关于斜拉桥的中英文翻译.docx

关于斜拉桥的中英文翻译.docx

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关于斜拉桥的中英文翻译

Studyonnonlinearanalysisofahighlyredundantcable-stayedbridge

1．Abstract

Acomparisononnonlinearanalysisofahighlyredundantcable-stayedbridgeisperformedinthestudy.Theinitialshapesincludinggeometryandprestressdistributionofthebridgearedeterminedbyusingatwo-loopiterationmethod,i.e.,anequilibriumiterationloopandashapeiterationloop.Fortheinitialshapeanalysisalinearandanonlinearcomputationprocedurearesetup.Intheformerallnonlinearitiesofcable-stayedbridgesaredisregarded,andtheshapeiterationiscarriedoutwithoutconsideringequilibrium.Inthelatterallnonlinearitiesofthebridgesaretakenintoconsiderationandboththeequilibriumandtheshapeiterationarecarriedout.Basedontheconvergentinitialshapesdeterminedbythedifferentprocedures,thenaturalfrequenciesandvibrationmodesarethenexaminedindetails.Numericalresultsshowthataconvergentinitialshapecanbefoundrapidlybythetwo-loopiterationmethod,areasonableinitialshapecanbedeterminedbyusingthelinearcomputationprocedure,andalotofcomputationeffortscanthusbesaved.Thereareonlysmalldifferencesingeometryandprestressdistributionbetweentheresultsdeterminedbylinearandnonlinearcomputationprocedures.However,fortheanalysisofnaturalfrequencyandvibrationmodes,significantdifferencesinthefundamentalfrequenciesandvibrationmodeswilloccur,andthenonlinearitiesofthecable-stayedbridgeresponseappearonlyinthemodesdeterminedonbasisoftheinitialshapefoundbythenonlinearcomputation.

2.Introduction

Rapidprogressintheanalysisandconstructionofcable-stayedbridgeshasbeenmadeoverthelastthreedecades.Theprogressismainlyduetodevelopmentsinthefieldsofcomputertechnology,highstrengthsteelcables,orthotropicsteeldecksandconstructiontechnology.Sincethefirstmoderncable-stayedbridgewasbuiltinSwedenin1955,theirpopularityhasrapidlybeenincreasingallovertheworld.Becauseofitsaestheticappeal,economicgroundsandeaseoferection,thecable-stayedbridgeisconsideredasthemostsuitableconstructiontypeforspansrangingfrom200toabout1000m.Theworld’slongestcable-stayedbridgetodayistheTatarabridgeacrosstheSetoInlandSea,linkingthemainislandsHonshuandShikokuinJapan.TheTataracable-stayedbridgewasopenedin1May,1999andhasacenterspanof890mandatotallengthof1480m.Acable-stayedbridgeconsistsofthreeprincipalcomponents,namelygirders,towersandinclinedcablestays.Thegirderissupportedelasticallyatpointsalongitslengthbyinclinedcablestayssothatthegirdercanspanamuchlongerdistancewithoutintermediatepiers.Thedeadloadandtrafficloadonthegirdersaretransmittedtothetowersbyinclinedcables.Hightensileforcesexistincable-stayswhichinducehighcompressionforcesintowersandpartofgirders.Thesourcesofnonlinearityincable-stayedbridgesmainlyincludethecablesag,beam-columnandlargedeflectioneffects.Sincehighpretensionforceexistsininclinedcablesbeforeliveloadsareapplied,theinitialgeometryandtheprestressofcable-stayedbridgesdependoneachother.Theycannotbespecifiedindependentlyasforconventionalsteelorreinforcedconcretebridges.Thereforetheinitialshapehastobedeterminedcorrectlypriortoanalyzingthebridge.Onlybasedonthecorrectinitialshapeacorrectdeflectionandvibrationanalysiscanbeachieved.Thepurposeofthispaperistopresentacomparisononthenonlinearanalysisofahighlyredundantstiffcable-stayedbridge,inwhichtheinitialshapeofthebridgewillbedeterminediterativelybyusingbothlinearandnonlinearcomputationprocedures.Basedontheinitialshapesevaluated,thevibrationfrequenciesandmodesofthebridgeareexamined.

3.Systemequations

3.1.Generalsystemequation

Whenonlynonlinearitiesinstiffnessaretakenintoaccount,andthesystemmassanddampingmatricesareconsideredasconstant,thegeneralsystemequationofafiniteelementmodelofstructuresinnonlineardynamicscanbederivedfromtheLagrange’svirtualworkprincipleandwrittenasfollows:

Kjbαj-∑Sjajα=Mαβqβ”+Dαβqβ’

3.2.Linearizedsystemequation

Inordertoincrementallysolvethelargedeflectionproblem,thelinearizedsystemequationshastobederived.BytakingthefirstordertermsoftheTaylor’sexpansionofthegeneralsystemequation,thelinearizedequationforasmalltime（orload）intervalisobtainedasfollows:

MαβΔqβ”+ΔDαβqβ’+2KαβΔqβ=Δpα-upα

3.3.Linearizedsystemequationinstatics

Innonlinearstatics,thelinearizedsystemequationbecomes

2KαβΔqβ=Δpα-upα

4.Nonlinearanalysis

4.1.Initialshapeanalysis

Theinitialshapeofacable-stayedbridgeprovidesthegeometricconfigurationaswellastheprestressdistributionofthebridgeunderactionofdeadloadsofgirdersandtowersandunderpretensionforceininclinedcablestays.Therelationsfortheequilibriumconditions,thespecifiedboundaryconditions,andtherequirementsofarchitecturaldesignshouldbesatisfied.Forshapefindingcomputations,onlythedeadloadofgirdersandtowersistakenintoaccount,andthedeadloadofcablesisneglected,butcablesagnonlinearityisincluded.Thecomputationforshapefindingisperformedbyusingthetwo-loopiterationmethod,i.e.,equilibriumiterationandshapeiterationloop.Thiscanstartwithanarbitrarysmalltensionforceininclinedcables.Basedonareferenceconfiguration（thearchitecturaldesignedform）,havingnodeflectionandzeroprestressingirdersandtowers,theequilibriumpositionofthecable-stayedbridgesunderdeadloadisfirstdeterminediteratively（equilibriumiteration）.Althoughthisfirstdeterminedconfigurationsatisfiestheequilibriumconditionsandtheboundaryconditions,therequirementsofarchitecturaldesignare,ingeneral,notfulfilled.Sincethebridgespanislargeandnopretensionforcesexistininclinedcables,quitelargedeflectionsandverylargebendingmomentsmayappearinthegirdersandtowers.Anotheriterationthenhastobecarriedoutinordertoreducethedeflectionandtosmooththebendingmomentsinthegirderandfinallytofindthecorrectinitialshape.Suchaniterationprocedureisnamedherethe‘shapeiteration’.Forshapeiteration,theelementaxialforcesdeterminedinthepreviousstepwillbetakenasinitialelementforcesforthenextiteration,andanewequilibriumconfigurationundertheactionofdeadloadandsuchinitialforceswillbedeterminedagain.Duringshapeiteration,severalcontrolpoints（nodesintersectedbythegirderandthecable）willbechosenforcheckingtheconvergencetolerance.Ineachshapeiterationtheratiooftheverticaldisplacementatcontrolpointstothemainspanlengthwillbechecked,i.e.,

Theshapeiterationwillberepeateduntiltheconvergencetoleranceε,say10-4,isachieved.Whentheconvergencetoleranceisreached,thecomputationwillstopandtheinitialshapeofthecable-stayedbridgesisfound.Numericalexperimentsshowthattheiterationconvergesmonotonouslyandthatallthreenonlinearitieshavelessinfluenceonthefinalgeometryoftheinitialshape.Onlythecablesageffectissignificantforcableforcesdeterminedintheinitialshapeanalysis,andthebeam-columnandlargedeflectioneffectsbecomeinsignificant.

Theinitialanalysiscanbeperformedintwodifferentways:

alinearandanonlinearcomputationprocedure.1.Linearcomputationprocedure:

Tofindtheequilibriumconfigurationofthebridge,allnonlinearitiesofcablestayedbridgesareneglectedandonlythelinearelasticcable,beam-columnelementsandlinearconstantcoordinatetransformationcoefficientsareused.Theshapeiterationiscarriedoutwithoutconsideringtheequilibriumiteration.Areasonableconvergentinitialshapeisfound,andalotofcomputationeffortscanbesaved.

2.Nonlinearcomputationprocedure:

Allnonlinearitiesofcable-stayedbridgesaretakenintoconsiderationduringthewholecomputationprocess.Thenonlinearcableelementwithsageffectandthebeam-columnelementincludingstabilitycoefficientsandnonlinearcoordinatetransformationcoefficientsareused.Boththeshapeiterationandtheequilibriumiterationarecarriedoutinthenonlinearcomputation.Newton–Raphsonmethodisutilizedhereforequilibriumiteration.

4.2.Staticdeflectionanalysis

Basedonthedeterminedinitialshape,thenonlinearstaticdeflectionanalysisofcable-stayedbridgesunderliveloadcanbeperformedincrementwiseoriterationwise.Itiswellknownthattheloadincrementmethodleadstolargenumericalerrors.Theiterationmethodwouldbepreferredforthenonlinearcomputationandadesiredconvergencetolerancecanbeachieved.Newton–Raphsoniterationprocedureisemployed.Fornonlinearanalysisoflargeorcomplexstructuralsystems,a‘full’iterationprocedure（iterationperformedforasinglefullloadstep）willoftenfail.Anincrement–iterationprocedureishighlyrecommended,inwhichtheloadwillbeincremented,andtheiterationwillbecarriedoutineachloadstep.Thestaticdeflectionanalysisofthecablestayedbridgewillstartfromtheinitialshapedeterminedbytheshapefindingprocedureusingalinearornonlinearcomputation.Thealgorithmofthestaticdeflectionanalysisofcable-stayedbridgesissummarizedinSection4.4.2.

4.3.Linearizedvibrationanalysis

Whenastructuralsystemisstiffenoughandtheexternalexcitationisnottoointensive,thesystemmayvibratewithsmallamplitudearoundacertainnonlinearstaticstate,wherethechangeofthenonlinearstaticstateinducedbythevibrationisverysmallandne