A coupled model for train-track-bridge stochastic analysis with consideration of spatial variation and temporal evolution

A coupled model for train-track-bridge stochastic analysis with consideration of spatial

variation and temporal evolution

Xu, Lei; Zhai, Wanming; Li, Zili

DOI

10.1016/j.apm.2018.07.001

Publication date

2018

Document Version

Final published version

Published in

Applied Mathematical Modelling

Citation (APA)

Xu, L., Zhai, W., & Li, Z. (2018). A coupled model for train-track-bridge stochastic analysis with

consideration of spatial variation and temporal evolution. Applied Mathematical Modelling, 63, 709-731.

https://doi.org/10.1016/j.apm.2018.07.001

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ContentslistsavailableatScienceDirect

Applied

Mathematical

Modelling

journalhomepage:www.elsevier.com/locate/apm

A

coupled

model

for

train-track-bridge

stochastic

analysis

with

consideration

of

spatial

variation

and

temporal

evolution

Lei

Xu

a,b,∗

_,

_Wanming

_Zhai

a

_,

_Zili

_Li

b

a Train and track research institute, State-Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China b The Section of Road and Railway Engineering, Delft University of Technology, Delft 2628, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history:

Received 30 September 2017 Revised 2 June 2018 Accepted 4 July 2018 Available online 29 July 2018 Keywords: Train-track-bridge interactions Random vibrations Monte-Carlo method Karhunen–Loève expansion Track irregularities

a

b

s

t

r

a

c

t

Due to random characteristics of system parameters and excitations, the dynamic as-sessment and prediction for the train-track-bridge interaction systems become rather complexissuesneedingtobeaddressed,especiallyconsideringthelongitudinal inhomo-geneityanduncertaintyofdynamicpropertiesinphysicsandcorrespondinglytheir tem-poralevolutions. In this paper, atemporal-spatial coupledmodel is developed to fully dealwiththedeterministically/non-deterministicallycomputationalandanalyticalmatters inthe train-track-bridgeinteractions withanovelty, where atrain-track-bridge interac-tionmodel is newly developed by effectivelycoupling the three-dimensional nonlinear wheel-railcontactmodelandthefiniteelementtheory,moreover,theMonte-Carlomethod (MCM)andKarhunen–Loèveexpansion(KLE)areeffectivelyunitedtomodeltherandom fieldoftrack-bridgesystems,andaspectralevolutionmethodaccompaniedbyatrack ir-regularityprobabilisticmodelareintroducedtoselectthemostrepresentativetrack irregu-laritysetsandtocharacterizetheirrandomevolutionsintemporaldimension.Intermsof randomvibrationanalysis,thehigh-efficiencyandeffectivenessofthisdevelopedmodel isvalidatedbycomparingtoarobustmethod,i.e., MCM.Apartfromvalidations, multi-applicationsofthetemporal-spatialcoupledmodelfromaspectsofdeterministic compu-tation,randomvibration,resonant analysisand long-termdynamicprediction,etc.,have beenfullypresentedtoillustratetheuniversalityoftheproposedmodel.

© 2018ElsevierInc.Allrightsreserved.

1. Introduction

Railwaybridges, asa kindofinfrastructure,are becoming increasingly importantinsupporting andguidingthe train-tracksystems.Especiallyinhigh-speedlines,theproportionofbridgesismuchhigherthancommonrailwaylinesthat are mainlysupported by the subgrade layer. Tospecific lines,even 90% over isoccupied by bridges forconservationof land andenvironmentprotection.Hencethetheoreticalmethodsandappliedtechnologiesrelatedtotheassessmentofdynamic performance ofrailway systems,when a train passesthrough the track/bridgestructures, have attractedmore andmore attentionsinlasttwodecades.

Comparing to expensively experimental studies in situ or lab, the dynamic simulations actualized by the computer programhavebecomeadominantstrategyinmostsituations.Tillnow,thedynamicmodelsdevelopedtocharacterizethe train-track-bridgeinteractionsarenumerous,butmostlyconcentratedonverticalvibrations[1–8],obviously,itlimitsthe

re-∗ _{Corresponding author at: Train and track research institute, State-Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031,} China.

E-mail address: leix_2014@my.swjtu.edu.cn (L. Xu).

https://doi.org/10.1016/j.apm.2018.07.001

Nomenclature

m r therailmassperunitlength

A r thecross-sectionalareaoftherail

W r thepolarmomentofinertiaoftherailcross-section

L thelengthofthebeamelement

l r thedistancebetweentwoadjacentrailpads

E r theYoung’smodulusoftherail

(I ry,I rz) theflexuralmomentofinertiaaboutthe Y -axisand Z- axisofthecrosssectionoftherailrespectively

k rt thetorsionalrigidityoftherailcrosssectionaroundthe X -axis

m s themassoftheslabperunitvolume

(h s,b s,l s) theheight,widthandlengthoftheslabtrackelementrespectively

E s theYoung’smodulusoftheslabtrack

μ

s thePoissonratiooftheslabtrack

I sz themomentofinertiaofthetrackslabaround Z -axis

m p themassofthepier

(k px,k px,k px) thesupportingstiffnesscoefficientsofthesubgradein X -, Y -and Z -axis,respectively

(c px,c px,c px) thesupportingdampingcoefficientsofthesubgradein X -, Y -and Z -axis,respectively

(k rp,z,k rp,y) theverticalandlateralstiffnessoftherailpadrespectively

(c rp,z,c rp,y) theverticalandlateraldampingoftherailpadrespectively

b thelateraldistancebetweenthecontactpointofrailpad-slabtrackandtheleft-sideborderoftheslab along X -axis

(k ca,z,k ca,y) theverticalandlateralstiffnesscoefficientsoftheCAM

(B r,H r) thecentrallateralandverticaldistancesbetweentheslabtrackandthegirder

(k p,z,k p,y) theverticalandlateralstiffnessreflectingthepropertiesofthebearing

a 0 thehalfofthehorizontaldistancebetweentwocontactpoints

(r li,r ri) therollingradiusoftheleftandrightwheelsetrespectively



thenominalrollingangularvelocityofthewheelset (I wY,I wZ) themomentofinertiaofthewheelsetaround Y -and Z -axis

M w themassofthewheelset

R _i theradiusofthecurvatureoftherailcorrespondingtothe i thwheelset

φ

wi theangleofsuperelevationcorrespondingtothecenterofthe i thwheelset

(

ϕ

˙wi,

ϕ

¨wi) thefirst-orderandsecond-orderderivativesof

φ

i

r0 thenominalrollingradiusofthewheelset

˙

λ

wi thefirst-orderderivativeofthecurvature

¯g theaccelerationofgravity V therunningspeedofthevehicle M c themassofthecarbody

(I cX,I cZ) themomentofinertiaofthecarbodyaround X -and Z -axisrespectively

R c theradiusofcurvaturewithregardtothecentroidofthecarbody

φ

c theangleofsuperelevationcorrespondingtothecentroidofthecarbody

¨

ϕ

c thesecond-orderderivativeof

φ

c

˙

ς

c thefirst-orderderivativeofthetrackcurvature

H tw theverticaldistancebetweenthecentroidofthebogieframeandthecenterofthewheelset

H bt thevertical distance between thecentroid of the bogie frame andthe bottom plane of the secondary

suspension

H _cb theverticaldistancebetweenthecentroidofthebogieframeandtheupperplane ofthesecondary sus-pension

(k sz,k sy,k sx) thesecondarysuspensionstiffnessinvertical,lateralandlongitudinaldirections

(k pz,k py,k px) theprimarysuspensionstiffnessinvertical,lateralandlongitudinaldirections

(d s,d p) thesemi-horizontaldistanceofthesecondaryandprimarysuspensionrespectively

l c thesemi-longitudinaldistancebetweenbogies

l t thesemi-longitudinaldistancebetweenwheelsetsinabogie

(l h,l v) thelateralandverticaldistancebetweenthewheel-railcontactpointandthecentroidoftherail

searchscopessincetheissuesonlateralstabilityandsafetyofsystemcomponentshavegraduallybecomenotableconcerns inrailwayengineering, especially inconditionsofhighspeed operations.Accounting forthis, moreandmoreresearchers starttofocusonbuildingthethree-dimensionaltrain-track/bridgeinteractionmodels,seeforexample,Zhaietal.[9]made a pioneeringworkin comprehensivelyconsidering thecoupleddynamics betweenavehicle andthetracks,in whichthe

three-dimensional(3-D)nonlinearcontact/creepinwheel-railinteractionsareintroducedinfocusedmanner,later,by adopt-ingthesamefundamentals,extensiveresearchesintrain-track-bridgecoupleddynamicsareputintopractice[10,11]; more-over,Refs.[12–17]conductedsignificantworkon3-Dtrain-bridgeinteractions,butneglectingtheeffectsoftrackstructural participationsonsyntheticvibrations,additionally,onlylinearwheel-railinteractionsareaccountedfor;Zengetal.[18] con-structed arather comprehensivemodelfor train-slabtrack-bridge interactions, butstill, thecoremechanismhighlighting thecomplexwheel-railinteractionsislinearizedtounifythesystemcomponentsbyenergy-variationalprinciple[19].

Apart from the developments on constructing train-bridge dynamic simulation models, researchers gradually ex-tend the general deterministic computations into random vibration analysis in light of the random characteristics of the train/track/bridge systems on physical properties and mechanical status. With regard to the random analysis of train-track/bridge interactions, there are mainly three types of methods, i.e., Monte-Carlo method (MCM) [6,20,21], pseudo-excitationmethod(PEM)[17,18,22,23]andprobability densityevolutionmethod(PDEM)[16,24],wherethesystem parametersorexcitations,e.g,trackirregularitiesandseismicmotions,etc.,areassumedtoberandomprocessesstatically characterized by probability densityfunction (PDF) and spectral densities. Undoubtedly,thesework has takensignificant stepstowardstheroadofrandomanalysis,however,farmorescientificresearchesneedtobeprobedinto,e.g.,

• Randomsimulationinvolvingwiththelongitudinaluncertaintyofdynamicpropertiesoftrack/bridgesystemsatspatial dimension.Thoughfragmentaryreportsinvestigatingtheeffectsofrandomtrackbedstiffnessonvehicle-track interac-tionshavebeenpresented[25],aunitedandhigh-efficientmethoddealingwithrandomsimulationandcombinationof multiplevariablesisrarelypresented;

• Ergodiccharacterizationofthesystemexcitations.Forexample,trackirregularities,perhapsthemostimportant excita-tion oftrain-track/bridge systems,holdrandom nature.InRefs.[16,18],onlytheexcitationofspecific trackirregularity spectrums, whicharejuststatisticalstatusoftheraildeformation atoneprobability,isconsidered,certainly,basedon whichthefullresponsesofsystemcomponentscannotberevealed;

• Thestrategiestocharacterizetheevolutionofsystemdynamiccharacteristicsaffectingthelong-termbehaviorsof train-track/bridgesystems.

Previously,XuandZhai[26]hadproposedatheoreticalprototypetoexpand thegeneralrandomvibrationanalysisinto temporal-spatial stochastic analysis withrespect to the vehicle-track systems, in which the randomness andcorrelation betweenlongitudinal system parameters are considered but being limited to normal distribution, moreover, the ergodic simulationoftrackrandomirregularitiesaresolvedbyaprobabilisticmethod.Thisresearchnoticesthefeaturesofrailway linesinlarge-scaleconstructionandlongitudinalunevenness/randomnessinsystemparametersandtrackirregularities,the temporal-spatial stochastic model is therefore established to wholly consider the possibility ofcombination over system parameters and excitations, and achieving far more comprehensive dynamics results that are necessities in reliability assessment,trackmaintenanceanddesign,etc.

Basedon thefundamentals ofthework inRef. [26], thestudies aimingat constructinga generalmodelforachieving the temporal-stochastic analysis of train-track-bridge systems will be further extended and perfected. The main content of this paper can be illustrated by: firstly, a more advanced 3-D nonlinear train-track-bridge dynamic model will be developed by fully modellingthe tracks andthe bridges asan integratedone, while the interactions between the train and the track-bridge systems are characterized by the complex nonlinear wheel-rail forces by correcting the deficiency ofRef.[18,27]; secondly,a unitedmethodwillbe providedto randomlysimulatethesystemvariables thatare correlated andfollowingarbitraryprobabilitydistribution, basedonMCMandKarhunen–Loève expansion(KLE);thirdly,a statistical strategyisdevelopedtodescribetheevolutionoftrackrandomirregularitiesaccompaniedbya simpleintroductionofthe trackirregularityprobabilistic model;fourthly, the theoretical framework forconstructing the temporal-spatial stochastic modelfortrain-track-bridgeinteractionsispresented;finally,numericalstudiesandconclusionswillbefurtherpresented.

2. Modellingfortrain-track-bridgeinteractions 2.1. Basic instructions

Tothethree-dimensional(3-D)train-track-bridgemodel(Fig.1),thefollowinginstructionsshouldbe madeinadvance, namely

(1) Thetrainconsistsofthefrontandrearmotorcarandseveraltrailercarsmodelledasrigidbodiesofacarbody,two bogieframe,fourwheelsetsandlinearsuspensionsystems,andthevehiclesmoveataconstantvelocity V alongthe tracks;

(2) Thebridgeismodelledassimplysupportedgirderbridges;thetracksrestingonthebridgesarechosenasslabtracks modelledbythin-plateelement,whilethetracksonrigid-subgradesaremodelledbycommonlyusedballastedtracks. (3) Onlythepresentationoverthedynamicequationsofmotionforthetrain-track-bridgeinteractionswillbeillustrated

Fig. 1. Three-dimensional train-track-bridge interaction model (end view). 2.2. Dynamic equations of motion for train-track-bridge interactions

Based onfundamentals offiniteelement theory andenergy-variational principle,one can derive the3-D equationsof motioninsub-matrixformforthetrain-trackbridgeinteractionsystemas

⎡

⎢

⎢

⎣

Mtt 0 0 0 0 0 Mrr 0 0 0 0 0 Mss 0 0 0 0 0 Mbb 0 0 0 0 0 Mpp

⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎣

¨ Xt ¨ Xr ¨ Xs ¨ Xb ¨ Xp

⎤

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎣

Ctt Ctr 0 0 0 Crt Crr Crs 0 0 0 Crs Css Csb 0 0 0 Cbs Cbb Cbp 0 0 0 Cpb Cpp

⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎢

⎣

˙ Xt ˙ Xr ˙ Xs ˙ Xb ˙ Xp

⎤

⎥

⎥

⎥

⎦

+

⎡

⎢

⎢

⎣

Ktt Ktr 0 0 0 Krt Krr Krs 0 0 0 Krs Kss Ksb 0 0 0 Kbs Kbb Kbp 0 0 0 Kpb Kpp

⎤

⎥

⎥

⎦

⎡

⎢

⎢

⎣

Xt Xr Xs Xb Xp

⎤

⎥

⎥

⎦

=

⎡

⎢

⎢

⎣

Ft Fr Fs Fb Fp

⎤

⎥

⎥

⎦

(1)

wherethesubscripts“t”,“r”,“s”,“b” and“p” denotethetrain,rail,slabtrack,girderandpier,respectively. 2.2.1. The methodologies used to couple the train-track-bridge systems

Surveying this train-track-bridge coupled system, one can discover that the connection of all system components includingthewheel-railinterfacialinteractioncanbeequivalentlytreatedasaformofspring-dashpotcontact.

Obviously the dynamicproblems of elastic systems can be also transformed asstatic equilibrium problemswhere d’ Alembertprincipleandtheeffectsofdampingforcesisrespectivelyintroducedandconsideredbyageneralform[19]

fe= fρ+fc− Fsign

(

u

)

+p

(

t

)

+G (2)

with f_ρ=_υ

(

−

ρ

u¨

)

d

υ

; fc=_υ

(

−cu˙

)

d

υ

,wherefe,fρ,fc,F,p(t ) andGare thevectorsofsystemelastic force,inertiaforce,

damping force,Coulomb’s frictionforce, external interfereforce andgravity, respectively; u¨, u˙ and u are respectivelythe acceleration,velocityanddisplacementquantity;

ρ

and c respectivelydenotetheparametersofmassdensityanddamping coefficient.

Based on the principle of virtual work, the equation below can be derived by accounting for the system virtual displacement

δ

u,thatis[19]

δ

Ue=− υ

δ

u T

_ρ

_u¨d

_υ

₋ υ

δ

u T_cu˙d

_υ

₋

_δ

_uT_Fsign

₍

_u

₎

₊

_δ

_uT_p

₍

_t

₎

₊

_δ

_uT_{G (3)} withU _ρ=_υ

δ

uT

_ρ

_u¨d

_υ

_{, U}

c=_υ

δ

uTc u˙d

υ

, V F=uTFsign(u), V p=−uTp(t ), V g=−uTG,inwhich U eistheelasticstrainenergyof

force, Coulomb’s friction force and external interfere force, respectively; V g is the gravitational potential energy of the

systems.

Followingtheprincipleoffiniteelementmethod(FEM),itisknownthatu= NX,u˙=NX˙ andu¨=NX¨,inwhichNisthe equivalentshapefunction, X,X˙ andX¨ are respectivelythe displacement,velocity andaccelerationvector ofthe systems, thusEq.(3)canbetransformedas

δ

Ue+

δ

XTMX¨+

δ

XTCX˙ =

δ

XTFs (4)

with M=_υ

ρ

NT

mNmd

υ

, C=_υc NcTNcd

υ

, Fs=−NTFFsign

(

NFX

)

− NTpp

(

t

)

− NTGG, where Nm and Nc denote the equivalent

shapefunctionsforthemassmatrixanddampingmatrixrespectively;NF,NpandNGdenotetheequivalentshapefunctions

forloaddistribution;

δ

U eisthevariationoftheelasticstrainenergy,whichhasdifferentexpressionsaccordingtothemotion

modes,forexample,

δ

U e=

δ

XT_υEI NTNd

υ

X¨ fortheelastic deformationenergyofa beam,inwhich E and I denotethe

modulusofelasticityandmomentofinertiarespectively,N denotesthesecondderivativeoftheequivalentshapefunction. From Eq.(4)one can observe that thedynamicmatrices canbe obtained by removing thevirtual displacement term

δ

X, thus thekey work in buildingthe equationsof motionfora dynamical systemlies inclarifying theshape functions andloadingvectors. Thedetailmethodologyandderivation processfromthepotential energyofa dynamicsystemtothe correspondinglydynamicmatriceshavebeenpresentedintheworkofXuetal.[28],whichcanbeconsultedforreference. Inthefollowingparts,thedisplacementvectorsandsub-matriceswillbegivendirectlywithoutfurtherderivationsfor brevity.

2.2.2. Displacement vector

Thetotaltraindisplacementvector,Xt,canbeassembledas

Xt =

XMo,1 XTr,1 XTr,2 ... XTr,Nv XMo,2



T

(5)

wherethesuperscript“T” denotesthetranspositionofthematrix;thesubscripts“Mo”,“Tr” denotethemotorcarandtrailer carrespectively,andXMo,i, i =1,2andXTr,j, j =1,2,..., N vdenotethesub-vectorsofthe i thmotorcarandthe j thtrailercar,

allwith35degreesoffreedom(DOF’s),asshowninAppendixA-TableA1.

Thedisplacementvectoroftherail,Xr,withorderof(2× Nr× nr)× 1,where N risthetotalnumberofrailbeamelement,

n risthenumberofDOFsofarail beamandthenumber“2” originatesfromtheleft- andright-side oftherails,can be

writtenas

Xr=

XLr,1 XLr,2 ... XLr,nr XRr,1 XRr,2 ... XRr,nr



T

(6)

wherethesubscript“Lr” and“Rr” denotestheleft-andright-siderailrespectively.TheDOFforeachnodeoftherailbeam elementhasbeenlistedinAppendixA-TableA2.

Thedisplacementvectoroftheslabtrack,Xs,withorderof(N s× ns)× 1,where N sisthetotalnumberofslabelement,

n sisthenumberofDOFsofaslabthin-plateelement,canbewrittenas

Xs=

Xs,1 Xs,2 ... Xs,Ns



T

(7)

wherethe Xs,l, l =1,2,..., N s,denotes the displacementvector of the l thslab trackelement.The DOFsfor thethin-plate

elementoftheslabtrackshavebeenlistedinAppendixA-TableA3.

The displacement vector of the girder, Xb, with orderof (N b× nb)× 1, where N b is thetotal number of bridge girder

elements, n b isthenumberofDOFsofagirderelement,canbewrittenas

Xb=

Xb,1 Xb,2 ... Xb,Nb



T

(8)

wheretheXb,k, k =1,2,..., N bdenotesthedisplacementvectorofthe k thgirderelement.TheDOFsforthegirderelement

arethesameastherailbeamelement.

The displacement vector of the pier, Xp, with order of (N p× np)× 1, where N p is the total number of bridge girder

elements, n p isthenumberofDOFsofagirderelement,canbewrittenas

Xp=

Xp,1 Xp,2 ... Xp,Nb



T

(9)

wheretheXp,q, q =1,2,..., N pdenotesthedisplacementvectorofthe q thgirderelement.TheDOFsforthepierelementare

lineardisplacementsalong X -, Y -and Z -axissincebeingassumedasmasselement. 2.2.3. Sub-matrices for the train

Themassmatrixofthetrain,Mtt,canbeassembledas

Mtt = Nv  Tr=1 MTr,c

(



Tr,c,



Tr,c

)

+MTr,Gq

(



Tr,Gq,



Tr,Gq

)

+MTr,Gh

(



Tr,Gh,



Tr,Gh

)

+ nwr  i=1 MTr,w,i

(



Tr,w,i,



Tr,w,i

)

+ 2  Mo=1 MMo,c

(



Mo,c,



Mo,c

)

+MMo,Gq

(



Mo,Gq,



Mo,Gq

)

+MMo,Gh

(



Mo,Gh,



Mo,Gh

)

+ nwr



i=1

M_Mo,w,i

(



Mo,w,i,



Mo,w,i

)

(10)

with



u=

yu zu

ψ

u

β

u

φ

u



, Mu

(



u,



u

)

=diag

(

[mu mu Iu,z Iu,y Iu,x ]

)

,



w,g=

yw,g zw,g

ψ

w,g

β

w,g

φ

w,g



, Mw,g

(



w,g,



w,g

)

=diag

(

[mw,g mw,g Iw,z,g Iw,y,g Iw,x,g ]

)

,

where thesubscript “u” will be substitutedby “c”, “Gq” and“Gh” representing the carbody,front bogie frame andrear bogieframerespectively;thesubscript“w” denotesthewheelset,“g” denotesthe g thwheelset; m isthemass, I x, I yand I z

denotemomentofinertiaaround X -, Y -and Z -axis;



[•] isageneraloperatorindicatingtheDOFnumbercorresponding tothematrices.

Thestiffnessmatrixofthetraincanbederivedas

Ktt= Nv  Tr=1 KTr+ 2  Mo=1 KMo (11)

ThedetailedexpressionofEq.(11)hasbeenelaboratedinAppendixB[29].Additionally,thedampingmatrixofthetrain,

Ctt,hasalmostthesameexpressionasKtt.Itisjustneededtosubstitutethestiffnesscoefficient k withdampingcoefficient c .

2.2.4. Sub-matrices for the rail

Themassmatrixoftherail,Mrr,canbewrittenas

Mrr= Nr  h=1 2  g=1 mr

lr 0 NT XrNXrd

ξ

+ lr 0 NT YrNYrd

ξ

+ lr 0 NT ZrNZrd

ξ

+ lr 0 Wr Ar NT θXrNθXrd

ξ



(12) with NXr=NθXr=

... 1−

ξ

/L ...

ξ

/L ...



_1×n r, NYr=



... 1− 3



ξ_L



2+2



ξ_L



3



ξ L− 2



_ξ L



2 +



ξ_L



3



L 3



ξ_L



2− 2



ξ_L



3



ξL



3 −



ξ_L



2



L ...



1×nr , NZr=



... 1− 3



ξL



2 +2



ξ_L



3 ₋



ξ_{L − 2}



ξ_L



2₊



ξ_L



3



L 3



ξ_L



2− 2



ξL



3 −



ξL



3 −



ξL



2



L ...



1×nr ,

where N Xr, N Yr, N Zr and N θ_Xr arethe shapefunctionsfordisplacement along X -, Y -and Z -axisandanglearound X –axis;

g =1,2denotetheleftandrightsiderailalong X -axis; N risthetotalnumberofrailbeamelements;

ζ

isthelocaldistance

fromtheleftnodeoftherailbeamtoarbitrarypointswithinthebeamelementalong X -axis. Thestiffnessmatrixoftherail,Krr,canbewrittenas

Krr= Nr  h=1

⎡

⎣

2 g=1



ErAr lr 0 [N  Xr] T [N_Xr ]d

ξ

+ErIry lr 0 [N  Zr] T [N_Zr]d

ξ



+2 g=1



ErIrz lr 0 [N  Yr] T [N_Y_r]d

ξ

+krt lr 0 [N  θXr] T [N_θ Xr]d

ξ



⎤

⎦

(13)

where N and N  arethefirstandsecondderivativeoftheshapefunctionrespectively. 2.2.5. Sub-matrices for the slab tracks

Themassmatrixoftheslabtrack,Mss,canbewrittenas

Mss= Ns  h=1 mshs

(

bs/2 −bs/2 ls/2 −ls/2 NT ZsNZsd

ξ

d

ζ

+bs ls 0 NT YsNYsd

ξ

)

(14)

where

ζ

is the local coordinatealong the Y -axis; N Ys has the sameexpression as N Yr, while N Zs is chosen asthe shape

functionofarectangularthin-plateelement.

Thestiffnessmatrixoftheslabtrack,Kss,canbewrittenas

Kss= Ns  h=1



ls 0 bs 0 BT_DBd

_ζ

_d

_ξ



+EsIsz ls 0 N_ZsTN_Zsd

ζ

] (15) with B=−



∂

2_N Zs

∂

ξ

2 ;

∂

2_N Zs

∂

ζ

2 ;

∂

2_N Zs

∂

ξ∂

ζ



T ,D= Esh3s 12

(

1−

μ

s2

)



₁

_μ

s 0

μ

s 1 0 0 0

(

1−

μ

s

)

/2



,

2.2.6. Sub-matrices for the girder

Becausethegirderisassumedtobebeamelementastherails,thusitssub-matriceshavethesameexpressionsasthe rail.

Tothemassmatricesofthegirder,M_bb,onecanderiveitbysubstitutingthe“mr”,“Ar” and“Wr” inEq.(12)with“mb”,

“Ab” and“Wb”,namely

Mbb= Ns  h=1 mb

lr 0 NT XrNXrd

ξ

+ lr 0 NT YrNYrd

ξ

+ lr 0 NT ZrNZrd

ξ

+ lr 0 Wb Ab NT θXrNθXrd

ξ



(16)

Tothe stiffnessmatrix of thegirder, Kbb, it can be obtainedby substituting the “Er”, “Ar”, “Iry”, “Irz” and “krt” in Eq.

(13)with“E_b”,“A_b”,“I_by”,“I_bz” and“k_bt”,namely

Kbb= Ns  h=1

⎡

⎣



EbAb lr 0 [N  Xr] T [N_Xr]d

ξ

+EbIby lr 0 [N  Zr] T [N_Zr]d

ξ



+



EbIbz lr 0 [N  Yr] T [N_Y_r]d

ξ

+kbt lr 0 [N  θXr] T [N_θ Xr]d

ξ



⎤

⎦

(17)

BasedontheassumptionofRayleighdamping,thedampingmatrixCbbcanbecomputedby

Cbb=

α

Mbb+

β

Kbb (18)

with

α

₌2

ζ

_b

ω

1

ω

2/(

ω

1+

ω

2)and

β

=2

ζ

b/(

ω

1+

ω

2).

Where

ζ

bisthedampingratio,

ω

1 and

ω

2arethefirsttwocircularfrequenciesofvibrationofthegirder.

2.2.7. Sub-matrices for the pier

The pier is simplified asmass element, the mass, stiffnessand damping matrix of which, Mpp, Kpp and Cpp, can be

writtenas Mpp= Np  h=1 diag

(

[mp mp mp ]

)

(19) Kpp= Np  h=1 diag

(

[kpx kpy kpz ]

)

(20) Cpp= Np  h=1 diag

(

[cpx cpy cpz ]

)

(21)

2.2.8. Sub-matrices for the rail-slab track interactions

Thematricesforrail-slabinteractionsaremarkedwiththesubscripts“rs” and“sr”,whicharederivedbytheinteractions betweentherailsandtheslabtracksconnectedbytherailpads.

Thestiffnessmatrix,K˜r−s,withorderof(2× Nr× nr+N s× ns)× (2× Nr× nr+N s× ns),canbewrittenas

˜ Kr−s= 2(Nr+1) h=1

⎡

⎣

krp,z[

(

[NZr] T ξ=0− [NZs]T_ξ=0,ζ=b

)(

[NZr]_ξ=0− [NZs]_ξ=0,ζ=b

)

+

(

[NZr]T_ξ=0− [NZs]T_{ξ=0,ζ=bs−b}

)(

[NZr]_ξ=0− [NZs]_{ξ=0,ζ=bs−b}

)

] +krp,y[

(

[NYr]T_ξ=0− [NYs]T_ξ=0

)(

[NYr]_ξ=0− [NYs]_ξ=0

)

]

⎤

⎦

(22)

ThedampingmatrixC˜r−s canbe obtainedby substitutingthe stiffnesscoefficients k rp,z and k rp,y inK˜r−swithdamping

coefficients c rp,z and c rp,y.

It should be noted that the matrices of stiffness and damping are actually assembled ones, which will be further partitionedas



¯ Krr Krs Ksr K¯ss,1



=K˜r−s,



¯ Crr Crs Csr C¯ss1



=C˜r−s, (23)

ThusthestiffnessanddampingmatrixoftherailwillbereplacedbyKrr+K¯rr andCrr+C¯rr.

2.2.9. Sub-matrices for the slab track-girder interactions

Thematricesforslab track-girderinteractions aremarked withthesubscripts“sb” and“bs”,which arederived by the interactionsbetweentheslabtracksandthegirderconnectedbythecementasphaltmortar(CAM).

Thestiffnessmatrix,K˜s−b,withorderof(N s× ns+N b× nb)× (N s× ns+N b× nb),canbewrittenas

˜ Ks−b= Ns  h=1



kca,z ls 0 bs 0 NT Z,sbNZ,sbd

ζ

d

ξ

+kca,y ls 0 NT Y,sbNY,sbd

ξ



(24)

Fig. 2. Wheel-rail coupled model [24] . with N_Z,sb=NZs− NZb, NZb=



NZr

Br− bs 2 +

ζ



N_θXr



, NY,sb=NYs− NYb, NYb=

NYr Hr N_θXr



,

The dampingmatrix,C˜s−b,has almostthe sameexpression asK˜s−b justby substitutingthe stiffnesscoefficients“kca,z”

and“kca,y” withdampingcoefficients“cca,z” and“cca,y” .

InthesamemannerasEq.(23),thematricesofstiffnessanddampingshouldbepartitionedas



¯ K_ss,2 Ksb Kbs K¯bb,1



=K˜s−b;



¯ C_ss,2 Csb Cbs C¯bb,1



=C˜s−b, (25)

ThusthestiffnessanddampingmatrixoftheslabwillbereplacedbyKss+K¯ss,1+K¯ss,2 andCss+C¯ss,1+C¯ss,2.

2.2.10. Sub-matrices for the Girder–Pier interactions

The matrices for girder-pier interactions are marked with the subscripts “bp” and “pb”, which are derived by the interactionsbetweenthegirderandthepierconnectedbythecementasphaltmortar(CAM).

Thestiffnessmatrix,K˜b−p,withorderof(N b× nb+N p× np)× (N b× nb+N p× np),canbewrittenas

˜ Kb−p= Np  h=1

kp,zNZ,bpT NZ,bp+kp,yNY,bpT NY,bp



(26) with N Z,bp=N Zs− NZp, N Zb=1; N Y,bp=N Yb− NYp, N Yp=1;

Thedampingmatrix,C˜b−p,can beobtainedbyreplacing thestiffnesscoefficients“kp,z” and“kp,y” in thecorresponding

stiffnessmatrixwithdampingcoefficients“cp,z” and“cp,y” .

Thematricesofstiffnessanddampingshouldbepartitionedas



_¯ Kbb,2 Kbp Kpb K¯pp,1



=_K˜_b−p;



_¯ Cbb,2 Cbp Cpb C¯pp,1



=_C˜_{b−p, (27)}

Accordingly, thestiffness anddampingmatrixof thegirder willbe assembled asK_bb+K¯_bb,1+K¯_bb,2 and C_bb+C¯_bb,1+C¯_bb,2, andthestiffnessanddampingmatrixofthepierwillbeobtainedbyKpp+K¯pp,1 and Cpp+C¯pp,1.

2.2.11. Load vectors

The wheel-rail contact geometries are almost the most important and complex part in the whole train-track-bridge dynamicmodel.Inthewheel-railcoupledmodel[30],theyaw,pitch,roll,transverseandbouncemotionsofthewheelset andthetransverse,bounceandtorsionalmotionsoftherailareconsideredwithacomprehensiveway,asshowninFig.2, thewheel-railcontactswillthereforebetreatedasathree-dimensional(3-D)nonlinearandasymmetryprobleminevitably.

Thedynamicloadactingonthewheelsetisderived bythework oftheforce/momentontransverse,bounce,roll,yaw andpitchmotionsofthewheelsetrespectively,thatis[24],

Fw= 2+Nv j=1 4  i=1 Fw,i (28) with Fw,i=

⎡

⎢

⎢

⎣

Fw,Y i Fw,Zi Fw,φi Fw,ψi Fw,βi

⎤

⎥

⎥

⎦

=

⎡

⎢

⎢

⎢

⎣

FYli+FYri+Mw¯g

ϕ

i− MwV2/Ri−Mwr0

ϕ

¨wi FZli+FZri+Mw¯g+Mwa0

ϕ

¨wi+MwV2/Ri

ϕ

wi a0

(

FZri− FZli

)

− rliFYli− rriFYri+IwY

(

β

˙wi−

)(

ψ

˙wi+V/Ri

)

−IwX

ϕ

¨wi a0

(

FXli− FXri

)

− a0

ψ

wi

(

FYli− FYri

)

+MZli+MZri+IwY

(

β

˙wi−

)(

φ

˙wi+

ϕ

˙wi

)

−IwZV

λ

˙wi rliFXli+rriFXri+

ψ

wi

(

rliFYli+rriFYri

)

+MYli+MYri

⎤

⎥

⎥

⎥

⎦

,

wherethesubscript“i”, i =1,2,3,4,denotesthe i thwheelset; F Xliand F Xridenotethelongitudinalforcesactingonthe

left-andright-sideofwheel/railcontactinterface,respectively; F _Yliand F _Yridenotethetransverseforcesactingontheleft-and right-sideofwheel/railcontactinterface,respectively; F Zliand F Zridenotetheverticalforcesactingontheleft-and

right-sideofthewheel/railcontactinterface,respectively; M Zliand M Zriarethemomentofforcearound Z -axis,respectively; M Yli

and M _Yriarethemomentofforce around Y -axis,respectively;

β

˙_wi,

ψ

˙_wiand

φ

˙_wi denotethepitch,yawandrollingvelocity ofthe i thwheelset;

ψ

wiistheyawangle.

Theforcesactingonthecentroidofmassofthecarbodyaremainlyderivedfromtheworkofforce/momentatmotions oftransverse,bounce,rollingandyawwhenthevehiclepassesthroughthecurvedtracks,namely

Fc= 2+Nv i=1 Fc,i (29) with Fc,i=

⎡

⎢

⎣

Fc,Y i Fc,Zi Fc,φi Fc,ψi

⎤

⎥

⎦

=

⎡

⎢

⎣

Mcg

ϕ

c−McV2/Rc−Mc

(

r0+Htw+Hbt+Hcb

)

ϕ

¨c Mcg+McV2/Rc

ϕ

c+Mca0

ϕ

¨c −IcX

ϕ

¨c −IcZV

ς

˙c

⎤

⎥

⎦

,

Theforces actingon the centroidof massofthe bogie framesare mainly derived fromthe work offorce/momentat motionsoftransverse,bounce,rollingandyawwhenthevehiclepassesthroughthecurvedtracks,namely

Fb= 2+Nv i=1 2  k=1 Fk (30) with

⎡

⎢

⎣

Fk,Y Fk,Z Fk,φ Fk,ψ

⎤

⎥

⎦

=

⎡

⎢

⎣

Mkg

ϕ

k−MkV2/Rk−Mk

(

r0+Htw

)

ϕ

¨k Mkg+MkV2/Rk

ϕ

k+Mka0

ϕ

¨k −IkX

ϕ

¨k −IkZV

ς

˙k

⎤

⎥

⎦

,

where k =1and2representthefrontbogieframe,“Gq”,andrearbogieframe,“Gh”,respectively.

Withregard tothetracksystems,rail isthecomponent thatdirectly bearsthedynamicloadsfromthe train,andthe dynamicforceactingonthetransverse,verticalandtorsionaldisplacementcanbeexpressedas

Fr= 2+Nv i=1 4  j=1 2  l=1 Fk,l, j (31) with Fk,l, j=



_F Y,l j FZ,l j MT,l j



=



_F YlNYr, j FZlNZr, j

(

FZllhi+FYllvi

)

NθXT, j



,

wherethesubscript “l” denotestheleft orrightsideofthe rail;thesubscript “j” denotesthe j thwheel-railcontactpair; F Y,lj, F Z,ljand M T,ljdenotetheequivalentlateral,verticalandtorsionalforcesactingonthecentroidoftherail.

Accordingtoabovepresentations,onecanobtainedthatFt=Fc+Fb+Fw,Fs=0,Fb=0,Fp=0.

Tillnow,thedynamicsub-matricesandloadingvectorsinEq.(1)havebeenfullyrevealedtoeffectivelycouplethetrain, trackandbridgesystems asan entiresystem. Byadoptingnumericalintegration methods,thesystem responsesof train, trackandbridgecanbeobtainedsimultaneously.

Fig. 3. Comparisons with the model of Zhai et al. [1] on car body accelerations and the vibrations at the mid-span of the bridge (a. lateral acceleration of the car body; b. vertical acceleration of the car body; c. lateral displacement of the girder at the mid-span; d. vertical displacement of the girder at the mid-span).

2.2.12. Comparison with general solutions

For validating the practicality of this dynamical model compiled in a computer program, a comparison to a three-dimensionalmodelbyZhaietal.[10]isconducted. Itisassumedthatanentiretraingroupedbyfivevehiclesrunsonthe bridge(five-spansimplysupportedbeam-bridge)withaconstantvelocity(300km/h).

Fig.3presentsthe comparisons on the dynamic responses of the bridge at thesecond mid-span and car body accelera-tions,fromwhichonecanobservethatthedynamicresultsrespectivelyderivedfromthesetwomodelscoinciderelatively well witheach other.Theaccuratenessandpracticalityofthismodelisthereforeillustratedclearly.Howeverit shouldbe notedthat thereinevitablyexist slightdeviations betweentheresults becauseofthecompletelydifferentmethodsinthe modelconstructions,seeforexample:thecoupleranddraftsystembetweenthecarbodiesofvehicles ignoredin[10]are consideredeffectivelyinthismodel;inthismodel,thetrackandthebridgehavebeenunitedanentiresystem,whilethey aresolvedseparatelyin[10].

3. Characterizationofthetemporal-spatialstochasticityforthecoupledsystem

The model constructed in Section 2 is developed to characterize the train-track-bridge interactions with respect to deterministic systemparameters andexcitations, e.g.,track randomirregularities. Ithasbeen pointedout previously that thetrain-track-bridgesystemsareessentiallyrandominnature,especiallythetrack-bridgesystem,whichholdssignificantly inhomogeneous characteristics in the spatial domain, moreover, the random field of dynamic parameters is constantly evolvedcorrespondingtothecyclictrain-track-bridgeinteractions.

3.1. Joint simulation based on Monte-Carlo Method (MCM) and Karhunen-Loève Expansion (KLE) 3.1.1. Random field representation

Byneglecting thestochasticity ofthesystemcomponents in Y direction, therandom variables oftrack-bridgesystems canbeexpressedas



(x, z, t ),where x representsthelongitudinalabscissaofthetracks, z representstheverticalcoordinate and t representsthetimeprocess.Thustheunifiedfieldofsystemstructurescanbeexpressedby

T=T

((

x,z,t

))

,



∈R3 ₍₃₂₎

where R 3 _{representsthethree-dimensionalEuclideanspace,}

_

_{representsthe definitiondomainofthe randomfield, and}

the variable z is mainly applied to express the vertically layered characteristics of track and bridge structures, namely differentphysicalandmechanicalparametersofstructures.

Withrespecttothedeterminedstructuralparameters z _υ,

υ

=1, 2, ..., M and thetime point t k, k =1,2,..., N ,thespatial

distributionof



(x, z, t ),whichisregardedtofollowacertaintypeofprobabilitydistribution,canbecharacterizedby

whereP(•)representsthepossibleprobability distributiontype,andV ˜ representsthecorresponding characteristic param-eters,e.g.,thestatisticalmeanandstandarddeviationofnormaldistribution,thedegreeofchisquaredistribution,etc.

Generally,thelongitudinalcoordinateofthe raillinerepresentedby parameter x will bewithahuge magnitudeonce considering the whole bridge sections existing in a railway line. Therefore, a longitudinal division towards the random spaceisconductedbyconsideringthelimitlengthofnumericalcalculations,namely

x=∩xl,xl∈

(

0,Stot/N] (34)

where S tot_{representsthetotalrailwaylength, N representsthenumberofdivision,and l =1,2,..., N .}

Insummary,therandomfieldoftrack-bridgesystemscanbefurthertransformedfromEq.(32)as

T_=T

((

xl,zυ,tκ

))

(35)

3.1.2. Monte-Carlo simulation

Innumericalstudies, therandomfieldrepresentedby Eq.(35)isconventionallymodelledby MCMthat isperhapsthe mostrobust approachinrandom analysis.Foracertaintime point t _κ,thespacevectorinEq.(35)canbe expoundedasa two-dimensionalmatrixform

T_=

⎛

⎜

⎜

⎜

⎜

⎜

⎝

T

(

u

(

x1,z1,tκ

))

T

(

u

(

x2,z1,tκ

))

· · · T

(

u

(

xN,z1,tκ

))

. . . ... ... ... T

(

u

(

x1,z1×H,tκ

))

T

(

u

(

x2,z1×H,tκ

))

· · · T

(

u

(

xN,z1×H,tκ

))

. . . ... ... ... T

(

u

(

x1,zM×H,tκ

))

T

(

u

(

x2,zM×H,tκ

))

· · · T

(

u

(

xN,zM×H,tκ

))

⎞

⎟

⎟

⎟

⎟

⎟

⎠

(M×H)×N (36)

wherethesubscript“H” denotesthenumberforaspecificvariableinonesample.

Tothe actualrandom variables z _υ,which areassumed tofollow an alreadyknownprobability densityfunction(PDF) f Z

(

z υ

)

,thecumulativedistributionfunction(CDF),amonotonicfunction,canbederivedby

FZ

(

zυ

)

=

zυ

−∞fZ

(

zυ

)

dzυ,

υ

=1,2,...,M, (37)

Thecorrelationmatrixbetween z _υ canbeexpressedby

RM×M=

⎛

⎜

⎜

⎝

1 r12 ... r1M r21 1 ... r2M . . . ... ... ... rM1 rM2 ... 1

⎞

⎟

⎟

⎠

(38)

Forobtainingrandom seriesof z _υ that satisfiesEqs.(37) and(38),based onalinear-nonlinear transformationmethod [31],thestepsforachievingthisgoalcanbefollowedby

(1)Let g _υ bethevariablefollowingGaussiandistribution,thatis,

fG

(

gυ

)

= 1



2

πσ

gυ exp

−

(

z−

μ

gυ

)

2 2

σ

2 gυ



(39)

where

μ

g_υ and

σ

g_υ respectivelyrepresentthestatisticalaverageandstandarddeviationwithrespecttothevariable g υ.For

simplifyingthetransformation,itispracticaltoset

μ

g_υ =0and

σ

g_υ =1.

InthesamemannerasEq.(37),theCDFof g _υcanbeobtainedby

FG

(

gυ

)

=

gi

−∞fG

(

gυ

)

dgυ (40)

BasedonEqs.(37)and(40),thenonlineartransformationbetween x _υ and g _υ canbeestimatedas

x_υ=¯h_υ

(

g_υ

)

=F_Z−1

(

FG

(

gυ

))

(41)

where



(•)denotesthenonlineartransformationoperatorforargument(•); F -1_{(•)denotestheinversefunction.}

(2)Derivingthecorrelationcoefficient r g_ig_j through r x_ix_j, i, j =1,2,..., M, i = j ,thatis[30,31],

rxixj = E

(

xi−

μ

xi

)(

xj−

μ

xj

)



σ

xi

σ

xj = _∞ −∞ _∞ −∞

(

i

(

gi

)

−

μ

xi

)

σ

xi

(

j

(

gj

)

−

μ

xj

)

σ

xj fGG

(

gi,gj

)

dgidgj (42) with

Fig. 4. PDF comparison between the results of MCM and MCM-KLE. fGG

(

gi,gj

)

= 1 2

π



1− r2 gigj exp



−

(

g 2 i− 2rgigjgigj+g 2 j

)

2

(

1− r2 gigj

)



;

μ

x=E

(

x

)

;

σ

x=E

(

x2

)

− [E

(

x

)

]2,

inEq.(42), E denotesthemathematicalexpectation, andtherelationshipbetween r g_ig_j and r x_ix_j hasbeenconnected

defi-nitely.ForsolvingtheintegrationofEq.(42),Newton–Raphsoniterationmethodisapplied[32],andthen,thecorrelation matrixR ˜M×Mwithregardto g icanbeconfirmedaccordingly.

(3)UsingMonte-Carlomethod,therandomvectorof g icanbeassembledas

(

G=

(

G1,G2,...,GM

)

L×M

)

(43)

Andthen,theCholeskydecompositionisconductedonR ˜M×M,andtheGwillbeupdatedby

G=GL×MCr,M×M (44)

withR ˜M×M=C gTC g,theupdatedrandomvectorGwillsatisfythecorrelationmatrixR ˜M×M.

(4)Finally,throughthenonlinear transformation,thestandardized randomvector Gcan betransformed intoarbitrary correlatedpseudorandomvariables,namelythespacevectoroftherandomfiledby

T_=¯h

(

G

)

(45)

3.1.3. Dimension reduction based on Karhunen–Loève Expansion (KLE)

MCMisgenerallyaccompaniedbyratherlow computationalefficiencyandconvergenceintherandomanalysis.Inthe currentwork,Karhunen–LoèveExpansion(KLE)isfurtherappliedtoachieve adimensionalreductionfortherandom filed oftrack-bridgesystems.

BasedonthedefinitionofKLE[33],therandomfieldoftrackstructurescanbefurtherexpressedby

T_≈ E[T_]+ Ns  i=1



λ

iui

ζ

i (46)

with E[

ζ

i

ζ

j]=

δ

i j, where

δ

ij is the Kronecker delta, E[· ] denotes the expectation operation; E[ T] is the mean of the

stochasticprocess,

ζ

iareorthonormalrandomvariables.

Thesetofdeterministicfunctions

λ

_iandu_iareeigenvaluesandeigenvectors,respectively,withsatisfyingthefollowing equation,namely

CXX

μ

=

λμ

(47)

where CXX represents an bounded, symmetric and positive definite auto-covariance function of T. The projection basis

{

μ

1,

μ

2,...,

μ

Ns

}

ischosenorthonormalinthesensethat,forall

γ

and:

(

μ

γ

₎

T

_μ

₌

_δ

_γ

 (48)

TheKLEiscapableofcapturingthespatialcorrelationofthefields,guaranteeingtherandomicityofsignalsandgreatly reducingthecalculationsamplesby N s N.

FormoredetailsaboutKLE,Refs.[34,35]canbeconsultedforreferences.

Forvalidatingthe effectivenessofthe MCM-KLEmethod,Fig. 4plots aPDF comparisonbetweenthe randomsamples respectivelysimulatedby MCMandMCM-KLEwithregard totherandomvariable,i.e.,lateralstiffnessoffastener.Asseen fromFig.4,thestatisticalpropertiesforthesimulatingresultsofthesetwomethodcoincidewellwitheachother,however, thereare 2000samplesintheMCM, butbyapplyingMCM-KLE,the samplingnumberis reducedto174,thusthe united MCM-KLEmethodgreatlyincreasesthecomputationalefficiency.

3.2. Track irregularity probabilistic model

Xu and Zhai [36] proposed a track irregularity probabilistic model (TIPM) to select representative and realistic track irregularitysetsforahigh-efficientcharacterizationofthestatisticalpropertiesoftrackgeometries.

Theabbreviatedproceduresareillustratedas

1) DividethemeasuredtrackirregularitiesI(s )into N segments,namely

I

(

s

)

=∪

(

I

(

s˜k

))

(49)

where s denotesthedistancealongthetrack, k ₌1,2,..., N ,s ˜_k∈

(

S tot

(

k − 1

)

/N,S totk/N],

2) Set  (•)asthepowerspectraldensity(PSD)operator,andthePSDofI

(

s ˜k

)

canbeobtainedby P k

(

ω

)

=

(

I

(

s ˜k

))

;

3) Withaseriesofderivations,theprobabilitydensityfunction(PDF)of P k(

ω

)denotedby



(P k(

ω

))canbederived.

4) Basedontime-frequencytransformationapproach[35],theequivalenttimedomaintrackirregularitiesI˜

(

s ˜_k

)

usedasthe excitinginputsoftheinteractionmodelcanbeperformedon P (k ),andassuming

(

P _k

(

ω

))

=

(

I˜

(

s ˜_k

))

.

5) Withcognitionoftheprobabilitydistributionof P k(

ω

),itisratherconvenienttoselectrepresentativePSDfunction,i.e.,

P r(

ω

),inwhich r istheselectedsampleofPSD, r =1,2,...,N ˜,usingrandomsimulationmethods.Generally,itisobvious

that N ˜ N,whilethestatisticalpropertiesofI(s ) inamplitudesandfrequenciesare definedwithoutinformationloss, andbeingcontainedin P r(

ω

).

3.3. Temporal evolution of system dynamic characteristics

Astotheevolutionofsystemparameters,theexperimentalstudiesinsituorlabhaverarelybeenreportedyet,thusthe formulaicexpressions usedto program thevariation ofparameters againsttime cannot be revealedin thepresentstudy, and accordingly, the coefficients of variation (COV) are simply adopted to characterize the evolution of random system parameters,asshowninRef.[26].

Tothetrackirregularities,trackmaintenancedepartmentwillperiodicallydetecttherailprofiledeformationsusingtrack inspection car. The trackirregularity probabilistic model mentioned inlast partis developed to extract the realistic and representativesamplesfromthemassivetrackirregularitydata,basedonwhichthestrategyonstatisticallycharacterizing thetemporalevolution oftrackrandomirregularities canbe furtherachievedby equivalentlytransforming theamplitude evolutionoftrackirregularitiesintoitspowerspectraldensityevolution,thedetailedprocedurescanbefollowedasbelow 1) Thepowerspectrumsoftrackirregularitiesattime t _iand t _i+1 denotedbyP_ς,ti

(

ω

)

andPς,ti+1

(

ω

)

respectively,wherethe

subscript “

ζ

” denotesthetrackirregularitytype,aresummarily obtainedby thespectral estimationmethod,indicating that theamplitude-frequencypropertiesofI_ς,t_i

(

s

)

willbeevolvedfromPς,t_i

(

ω

)

toPς,t_i+1

(

ω

)

duringthetimeperiodof

(t _i,t _i+1],thusthespectralevolutioncanbefiguredoutby

˜

P_ς,(ti,ti+1 ]

(

ω

)

=Pς,ti+1

(

ω

)

− Pς,ti

(

ω

)

(50)

2) Fortheanalyticalconvenience,aspectraldensitymatrixwithorderof E × Wcanbeassembledby



ς,i

(

q,

ω

)

=

∪Pς,ti,q

(

ω

d

)



E×W,q=1,2,...,E, (51)

wherethesubscript"i "and"q "denotesthetimeintervalofmeasurement(t _i,t _i+1]andthe q thtrackirregularitypower spectrumwithrespecttothe q thtrackportion, E isthetotalnumberofpowerspectrumsandthesubscript“W” denotes thetotalnumberoffrequencypoints.Inthesamemanner,aspectraldensityevolutionmatrix



˜_ς,i

(

q,

ω

)

canbeobtained by[



_ς,i+1(q ,

ω

)₋



_ς,i(q ,

ω

)].

3) Obviously, with large-scale and long-term measurements, namely i =1, 2, ..., T, T is a relatively large value, a com-pletely mapping relationbetween



_ς,i(q ,

ω

) and



˜ς,i

(

q,

ω

)

can be numericallymatched, where

ς

(q ,

ω

)=∪



ς,i(q ,

ω

)

and

ς

˜

(

q,

ω

)

=∪

_ς

˜

,i

(

q,

ω

)

.

4) LetP_ς,l(

ω

d)=min[

ς

(q ,

ω

d)]andPς,u(

ω

d)=max[

ς

(q ,

ω

d)],wheremin[•]andmax[•]denotetheminimumand

max-imumoperators respectively.Because thespectraldensities arediscretely distributed,the amplitudedomain ofP_ς(

ω

d)

canbedividedinto

P_ς,g

(

ω

d

)

∈[Pς,l

(

ω

d

)

+gP˜

(

ω

d

)

,Pς,l

(

ω

d

)

+

(

g+1

)

P˜

(

ω

d

)

] (52)

withP˜

(

ω

_d

)

=int[Pς,u(ωd)−Pς,l(ωd)

Z ],0≤ g≤ Z-1,

whereP˜isthediscreteintervalofspectraldensities; Z isthetotalpartitionnumber;int[_·]isanoperatorusedtoobtain thelargestintegerbeingsmallerthanthenumberinthebracket;thesubscript "

ς

,g"denotesthefield ofspectral

densitiessatisfyingEq.(52).

5) It is certain that the spectral densities P_ς,g

(

ω

_d

)

included inthe field of

ς

,g will correspond to different evolution

values, P˜_ς,g

(

ω

d

)

∈

ς

˜

(

ω

d

)

. Based on probabilistic statistics, the probability density function (PDF) of the evolutions

denotedby f _ς,ωd

(

P˜_ς,g

)

canbedeterminedbyP˜_ς,g

(

ω

d

)

.

6) Itisregardedthattheevolutionofthespectraldensitieswithin

ς

,gwillfollowthesamePDFcharacteristics.Therefore,

acumulativeprobabilityindex p _ς ∈(0,1)isintroducedtouniquelyconfirmtheevolutionvalueofspectraldensityover

P_ς,t_i

(

ω

)

,namely ˜ Pti

(

ω

; pς

)

=F −1 ς,ωd

(

pς

)

(53) with F _ς,ωd

(

P˜ς,g

)

= P˜_ς,g −∞ f ς,ωd

(

P˜ς,g

)

dP˜ς,g,

Fig. 5. Modelling framework for train-track-bridge stochastic model. where F _ς,ωd

(

·

)

istheCDFof f _ς,ωd

(

·

)

.

7) OnthebasisofEq.(53),theevolutionspectrumP_t

i+1

(

ω

; pς

)

canbeobtainedby

Pti+1

(

ω

; pς

)

=Pς,ti

(

ω

)

+P˜ti

(

ω

; pς

)

(54)

8) Itshouldbenotedthat P_t

i+1

(

ω

; pς

)

,whichisdeterminedby theprobabilityindex p ς andevolution domainP˜ς,g

(

ω

d

)

, isanuncertainquantity,thusitcanbeselectedbyrandomsimulationmethods.

By implementingthe time-frequency transformation process [36], the evolution spectrum P_t

i+1

(

ω

; pς

)

in Eq.(54) can be equivalentlytransformed intotime-domain track irregularitiesI_t

i+1

(

s

)

. Byloading Iti+1

(

s

)

intothe dynamicmodel,the dynamicperformanceoftrain-track-bridgesystemscanbeassessedinlightofdifferentevolutionarystatusofexcitations. 4. Constructionofthetemporal-spatialstochasticmodel

Till now, the work related to the establishment of the dynamic model, random simulation and characterization of the systemparameters andexcitationsin temporal-spatial domain have all been accomplishedproperly. The essencefor constructingthetrain-track-bridgetemporal-spatialstochasticmodellaysonunitingtherandomphysical-mechanicalstatus ofthesystemsandthedynamicalinteractionmechanisms.

Summarily,themodellingframeworkcanbeillustratedasFig.5 5. Casestudies

Inthenumericalstudies,it isassumedthatthereare twomotorcarsinthe frontandrearpartofthetrain,andwith three trailer cars containedin the middle,the relatedparameters havebeen presented inRef. [18]; besides, a fivespan simplesupportbeambridge isconstructedby 32-meterbeamelements,therelateddynamicparametersofthebridgeare listedinRef.[18].

5.1. Case 1: characteristics of system responses for train-track-bridge interactions

Ratherdifferent fromcommon infrastructural systems,the railwaybridges normally assumedasbeam elementsoffer moreflexiblesupportstothetrain-tracksystemsingeneral,thusthedynamicresponseoftrain-track-bridgesystemsmight possessuniquecharacteristics.

Fig. 6 plots the time-varyingrail displacements by tracking the wheel-railcontact points. It can be seen fromFig. 7 that the time processesof rail displacements show significantly non-stationaryandasymmetry characteristics; when the wheelsetsarriveatthemid-spanofthebridge,theverticaldynamicdeflections D zrofthetrack-bridgestructures reachthe

maximumresponses; astothe raillateral displacement D yr,though, withoutthat significantly asymmetry feature as D zr,

wecanstillobservethatthe D yrhasobviouslydeviatedfromthenormalcentralline.

Moreover, Fig.7 illustrates the representative responses ofthe displacement andthe accelerationwithspecific to the point atthemid-spanofthebridge girder,fromwhich onecan observethat whenthefirst motorcarapproachestothe specific point,theverticaldeflectionwillbe graduallyincreasedtoreachthefirst peakwithloadingofahalfweightofa car,andthenthedeflectionwillreachthemaximumafterafullweightofthecaractingonthegirderofthebridge,when thetrain gradually passesthrough themid-span, thedeflectionofthebridge structureswill accordinglydecayto original