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(asce)be.1943-5592.0000313

NonlinearAnalysisfortheLateralVibrationof

FootbridgesInducedbyPedestrians

ZhenBin1;XieWeiping2;andXuJian3

Abstract:Inthispaper,thefactorsandvariablesthataffecttheamplitudeofthelateralvibrationofthefootbridgeinducedbypedestriansinNakamura’smodelareinvestigatedintheaspectofnonlineardynamics.Nakamura’smodelisbasedonthemotionofequationsincludingcoef cientsoftherateofapedestrian’slateralforce,thepedestriandensity,therateofsynchronizedpedestrians,thefunctionGðfBÞdescribinghowpedestrianssynchronizewiththebridgenaturalfrequency,andthepedestrians’attitudetolargevibrationamplitude.Theimportantcoef cientGðfBÞcannotbedecidedinthismodelbecauseofthelackofdataavailableonthelateralloadbehaviorsofwalkingpeople.NakamuraassumedthatGðfBÞ51:0inhismodelbyconsideringthatpedestriansaremostlikelytosynchronizeatthebridgefrequencyaround1.0Hz.TheanalysisresultspresentedinthispaperdemonstratethattheamplitudeofthelateralvibrationincreaseswhenthefunctionGðfBÞincreases.ThissuggestsGðfBÞ51:0maynotbetheworstcasescenarioforthelateralvibrationoffootbridges.AssumingGðfBÞ51:0maybeonereasonwhymostofthepredictedresultsofNakamuraareonthelargeside.ThroughmodifyingNakamura’smodel,anequationofmotionincludingthetimedelayoftheinteractionbetweenthepedestriansandthefootbridgeisproposedtopreferablyagreewiththemeasuredata.Theanalysisshowsthattheamplitudeofthelateralvibrationofthefootbridgedecreasesasthetimedelayincreases.TheignoredtimedelayoftheinteractionbetweenpedestriansandthefootbridgeinNakamura’smodelisanotherreasonwhythenumericalresultsofNakamuraarelargerthanthemeasureddata.Fromtheanalysis,appropriatelyincreasingthetimedelayoftheinteractionbetweenpedestriansandthefootbridgeisapossiblewaytoreducethelateralvibrationinducedbypedestrians.DOI:10.1061/(ASCE)BE.1943-5592.0000313.©2013AmericanSocietyofCivilEngineers.CEDatabasesubjectheadings:Vibration;Footbridges;Pedestrians;Delaytime;Nonlinearanalysis.

Authorkeywords:Lateralvibration;Footbridge;Pedestrians;Nakamura’smodel;Timedelay;Nonlineardynamics.

Downloaded from http://www.wenkuxiazai.com by CHANG'AN UNIVERSITY on 03/14/14. Copyright ASCE. For personal use only; all rights reserved.

Introduction

Overthelastfewdecades,slenderfootbridgeswithlowstiffness,lowmass,lowdamping,andlownaturalfrequencieshavebecomeincreasinglypopularbecauseoftheemergenceofnewmaterialandadvancedengineeringtechnology.Asaconsequence,moreofthesefootbridgesshowgreatsensitivitytohuman-inducedvibrations,particularlyinthelateraldirection(Fujinoetal.1993;Dallardetal.2001a).Thusfar,thesekindsofdynamiclateralloadhaveneverinvolvedstructuralfailures,buttheyhaveoftencauseddiscomfortforthepedestriansandthetemporaryclosureofthefootbridges.TheexcessivelateralvibrationcausedbysynchronouslateralexcitationexertedbyacrowdwalkingacrosstheinfamousMillenniumBridgeinLondonistheprimeexampleofthevibrationserviceabilityproblemoffootbridges.Thebridgewasclosedfor20monthsuntilthevibrationwassuppressedbyTaylordampersandtunedmassdampers(Dallardetal.2001a).However,presentbridgedesigncodesworldwidedonotprovidesuf cientguidelinesandinformationtoaddresssuchvibrationsproblemsandtoensuresafetyandserviceabilityasaresult

Doctor,SchoolofCivilandArchitectureEngineering,WuhanUniv.ofTechnology,Wuhan430070,P.R.China(correspondingauthor).E-mail:zhenbin80@http://www.wenkuxiazai.com2

Professor,SchoolofCivilandArchitectureEngineering,WuhanUniv.ofTechnology,Wuhan430070,P.R.China.3

Professor,SchoolofAerospaceandMechanicsEngineering,TongjiUniv.,Shanghai200092,P.R.China.

Note.ThismanuscriptwassubmittedonFebruary15,2011;approvedonSeptember22,2011;publishedonlineonSeptember26,2011.DiscussionperiodopenuntilJuly1,2013;separatediscussionsmustbesubmittedforindividualpapers.ThispaperispartoftheJournalofBridgeEngineering,Vol.18,No.2,February1,2013.©ASCE,ISSN1084-0702/2013/2-122e130/$25.00.

122/JOURNALOFBRIDGEENGINEERING©ASCE/FEBRUARY2013

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ofthelackofknowledgeonthedynamicperformanceofsuchslendervibration-sensitivebridgestructures.

Duringwalkingonafootbridge,pedestriansinducehorizontaldynamictime-varyingforcesonthesurfaceofthestructure.Theseforcesareaconsequenceofalateraloscillationofthegravitycenterofthebody,andthelateraloscillationsareaconsequenceofbodymovementswhenpersonsstepwiththeirrightandleftfootinturn.Thefrequencyofthehorizontalforceforasinglepersonishalfthepacingfrequencyandthereforeisbetween0.7and1.2Hzforapacingfrequencyof1.4e2.4Hz(Bachmann2002).Onastationarysurface,thisforcehasbeenfoundtobeabout10%oftheverticalloading,whichisabout4%ofthepedestrian’sweight(Nakamura2004).Thesituationisverydifferentandcomplexforalargegroupofpedestrianswalkingacrossafootbridge.PiccardoandTubino(2008)classi edthemechanismsoflateralvibrationsofafootbridgeintroducedbyalargegroupofpedestriansintothreeclasses:directresonance,activatedifthepedestrianexcitationisinresonancewithamodeofvibrationofthebridge(e.g.,theT-bridgeinJapan)(Fujinoetal.1993);dynamicinteraction,basedonsuitablemodelsoftheinteractionbetweenthemotionofthebridgeandthemotionofpedestrianswalkingonit(e.g.,thewell-knownMillenniumFoot-bridgeinLondon)(Dallardetal.2001b);andinternalresonance,causedbystructuralnonlinearitiesgivingrisetointernalresonanceconditionsamongdifferentnaturalmodesofvibrationofthebridgeitself(e.g.,theSolferinoFootbridgeinParis)(DanbonandGrillaud2005).

Dynamicinteractionmechanismshavereceivedgreatattention,becausetheyallowtheintroductionoftheconceptofthecriticalnumberofpedestrians.TheresearchonthelateralvibrationthatoccurredontheMillenniumBridgeshowsthatthelateraldynamicforceisproducedbythezigzagmovementofpedestriansandisresonantwiththebridgenaturalfrequency.Ifthesmallmotionof

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