 02 04 06 08 10 12 14

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

Heavy-haul track irregularityChinarsquos three mail lines track irregularitySmooth rail

(b)

Figure 15 Dynamic stress at the subgrade surface under dierent irregular track spectra (a) below rail (b) below centerline

Adjacent bogies

Tension zone

Field 9 S22+2258 times 104

+1000 times 103

ndash3231 times 103

ndash7462 times 103

ndash1169 times 104

ndash1592 times 104

ndash2015 times 104

ndash2439 times 104

ndash3285 times 104ndash2862 times 104

ndash3708 times 104

ndash4131 times 104

ndash4554 times 104

ndash4977 times 104

ndash5400 times 104

ndash5823 times 104

ndash6246 times 104

ndash6670 times 104

ndash7093 times 104

ndash7516 times 104

ndash7939 times 104

ndash8362 times 104

Figure 16 Dynamic stress contour at the subgrade surface con-sidering the irregularity dierence of the left and right rails

0 1 2 3 4

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Distance (m)

Subgrade surface Bottom of SBSL

Bottom of SBBLEmbankment bottom

Embankmentcenterline

Figure 17 Transverse distributions of dynamic stresses at dierentdepths

Shock and Vibration 9

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

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treated as continuum media but with dierent mechanicalparameterse rail sleepers and ballast layer were modeledas elastic media while the substructures were simulated aselastoplastic media using the MohrndashCoulomb constitutivemodel e detailed mechanical properties of the FE modelare listed in Table 3 where E ] ρ φ c and D represent theelastic modulus Poissonrsquos ratio density frictional anglecohesion and damping ratio of the corresponding com-ponent respectively

e bottom width of the FE model was 55m and thedamping ratios of the FE model boundaries were set as 10 tomitigate the reshyection eects of the dynamic stress wavese horizontal displacements of the left right front andback boundaries were xed but vertical movement wasallowed while at the bottom both the vertical and horizontaldisplacements were xed e FE model was meshed using a

hexahedron element with eight nodes (C3D8) e elementsused tomodel the subgrade bed layers had a side length of 01to 02m e FE mesh of the entire train-track-subgrademodel is presented in Figure 5

3 Track Irregularity

Track irregularity is a key factor that intensies the dynamicresponse of a train-track-subgrade interaction system Track

Rail

05m06m

19m

35m

Embankment foundation

1175SBSL 06m

SBBL 19m

Layers below subgrade bed115

30m70m

SleeperBallast

25m

Figure 1 Embankment cross section

v

Bogie

Train carriage

RailWheel-rail contact

Spring

K1 D1

K2 D2 Damping

Bogie

First suspension

Second suspension

(a)

Bogie axle spacing

Bogie center spacing Adjacent bogies

(b)

Figure 2 Train vehicle model (a) carriage components (b) perspective view of the train model

Table 1 Main parameters of simulated trains

Trainmodel

Axleload (t)

Bogie axlespacing (mm)

Bogie centerspacing (mm)

Carriagelength (mm)

C80 25 1830 8200 12000C96 30 325 35 1860 9926 13726

Table 2 Parameters of the spring-damping units

K1 (kNm) K2 (kNm) C1 (kNmiddotsm) C2 (kNmiddotsm)17000 5900 90 100

(h4 p4)

(hn pn)

(h3 p3)(h2 p2)

Overclosure (h)

Cont

act p

ress

ure (

p)

(h1 0)

Figure 3 Soft contact mode between the train vehicle wheel andrail in the vertical direction

Shock and Vibration 3

irregularity consists of vertical irregularity horizontal ir-regularity direction irregularity and rail spacing irregu-larity e vertical and horizontal irregularities maysignicantly aect the vertical interaction of the system [29]is study mainly considers vertical irregularity

ere has not been an authoritative track spectrumavailable for describing the track irregularities of heavy-haulrailways in China Wei [30] described that the vertical trackirregularity of Americarsquos fth-grade spectrum is similar tothat of the Chinarsquos heavy-haul railways hence it was appliedas the heavy-haul track irregular spectrum in the establishedFE model e vertical track irregularity of Americarsquos fth-grade spectrum is expressed as

Sv(Ω) kFAvΩ2cΩ2 Ω2 +Ω2c( )

(2)

where Sv(Ω) is the power spectrum density of irregularityΩis the spatial frequency of irregularity Ωc is the cutofrequency Av is the roughness constant (02095) and kF isthe safety factor with a value of 025 generally used

e basic principle for modeling the random track ir-regularity is converting the irregular power spectrum densityto a time domain sample using the inverse fast Fouriertransform (IFFT) e relationship between the discretepower spectral density (Sx(k)) and time series (xs) in theIFFT is formulated as

Sx(k) 1N2 D xs( )∣∣∣∣

∣∣∣∣2 1N2 X

lowast(k)X(k)∣∣∣∣

∣∣∣∣ (3)

where N is the number of discrete sampling points D(xs)means conducting a discrete Fourier transform on xs (s 0to N minus 1) X(k) is the Fourier frequency spectrum ofthe time series xs and Xlowast(k) is the adjoint of X(k) (k 0to N minus 1)

According to equation (3) X(k) can be back-analyzedusing the available power spectral density of irregularityen the serialized time domain model for the track ir-regularity can be obtained by conducting an inverse discreteFourier transform on X(k)

x(n) 1NsumN

k1X(k)exp

2iπknN

( )

X(k) Nξ(k)Sx(kΔω)Δωradic

Δω t

(2π)

ξ(k) exp iϕk( )

(4)

where x(n) is the discrete time domain irregularity t is thetime i is the imaginary unit ξ(k) is an independent phaseseries and ϕk is a random variable obeying a uniformdistribution

Specically the discrete time domain track irregularitycan be obtained as follows (1) discretely sampling on thepower spectrum density of irregularity (2) bank-deducingX(k) using equation (3) and (3) conducting an inversediscrete Fourier transform on X(k) to obtain x(n) eMATLAB software was adopted to improve the analysiseiexclciency and the obtained vertical time domain irregularityof Americarsquos fth-grade spectrum is presented in Figure 6By comparison the vertical time domain irregularity of theleft rail of the Chinarsquos three main railway lines (Beijing-Shanghai railway Beijing-Guangzhou railway and Beijing-Jiulong railway) is also shown in Figure 6 It is found that thevertical irregularity of Chinarsquos three main railway lines is lessthan that of Americarsquos fth-grade spectrum hence it maynot be available for heavy-haul railways in China

It is assumed that the vertical track irregularity is fullyattributed to the surface undulation of the rail e nodecoordinates matrix of the smooth rail surface was rstexported from the ABAQUS FE model (INP le) thenthe vertical node coordinates were modied according tothe vertical track irregularity illustrated in Figure 6 usingthe MATLAB software Finally the modied node co-ordinate matrix was imported to the smooth rail surfaceFigure 7 presents the rail models before and after modi-cation It shows that the vertical track irregularity wasproperly simulated

Fasteners and cushion blocks

Rail

Rail

SleeperSleeper

Figure 4 Connection between the rail and sleeper

Table 3 Mechanical parameters of dierent parts of the FE model

Part E (MPa) ] ρ (kgm3) φ (deg) c (kPa) DTrain Rigid body mdash mdash mdash mdash mdashRail 21 times 105 025 7850 mdash mdash mdashSleeper 30000 02 2500 mdash mdash mdashBallast 200 025 2400 mdash mdash 005SBSL 180 03 2300 33 58 01SBBL 150 03 2200 31 45 01SBSB 70 035 1800 17 30 015Foundation 50 035 1700 15 25 015

4 Shock and Vibration

4 FEA Results and Analysis

41Verication of the FEModel Ding et al [31] reported themeasured maximum vertical displacements of the rail andsleeper and the maximum acceleration of the ballast at a testsite of the Shuo-Huang heavy-haul railway under a trainwith an axle load of 25 t the FE analyzed (FEA) results arecompared with the eld measurements as listed in Table 4 Itshows that the analyzed values agreed well with the eldmeasurements Miao et al [32] reported the dynamic stress-time curve measured at the subgrade surface under a trial-train with an axle load of 30 t the FE analyzed curve iscompared with themeasured curve as illustrated in Figure 8It is observed that the calculated curve well simulated themeasurements in both wave shape and peak valuese good

agreements between the FEA results and eld measurementsdemonstrated the eectiveness and reliability of the FEmodel

42 Dynamic Stress of the Subgrade Bed Layers e trainspeed of the Shuo-Huang heavy-haul railway is approxi-mately 80 kmh hence the default speed in the FEA was80 kmh e analyzed dynamic stress contour at the sub-grade surface induced by a heavy-haul train with an axle loadof 30 t is presented in Figure 9 In the specied case theanalyzed results all corresponded to a train axle load of 30 tin the following parts of this article It shows that the dy-namic stress is symmetrically distributed along track cen-terline the maximum dynamic stress appeared directlyunder the rail and was attenuated toward the track centerlineand the embankment shoulder the transverse distribution ofthe dynamic stress exhibited a bimodal pattern with twopeak points and a tensile stress appeared under the middlepart of the train carriage In addition the train ran smoothlyon a smooth track the dierence of the dynamic stress at thesubgrade surface under the four bogies was less than 10 kPaand the eects of a bogie were similar to a nonuniformrectangular pressure as shown in Figure 9(a) Converselythe train exhibited bouncing movements on the verticallyirregular track thus the maximum dynamic stress wasgreater than that under a smooth track condition noddingmotion appeared on the train carriages and bogies hence thedynamic stresses at the subgrade surface under the fourbogies appeared to be obviously dierent with a greaterdeviation of approximately 20 kPa (Figure 9(b))

Figures 10 and 11 present the dynamic stress contourat the bottom of the SBSL and SBBL under irregulartrack conditions respectively It is determined that thetransverse distribution of the dynamic stress varied from abimodal pattern to a unimodal pattern with an increase inthe depth from the subgrade surface (Figures 9(b) and 10)

0 50 100 150 200ndash10

ndash5

0

5

10

Ver

tical

irre

gula

rity

(mm

)

Distance (m)

Americarsquos fifth-grade track irregularityChinarsquos three mail lines track irregularity

Figure 6 Vertical track irregularity in the time domain

TrainRailBallast

72m

5m

55m

Subgrade bed layersEmbankment foundation

Figure 5 Train-track-subgrade interaction model

Shock and Vibration 5

0e dynamic stress directly beneath the rail was close to thatunder the track centerline at the bottom of the subgrade bedsurface layer as illustrated in Figure 10 0e distribution ofthe dynamic stress induced by adjacent bodies of adjacentcarriages exhibited an elliptical shape and the superimposeddynamic stress was apparently greater than that induced by asingle bogie (Figure 11) 0erefore the superposition effectsof the adjacent bogies could enlarge the influencing depth of

the dynamic stress because of the small bogie spacing of theheavy-haul train

Figures 12(a) and 12(b) present the subgrade surfacedynamic stress-time curves directly below the rail and trackcenterline under different train axle loads respectively Itshows that the maximum dynamic stresses (σdm) at thesubgrade surface directly below the rail and track centerlinehad increments of 338 kPa and 32 kPa (corresponding topercent increases of 398 and 473) when the train axleload increased from 25 t to 35 t respectively An increase inthe train axle load intensified the subgrade dynamic stressand the interaction between the train and track system Inaddition the subgrade surface dynamic stresses induced bythe adjacent bogies exhibited an obvious superposition ef-fect and the dynamic stresses were maintained around thepeak value during the pass of the adjacent bogies hence thepass of adjacent bogies of adjacent carriages may be sim-plified once loaded on the railway subgrade

0e relationships between the maximum dynamic stressand train axle load (P) at different locations below a sleeperare illustrated in Figure 13 It shows that the maximumdynamic stresses increased approximately linearly as thetrain axle load increased 0e average increase rate of thedynamic stress directly below the rail was 317 kPat373 kPat and 152 kPat at the subgrade surface and thebottoms of SBSL and SBBL respectively For the locationsdirectly below the track centerline the average increase rateswere 319 kPat 359 kPat and 178 kPat Generally thedynamic stress at the bottom of the SBSL was more sensitiveto the train axle load compared to that at the subgradesurface because the dynamic stresses induced by the two railswere gradually overlaid with an increase in the depth fromthe subgrade surface

Figure 14 presents the relationship between the maxi-mum dynamic stress and the depth from the subgradesurface σdm at the subgrade surface directly below the rail aswell as its attenuation rate was greater than that below the

(a) (b)

Figure 7 Rail model (a) before modification (b) after modification

Table 4 Measured and simulated vertical displacement of the railand sleeper

Parameter Measuredvalue

Calculatedvalue

Vertical displacement of rail (mm) 201 192ndash221Vertical displacement of sleeper (mm) 155 148ndash173Acceleration of ballast (ms2) 131 081ndash113

04 06 08 10 12

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

FE analyzedMeasured

Figure 8 Measured and calculated subgrade surface dynamicstress

6 Shock and Vibration

track centerline σdm below the rail became less than thatbelow the track centerline at a depth of approximately 03mσdm at the subgrade bed bottom was reduced by approxi-mately 70 compared with that at the subgrade surface eratio of σdm to the self-weight stress (σs) at the subgrade bedbottom (25m below the subgrade surface) under a train axle

load of 25 t was 021 and 022 for the locations directly belowthe rail and the track centerline respectively the σdmσsvalues increased to 034 and 036 under a train axle load of30 t respectively Zhou [33] suggested that the σdmσs valueshould be less than 020 to maintain the long-term stabilityof a railway subgrade Consequently the thickness of thesubgrade bed (25m a subgrade bed surface layer of 06mand a subgrade bed bottom layer of 19m) of existingrailways in China may meet the requirements of running aheavy-haul train with an axle load of 25 t but cannot enablethe operation of heavy-haul trains with axle loads greaterthan 25 t unless the subgrades have been reinforced

43 Eects of Irregular Track Spectrum Figure 15 illustratesthe subgrade surface dynamic stress-time curves underdierent track conditions e vertical track irregularityintensied the vertical excitation between the train wheelsand rails and induced a greater dynamic stress comparedwith the smooth rail condition e maximum dynamicstress at the subgrade surface increased by 23 and 158under the action of the heavy-haul irregular track spectrumand Chinarsquos three main railway lines irregular track spec-trum respectively

In China the irregular track spectra of the left and rightrails are dierent e dynamic stress contour at the sub-grade surface taking this dierence into account is shown inFigure 16 It is determined that the FE model eectivelyreshyected the irregularity dierence of the left and right railsand the dynamic stress was not symmetrically distributedalong the track centerline

Figure 17 presents the transverse distributions of thedynamic stress at dierent depths e subgrade surfacedynamic stress exhibited a bimodal pattern (or an ldquoMrdquoshape) and the value under the left rail was approximately19 greater than that under the right rail whereas theasymmetry of the dynamic stress gradually decreased as thedepth increased and was negligible at the subgrade bedbottom hence the irregularity dierence between the leftand right rails had a slight impact on the embankment soilsbelow the subgrade bed bottom

Adjacent b

ogies

Tension zone

Field 2 S22(avg 75)

+2359 times 104

+0000 times 100

ndash3793 times 103

ndash7586 times 103

ndash1138 times 104

ndash1517 times 104

ndash1896 times 104

ndash2276 times 104

ndash2655 times 104

ndash3034 times 104

ndash3414 times 104

ndash3793 times 104

ndash4172 times 104

ndash4552 times 104

ndash4931 times 104

ndash5310 times 104

ndash5689 times 104

ndash6069 times 104

ndash6448 times 104

ndash6827 times 104

ndash7207 times 104

ndash7586 times 104

(a)

Adjacent bogies

Tension zone

Field 3 S22(avg 75)

+2729 times 104

+0000 times 100

ndash5198 times 103

ndash1040 times 104

ndash1559 times 104

ndash2079 times 104

ndash2599 times 104

ndash3119 times 104

ndash3638 times 104

ndash4158 times 104

ndash4678 times 104

ndash5198 times 104

ndash5718 times 104

ndash6237 times 104

ndash6757 times 104

ndash7277 times 104

ndash7797 times 104

ndash8316 times 104

ndash8836 times 104

ndash9356 times 104

ndash9876 times 104

ndash1040 times 105

(b)

Figure 9 Dynamic stress contour at the subgrade surface (a) smooth track (b) vertically irregular track

Adjacent bogies

Field 3 S22(avg 75)

+1246 times 104

+0000 times 100

ndash3797 times 103

ndash7595 times 103

ndash1139 times 104

ndash1519 times 104

ndash1899 times 104

ndash2278 times 104

ndash2658 times 104

ndash3038 times 104

ndash3418 times 104

ndash3797 times 104

ndash4177 times 104

ndash4557 times 104

ndash4937 times 104

ndash5316 times 104

ndash5696 times 104

ndash6076 times 104

ndash6056 times 104

ndash6835 times 104

ndash7215 times 104

ndash7595 times 104

Figure 10 Dynamic stress contour at the bottom of the subgradebed surface layer

Adjacent bogies

Field 3 S22(avg 75)

+1487 times 100

+0000 times 100

ndash1945 times 103

ndash3891 times 103

ndash5836 times 103

ndash7781 times 103

ndash9727 times 103

ndash1167 times 104

ndash1362 times 104

ndash1556 times 104

ndash1945 times 104ndash1751 times 104

ndash2140 times 104

ndash2334 times 104

ndash2529 times 104

ndash2724 times 104

ndash2918 times 104

ndash3113 times 104

ndash3303 times 104

ndash3502 times 104

ndash3696 times 104

ndash3891 times 104

Figure 11 Dynamic stress contour at the bottom of the subgradebed bottom layer

Shock and Vibration 7

44 Statistical Analysis of theMaximumDynamic Stress at theSubgrade Surface e maximum dynamic stress at thesubgrade surface was randomly distributed under dierentsleepers because of the vertical track irregularity Figure 18illustrates the statistical results of σdm at the subgrade surfacedirectly below the intersection points of the rail and sleepersIt is observed that the track conditions signicantly aectedthe distribution of σdm σdm was relatively small and thedistribution was relatively concentrated when the rail trackwas smooth and σdm increased and the distribution rangebecame wider when the track condition worsened indicating

that the subgrade bed layers were more likely to withstand agreater dynamic stress which is harmful for maintaining thelong-term service performance of the subgrade

e statistical distributions of σdm under dierent ir-regular track conditions can be well-tted by normal dis-tribution curves as illustrated in Figure 18 and theexpectation (μ) and standard deviation (σ) increased as thetrack condition worsened

e possible maximum dynamic stress at the subgradesurface was predicted using the normal distribution theoryIf σdmsim(μ σ

2) then

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

140

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(a)

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(b)

Figure 12 Dynamic stress at the subgrade surface (a) below the rail (b) below track centerline

24 26 28 30 32 34 36

20

40

60

80

100

120

140

160

Bottom of SBSL rail

Subgrade surface railSubgrade surface centerline

Bottom of SBSL centerlineBottom of SBBL railBottom of SBBL centerline

Max

imum

dyn

amic

stre

ss (k

Pa)

Axle load (t)

Figure 13 Relationship between the maximum dynamic stress andtrain axle load

0 20 40 60 80 100

20

15

10

05

25σdmσs = 036

σdmσs = 034

σdmσs = 021

Maximum dynamic stress (kPa)

Dep

th (m

)

σdmσs = 022

30 t below rail

25 t below rail25 t below centerline

30 t below centerlineSelf-weight stress

Figure 14 Relationship between the maximum dynamic stress anddepth from the subgrade surface

8 Shock and Vibration

P σdm minus μ∣∣∣∣

∣∣∣∣ge 3σ( ) 1minusP σdm minus μ∣∣∣∣

∣∣∣∣lt 3σ( )

1minus 09973 00027lt 0003(5)

erefore the probability of σdm outside the interval of(μ minus 3σ μ + 3σ) was less than 0003 Statistically the intervalof (μ-3σ μ+3σ) is generally taken as the actual possible rangeof a normal distribution random variable namely the triplestandard deviation principle Consequently the possiblemaximum value of the dynamic stress at the subgradesurface can be estimated as μ + 3σ e analyzed possiblemaximum dynamic stresses at the subgrade surface underdierent irregular track spectra are listed in Table 5 emaximum dynamic stress at the subgrade surface of theShuo-Huang heavy-haul railway induced by a train with anaxle load of 30 t could reach 123 kPa [34] which was fa-vorably simulated with a predicted value of 1169 kPa (with adeviation less than 5)

5 Conclusions

A 3D train-track-subgrade interaction nite element (FE)model was constructed using the ABAQUS code (version614-5) where dierent vertical irregular track spectra weresimulated by modifying the rail node coordinates in theABAQUS INP le en the dynamic stress of the subgradebed layers of a heavy-haul railway embankment was ana-lyzed in detail using the 3D FE model e main conclusionsdrawn from this study are as follows

(1) e transverse distribution of the dynamic stress inthe subgrade bed layers gradually transformed from

00 02 04 06 08 10 12 14

20

40

60

80

100

120D

ynam

ic st

ress

(kPa

)

Time (s)

Heavy-haul track irregularityChinarsquos three main lines track irregularitySmooth rail

(a)

00 02 04 06 08 10 12 14

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

Heavy-haul track irregularityChinarsquos three mail lines track irregularitySmooth rail

(b)

Figure 15 Dynamic stress at the subgrade surface under dierent irregular track spectra (a) below rail (b) below centerline

Adjacent bogies

Tension zone

Field 9 S22+2258 times 104

+1000 times 103

ndash3231 times 103

ndash7462 times 103

ndash1169 times 104

ndash1592 times 104

ndash2015 times 104

ndash2439 times 104

ndash3285 times 104ndash2862 times 104

ndash3708 times 104

ndash4131 times 104

ndash4554 times 104

ndash4977 times 104

ndash5400 times 104

ndash5823 times 104

ndash6246 times 104

ndash6670 times 104

ndash7093 times 104

ndash7516 times 104

ndash7939 times 104

ndash8362 times 104

Figure 16 Dynamic stress contour at the subgrade surface con-sidering the irregularity dierence of the left and right rails

0 1 2 3 4

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Distance (m)

Subgrade surface Bottom of SBSL

Bottom of SBBLEmbankment bottom

Embankmentcenterline

Figure 17 Transverse distributions of dynamic stresses at dierentdepths

Shock and Vibration 9

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

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irregularity consists of vertical irregularity horizontal ir-regularity direction irregularity and rail spacing irregu-larity e vertical and horizontal irregularities maysignicantly aect the vertical interaction of the system [29]is study mainly considers vertical irregularity

ere has not been an authoritative track spectrumavailable for describing the track irregularities of heavy-haulrailways in China Wei [30] described that the vertical trackirregularity of Americarsquos fth-grade spectrum is similar tothat of the Chinarsquos heavy-haul railways hence it was appliedas the heavy-haul track irregular spectrum in the establishedFE model e vertical track irregularity of Americarsquos fth-grade spectrum is expressed as

Sv(Ω) kFAvΩ2cΩ2 Ω2 +Ω2c( )

(2)

where Sv(Ω) is the power spectrum density of irregularityΩis the spatial frequency of irregularity Ωc is the cutofrequency Av is the roughness constant (02095) and kF isthe safety factor with a value of 025 generally used

e basic principle for modeling the random track ir-regularity is converting the irregular power spectrum densityto a time domain sample using the inverse fast Fouriertransform (IFFT) e relationship between the discretepower spectral density (Sx(k)) and time series (xs) in theIFFT is formulated as

Sx(k) 1N2 D xs( )∣∣∣∣

∣∣∣∣2 1N2 X

lowast(k)X(k)∣∣∣∣

∣∣∣∣ (3)

where N is the number of discrete sampling points D(xs)means conducting a discrete Fourier transform on xs (s 0to N minus 1) X(k) is the Fourier frequency spectrum ofthe time series xs and Xlowast(k) is the adjoint of X(k) (k 0to N minus 1)

According to equation (3) X(k) can be back-analyzedusing the available power spectral density of irregularityen the serialized time domain model for the track ir-regularity can be obtained by conducting an inverse discreteFourier transform on X(k)

x(n) 1NsumN

k1X(k)exp

2iπknN

( )

X(k) Nξ(k)Sx(kΔω)Δωradic

Δω t

(2π)

ξ(k) exp iϕk( )

(4)

where x(n) is the discrete time domain irregularity t is thetime i is the imaginary unit ξ(k) is an independent phaseseries and ϕk is a random variable obeying a uniformdistribution

Specically the discrete time domain track irregularitycan be obtained as follows (1) discretely sampling on thepower spectrum density of irregularity (2) bank-deducingX(k) using equation (3) and (3) conducting an inversediscrete Fourier transform on X(k) to obtain x(n) eMATLAB software was adopted to improve the analysiseiexclciency and the obtained vertical time domain irregularityof Americarsquos fth-grade spectrum is presented in Figure 6By comparison the vertical time domain irregularity of theleft rail of the Chinarsquos three main railway lines (Beijing-Shanghai railway Beijing-Guangzhou railway and Beijing-Jiulong railway) is also shown in Figure 6 It is found that thevertical irregularity of Chinarsquos three main railway lines is lessthan that of Americarsquos fth-grade spectrum hence it maynot be available for heavy-haul railways in China

It is assumed that the vertical track irregularity is fullyattributed to the surface undulation of the rail e nodecoordinates matrix of the smooth rail surface was rstexported from the ABAQUS FE model (INP le) thenthe vertical node coordinates were modied according tothe vertical track irregularity illustrated in Figure 6 usingthe MATLAB software Finally the modied node co-ordinate matrix was imported to the smooth rail surfaceFigure 7 presents the rail models before and after modi-cation It shows that the vertical track irregularity wasproperly simulated

Fasteners and cushion blocks

Rail

Rail

SleeperSleeper

Figure 4 Connection between the rail and sleeper

Table 3 Mechanical parameters of dierent parts of the FE model

Part E (MPa) ] ρ (kgm3) φ (deg) c (kPa) DTrain Rigid body mdash mdash mdash mdash mdashRail 21 times 105 025 7850 mdash mdash mdashSleeper 30000 02 2500 mdash mdash mdashBallast 200 025 2400 mdash mdash 005SBSL 180 03 2300 33 58 01SBBL 150 03 2200 31 45 01SBSB 70 035 1800 17 30 015Foundation 50 035 1700 15 25 015

4 Shock and Vibration

4 FEA Results and Analysis

41Verication of the FEModel Ding et al [31] reported themeasured maximum vertical displacements of the rail andsleeper and the maximum acceleration of the ballast at a testsite of the Shuo-Huang heavy-haul railway under a trainwith an axle load of 25 t the FE analyzed (FEA) results arecompared with the eld measurements as listed in Table 4 Itshows that the analyzed values agreed well with the eldmeasurements Miao et al [32] reported the dynamic stress-time curve measured at the subgrade surface under a trial-train with an axle load of 30 t the FE analyzed curve iscompared with themeasured curve as illustrated in Figure 8It is observed that the calculated curve well simulated themeasurements in both wave shape and peak valuese good

agreements between the FEA results and eld measurementsdemonstrated the eectiveness and reliability of the FEmodel

42 Dynamic Stress of the Subgrade Bed Layers e trainspeed of the Shuo-Huang heavy-haul railway is approxi-mately 80 kmh hence the default speed in the FEA was80 kmh e analyzed dynamic stress contour at the sub-grade surface induced by a heavy-haul train with an axle loadof 30 t is presented in Figure 9 In the specied case theanalyzed results all corresponded to a train axle load of 30 tin the following parts of this article It shows that the dy-namic stress is symmetrically distributed along track cen-terline the maximum dynamic stress appeared directlyunder the rail and was attenuated toward the track centerlineand the embankment shoulder the transverse distribution ofthe dynamic stress exhibited a bimodal pattern with twopeak points and a tensile stress appeared under the middlepart of the train carriage In addition the train ran smoothlyon a smooth track the dierence of the dynamic stress at thesubgrade surface under the four bogies was less than 10 kPaand the eects of a bogie were similar to a nonuniformrectangular pressure as shown in Figure 9(a) Converselythe train exhibited bouncing movements on the verticallyirregular track thus the maximum dynamic stress wasgreater than that under a smooth track condition noddingmotion appeared on the train carriages and bogies hence thedynamic stresses at the subgrade surface under the fourbogies appeared to be obviously dierent with a greaterdeviation of approximately 20 kPa (Figure 9(b))

Figures 10 and 11 present the dynamic stress contourat the bottom of the SBSL and SBBL under irregulartrack conditions respectively It is determined that thetransverse distribution of the dynamic stress varied from abimodal pattern to a unimodal pattern with an increase inthe depth from the subgrade surface (Figures 9(b) and 10)

0 50 100 150 200ndash10

ndash5

0

5

10

Ver

tical

irre

gula

rity

(mm

)

Distance (m)

Americarsquos fifth-grade track irregularityChinarsquos three mail lines track irregularity

Figure 6 Vertical track irregularity in the time domain

TrainRailBallast

72m

5m

55m

Subgrade bed layersEmbankment foundation

Figure 5 Train-track-subgrade interaction model

Shock and Vibration 5

0e dynamic stress directly beneath the rail was close to thatunder the track centerline at the bottom of the subgrade bedsurface layer as illustrated in Figure 10 0e distribution ofthe dynamic stress induced by adjacent bodies of adjacentcarriages exhibited an elliptical shape and the superimposeddynamic stress was apparently greater than that induced by asingle bogie (Figure 11) 0erefore the superposition effectsof the adjacent bogies could enlarge the influencing depth of

the dynamic stress because of the small bogie spacing of theheavy-haul train

Figures 12(a) and 12(b) present the subgrade surfacedynamic stress-time curves directly below the rail and trackcenterline under different train axle loads respectively Itshows that the maximum dynamic stresses (σdm) at thesubgrade surface directly below the rail and track centerlinehad increments of 338 kPa and 32 kPa (corresponding topercent increases of 398 and 473) when the train axleload increased from 25 t to 35 t respectively An increase inthe train axle load intensified the subgrade dynamic stressand the interaction between the train and track system Inaddition the subgrade surface dynamic stresses induced bythe adjacent bogies exhibited an obvious superposition ef-fect and the dynamic stresses were maintained around thepeak value during the pass of the adjacent bogies hence thepass of adjacent bogies of adjacent carriages may be sim-plified once loaded on the railway subgrade

0e relationships between the maximum dynamic stressand train axle load (P) at different locations below a sleeperare illustrated in Figure 13 It shows that the maximumdynamic stresses increased approximately linearly as thetrain axle load increased 0e average increase rate of thedynamic stress directly below the rail was 317 kPat373 kPat and 152 kPat at the subgrade surface and thebottoms of SBSL and SBBL respectively For the locationsdirectly below the track centerline the average increase rateswere 319 kPat 359 kPat and 178 kPat Generally thedynamic stress at the bottom of the SBSL was more sensitiveto the train axle load compared to that at the subgradesurface because the dynamic stresses induced by the two railswere gradually overlaid with an increase in the depth fromthe subgrade surface

Figure 14 presents the relationship between the maxi-mum dynamic stress and the depth from the subgradesurface σdm at the subgrade surface directly below the rail aswell as its attenuation rate was greater than that below the

(a) (b)

Figure 7 Rail model (a) before modification (b) after modification

Table 4 Measured and simulated vertical displacement of the railand sleeper

Parameter Measuredvalue

Calculatedvalue

Vertical displacement of rail (mm) 201 192ndash221Vertical displacement of sleeper (mm) 155 148ndash173Acceleration of ballast (ms2) 131 081ndash113

04 06 08 10 12

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

FE analyzedMeasured

Figure 8 Measured and calculated subgrade surface dynamicstress

6 Shock and Vibration

track centerline σdm below the rail became less than thatbelow the track centerline at a depth of approximately 03mσdm at the subgrade bed bottom was reduced by approxi-mately 70 compared with that at the subgrade surface eratio of σdm to the self-weight stress (σs) at the subgrade bedbottom (25m below the subgrade surface) under a train axle

load of 25 t was 021 and 022 for the locations directly belowthe rail and the track centerline respectively the σdmσsvalues increased to 034 and 036 under a train axle load of30 t respectively Zhou [33] suggested that the σdmσs valueshould be less than 020 to maintain the long-term stabilityof a railway subgrade Consequently the thickness of thesubgrade bed (25m a subgrade bed surface layer of 06mand a subgrade bed bottom layer of 19m) of existingrailways in China may meet the requirements of running aheavy-haul train with an axle load of 25 t but cannot enablethe operation of heavy-haul trains with axle loads greaterthan 25 t unless the subgrades have been reinforced

43 Eects of Irregular Track Spectrum Figure 15 illustratesthe subgrade surface dynamic stress-time curves underdierent track conditions e vertical track irregularityintensied the vertical excitation between the train wheelsand rails and induced a greater dynamic stress comparedwith the smooth rail condition e maximum dynamicstress at the subgrade surface increased by 23 and 158under the action of the heavy-haul irregular track spectrumand Chinarsquos three main railway lines irregular track spec-trum respectively

In China the irregular track spectra of the left and rightrails are dierent e dynamic stress contour at the sub-grade surface taking this dierence into account is shown inFigure 16 It is determined that the FE model eectivelyreshyected the irregularity dierence of the left and right railsand the dynamic stress was not symmetrically distributedalong the track centerline

Figure 17 presents the transverse distributions of thedynamic stress at dierent depths e subgrade surfacedynamic stress exhibited a bimodal pattern (or an ldquoMrdquoshape) and the value under the left rail was approximately19 greater than that under the right rail whereas theasymmetry of the dynamic stress gradually decreased as thedepth increased and was negligible at the subgrade bedbottom hence the irregularity dierence between the leftand right rails had a slight impact on the embankment soilsbelow the subgrade bed bottom

Adjacent b

ogies

Tension zone

Field 2 S22(avg 75)

+2359 times 104

+0000 times 100

ndash3793 times 103

ndash7586 times 103

ndash1138 times 104

ndash1517 times 104

ndash1896 times 104

ndash2276 times 104

ndash2655 times 104

ndash3034 times 104

ndash3414 times 104

ndash3793 times 104

ndash4172 times 104

ndash4552 times 104

ndash4931 times 104

ndash5310 times 104

ndash5689 times 104

ndash6069 times 104

ndash6448 times 104

ndash6827 times 104

ndash7207 times 104

ndash7586 times 104

(a)

Adjacent bogies

Tension zone

Field 3 S22(avg 75)

+2729 times 104

+0000 times 100

ndash5198 times 103

ndash1040 times 104

ndash1559 times 104

ndash2079 times 104

ndash2599 times 104

ndash3119 times 104

ndash3638 times 104

ndash4158 times 104

ndash4678 times 104

ndash5198 times 104

ndash5718 times 104

ndash6237 times 104

ndash6757 times 104

ndash7277 times 104

ndash7797 times 104

ndash8316 times 104

ndash8836 times 104

ndash9356 times 104

ndash9876 times 104

ndash1040 times 105

(b)

Figure 9 Dynamic stress contour at the subgrade surface (a) smooth track (b) vertically irregular track

Adjacent bogies

Field 3 S22(avg 75)

+1246 times 104

+0000 times 100

ndash3797 times 103

ndash7595 times 103

ndash1139 times 104

ndash1519 times 104

ndash1899 times 104

ndash2278 times 104

ndash2658 times 104

ndash3038 times 104

ndash3418 times 104

ndash3797 times 104

ndash4177 times 104

ndash4557 times 104

ndash4937 times 104

ndash5316 times 104

ndash5696 times 104

ndash6076 times 104

ndash6056 times 104

ndash6835 times 104

ndash7215 times 104

ndash7595 times 104

Figure 10 Dynamic stress contour at the bottom of the subgradebed surface layer

Adjacent bogies

Field 3 S22(avg 75)

+1487 times 100

+0000 times 100

ndash1945 times 103

ndash3891 times 103

ndash5836 times 103

ndash7781 times 103

ndash9727 times 103

ndash1167 times 104

ndash1362 times 104

ndash1556 times 104

ndash1945 times 104ndash1751 times 104

ndash2140 times 104

ndash2334 times 104

ndash2529 times 104

ndash2724 times 104

ndash2918 times 104

ndash3113 times 104

ndash3303 times 104

ndash3502 times 104

ndash3696 times 104

ndash3891 times 104

Figure 11 Dynamic stress contour at the bottom of the subgradebed bottom layer

Shock and Vibration 7

44 Statistical Analysis of theMaximumDynamic Stress at theSubgrade Surface e maximum dynamic stress at thesubgrade surface was randomly distributed under dierentsleepers because of the vertical track irregularity Figure 18illustrates the statistical results of σdm at the subgrade surfacedirectly below the intersection points of the rail and sleepersIt is observed that the track conditions signicantly aectedthe distribution of σdm σdm was relatively small and thedistribution was relatively concentrated when the rail trackwas smooth and σdm increased and the distribution rangebecame wider when the track condition worsened indicating

that the subgrade bed layers were more likely to withstand agreater dynamic stress which is harmful for maintaining thelong-term service performance of the subgrade

e statistical distributions of σdm under dierent ir-regular track conditions can be well-tted by normal dis-tribution curves as illustrated in Figure 18 and theexpectation (μ) and standard deviation (σ) increased as thetrack condition worsened

e possible maximum dynamic stress at the subgradesurface was predicted using the normal distribution theoryIf σdmsim(μ σ

2) then

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

140

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(a)

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(b)

Figure 12 Dynamic stress at the subgrade surface (a) below the rail (b) below track centerline

24 26 28 30 32 34 36

20

40

60

80

100

120

140

160

Bottom of SBSL rail

Subgrade surface railSubgrade surface centerline

Bottom of SBSL centerlineBottom of SBBL railBottom of SBBL centerline

Max

imum

dyn

amic

stre

ss (k

Pa)

Axle load (t)

Figure 13 Relationship between the maximum dynamic stress andtrain axle load

0 20 40 60 80 100

20

15

10

05

25σdmσs = 036

σdmσs = 034

σdmσs = 021

Maximum dynamic stress (kPa)

Dep

th (m

)

σdmσs = 022

30 t below rail

25 t below rail25 t below centerline

30 t below centerlineSelf-weight stress

Figure 14 Relationship between the maximum dynamic stress anddepth from the subgrade surface

8 Shock and Vibration

P σdm minus μ∣∣∣∣

∣∣∣∣ge 3σ( ) 1minusP σdm minus μ∣∣∣∣

∣∣∣∣lt 3σ( )

1minus 09973 00027lt 0003(5)

erefore the probability of σdm outside the interval of(μ minus 3σ μ + 3σ) was less than 0003 Statistically the intervalof (μ-3σ μ+3σ) is generally taken as the actual possible rangeof a normal distribution random variable namely the triplestandard deviation principle Consequently the possiblemaximum value of the dynamic stress at the subgradesurface can be estimated as μ + 3σ e analyzed possiblemaximum dynamic stresses at the subgrade surface underdierent irregular track spectra are listed in Table 5 emaximum dynamic stress at the subgrade surface of theShuo-Huang heavy-haul railway induced by a train with anaxle load of 30 t could reach 123 kPa [34] which was fa-vorably simulated with a predicted value of 1169 kPa (with adeviation less than 5)

5 Conclusions

A 3D train-track-subgrade interaction nite element (FE)model was constructed using the ABAQUS code (version614-5) where dierent vertical irregular track spectra weresimulated by modifying the rail node coordinates in theABAQUS INP le en the dynamic stress of the subgradebed layers of a heavy-haul railway embankment was ana-lyzed in detail using the 3D FE model e main conclusionsdrawn from this study are as follows

(1) e transverse distribution of the dynamic stress inthe subgrade bed layers gradually transformed from

00 02 04 06 08 10 12 14

20

40

60

80

100

120D

ynam

ic st

ress

(kPa

)

Time (s)

Heavy-haul track irregularityChinarsquos three main lines track irregularitySmooth rail

(a)

00 02 04 06 08 10 12 14

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

Heavy-haul track irregularityChinarsquos three mail lines track irregularitySmooth rail

(b)

Figure 15 Dynamic stress at the subgrade surface under dierent irregular track spectra (a) below rail (b) below centerline

Adjacent bogies

Tension zone

Field 9 S22+2258 times 104

+1000 times 103

ndash3231 times 103

ndash7462 times 103

ndash1169 times 104

ndash1592 times 104

ndash2015 times 104

ndash2439 times 104

ndash3285 times 104ndash2862 times 104

ndash3708 times 104

ndash4131 times 104

ndash4554 times 104

ndash4977 times 104

ndash5400 times 104

ndash5823 times 104

ndash6246 times 104

ndash6670 times 104

ndash7093 times 104

ndash7516 times 104

ndash7939 times 104

ndash8362 times 104

Figure 16 Dynamic stress contour at the subgrade surface con-sidering the irregularity dierence of the left and right rails

0 1 2 3 4

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Distance (m)

Subgrade surface Bottom of SBSL

Bottom of SBBLEmbankment bottom

Embankmentcenterline

Figure 17 Transverse distributions of dynamic stresses at dierentdepths

Shock and Vibration 9

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

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Submit your manuscripts atwwwhindawicom

4 FEA Results and Analysis

41Verication of the FEModel Ding et al [31] reported themeasured maximum vertical displacements of the rail andsleeper and the maximum acceleration of the ballast at a testsite of the Shuo-Huang heavy-haul railway under a trainwith an axle load of 25 t the FE analyzed (FEA) results arecompared with the eld measurements as listed in Table 4 Itshows that the analyzed values agreed well with the eldmeasurements Miao et al [32] reported the dynamic stress-time curve measured at the subgrade surface under a trial-train with an axle load of 30 t the FE analyzed curve iscompared with themeasured curve as illustrated in Figure 8It is observed that the calculated curve well simulated themeasurements in both wave shape and peak valuese good

agreements between the FEA results and eld measurementsdemonstrated the eectiveness and reliability of the FEmodel

42 Dynamic Stress of the Subgrade Bed Layers e trainspeed of the Shuo-Huang heavy-haul railway is approxi-mately 80 kmh hence the default speed in the FEA was80 kmh e analyzed dynamic stress contour at the sub-grade surface induced by a heavy-haul train with an axle loadof 30 t is presented in Figure 9 In the specied case theanalyzed results all corresponded to a train axle load of 30 tin the following parts of this article It shows that the dy-namic stress is symmetrically distributed along track cen-terline the maximum dynamic stress appeared directlyunder the rail and was attenuated toward the track centerlineand the embankment shoulder the transverse distribution ofthe dynamic stress exhibited a bimodal pattern with twopeak points and a tensile stress appeared under the middlepart of the train carriage In addition the train ran smoothlyon a smooth track the dierence of the dynamic stress at thesubgrade surface under the four bogies was less than 10 kPaand the eects of a bogie were similar to a nonuniformrectangular pressure as shown in Figure 9(a) Converselythe train exhibited bouncing movements on the verticallyirregular track thus the maximum dynamic stress wasgreater than that under a smooth track condition noddingmotion appeared on the train carriages and bogies hence thedynamic stresses at the subgrade surface under the fourbogies appeared to be obviously dierent with a greaterdeviation of approximately 20 kPa (Figure 9(b))

Figures 10 and 11 present the dynamic stress contourat the bottom of the SBSL and SBBL under irregulartrack conditions respectively It is determined that thetransverse distribution of the dynamic stress varied from abimodal pattern to a unimodal pattern with an increase inthe depth from the subgrade surface (Figures 9(b) and 10)

0 50 100 150 200ndash10

ndash5

0

5

10

Ver

tical

irre

gula

rity

(mm

)

Distance (m)

Americarsquos fifth-grade track irregularityChinarsquos three mail lines track irregularity

Figure 6 Vertical track irregularity in the time domain

TrainRailBallast

72m

5m

55m

Subgrade bed layersEmbankment foundation

Figure 5 Train-track-subgrade interaction model

Shock and Vibration 5

0e dynamic stress directly beneath the rail was close to thatunder the track centerline at the bottom of the subgrade bedsurface layer as illustrated in Figure 10 0e distribution ofthe dynamic stress induced by adjacent bodies of adjacentcarriages exhibited an elliptical shape and the superimposeddynamic stress was apparently greater than that induced by asingle bogie (Figure 11) 0erefore the superposition effectsof the adjacent bogies could enlarge the influencing depth of

the dynamic stress because of the small bogie spacing of theheavy-haul train

Figures 12(a) and 12(b) present the subgrade surfacedynamic stress-time curves directly below the rail and trackcenterline under different train axle loads respectively Itshows that the maximum dynamic stresses (σdm) at thesubgrade surface directly below the rail and track centerlinehad increments of 338 kPa and 32 kPa (corresponding topercent increases of 398 and 473) when the train axleload increased from 25 t to 35 t respectively An increase inthe train axle load intensified the subgrade dynamic stressand the interaction between the train and track system Inaddition the subgrade surface dynamic stresses induced bythe adjacent bogies exhibited an obvious superposition ef-fect and the dynamic stresses were maintained around thepeak value during the pass of the adjacent bogies hence thepass of adjacent bogies of adjacent carriages may be sim-plified once loaded on the railway subgrade

0e relationships between the maximum dynamic stressand train axle load (P) at different locations below a sleeperare illustrated in Figure 13 It shows that the maximumdynamic stresses increased approximately linearly as thetrain axle load increased 0e average increase rate of thedynamic stress directly below the rail was 317 kPat373 kPat and 152 kPat at the subgrade surface and thebottoms of SBSL and SBBL respectively For the locationsdirectly below the track centerline the average increase rateswere 319 kPat 359 kPat and 178 kPat Generally thedynamic stress at the bottom of the SBSL was more sensitiveto the train axle load compared to that at the subgradesurface because the dynamic stresses induced by the two railswere gradually overlaid with an increase in the depth fromthe subgrade surface

Figure 14 presents the relationship between the maxi-mum dynamic stress and the depth from the subgradesurface σdm at the subgrade surface directly below the rail aswell as its attenuation rate was greater than that below the

(a) (b)

Figure 7 Rail model (a) before modification (b) after modification

Table 4 Measured and simulated vertical displacement of the railand sleeper

Parameter Measuredvalue

Calculatedvalue

Vertical displacement of rail (mm) 201 192ndash221Vertical displacement of sleeper (mm) 155 148ndash173Acceleration of ballast (ms2) 131 081ndash113

04 06 08 10 12

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

FE analyzedMeasured

Figure 8 Measured and calculated subgrade surface dynamicstress

6 Shock and Vibration

track centerline σdm below the rail became less than thatbelow the track centerline at a depth of approximately 03mσdm at the subgrade bed bottom was reduced by approxi-mately 70 compared with that at the subgrade surface eratio of σdm to the self-weight stress (σs) at the subgrade bedbottom (25m below the subgrade surface) under a train axle

load of 25 t was 021 and 022 for the locations directly belowthe rail and the track centerline respectively the σdmσsvalues increased to 034 and 036 under a train axle load of30 t respectively Zhou [33] suggested that the σdmσs valueshould be less than 020 to maintain the long-term stabilityof a railway subgrade Consequently the thickness of thesubgrade bed (25m a subgrade bed surface layer of 06mand a subgrade bed bottom layer of 19m) of existingrailways in China may meet the requirements of running aheavy-haul train with an axle load of 25 t but cannot enablethe operation of heavy-haul trains with axle loads greaterthan 25 t unless the subgrades have been reinforced

43 Eects of Irregular Track Spectrum Figure 15 illustratesthe subgrade surface dynamic stress-time curves underdierent track conditions e vertical track irregularityintensied the vertical excitation between the train wheelsand rails and induced a greater dynamic stress comparedwith the smooth rail condition e maximum dynamicstress at the subgrade surface increased by 23 and 158under the action of the heavy-haul irregular track spectrumand Chinarsquos three main railway lines irregular track spec-trum respectively

In China the irregular track spectra of the left and rightrails are dierent e dynamic stress contour at the sub-grade surface taking this dierence into account is shown inFigure 16 It is determined that the FE model eectivelyreshyected the irregularity dierence of the left and right railsand the dynamic stress was not symmetrically distributedalong the track centerline

Figure 17 presents the transverse distributions of thedynamic stress at dierent depths e subgrade surfacedynamic stress exhibited a bimodal pattern (or an ldquoMrdquoshape) and the value under the left rail was approximately19 greater than that under the right rail whereas theasymmetry of the dynamic stress gradually decreased as thedepth increased and was negligible at the subgrade bedbottom hence the irregularity dierence between the leftand right rails had a slight impact on the embankment soilsbelow the subgrade bed bottom

Adjacent b

ogies

Tension zone

Field 2 S22(avg 75)

+2359 times 104

+0000 times 100

ndash3793 times 103

ndash7586 times 103

ndash1138 times 104

ndash1517 times 104

ndash1896 times 104

ndash2276 times 104

ndash2655 times 104

ndash3034 times 104

ndash3414 times 104

ndash3793 times 104

ndash4172 times 104

ndash4552 times 104

ndash4931 times 104

ndash5310 times 104

ndash5689 times 104

ndash6069 times 104

ndash6448 times 104

ndash6827 times 104

ndash7207 times 104

ndash7586 times 104

(a)

Adjacent bogies

Tension zone

Field 3 S22(avg 75)

+2729 times 104

+0000 times 100

ndash5198 times 103

ndash1040 times 104

ndash1559 times 104

ndash2079 times 104

ndash2599 times 104

ndash3119 times 104

ndash3638 times 104

ndash4158 times 104

ndash4678 times 104

ndash5198 times 104

ndash5718 times 104

ndash6237 times 104

ndash6757 times 104

ndash7277 times 104

ndash7797 times 104

ndash8316 times 104

ndash8836 times 104

ndash9356 times 104

ndash9876 times 104

ndash1040 times 105

(b)

Figure 9 Dynamic stress contour at the subgrade surface (a) smooth track (b) vertically irregular track

Adjacent bogies

Field 3 S22(avg 75)

+1246 times 104

+0000 times 100

ndash3797 times 103

ndash7595 times 103

ndash1139 times 104

ndash1519 times 104

ndash1899 times 104

ndash2278 times 104

ndash2658 times 104

ndash3038 times 104

ndash3418 times 104

ndash3797 times 104

ndash4177 times 104

ndash4557 times 104

ndash4937 times 104

ndash5316 times 104

ndash5696 times 104

ndash6076 times 104

ndash6056 times 104

ndash6835 times 104

ndash7215 times 104

ndash7595 times 104

Figure 10 Dynamic stress contour at the bottom of the subgradebed surface layer

Adjacent bogies

Field 3 S22(avg 75)

+1487 times 100

+0000 times 100

ndash1945 times 103

ndash3891 times 103

ndash5836 times 103

ndash7781 times 103

ndash9727 times 103

ndash1167 times 104

ndash1362 times 104

ndash1556 times 104

ndash1945 times 104ndash1751 times 104

ndash2140 times 104

ndash2334 times 104

ndash2529 times 104

ndash2724 times 104

ndash2918 times 104

ndash3113 times 104

ndash3303 times 104

ndash3502 times 104

ndash3696 times 104

ndash3891 times 104

Figure 11 Dynamic stress contour at the bottom of the subgradebed bottom layer

Shock and Vibration 7

44 Statistical Analysis of theMaximumDynamic Stress at theSubgrade Surface e maximum dynamic stress at thesubgrade surface was randomly distributed under dierentsleepers because of the vertical track irregularity Figure 18illustrates the statistical results of σdm at the subgrade surfacedirectly below the intersection points of the rail and sleepersIt is observed that the track conditions signicantly aectedthe distribution of σdm σdm was relatively small and thedistribution was relatively concentrated when the rail trackwas smooth and σdm increased and the distribution rangebecame wider when the track condition worsened indicating

that the subgrade bed layers were more likely to withstand agreater dynamic stress which is harmful for maintaining thelong-term service performance of the subgrade

e statistical distributions of σdm under dierent ir-regular track conditions can be well-tted by normal dis-tribution curves as illustrated in Figure 18 and theexpectation (μ) and standard deviation (σ) increased as thetrack condition worsened

e possible maximum dynamic stress at the subgradesurface was predicted using the normal distribution theoryIf σdmsim(μ σ

2) then

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

140

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(a)

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(b)

Figure 12 Dynamic stress at the subgrade surface (a) below the rail (b) below track centerline

24 26 28 30 32 34 36

20

40

60

80

100

120

140

160

Bottom of SBSL rail

Subgrade surface railSubgrade surface centerline

Bottom of SBSL centerlineBottom of SBBL railBottom of SBBL centerline

Max

imum

dyn

amic

stre

ss (k

Pa)

Axle load (t)

Figure 13 Relationship between the maximum dynamic stress andtrain axle load

0 20 40 60 80 100

20

15

10

05

25σdmσs = 036

σdmσs = 034

σdmσs = 021

Maximum dynamic stress (kPa)

Dep

th (m

)

σdmσs = 022

30 t below rail

25 t below rail25 t below centerline

30 t below centerlineSelf-weight stress

Figure 14 Relationship between the maximum dynamic stress anddepth from the subgrade surface

8 Shock and Vibration

P σdm minus μ∣∣∣∣

∣∣∣∣ge 3σ( ) 1minusP σdm minus μ∣∣∣∣

∣∣∣∣lt 3σ( )

1minus 09973 00027lt 0003(5)

erefore the probability of σdm outside the interval of(μ minus 3σ μ + 3σ) was less than 0003 Statistically the intervalof (μ-3σ μ+3σ) is generally taken as the actual possible rangeof a normal distribution random variable namely the triplestandard deviation principle Consequently the possiblemaximum value of the dynamic stress at the subgradesurface can be estimated as μ + 3σ e analyzed possiblemaximum dynamic stresses at the subgrade surface underdierent irregular track spectra are listed in Table 5 emaximum dynamic stress at the subgrade surface of theShuo-Huang heavy-haul railway induced by a train with anaxle load of 30 t could reach 123 kPa [34] which was fa-vorably simulated with a predicted value of 1169 kPa (with adeviation less than 5)

5 Conclusions

A 3D train-track-subgrade interaction nite element (FE)model was constructed using the ABAQUS code (version614-5) where dierent vertical irregular track spectra weresimulated by modifying the rail node coordinates in theABAQUS INP le en the dynamic stress of the subgradebed layers of a heavy-haul railway embankment was ana-lyzed in detail using the 3D FE model e main conclusionsdrawn from this study are as follows

(1) e transverse distribution of the dynamic stress inthe subgrade bed layers gradually transformed from

00 02 04 06 08 10 12 14

20

40

60

80

100

120D

ynam

ic st

ress

(kPa

)

Time (s)

Heavy-haul track irregularityChinarsquos three main lines track irregularitySmooth rail

(a)

00 02 04 06 08 10 12 14

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

Heavy-haul track irregularityChinarsquos three mail lines track irregularitySmooth rail

(b)

Figure 15 Dynamic stress at the subgrade surface under dierent irregular track spectra (a) below rail (b) below centerline

Adjacent bogies

Tension zone

Field 9 S22+2258 times 104

+1000 times 103

ndash3231 times 103

ndash7462 times 103

ndash1169 times 104

ndash1592 times 104

ndash2015 times 104

ndash2439 times 104

ndash3285 times 104ndash2862 times 104

ndash3708 times 104

ndash4131 times 104

ndash4554 times 104

ndash4977 times 104

ndash5400 times 104

ndash5823 times 104

ndash6246 times 104

ndash6670 times 104

ndash7093 times 104

ndash7516 times 104

ndash7939 times 104

ndash8362 times 104

Figure 16 Dynamic stress contour at the subgrade surface con-sidering the irregularity dierence of the left and right rails

0 1 2 3 4

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Distance (m)

Subgrade surface Bottom of SBSL

Bottom of SBBLEmbankment bottom

Embankmentcenterline

Figure 17 Transverse distributions of dynamic stresses at dierentdepths

Shock and Vibration 9

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

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0e dynamic stress directly beneath the rail was close to thatunder the track centerline at the bottom of the subgrade bedsurface layer as illustrated in Figure 10 0e distribution ofthe dynamic stress induced by adjacent bodies of adjacentcarriages exhibited an elliptical shape and the superimposeddynamic stress was apparently greater than that induced by asingle bogie (Figure 11) 0erefore the superposition effectsof the adjacent bogies could enlarge the influencing depth of

the dynamic stress because of the small bogie spacing of theheavy-haul train

Figures 12(a) and 12(b) present the subgrade surfacedynamic stress-time curves directly below the rail and trackcenterline under different train axle loads respectively Itshows that the maximum dynamic stresses (σdm) at thesubgrade surface directly below the rail and track centerlinehad increments of 338 kPa and 32 kPa (corresponding topercent increases of 398 and 473) when the train axleload increased from 25 t to 35 t respectively An increase inthe train axle load intensified the subgrade dynamic stressand the interaction between the train and track system Inaddition the subgrade surface dynamic stresses induced bythe adjacent bogies exhibited an obvious superposition ef-fect and the dynamic stresses were maintained around thepeak value during the pass of the adjacent bogies hence thepass of adjacent bogies of adjacent carriages may be sim-plified once loaded on the railway subgrade

0e relationships between the maximum dynamic stressand train axle load (P) at different locations below a sleeperare illustrated in Figure 13 It shows that the maximumdynamic stresses increased approximately linearly as thetrain axle load increased 0e average increase rate of thedynamic stress directly below the rail was 317 kPat373 kPat and 152 kPat at the subgrade surface and thebottoms of SBSL and SBBL respectively For the locationsdirectly below the track centerline the average increase rateswere 319 kPat 359 kPat and 178 kPat Generally thedynamic stress at the bottom of the SBSL was more sensitiveto the train axle load compared to that at the subgradesurface because the dynamic stresses induced by the two railswere gradually overlaid with an increase in the depth fromthe subgrade surface

Figure 14 presents the relationship between the maxi-mum dynamic stress and the depth from the subgradesurface σdm at the subgrade surface directly below the rail aswell as its attenuation rate was greater than that below the

(a) (b)

Figure 7 Rail model (a) before modification (b) after modification

Table 4 Measured and simulated vertical displacement of the railand sleeper

Parameter Measuredvalue

Calculatedvalue

Vertical displacement of rail (mm) 201 192ndash221Vertical displacement of sleeper (mm) 155 148ndash173Acceleration of ballast (ms2) 131 081ndash113

04 06 08 10 12

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

FE analyzedMeasured

Figure 8 Measured and calculated subgrade surface dynamicstress

6 Shock and Vibration

track centerline σdm below the rail became less than thatbelow the track centerline at a depth of approximately 03mσdm at the subgrade bed bottom was reduced by approxi-mately 70 compared with that at the subgrade surface eratio of σdm to the self-weight stress (σs) at the subgrade bedbottom (25m below the subgrade surface) under a train axle

load of 25 t was 021 and 022 for the locations directly belowthe rail and the track centerline respectively the σdmσsvalues increased to 034 and 036 under a train axle load of30 t respectively Zhou [33] suggested that the σdmσs valueshould be less than 020 to maintain the long-term stabilityof a railway subgrade Consequently the thickness of thesubgrade bed (25m a subgrade bed surface layer of 06mand a subgrade bed bottom layer of 19m) of existingrailways in China may meet the requirements of running aheavy-haul train with an axle load of 25 t but cannot enablethe operation of heavy-haul trains with axle loads greaterthan 25 t unless the subgrades have been reinforced

43 Eects of Irregular Track Spectrum Figure 15 illustratesthe subgrade surface dynamic stress-time curves underdierent track conditions e vertical track irregularityintensied the vertical excitation between the train wheelsand rails and induced a greater dynamic stress comparedwith the smooth rail condition e maximum dynamicstress at the subgrade surface increased by 23 and 158under the action of the heavy-haul irregular track spectrumand Chinarsquos three main railway lines irregular track spec-trum respectively

In China the irregular track spectra of the left and rightrails are dierent e dynamic stress contour at the sub-grade surface taking this dierence into account is shown inFigure 16 It is determined that the FE model eectivelyreshyected the irregularity dierence of the left and right railsand the dynamic stress was not symmetrically distributedalong the track centerline

Figure 17 presents the transverse distributions of thedynamic stress at dierent depths e subgrade surfacedynamic stress exhibited a bimodal pattern (or an ldquoMrdquoshape) and the value under the left rail was approximately19 greater than that under the right rail whereas theasymmetry of the dynamic stress gradually decreased as thedepth increased and was negligible at the subgrade bedbottom hence the irregularity dierence between the leftand right rails had a slight impact on the embankment soilsbelow the subgrade bed bottom

Adjacent b

ogies

Tension zone

Field 2 S22(avg 75)

+2359 times 104

+0000 times 100

ndash3793 times 103

ndash7586 times 103

ndash1138 times 104

ndash1517 times 104

ndash1896 times 104

ndash2276 times 104

ndash2655 times 104

ndash3034 times 104

ndash3414 times 104

ndash3793 times 104

ndash4172 times 104

ndash4552 times 104

ndash4931 times 104

ndash5310 times 104

ndash5689 times 104

ndash6069 times 104

ndash6448 times 104

ndash6827 times 104

ndash7207 times 104

ndash7586 times 104

(a)

Adjacent bogies

Tension zone

Field 3 S22(avg 75)

+2729 times 104

+0000 times 100

ndash5198 times 103

ndash1040 times 104

ndash1559 times 104

ndash2079 times 104

ndash2599 times 104

ndash3119 times 104

ndash3638 times 104

ndash4158 times 104

ndash4678 times 104

ndash5198 times 104

ndash5718 times 104

ndash6237 times 104

ndash6757 times 104

ndash7277 times 104

ndash7797 times 104

ndash8316 times 104

ndash8836 times 104

ndash9356 times 104

ndash9876 times 104

ndash1040 times 105

(b)

Figure 9 Dynamic stress contour at the subgrade surface (a) smooth track (b) vertically irregular track

Adjacent bogies

Field 3 S22(avg 75)

+1246 times 104

+0000 times 100

ndash3797 times 103

ndash7595 times 103

ndash1139 times 104

ndash1519 times 104

ndash1899 times 104

ndash2278 times 104

ndash2658 times 104

ndash3038 times 104

ndash3418 times 104

ndash3797 times 104

ndash4177 times 104

ndash4557 times 104

ndash4937 times 104

ndash5316 times 104

ndash5696 times 104

ndash6076 times 104

ndash6056 times 104

ndash6835 times 104

ndash7215 times 104

ndash7595 times 104

Figure 10 Dynamic stress contour at the bottom of the subgradebed surface layer

Adjacent bogies

Field 3 S22(avg 75)

+1487 times 100

+0000 times 100

ndash1945 times 103

ndash3891 times 103

ndash5836 times 103

ndash7781 times 103

ndash9727 times 103

ndash1167 times 104

ndash1362 times 104

ndash1556 times 104

ndash1945 times 104ndash1751 times 104

ndash2140 times 104

ndash2334 times 104

ndash2529 times 104

ndash2724 times 104

ndash2918 times 104

ndash3113 times 104

ndash3303 times 104

ndash3502 times 104

ndash3696 times 104

ndash3891 times 104

Figure 11 Dynamic stress contour at the bottom of the subgradebed bottom layer

Shock and Vibration 7

44 Statistical Analysis of theMaximumDynamic Stress at theSubgrade Surface e maximum dynamic stress at thesubgrade surface was randomly distributed under dierentsleepers because of the vertical track irregularity Figure 18illustrates the statistical results of σdm at the subgrade surfacedirectly below the intersection points of the rail and sleepersIt is observed that the track conditions signicantly aectedthe distribution of σdm σdm was relatively small and thedistribution was relatively concentrated when the rail trackwas smooth and σdm increased and the distribution rangebecame wider when the track condition worsened indicating

that the subgrade bed layers were more likely to withstand agreater dynamic stress which is harmful for maintaining thelong-term service performance of the subgrade

e statistical distributions of σdm under dierent ir-regular track conditions can be well-tted by normal dis-tribution curves as illustrated in Figure 18 and theexpectation (μ) and standard deviation (σ) increased as thetrack condition worsened

e possible maximum dynamic stress at the subgradesurface was predicted using the normal distribution theoryIf σdmsim(μ σ

2) then

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

140

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(a)

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(b)

Figure 12 Dynamic stress at the subgrade surface (a) below the rail (b) below track centerline

24 26 28 30 32 34 36

20

40

60

80

100

120

140

160

Bottom of SBSL rail

Subgrade surface railSubgrade surface centerline

Bottom of SBSL centerlineBottom of SBBL railBottom of SBBL centerline

Max

imum

dyn

amic

stre

ss (k

Pa)

Axle load (t)

Figure 13 Relationship between the maximum dynamic stress andtrain axle load

0 20 40 60 80 100

20

15

10

05

25σdmσs = 036

σdmσs = 034

σdmσs = 021

Maximum dynamic stress (kPa)

Dep

th (m

)

σdmσs = 022

30 t below rail

25 t below rail25 t below centerline

30 t below centerlineSelf-weight stress

Figure 14 Relationship between the maximum dynamic stress anddepth from the subgrade surface

8 Shock and Vibration

P σdm minus μ∣∣∣∣

∣∣∣∣ge 3σ( ) 1minusP σdm minus μ∣∣∣∣

∣∣∣∣lt 3σ( )

1minus 09973 00027lt 0003(5)

erefore the probability of σdm outside the interval of(μ minus 3σ μ + 3σ) was less than 0003 Statistically the intervalof (μ-3σ μ+3σ) is generally taken as the actual possible rangeof a normal distribution random variable namely the triplestandard deviation principle Consequently the possiblemaximum value of the dynamic stress at the subgradesurface can be estimated as μ + 3σ e analyzed possiblemaximum dynamic stresses at the subgrade surface underdierent irregular track spectra are listed in Table 5 emaximum dynamic stress at the subgrade surface of theShuo-Huang heavy-haul railway induced by a train with anaxle load of 30 t could reach 123 kPa [34] which was fa-vorably simulated with a predicted value of 1169 kPa (with adeviation less than 5)

5 Conclusions

A 3D train-track-subgrade interaction nite element (FE)model was constructed using the ABAQUS code (version614-5) where dierent vertical irregular track spectra weresimulated by modifying the rail node coordinates in theABAQUS INP le en the dynamic stress of the subgradebed layers of a heavy-haul railway embankment was ana-lyzed in detail using the 3D FE model e main conclusionsdrawn from this study are as follows

(1) e transverse distribution of the dynamic stress inthe subgrade bed layers gradually transformed from

00 02 04 06 08 10 12 14

20

40

60

80

100

120D

ynam

ic st

ress

(kPa

)

Time (s)

Heavy-haul track irregularityChinarsquos three main lines track irregularitySmooth rail

(a)

00 02 04 06 08 10 12 14

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

Heavy-haul track irregularityChinarsquos three mail lines track irregularitySmooth rail

(b)

Figure 15 Dynamic stress at the subgrade surface under dierent irregular track spectra (a) below rail (b) below centerline

Adjacent bogies

Tension zone

Field 9 S22+2258 times 104

+1000 times 103

ndash3231 times 103

ndash7462 times 103

ndash1169 times 104

ndash1592 times 104

ndash2015 times 104

ndash2439 times 104

ndash3285 times 104ndash2862 times 104

ndash3708 times 104

ndash4131 times 104

ndash4554 times 104

ndash4977 times 104

ndash5400 times 104

ndash5823 times 104

ndash6246 times 104

ndash6670 times 104

ndash7093 times 104

ndash7516 times 104

ndash7939 times 104

ndash8362 times 104

Figure 16 Dynamic stress contour at the subgrade surface con-sidering the irregularity dierence of the left and right rails

0 1 2 3 4

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Distance (m)

Subgrade surface Bottom of SBSL

Bottom of SBBLEmbankment bottom

Embankmentcenterline

Figure 17 Transverse distributions of dynamic stresses at dierentdepths

Shock and Vibration 9

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

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track centerline σdm below the rail became less than thatbelow the track centerline at a depth of approximately 03mσdm at the subgrade bed bottom was reduced by approxi-mately 70 compared with that at the subgrade surface eratio of σdm to the self-weight stress (σs) at the subgrade bedbottom (25m below the subgrade surface) under a train axle

load of 25 t was 021 and 022 for the locations directly belowthe rail and the track centerline respectively the σdmσsvalues increased to 034 and 036 under a train axle load of30 t respectively Zhou [33] suggested that the σdmσs valueshould be less than 020 to maintain the long-term stabilityof a railway subgrade Consequently the thickness of thesubgrade bed (25m a subgrade bed surface layer of 06mand a subgrade bed bottom layer of 19m) of existingrailways in China may meet the requirements of running aheavy-haul train with an axle load of 25 t but cannot enablethe operation of heavy-haul trains with axle loads greaterthan 25 t unless the subgrades have been reinforced

43 Eects of Irregular Track Spectrum Figure 15 illustratesthe subgrade surface dynamic stress-time curves underdierent track conditions e vertical track irregularityintensied the vertical excitation between the train wheelsand rails and induced a greater dynamic stress comparedwith the smooth rail condition e maximum dynamicstress at the subgrade surface increased by 23 and 158under the action of the heavy-haul irregular track spectrumand Chinarsquos three main railway lines irregular track spec-trum respectively

In China the irregular track spectra of the left and rightrails are dierent e dynamic stress contour at the sub-grade surface taking this dierence into account is shown inFigure 16 It is determined that the FE model eectivelyreshyected the irregularity dierence of the left and right railsand the dynamic stress was not symmetrically distributedalong the track centerline

Figure 17 presents the transverse distributions of thedynamic stress at dierent depths e subgrade surfacedynamic stress exhibited a bimodal pattern (or an ldquoMrdquoshape) and the value under the left rail was approximately19 greater than that under the right rail whereas theasymmetry of the dynamic stress gradually decreased as thedepth increased and was negligible at the subgrade bedbottom hence the irregularity dierence between the leftand right rails had a slight impact on the embankment soilsbelow the subgrade bed bottom

Adjacent b

ogies

Tension zone

Field 2 S22(avg 75)

+2359 times 104

+0000 times 100

ndash3793 times 103

ndash7586 times 103

ndash1138 times 104

ndash1517 times 104

ndash1896 times 104

ndash2276 times 104

ndash2655 times 104

ndash3034 times 104

ndash3414 times 104

ndash3793 times 104

ndash4172 times 104

ndash4552 times 104

ndash4931 times 104

ndash5310 times 104

ndash5689 times 104

ndash6069 times 104

ndash6448 times 104

ndash6827 times 104

ndash7207 times 104

ndash7586 times 104

(a)

Adjacent bogies

Tension zone

Field 3 S22(avg 75)

+2729 times 104

+0000 times 100

ndash5198 times 103

ndash1040 times 104

ndash1559 times 104

ndash2079 times 104

ndash2599 times 104

ndash3119 times 104

ndash3638 times 104

ndash4158 times 104

ndash4678 times 104

ndash5198 times 104

ndash5718 times 104

ndash6237 times 104

ndash6757 times 104

ndash7277 times 104

ndash7797 times 104

ndash8316 times 104

ndash8836 times 104

ndash9356 times 104

ndash9876 times 104

ndash1040 times 105

(b)

Figure 9 Dynamic stress contour at the subgrade surface (a) smooth track (b) vertically irregular track

Adjacent bogies

Field 3 S22(avg 75)

+1246 times 104

+0000 times 100

ndash3797 times 103

ndash7595 times 103

ndash1139 times 104

ndash1519 times 104

ndash1899 times 104

ndash2278 times 104

ndash2658 times 104

ndash3038 times 104

ndash3418 times 104

ndash3797 times 104

ndash4177 times 104

ndash4557 times 104

ndash4937 times 104

ndash5316 times 104

ndash5696 times 104

ndash6076 times 104

ndash6056 times 104

ndash6835 times 104

ndash7215 times 104

ndash7595 times 104

Figure 10 Dynamic stress contour at the bottom of the subgradebed surface layer

Adjacent bogies

Field 3 S22(avg 75)

+1487 times 100

+0000 times 100

ndash1945 times 103

ndash3891 times 103

ndash5836 times 103

ndash7781 times 103

ndash9727 times 103

ndash1167 times 104

ndash1362 times 104

ndash1556 times 104

ndash1945 times 104ndash1751 times 104

ndash2140 times 104

ndash2334 times 104

ndash2529 times 104

ndash2724 times 104

ndash2918 times 104

ndash3113 times 104

ndash3303 times 104

ndash3502 times 104

ndash3696 times 104

ndash3891 times 104

Figure 11 Dynamic stress contour at the bottom of the subgradebed bottom layer

Shock and Vibration 7

44 Statistical Analysis of theMaximumDynamic Stress at theSubgrade Surface e maximum dynamic stress at thesubgrade surface was randomly distributed under dierentsleepers because of the vertical track irregularity Figure 18illustrates the statistical results of σdm at the subgrade surfacedirectly below the intersection points of the rail and sleepersIt is observed that the track conditions signicantly aectedthe distribution of σdm σdm was relatively small and thedistribution was relatively concentrated when the rail trackwas smooth and σdm increased and the distribution rangebecame wider when the track condition worsened indicating

that the subgrade bed layers were more likely to withstand agreater dynamic stress which is harmful for maintaining thelong-term service performance of the subgrade

e statistical distributions of σdm under dierent ir-regular track conditions can be well-tted by normal dis-tribution curves as illustrated in Figure 18 and theexpectation (μ) and standard deviation (σ) increased as thetrack condition worsened

e possible maximum dynamic stress at the subgradesurface was predicted using the normal distribution theoryIf σdmsim(μ σ

2) then

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

140

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(a)

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(b)

Figure 12 Dynamic stress at the subgrade surface (a) below the rail (b) below track centerline

24 26 28 30 32 34 36

20

40

60

80

100

120

140

160

Bottom of SBSL rail

Subgrade surface railSubgrade surface centerline

Bottom of SBSL centerlineBottom of SBBL railBottom of SBBL centerline

Max

imum

dyn

amic

stre

ss (k

Pa)

Axle load (t)

Figure 13 Relationship between the maximum dynamic stress andtrain axle load

0 20 40 60 80 100

20

15

10

05

25σdmσs = 036

σdmσs = 034

σdmσs = 021

Maximum dynamic stress (kPa)

Dep

th (m

)

σdmσs = 022

30 t below rail

25 t below rail25 t below centerline

30 t below centerlineSelf-weight stress

Figure 14 Relationship between the maximum dynamic stress anddepth from the subgrade surface

8 Shock and Vibration

P σdm minus μ∣∣∣∣

∣∣∣∣ge 3σ( ) 1minusP σdm minus μ∣∣∣∣

∣∣∣∣lt 3σ( )

1minus 09973 00027lt 0003(5)

erefore the probability of σdm outside the interval of(μ minus 3σ μ + 3σ) was less than 0003 Statistically the intervalof (μ-3σ μ+3σ) is generally taken as the actual possible rangeof a normal distribution random variable namely the triplestandard deviation principle Consequently the possiblemaximum value of the dynamic stress at the subgradesurface can be estimated as μ + 3σ e analyzed possiblemaximum dynamic stresses at the subgrade surface underdierent irregular track spectra are listed in Table 5 emaximum dynamic stress at the subgrade surface of theShuo-Huang heavy-haul railway induced by a train with anaxle load of 30 t could reach 123 kPa [34] which was fa-vorably simulated with a predicted value of 1169 kPa (with adeviation less than 5)

5 Conclusions

A 3D train-track-subgrade interaction nite element (FE)model was constructed using the ABAQUS code (version614-5) where dierent vertical irregular track spectra weresimulated by modifying the rail node coordinates in theABAQUS INP le en the dynamic stress of the subgradebed layers of a heavy-haul railway embankment was ana-lyzed in detail using the 3D FE model e main conclusionsdrawn from this study are as follows

(1) e transverse distribution of the dynamic stress inthe subgrade bed layers gradually transformed from

00 02 04 06 08 10 12 14

20

40

60

80

100

120D

ynam

ic st

ress

(kPa

)

Time (s)

Heavy-haul track irregularityChinarsquos three main lines track irregularitySmooth rail

(a)

00 02 04 06 08 10 12 14

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

Heavy-haul track irregularityChinarsquos three mail lines track irregularitySmooth rail

(b)

Figure 15 Dynamic stress at the subgrade surface under dierent irregular track spectra (a) below rail (b) below centerline

Adjacent bogies

Tension zone

Field 9 S22+2258 times 104

+1000 times 103

ndash3231 times 103

ndash7462 times 103

ndash1169 times 104

ndash1592 times 104

ndash2015 times 104

ndash2439 times 104

ndash3285 times 104ndash2862 times 104

ndash3708 times 104

ndash4131 times 104

ndash4554 times 104

ndash4977 times 104

ndash5400 times 104

ndash5823 times 104

ndash6246 times 104

ndash6670 times 104

ndash7093 times 104

ndash7516 times 104

ndash7939 times 104

ndash8362 times 104

Figure 16 Dynamic stress contour at the subgrade surface con-sidering the irregularity dierence of the left and right rails

0 1 2 3 4

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Distance (m)

Subgrade surface Bottom of SBSL

Bottom of SBBLEmbankment bottom

Embankmentcenterline

Figure 17 Transverse distributions of dynamic stresses at dierentdepths

Shock and Vibration 9

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

44 Statistical Analysis of theMaximumDynamic Stress at theSubgrade Surface e maximum dynamic stress at thesubgrade surface was randomly distributed under dierentsleepers because of the vertical track irregularity Figure 18illustrates the statistical results of σdm at the subgrade surfacedirectly below the intersection points of the rail and sleepersIt is observed that the track conditions signicantly aectedthe distribution of σdm σdm was relatively small and thedistribution was relatively concentrated when the rail trackwas smooth and σdm increased and the distribution rangebecame wider when the track condition worsened indicating

that the subgrade bed layers were more likely to withstand agreater dynamic stress which is harmful for maintaining thelong-term service performance of the subgrade

e statistical distributions of σdm under dierent ir-regular track conditions can be well-tted by normal dis-tribution curves as illustrated in Figure 18 and theexpectation (μ) and standard deviation (σ) increased as thetrack condition worsened

e possible maximum dynamic stress at the subgradesurface was predicted using the normal distribution theoryIf σdmsim(μ σ

2) then

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

140

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(a)

00 02 04 06 08 10 12 14 16 18

20

40

60

80

100

120

Adjacent bogies passing

Dyn

amic

stre

ss (k

Pa)

Time (s)

25 t30 t

325 t35 t

(b)

Figure 12 Dynamic stress at the subgrade surface (a) below the rail (b) below track centerline

24 26 28 30 32 34 36

20

40

60

80

100

120

140

160

Bottom of SBSL rail

Subgrade surface railSubgrade surface centerline

Bottom of SBSL centerlineBottom of SBBL railBottom of SBBL centerline

Max

imum

dyn

amic

stre

ss (k

Pa)

Axle load (t)

Figure 13 Relationship between the maximum dynamic stress andtrain axle load

0 20 40 60 80 100

20

15

10

05

25σdmσs = 036

σdmσs = 034

σdmσs = 021

Maximum dynamic stress (kPa)

Dep

th (m

)

σdmσs = 022

30 t below rail

25 t below rail25 t below centerline

30 t below centerlineSelf-weight stress

Figure 14 Relationship between the maximum dynamic stress anddepth from the subgrade surface

8 Shock and Vibration

P σdm minus μ∣∣∣∣

∣∣∣∣ge 3σ( ) 1minusP σdm minus μ∣∣∣∣

∣∣∣∣lt 3σ( )

1minus 09973 00027lt 0003(5)

erefore the probability of σdm outside the interval of(μ minus 3σ μ + 3σ) was less than 0003 Statistically the intervalof (μ-3σ μ+3σ) is generally taken as the actual possible rangeof a normal distribution random variable namely the triplestandard deviation principle Consequently the possiblemaximum value of the dynamic stress at the subgradesurface can be estimated as μ + 3σ e analyzed possiblemaximum dynamic stresses at the subgrade surface underdierent irregular track spectra are listed in Table 5 emaximum dynamic stress at the subgrade surface of theShuo-Huang heavy-haul railway induced by a train with anaxle load of 30 t could reach 123 kPa [34] which was fa-vorably simulated with a predicted value of 1169 kPa (with adeviation less than 5)

5 Conclusions

A 3D train-track-subgrade interaction nite element (FE)model was constructed using the ABAQUS code (version614-5) where dierent vertical irregular track spectra weresimulated by modifying the rail node coordinates in theABAQUS INP le en the dynamic stress of the subgradebed layers of a heavy-haul railway embankment was ana-lyzed in detail using the 3D FE model e main conclusionsdrawn from this study are as follows

(1) e transverse distribution of the dynamic stress inthe subgrade bed layers gradually transformed from

00 02 04 06 08 10 12 14

20

40

60

80

100

120D

ynam

ic st

ress

(kPa

)

Time (s)

Heavy-haul track irregularityChinarsquos three main lines track irregularitySmooth rail

(a)

00 02 04 06 08 10 12 14

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

Heavy-haul track irregularityChinarsquos three mail lines track irregularitySmooth rail

(b)

Figure 15 Dynamic stress at the subgrade surface under dierent irregular track spectra (a) below rail (b) below centerline

Adjacent bogies

Tension zone

Field 9 S22+2258 times 104

+1000 times 103

ndash3231 times 103

ndash7462 times 103

ndash1169 times 104

ndash1592 times 104

ndash2015 times 104

ndash2439 times 104

ndash3285 times 104ndash2862 times 104

ndash3708 times 104

ndash4131 times 104

ndash4554 times 104

ndash4977 times 104

ndash5400 times 104

ndash5823 times 104

ndash6246 times 104

ndash6670 times 104

ndash7093 times 104

ndash7516 times 104

ndash7939 times 104

ndash8362 times 104

Figure 16 Dynamic stress contour at the subgrade surface con-sidering the irregularity dierence of the left and right rails

0 1 2 3 4

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Distance (m)

Subgrade surface Bottom of SBSL

Bottom of SBBLEmbankment bottom

Embankmentcenterline

Figure 17 Transverse distributions of dynamic stresses at dierentdepths

Shock and Vibration 9

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

P σdm minus μ∣∣∣∣

∣∣∣∣ge 3σ( ) 1minusP σdm minus μ∣∣∣∣

∣∣∣∣lt 3σ( )

1minus 09973 00027lt 0003(5)

erefore the probability of σdm outside the interval of(μ minus 3σ μ + 3σ) was less than 0003 Statistically the intervalof (μ-3σ μ+3σ) is generally taken as the actual possible rangeof a normal distribution random variable namely the triplestandard deviation principle Consequently the possiblemaximum value of the dynamic stress at the subgradesurface can be estimated as μ + 3σ e analyzed possiblemaximum dynamic stresses at the subgrade surface underdierent irregular track spectra are listed in Table 5 emaximum dynamic stress at the subgrade surface of theShuo-Huang heavy-haul railway induced by a train with anaxle load of 30 t could reach 123 kPa [34] which was fa-vorably simulated with a predicted value of 1169 kPa (with adeviation less than 5)

5 Conclusions

A 3D train-track-subgrade interaction nite element (FE)model was constructed using the ABAQUS code (version614-5) where dierent vertical irregular track spectra weresimulated by modifying the rail node coordinates in theABAQUS INP le en the dynamic stress of the subgradebed layers of a heavy-haul railway embankment was ana-lyzed in detail using the 3D FE model e main conclusionsdrawn from this study are as follows

(1) e transverse distribution of the dynamic stress inthe subgrade bed layers gradually transformed from

00 02 04 06 08 10 12 14

20

40

60

80

100

120D

ynam

ic st

ress

(kPa

)

Time (s)

Heavy-haul track irregularityChinarsquos three main lines track irregularitySmooth rail

(a)

00 02 04 06 08 10 12 14

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Time (s)

Heavy-haul track irregularityChinarsquos three mail lines track irregularitySmooth rail

(b)

Figure 15 Dynamic stress at the subgrade surface under dierent irregular track spectra (a) below rail (b) below centerline

Adjacent bogies

Tension zone

Field 9 S22+2258 times 104

+1000 times 103

ndash3231 times 103

ndash7462 times 103

ndash1169 times 104

ndash1592 times 104

ndash2015 times 104

ndash2439 times 104

ndash3285 times 104ndash2862 times 104

ndash3708 times 104

ndash4131 times 104

ndash4554 times 104

ndash4977 times 104

ndash5400 times 104

ndash5823 times 104

ndash6246 times 104

ndash6670 times 104

ndash7093 times 104

ndash7516 times 104

ndash7939 times 104

ndash8362 times 104

Figure 16 Dynamic stress contour at the subgrade surface con-sidering the irregularity dierence of the left and right rails

0 1 2 3 4

20

40

60

80

100

Dyn

amic

stre

ss (k

Pa)

Distance (m)

Subgrade surface Bottom of SBSL

Bottom of SBBLEmbankment bottom

Embankmentcenterline

Figure 17 Transverse distributions of dynamic stresses at dierentdepths

Shock and Vibration 9

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

a bimodal pattern to a unimodal pattern with anincrease in the depth from the subgrade surface

(2) e dynamic stress at the subgrade surface induced bythe adjacent bogies of adjacent carriages exhibitedobvious superposition eects because of the smalladjacent bogie spacing as well as the dynamic stressesmaintained around the peak value during the pass ofthe adjacent bogies hence the pass of adjacent bogiescan be simplied once loaded on the subgrade

(3) e maximum dynamic stress at dierent locationsincreased approximately linearly as the train axleload increased e dynamic stress at the bottom ofthe subgrade bed surface layer was more sensitive tothe train axle load compared with that at the sub-grade surface because the dynamic stresses inducedby the two rails were gradually overlaid with anincrease in the depth from the subgrade surface

(4) e maximum dynamic stress at the subgrade bedbottom was reduced by approximately 70 comparedwith that at the subgrade surface e ratio of max-imum dynamic stress to the self-weight stress at thesubgrade bed bottom under a train axle load of 25 texceeded the recommended value of existing designstandards (02) consequently the existing railways inChina cannot enable the operation of heavy-haultrains with axle loads greater than 25 t unless therailway subgrades have been reinforced

(5) e vertical track irregularity intensied the verticalexcitation between the train wheels and rails and themaximum dynamic stress at the subgrade surface

30 40 50 60 70 80 90 100 1100

5

10

15

20

25Co

unt

Maximum dynamic stress (kPa)

(a)

Fitting resultsσdm = 744 kPaσ = 81 kPa

40 60 80 100 1200

4

8

12

16

Coun

t

Maximum dynamic stress (kPa)

(b)

40 60 80 100 120 1400

2

4

6

8

10

12

Coun

t

Maximum dynamic stress (kPa)

Fitting resultsσdm = 866 kPaσ = 101 kPa

(c)

Figure 18 Statistical graph of the maximum dynamic stress at the subgrade surface (a) smooth rail (b) irregularity in Chinarsquos three mainlines (c) heavy-haul track irregularity

Table 5 Statistical results of the maximum dynamic stress at thesubgrade surface

Statistical parameter

Axle load (t) forheavy-haul track

irregularity

Axle load (t) forChinarsquos three main

lines trackirregularity

30 325 35 30 325 35μ (kPa) 866 973 1103 744 821 910σ (kPa) 101 982 1023 81 834 891μ + 3σ (kPa) 1169 1268 1410 987 1071 1177

10 Shock and Vibration

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

under the action of the irregular heavy-haul trackspectrum increased by 23 0e maximum dynamicstress and its distribution range increased with thetrack irregularity increasing which was harmful formaintaining the long-term service performance ofthe subgrade

(6) Based on the normal distribution theory the possiblemaximum dynamic stress (σdm) at the subgradesurface under the action of an irregular trackspectrum can be predicted using the triple standarddeviation principle ie σdm μ + 3σ (where μ and σare the expectation and standard deviation of σdmrespectively)

Data Availability

0e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

0e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

0e work presented in this paper was supported by theNational Natural Science Foundation of China (Grant nos51678572 51709284 and 51878666) and the PostdoctoralScience Foundation of Hunan Communications ResearchInstitute Co Ltd

References

[1] H L Yao Z Hu Z Lu and H Wang ldquoAnalytical model topredict dynamic responses of railway subgrade due to high-speed trains considering wheel-track interactionrdquo In-ternational Journal of Geomechanics vol 16 no 2 Article ID0401506 15 pages 2016

[2] L A Yang W Powrie and J A Priest ldquoDynamic stressanalysis of a ballasted railway track bed during train passagerdquoJournal of Geotechnical and Geoenvironmental Engineeringvol 135 no 5 pp 680ndash689 2009

[3] Z L Han and Q L Zhang ldquoDynamic stress analysis on speed-increase subgrade of existing railwayrdquo China Railway Sciencevol 26 no 5 pp 1ndash5 2005

[4] J F Wei Y G Wei and Y Y Bai ldquoAdaptability of 27 t axleload train and subgrade bridge and culvert of existing railwaylinesrdquo in Proceedings of Communication Conference ontransportation technologies of heavy-haul railways BeijingChina 2014

[5] H G Di W M Leng J L Xue and S H Zhou ldquoAssessmentof subgrade strength for transport capacity enlargement ofshuo-huang heavy-haul railwayrdquo Journal of China RailwaySociety vol 36 no 8 pp 84ndash90 2014

[6] F Lamas-Lopez Y-J Cui N Calon S Costa DrsquoAguiarM Peixoto De Oliveira and T Zhang ldquoTrack-bed mechanicalbehaviour under the impact of train at different speedsrdquo Soilsand Foundations vol 56 no 4 pp 627ndash639 2016

[7] M T Hendry C D Martin and S L Barbour ldquoMeasurementof cyclic response of railway embankments and underlying

soft peat foundations to heavy axle loadsrdquo Canadian Geo-technical Journal vol 50 no 5 pp 467ndash480 2013

[8] J A Priest W Powrie L Yang P J Grabe andC R I Clayton ldquoMeasurements of transient ground move-ments below a ballasted railway linerdquo Geotechnique vol 60no 9 pp 667ndash677 2010

[9] X Bian H Jiang C Cheng Y Chen R Chen and J JiangldquoFull-scale model testing on a ballastless high-speed railwayunder simulated train moving loadsrdquo Soil Dynamics andEarthquake Engineering vol 66 pp 368ndash384 2014

[10] R Chen X Zhao Z Wang H Jiang and X Bian ldquoExper-imental study on dynamic load magnification factor forballastless track-subgrade of high-speed railwayrdquo Journal ofRock Mechanics and Geotechnical Engineering vol 5 no 4pp 306ndash311 2013

[11] A Al Shaer D Duhamel K Sab G Foret and L SchmittldquoExperimental settlement and dynamic behavior of a portionof ballasted railway track under high speed trainsrdquo Journal ofSound and Vibration vol 316 no 1 pp 211ndash233 2008

[12] J H Xiao ldquoStudy on the dynamic properties of silt andsubgrade stability subjected to train moving loadingrdquo PhDthesis Beijing Jiaotong University Beijing China 2008

[13] X Bian C Cheng J Jiang R Chen and Y Chen ldquoNumericalanalysis of soil vibrations due to trains moving at criticalspeedrdquo Acta Geotechnica vol 11 no 2 pp 281ndash294 2014

[14] Y Shan B Albers and S A Savidis ldquoInfluence of differenttransition zones on the dynamic response of track-subgradesystemsrdquo Computers and Geotechnics vol 48 pp 21ndash282013

[15] K K Ang and J Dai ldquoResponse analysis of high-speed railsystem accounting for abrupt change of foundation stiffnessrdquoJournal of Sound and Vibration vol 332 no 12 pp 2954ndash2970 2013

[16] A El Kacimi P K Woodward O Laghrouche andG Medero ldquoTime domain 3D finite element modelling oftrain-induced vibration at high speedrdquo Computers andStructures vol 118 pp 66ndash73 2012

[17] L Hall ldquoSimulations and analyses of train-induced groundvibrations in finite element modelsrdquo Soil Dynamics andEarthquake Engineering vol 23 no 5 pp 403ndash413 2003

[18] Y Ma V L Markine A A Mashal and M Ren ldquoEffect ofwheel-rail interface parameters on contact stability in explicitfinite element analysisrdquo Proceedings of the Institution ofMechanical Engineers Part F Journal of Rail and RapidTransit vol 232 no 6 pp 1879ndash1894 2018

[19] J Giner-Navarro J Martınez-Casas F D Denia andL Baeza ldquoStudy of railway curve squeal in the time domainusing a high-frequency vehicletrack interaction modelrdquoJournal of Sound and Vibration vol 431 pp 177ndash191 2018

[20] Y Cai Z Cao H Sun and C Xu ldquoEffects of the dynamicwheel-rail interaction on the ground vibration generated by amoving trainrdquo International Journal of Solids and Structuresvol 47 no 17 pp 2246ndash2259 2010

[21] G Kouroussis D P Connolly K Vogiatzis andO Verlinden ldquoModelling the environmental effects of railwayvibrations from different types of rolling stock a numericalstudyrdquo Shock and Vibration vol 2015 Article ID 14280715 pages 2015

[22] C H Chiang and P H Tsai ldquoA numerical study of thescreening effectiveness of open trenches for high-speed train-induced vibrationrdquo Shock and Vibration vol 2014 Article ID489090 15 pages 2014

[23] G Kouroussis and O Verlinden ldquoPrediction of railway in-duced ground vibration throughmultibody and finite element

Shock and Vibration 11

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

modellingrdquo Mechanical Sciences vol 4 no 1 pp 167ndash1832013

[24] P Alves Costa R Calccedilada and A Silva Cardoso ldquoTrack-ground vibrations induced by railway traffic in-situ mea-surements and validation of a 25 D FEM-BEM modelrdquo SoilDynamics and Earthquake Engineering vol 32 no 1pp 111ndash128 2011

[25] W Zhai Z He X Song and X L ldquoPrediction of high-speedtrain induced ground vibration based on train-track-groundsystem modelrdquo Earthquake Engineering and EngineeringVibration vol 9 no 4 pp 545ndash554 2011

[26] P Galvın A Romero and J Domınguez ldquoVibrations inducedby HST passage on ballast and non-ballast tracksrdquo Soil Dy-namics and Earthquake Engineering vol 30 no 9 pp 862ndash873 2010

[27] W Zhai and Z Cai ldquoDynamic interaction between a lumpedmass vehicle and a discretely supported continuous rail trackrdquoComputers and Structures vol 63 no 5 pp 987ndash997 1997

[28] Dassault Systemes Simulia Corp AbaqusCAE 614 UserrsquosGuide Dassault Systemes Simulia Corp Providence RI USA2014

[29] X M Liu ldquoResearch on simulation and analysis of typicaltrack spectrumrdquo MS thesis Jilin University ChangchunChina 2009

[30] C F Wei ldquoTime domain conversion and the applicationresearch of track irregularity power spectrumrdquo MS thesisSouthwest Jiaotong University Chengdu China 2009

[31] R Ding J F Shen X Xiao and Y D Xu ldquoSimulation analysison mechanical characteristics of heavy haul railway trackstructurerdquo Journal of East China Jiaotong University vol 32no 3 pp 36ndash41 2015

[32] L Q Miao H P Feng Z R Yue and X G Miao ldquoDe-formation characteristics of heavy-haul railway subgradeunder large axle loadrdquo Subgrade Engineering vol 2015 no 3pp 35ndash37 2015

[33] X Zhou ldquoStudy on settlement control of composite foun-dation of heavy-haul railway high embankment in collapsibleloess areardquo MS thesis Beijing Jiaotong University BeijingChina 2015

[34] J Z Jia ldquoDynamic stress characteristics of heavy-haul railwaysubgradesrdquo Railway Engineering vol 2014 no 7 pp 89ndash912014

12 Shock and Vibration

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Journal ofEngineeringVolume 2018

SensorsJournal of

Hindawiwwwhindawicom Volume 2018

International Journal of

RotatingMachinery

Hindawiwwwhindawicom Volume 2018

Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Navigation and Observation

International Journal of

Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Active and Passive Electronic Components

VLSI Design

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Shock and Vibration

Hindawiwwwhindawicom Volume 2018

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

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