Influence of soil non-linearity on the dynamic response of high-speed railway tracks

2.5D维有限元经典文献,把等效线性化模型应用到其中。

SoilDynamicsandEarthquakeEngineering30(2010)221–235

ContentslistsavailableatScienceDirect

SoilDynamicsandEarthquakeEngineering

journalhomepage:http://wendang.chazidian.com/locate/soildyn

In?uenceofsoilnon-linearityonthedynamicresponseofhigh-speedrailwaytracks

´nioSilvaCardosoa,AndersBodarebPedroAlvesCostaa,RuiCalc-adaa,n,Anto

ab

FacultyofEngineeringoftheUniversityofPorto,RuaDr.RobertoFrias,4200-465Porto,Portugal

GeoRisk&VibrationScandinaviaAB,SolnaTorg13,3tr-17145Solna,Sweden

articleinfo

Articlehistory:

Received14August2009Receivedinrevisedform19November2009

Accepted20November2009Keywords:

High-speedrailwaytrackSoftsoil

Equivalentlinearanalysis2.5D?niteelementmethodFrequency-domainanalysis

abstract

Themainobjectivesofthispaperaretheevaluationoftherelevanceofthenon-linearbehaviourofthesoilonthetrackresponseandthevalidationofamethodology,whichincludestheseeffectsthroughanequivalentlinearanalysis.Theproposednumericalmodelisbasedon2.5D?nite/in?niteelementsmethod,coupledwithaniterativeprocedureinordertoobtainanagreementbetweenthestrainlevelsandthedynamicpropertiesofthematerials.Inordertovalidatethemodel,thecasestudyofLedsgardwassimulated,andtheexperimentalandnumericalresultsofdisplacementsofthetrackwerecompared,consideringseveralcirculationspeedsfortheX2000train.Fromtheresults,itispossibletorecognizethatthestiffnessdegradation,functionofthestrainlevel,playsarelevantroleforthecaseofhigh-speedrailwaylinesonsoftground.Moreover,thesimulationsdevelopedwiththeproposedmethodologyprovidedsimilarresultstothoseobserved,independentlyofthetrainspeed,contrarytowhatwasobtainedwhentheelasticlinearmodelwasused.

&2009ElsevierLtd.Allrightsreserved.

1.Introduction

Theassessmentofvibrationsatandalongsiderailwaytracksisbecomingasubjectofprimeimportance,mainlyduetotheincreasingnumberofhigh-speedrailwaylinesbeingbuiltthroughoutEuropeandEastAsia.Whenthetrackcrossesregionsofsoftsoils,theassessmentofthevibrationsinducedbytraf?cisparticularlyimportant.Inthesecases,thisissuehasnotonlyenvironmentalimplications,relatedwithnuisancetoinhabitantsofbuildingsinthevicinityofhigh-speedlines,butalsostructuralconsequences,sincetheampli?cationofthedynamicresponseofthetrackmaycompromiseitsstabilityandsafety.

http://wendang.chazidian.comuallyadistinctionismadebetweenquasi-staticanddynamicexcitationmechanisms[1,2].Thedynamicexcitation,ofgreatimportanceinenvironmentalvibrationassessment,iscausedbythetrain–trackinteractionduetoseveralmechanismsthatinduceverticalaccelerationontherolling-stock[3].Ontheotherhand,thequasi-staticexcitationisrelatedwiththemagnitudeofthemovingloadscorrespondingtothedistributionofthetrainweightbyitsaxles.Whenthetrainspeedislowincomparisonwiththecriticalphasevelocityofthetrack-groundsystem(minimumphasevelocityofthe1stRayleighmodeofthesoilandembankmentpro?leatthesite[4])the

Correspondingauthor.

E-mailaddresses:pacosta@fe.up.pt(P.AlvesCosta),ruiabc@fe.up.pt(R.Calc-ada),scardoso@fe.up.pt(A.SilvaCardoso),bodare@georisk.se(A.Bodare).0267-7261/$-seefrontmatter&2009ElsevierLtd.Allrightsreserved.doi:10.1016/j.soildyn.2009.11.002

n

quasi-staticexcitationpresentsareducedcontributiontotheresponseatthefree-?eld,althoughthetrackresponseisconditionedbythismechanism.However,theimportanceofthequasi-staticexcitationincreaseswhenthetrackcrossesregionsofsoftsoilsandthetrainspeedreachesthephasevelocityofthesystem.Inthesecases,highampli?cationsareidenti?ed[3,5],whicharenotcompatiblewiththeassumptionofsmall-strainsintheground,asusuallyconsideredintheanalysisofvibrationsinducedbytraf?c.

Duringthecurrentdecade,fewpredictionmodelsweredevelopedwithdifferentdegreesofcomplexityandaccuracy.However,inauthors’knowledge,almostalltheanalysesdevel-opeduntilnowwerebasedonelasto-dynamictheory,undertheassumptionthatstrainsinducedbythetraf?careboundedintherangeofverysmallstrains,thusnotinvolvingsoilstiffnessdegradation.

Thispaperisfocusedonthepredictionofthedynamicresponseofthetrack,includingtheeffectsofsoilnon-linearity.Intheauthors’opinion,theeffectofstiffnessdegradationwiththestrainlevelhasnotbeenappropriatelytakenintoaccountyet.Inthiscontext,thefewexperimentalcasestudiesperformeduntilnow,whereseveralmeasurementshavebeenobtained,areveryusefultoidentifythemaintrendsoftheproblemandtocalibrateandvalidatethenumericalmodels.Oneofthosecasesisthewell-knowncaseofLedsgard,atSweden,whereseveraltestswereperformedfordistincttrainspeeds,someofthemclosetothecriticalspeedofthesystem[6,7].Thiscasestudyhasbeenusedasbenchmarkbyseveralresearchers[4,7–11].Inthestudycon-ductedbyMadshusandKaynia[4]andalsobyHall[7],the

2.5D维有限元经典文献,把等效线性化模型应用到其中。

222

P.AlvesCostaetal./SoilDynamicsandEarthquakeEngineering30(2010)221–235

authorsrecognizedtherelevanceoftheconsiderationofthenon-linearbehaviourofthesoilandtriedtocontemplatethiseffectusingasimpli?edapproach.

Thestudiespresentedinthispaperrepresentanadvanceinknowledgeinthis?eldofresearch,namelywhencomparedwiththestudiespresentedbyMadshusandKaynia[http://wendang.chazidian.comparingbothstudies,severalimprovementscanbepointedout:(i)thegroundismodelledbya2.5D?niteelementapproach;ina2.5D?niteelementapproach,the3Dresponseofthegroundiscomputedassumingthatthestructureisinvariantalongthedevelopmentdirectionofthetrack;(ii)intheproposedapproach,thetrackandtheembankmentaresimulatedattendingtotheirspeci?ccharacteristics;theembankment,aswellastheground,issimulatedbymeansof2.5D?niteelements,whichallowfortheinclusionoftherealgeometryoftheembankmentanditsmechanicalbehaviour;(iii)thenon-lineareffectsarehandledintheproposedmodelthrougha‘‘true’’equivalentlinearanalysis,i.e.,theconvergenceisobtainedelement-by-element,whichinducesatransverseinhomogeneityinthegroundduetothefactthatthenon-lineareffectsaremorepronouncedalongtherailwaytrackthanatthefar-?eld.Inthefollowingsections,themaincharacteristicsoftheproposedmodelareoutlined.

Followingacommonstrategyforresolutionofsoildynamicsproblems,themodelpresentedinthispaperconsidersthenon-linearsoilbehaviourthroughtheequivalentlinearapproach,whichconstitutesacompromiseoptionbetweenaccuracyandcomplexity.Afteridentifyingthemechanismsofvibrationgen-erationandtheirimportancefortheresponseofrailwaytrackanditssurroundingsoilmass,itisassumed,forthepresentcasestudy,thatthemainmechanismofvibrationisquasi-static,forthereasonspointedoutabove.Theextensionofthemodeltowardsincludingthedynamicexcitationisasimplestepfromthepointofviewoftheformulation;however,thecomputationaleffortisdrasticallyincreased.

Asinothernumericalmodels,thepresentmodelisbasedonthe‘‘two-and-a-half’’dimensionconcept.Thismethodologyleadstoanef?cientcomputationscheme,usingFouriertransformswithrespecttothespatialcoordinatealongthetrack.Thisisthemainadvantageofthiskindofmodelsbutitisalsoitsmaindrawback,sinceitimpliestheassumptionofinvariabilityofgeometryandofmechanicalpropertiesalongthatdirection.Concerningthenumericalmethoditself,the2.5D?niteelement(2.5DFEM)formulationadoptedissimilartotheoneusedbyMuller[12],YangandHung[13]andShengetal.[14].Toavoidspuriousre?ectionsonarti?cialboundaries,2.5Din?niteelements(2.5DIEM)areused.ThismodelismoreversatilethanthemodelsbasedonGreen’sfunctionsconcept,sinceitallowsfortheconsiderationofanarbitrarycross-sectionforthetrack/groundsystem.

The2.5DFEM–IEMiscoupledwithaniterative,element-by-element,methodology;asaresult,successivelinearelasticequivalentanalysesareperformed,wherethestiffnesspropertiesareadjustedtomatchthemechanicalpropertiesandthestrainlevel,untiltheconvergencecriterionisreached.

Inthepresentstudy,theresultsofseveralsimulationsarecomparedwith?eldmeasurementsandthein?uenceofthenon-linearityofthesoilontheresponseofthetrackisanalyzedanddiscussed.

2.Numericalmodel

2.1.2.5D?niteelementmethodformodellingboundedregionsTheapplicationofthe2.5D?niteelementsiscon?nedto

structureswhichcanbeassumedtohavein?nitedevelopmentandinvariantpropertiesinonedirection,asillustratedinFig.1.In

thesecasesthestructureis2D,sincethecross-sectionremainsinvariableinthelongitudinaldirection,buttheloadingis3D.Themainconceptbehindtheproposedsolutiontotheproblemistheuseofamethodwhichisbetweenthetwo-dimensionalandthethree-dimensionaldomain.Thismethodwas?rstlyproposedbyHwangandLysmer[15]forthestudyofundergroundstructuresundertravellingseismicwaves.Subsequently,themethodhasbeenappliedbyafewresearcherstothestudyofvibrationsinducedbytraf?c.Inthis?eld,specialattentionshouldbededicatedtotheworksofMuller[12],YangandHung[13],Shengetal.[14]andAlvesCosta[16].

Assumingthattheresponseofthestructureislinear,theanalysiscanbecarriedoutonthewavenumber/frequencydomain.Allthevariables,i.e.,loads(action)anddisplacements(response),mustbetransformedtothewavenumber/frequencydomainbymeansofadoubleFouriertransform,relatedwiththedirectionalongthetrack(xdirection)andwithtime.TransformedquantitieswillbedenotedasfunctionsoftheFourierimagesofxandt,de?nedaswavenumberandfrequency,arerepresentedbyk1ando,respectively.

Followingtheusualstepsofthe?niteelementprocedure,namelythestrongandweakformulations,thefollowingequili-briumequationcanbederivedforanypointofathree-dimensionaldomain:Z

desdVþZ

dur@2uiðx;tÞ

dVZ

dupdSð1Þ

VV

@t2¼

S

wheredeisthevirtualstrain?eld,srepresentsthestress?eld,du

isthevirtualdisplacement?eld,uisthedisplacement?eld,risthemassdensityandprepresentstheappliedloads.

Afterthetransformation,thecross-sectionofthedomainremainsontheuntransformeddomainandisdiscretizedinto?niteelements.ThisapproachenablestorewriteEq.(1)intermsofnodalvariables.

Inordertoapplytheconceptofvirtualworkonthetransformeddomain,someconsiderationsmustbeattended,namelytheParserval’stheorem[12,17,18]:Z

dfðxÞpðxÞdx¼

Z

dfðÀk1Þpðk1Þdk1

ð2Þ

Eq.(2)providestheformulationoftheprincipleofvirtualworksinthetransformeddomain.ConsideringEq.(1),thevirtualworkoftheinternalstressesandinertialforcesinthetransformeddomainisgivenby,respectively:Z

desdV¼

Z

duTÞ

ZZ

BTðÀk1ÞDBðk1Þdydzunðk1;oÞdk1

V

knðÀk1;

o1

z

y

ð3

Þ

Fig.1.In?niteandinvariantstructureinonedirection(after[13]).

2.5D维有限元经典文献,把等效线性化模型应用到其中。

P.AlvesCostaetal./SoilDynamicsandEarthquakeEngineering30(2010)221–235

223

Z

dur@2uðx;tÞZ

@tdV¼Ào2

duTZ

1;

oÞ

ZNTrNdydzunðk1;oÞdk1

V

knðÀk1

z

y

ð4Þ

whereBisthematrixwiththederivativesoftheshapefunctions;Nistheshapefunctionmatrix;Disthestrain–stressmatrix;unisthevectorofnodaldisplacements(inthetransformeddomain).

ThevirtualworkdonebytheexternalloadsiscomputedtakingadvantageofthefactthatthegeometryisonlydiscretizedontheZYplane.So,consideringacoordinates,paralleltotheedgeoftheelementwheretractionisapplied,thevirtualworkdevelopedbytheloadsystemisgivenby

Z

dupdS¼

Z

duTZ

T

Z

1;oÞdsdk1¼

duTÞpnðk1;oÞdk1

S

knðÀk1;oÞNpðk1

s

knðÀk11

ð5Þ

ReplacingandrearrangingEqs.(3)–(5)onEq.(1)yields

??ZZBTðÀk1ÞDBðk1ÞdydzÀo2ZZ

NTrNdydz unðk1;oÞ¼pnðk1;oÞ

z

y

z

y

ð6Þ

Adoptingtheclassic?niteelementnotation,resultson½K ¼ZZBTðÀk1ÞDBðk1Þdydz

ð7Þ

z

y

and½M ¼

ZZ

NTrNdydz

ð8Þ

z

y

where[K]and[M]arethestiffnessandmassmatrices,respectively.

Asusual,thematrix[B]isderivedfromtheproductofthedifferentialoperatormatrix[L](onthetransformeddomain)withthematrix[N].Sincethedirectionxistransformedtothewavenumberdomain,thederivativesinordertok1areanalyti-callycomputed,aspresentedinthefollowingexpression:

2@3T6ik@1

00607½L ¼666@@760

6@y0ik1@z0777ð9Þ

40

@@7@z

ik7@y

1

5Dampingisintroducedbyahystereticdampingmodel,i.e.,consideringcomplexstiffnessparameters.

Thecomputationalef?ciencycanbeimprovedbydividingmatrix[K]intosub-matrices,independentofthewavenumberandfrequency.Thisstepisdevelopedconsideringthematrix[B]astheresultoftheadditionoftwomatrices,wherethenumericalandanalyticalderivativesareseparated.Inthatcase,Eq.(6)canbereplacedby

ð½K 1þik1½K 2þk22

1½K 3Ào½M Þunðk1Þ¼pnðk1Þ

ð10Þ

Theglobalsystemofequationsiscompletelyde?nedaftertheassemblyoftheindividualmatricesofeachelementandthede?nitionoftheNewmanandDiricheletboundaryconditions.Theresultsobtainedaftersolvingthesystemofequationsareinthetransformeddomain,requiringadoubleinverseFouriertransform,inordertode?neasolutioninthespace/timedomain.

Theadvantageofthemethodinrelationtothefullythree-dimensional?niteelementmethodisevident:insteadofsolvingasystemofequationswithahighnumberofdegreesoffreedom,asmallersystemofequationsissolvedmanytimes,correspondingtoarangeofwavenumbers.Thisprocedurerepresentsagreatreductionofcomputationaltime.

2.2.Modellingunboundedregionsthroughin?niteelementsManycivilengineeringstructurescanbeidealizedasresting

ontheground,which,forpracticalpurposes,canbeconsideredunbounded.Forstaticproblems,thecontributionofthegroundisre?ectedintermsofstiffness,soitispossibletotruncatethedomainatasuf?cientlyfardistancewherethegrounddeforma-tionissosmallthatitcanbeneglected.However,indynamicanalyses,thegroundmodelshouldful?ltherequirementsofrepresentingthedynamicgroundstiffnessbutalsotheradiationconditions.Thelatterrequirementdemandsaspecialtreatmentoftheboundaryconditions,sincespuriousre?ectionofthewavesattheboundariesshouldnotoccur.Arigorousapproachcanbereachedusing?niteelementstorepresentthenear-?elddomainandboundaryelementstosimulatethefar-?elddomain.Thisapproachhasbeenusedinthecontextof2.5Dmodellingbyseveralresearcherswithsatisfactoryresults[12–14,16].Anotherapproachconsistsontheuseofabsorbingboundaryconditionsorin?niteelements.

Inthispaper,theauthorsoptedforusingin?niteelementsduetothesimplicityofitsnumericalimplementation.Inthein?niteelementsformulation,similarlytothe?niteelementsformula-tion,the?eldvariableisapproximatedbyshapefunctions.However,theshapefunctionsforthein?niteelementsmustbemoreelaborated,sincethesehavetorepresenta‘‘reasonable’’behaviourofthe?eldvariabletowardsin?nite[19].Inelasto-dynamicharmonicconditions,thispurposecanbereachedbythecombinationofthreefunctions:(i)astandardshapefunction,(ii)adecayfunction,and(iii)anoscillatoryfunction.Theissueiscomplexinelasto-dynamicproblemssincethereisnolongerauniquewavespeed,evenforahalf-spaceproblem.Thisproblemcanbeovercomebymeansofspecialin?niteelements,whichrepresentthecharacteristicsofmultiplewavespropagatingintotheunboundouterdomainofthemedia[20];however,thisprocedureincreasesitscomplexity.Alternatively,asdemon-stratedbyYangandHung[13],theuseofconventionalin?niteelementscombinedwithcriteriaforthechoiceofthedecayandoscillatoryfactorscanleadtoaccurateresultsevenformovingloadproblems.So,inthepresentwork,theauthorsdecidedontheuseofthein?niteelementsproposedbytheaforementionedauthors.

Aschematicrepresentationoftheadoptedin?niteelementsispresentedinFig.2.Thedisplacementshapefunctionsoftheelementarede?nedbyN1¼

12

ZðZÀ1ÞeÀaxeik0

xð11Þ

NÞðZþ1ÞeÀaxeik0

2¼ÀðZÀ1x

ð12

Þ

Fig.2.In?niteelements:(a)globalcoordinatesand(b)localcoordinates(after[13]).

2.5D维有限元经典文献,把等效线性化模型应用到其中。

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P.AlvesCostaetal./SoilDynamicsandEarthquakeEngineering30(2010)221–235

N3¼

1ZðZþ1ÞeÀaxeik0

xð13Þ

whereaandk0arethedecayandoscillatoryfactors,respectively.Fordetailsregardingtheselectionprocedureofvaluesforthesefactors,readersshouldrefertotheworksofYangetal.[13,21].

Havingde?nedtheshapefunctionsofthein?niteelements,theusual?niteelementprocedureisapplied,i.e.,thestiffnessandmassmatricesofeachin?niteelementarecomputedandaddedtotheglobalmatrices,formingtheglobalsystemofequations.2.3.Modellingparticularelements:sleepersandrails

Thesleepersarediscontinuouselementsandtheconsiderationofthisdiscontinuityincreasesthedif?cultyoftheirimplementa-tionona2.5Dmodel.However,someauthorshavemodelledtheinherentinhomogeneityofthesleepers,havingconcludedthatthecomputationaleffortwasdisproportionatewhencomparedwiththeobtainedincreaseofaccuracy[11,22,23].

Inthiswork,thesleepersaremodelledconsideringanequivalentcontinuousformulation,whichrepresentsacompro-misesolutionbetweencomplexity/computationalcostandaccu-racy.Theadoptedformulationisthemostusualforthistypeofproblems;nevertheless,thecomparisonbetweencomputedandexperimentalresultsprovedtobesatisfactory,mainlywhentherelevantfrequencieswerebelow500Hz,asshowedinthereceptancetestsperformedbyKnotheandGrassie[24]andKnotheandWu[25].Sheng[23]alsodedicatedattentiontothisissueand,aftercomparingtheoreticalresultsoftheverticaldynamicresponseinducedonaperiodicmodelandonanequivalentcontinuousmodel,concludedthatforfrequenciesupto250Hztheresultsobtainedbythetwodistinctformulationsareverysimilar.TakemiyaandBian[11]developedaninhomo-geneoustrackmodelwherethesleepersweresimulatedusingadiscreteformulationandalsoconcludedthat,forlow-frequencyanalyses,theuseofadiscretemodelforthesleeperswouldnotbeadvantageous,evenwhenthesoilbeneaththetrackpresentspoormechanicalproperties.Therefore,equivalentcontinuoustrackmodelsprovideareliablepredictionofthetrackreceptance,butdonotaccountforparametricexcitationasthesmallspatialvariationofthesupportstiffnessofthediscretelysupportedtrackisdisregarded[5,26].

Thedevelopedmethodpresentssomeimprovementsinrelationtootherequivalentcontinuousmethodspreviouslyproposedbyotherauthors,sincealmostallotherstudiesconsidertheinertialeffectofthesleepersbutdisregarditsstiffnessinthecrossdirection.Anexceptionmustbementioned,relativelytotheworkdevelopedbyKarlstromandBostrom[10]whereanisotropicKirshoffplateswereusedtosimulatethesleepers.

Inthispaper,themethodproposedbyKarlstromandBostrom[10]isusedasastartingpointforthedevelopmentofthesleepersmodel.InsteadofKirshoffplates,assuggestedbytheabovereferredauthors,theimplementedformulationmakesuseofvolumetricelementsinthe2.5Dsense.Theconstitutivemodelisbasedonananisotropicformulation,wherethephysicalproper-tiesofthesleepersareappliedtode?nethecontinuousequivalentstiffnesspropertiesofthemodelontheplaneperpendiculartothetrackdirection(xdirection).Inturn,regardingthepropertiesonthexdirection,theirstiffnesscanbeassumedeitherclosetozeroorclosetothestiffnessoftheballast,sincethesleepersareusuallyembeddedintheballast.Inwhatconcernstherails,Euler–Bernoullibeamsconnectedtotherestofthe?eldbyrailpadsareused,asillustratedinFig.3.TheselectedoptionofEuler–BernoullibeamsinsteadofTimoshenkobeamsresidesonthefactthat,forfrequenciesupto500Hz,theresultsprovidedbybothformulationsareidentical[27],buttheEuler–

Fig.3.Rail–sleeperconnection.

Bernoullibeammodeliseasiertoimplementinanumericalcode.Somesimpli?cationsareassumed,suchas,forexample,theconnectionbetweentherailandthesleepersisassumedtobecontinuousandonlyonedegreeoffreedomisconsidered,correspondingtotheverticaldisplacement.Thesesimpli?cationsareperfectlyadmissibleforrailwaytrackmodels,asexplainedin[5,24,25].

Themotionoftherailinthetransformeddomainisexpressedbythefollowingsystemofequations:

B"#BEIk4þkÃpÀkÃ

1p??mr??1@20CÀkÃ|???????????????pkÃÀop0CA(?{z????????????????}½K rail

|??????{z??????}0

½M railurail

)

u¼fPgð14Þ

|???????sleeper?{z????????}|{z}fuwhereEIisthebendingstiffnessoftherail;kpisthecomplexstiffnessoftherailpad,mristhemassperunitlengthoftherail;andthevectorsuandfarethevectorsofdisplacementsandloads,respectively.Thesuperscriptsymbolninkprepresentscomplexstiffnessinordertohaverailpad’sdampingintoaccount.Inthiscase,kÃp¼kpþiocp,wherecpistheviscousdampingfactorandkpisthestiffnessoftherailpad.

Thestiffnessandmassmatricesoftherailarethenassembledtotheglobaldynamicsystemofequations.Animprovednumericalperformanceisobtainedbythesubdivisionofmatrix[K]railintosub-matrices,totallyindependentofthewavenumber;thiscanbeachievedbyfollowingtheprocedureexplainedabove.2.4.Equivalentlinearanalysis

Whenlargestrains,i.e.between10À4and10À2,canbeexpected,theuseofanequivalentlinearmodelmaybenecessary.Withtheincreaseofthestrainlevel,thestiffnesstendstodecreaseandthedissipationofenergytendstoincrease,asillustratedinFig.4forasymmetriccyclicloadingcondition[28,29].

ThehystereticloopsrepresentedinFig.4canbedescribedby:(i)thepathofloopitselfand(ii)theparametersdescribingthegeneralshapeoftheloopanditsevolutionwiththestrainlevel.Todescribethepathofloopsanditsevolutionwithstrainlevel,acyclicnon-linearmodelmaybeused[30].However,anaveragebehaviourcanbedescribedbytwoimportantparametersofeachloop:itsinclination(stiffness)anditsbreadth(damping).Thelatterapproachisusedinthispaper.Forthedevelopmentofanequivalentlinearanalysis,twomainparametersareneeded:(i)theelasticpropertiesforverysmallstrains(usuallyobtainedfromthepropagationvelocityofSandPwaves)and(ii)thelawsdescribingstiffnessdegradationanddampingincreasewith

the

2.5D维有限元经典文献,把等效线性化模型应用到其中。

P.AlvesCostaetal./SoilDynamicsandEarthquakeEngineering30(2010)221–235

225

increaseofthestrainlevels.Theselawsshouldbeobtainedfromlaboratorytestsor,initsabsence,fromcorrelationswithsomephysicalparametersofthesoil[31,32].Fig.5showstheglobalaspectofthetypicaldegradationlawsandtheirdependenceontheplasticityindex(PI)andmeancon?ningstress(s0m)[32].

Inordertoaddressthisproblem,anobjectivede?nitionofstrainlevelateach?niteelementisrequired.Inthree-dimen-sionalproblems,thestrainlevelisusuallyde?nedbytheeffectiveoctahedralshearstrainasproposedbyHalabianandNaggar[33]andLysmeretal.[34].Theeffectiveoctahedralshearstrainiscalculatedby

q????????????????????????????????????????????????????????????????????????????????????????????????????122geff¼að15ÞðexÀeyÞ2þðexÀezÞ2þðeyÀezÞ2þ6ðg2xyþgxzþgyzÞ3whereaisaparameterinthe0.5–0.7interval.Inthiswork,ais

assumedequalto0.65asitisusuallyconsideredinseismicanalysis.Thevariableseiandgicorrespondtothestrainsofthestraintensor.

Regardingthenumericalprocedure,itispossible,foradiscretizedmediumthroughanelement-by-elementprocedure,tocomputetheinducedstrainateachelementandtomakecorrectionstothepropertiesuntilanagreementisobtainedwith

theinvolvedstrains.Intheimplementedmodel,itisassumedthatthepropertiesareconstantinsideeach?niteelement,sothestrainsinthecentrepointofeachelementareconsideredrepresentativeofthestrainsinwholeelementdomain.

Fortheupdateofthepropertiesoftheelements,anumberoflinearproblemsaresuccessivelysolveduntilamatchbetweenthestrainlevelandthedynamicpropertiesofthesoilisobtained.Thecomputationalprocedurecanbesummarizedasfollows:1.Assumelow-strainpropertiesforallelements.

http://wendang.chazidian.comingthevalueofgieff,choosenewequivalentlinearvalues,

i+1i+1

Gsecandx,forthenextiteration.

4.Repeatsteps2and3untilthedifferencesbetweencomputedshearmodulusanddampingintwosuccessiveiterationsfallbelowapreviouslyestablishedtoleranceforall?niteelements.Concerningtheconvergencetolerance,avalueof3%is,intheauthors’opinion,consideredacceptable,sincethelinearequiva-lentmodelisanapproximationtotherealproblem.

Itisimportanttobearinmindthateventhoughthelinearequivalentanalysisallowsforanapproximationtothenon-linearsoilbehaviour,theresponseisperformedbyalinearelasticmethod.Thismeansthatthemethodisincapableofrepresentingthechangeofthesoilpropertiesthatactuallyoccurduringthepassageofthetrain[30].Moreover,theapplicationofthismodelinthecontextof2.5D?niteelementsdealswithanapproxima-tioninherenttothemodel:thepropertiesremainconstantinthedirectionalongthetrack,meaningthattheconsidereddegrada-tionprocessisindependentofthetrainposition.

3.Theoreticalvalidation

ThenumericalmodelpresentedintheprevioussectionwasimplementedbytheauthorsinthenumericalplatformMatlab.Sincethemodelisdevelopedinthefrequencydomain,itispossibletotakeadvantageofthenumericalcomputationaltoolsavailableinMatlab2009forparallelprocessing,whichallowforaconsiderablereductionofthecomputationaltime.

Inordertocheckthereliabilityandaccuracyoftheproposedapproachforthedynamicanalysisofthegroundresponseundermovingloadactions,asmallexample,whichwaspreviously

Fig.4.Stress–strainpathduringcyclic

loading.

10.90.80.7Gsec/Gmax0.60.50.40.30.20.1010-6

10-5

10-4

10-3

10-2

10-1

Cyclic shear strain amplitude

1

0.90.80.7Gsec/Gmax0.60.50.40.30.20.1010-6

10-510-410-310-210-1Cyclic shear strain amplitude

Fig.5.Modulusreductioncurvesfor:(a)non-plasticsoilsand(b)plasticsoils(after[32],adapted).