關(guān)于斜拉橋的中英文翻譯

Studyonnonlinearanalysisofahighlyredundantcable-stayedbridge1．AbstractAcomparisononnonlinearanalysisofahighlyredundantcable-stayedbridgeisperformedinthestudy.Theinitialshapesincludinggeometryandprestressdistributionofthebridgearedeterminedbyusingatwo-loopiterationmethod,i.e.,anequilibriumiterationloopandashapeiterationloop.Fortheinitialshapeanalysisalinearandanonlinearcomputationprocedurearesetup.Intheformerallnonlinearitiesofcable-stayedbridgesaredisregarded,andtheshapeiterationiscarriedoutwithoutconsideringequilibrium.Inthelatterallnonlinearitiesofthebridgesaretakenintoconsiderationandboththeequilibriumandtheshapeiterationarecarriedout.Basedontheconvergentinitialshapesdeterminedbythedifferentprocedures,thenaturalfrequenciesandvibrationmodesarethenexaminedindetails.Numericalresultsshowthataconvergentinitialshapecanbefoundrapidlybythetwo-loopiterationmethod,areasonableinitialshapecanbedeterminedbyusingthelinearcomputationprocedure,andalotofcomputationeffortscanthusbesaved.Thereareonlysmalldifferencesingeometryandprestressdistributionbetweentheresultsdeterminedbylinearandnonlinearcomputationprocedures.However,fortheanalysisofnaturalfrequencyandvibrationmodes,significantdifferencesinthefundamentalfrequenciesandvibrationmodeswilloccur,andthenonlinearitiesofthecable-stayedbridgeresponseappearonlyinthemodesdeterminedonbasisoftheinitialshapefoundbythenonlinearcomputation.2.IntroductionRapidprogressintheanalysisandconstructionofcable-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.Systemequations3.1.GeneralsystemequationWhenonlynonlinearitiesinstiffnessaretakenintoaccount,andthesystemmassanddampingmatricesareconsideredasconstant,thegeneralsystemequationofafiniteelementmodelofstructuresinnonlineardynamicscanbederivedfromtheLagrange’svirtualworkprincipleandwrittenasfollows:Kjbαj-∑Sjajα=Mαβqβ”+Dαβqβ’3.2.LinearizedsystemequationInordertoincrementallysolvethelargedeflectionproblem,thelinearizedsystemequationshastobederived.BytakingthefirstordertermsoftheTaylor’sexpansionofthegeneralsystemequation,thelinearizedequationforasmalltime(orload)intervalisobtainedasfollows:MαβΔqβ”+ΔDαβqβ’+2KαβΔqβ=Δpα-upα3.3.LinearizedsystemequationinstaticsInnonlinearstatics,thelinearizedsystemequationbecomes2KαβΔqβ=Δpα-upα4.Nonlinearanalysis4.1.InitialshapeanalysisTheinitialshapeofacable-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.StaticdeflectionanalysisBasedonthedeterminedinitialshape,thenonlinearstaticdeflectionanalysisofcable-stayedbridgesunderliveloadcanbeperformedincrementwiseoriterationwise.Itiswellknownthattheloadincrementmethodleadstolargenumericalerrors.Theiterationmethodwouldbepreferredforthenonlinearcomputationandadesiredconvergencetolerancecanbeachieved.Newton–Raphsoniterationprocedureisemployed.Fornonlinearanalysisoflargeorcomplexstructuralsystems,a‘full’iterationprocedure(iterationperformedforasinglefullloadstep)willoftenfail.Anincrement–iterationprocedureishighlyrecommended,inwhichtheloadwillbeincremented,andtheiterationwillbecarriedoutineachloadstep.Thestaticdeflectionanalysisofthecablestayedbridgewillstartfromtheinitialshapedeterminedbytheshapefindingprocedureusingalinearornonlinearcomputation.Thealgorithmofthestaticdeflectionanalysisofcable-stayedbridgesissummarizedinSection.3.LinearizedvibrationanalysisWhenastructuralsystemisstiffenoughandtheexternalexcitationisnottoointensive,thesystemmayvibratewithsmallamplitudearoundacertainnonlinearstaticstate,wherethechangeofthenonlinearstaticstateinducedbythevibrationisverysmallandnegligible.Suchvibrationwithsmallamplitudearoundacertainnonlinearstaticstateistermedlinearizedvibration.Thelinearizedvibrationisdifferentfromthelinearvibration,wherethesystemvibrateswithsmallamplitudearoundalinearstaticstate.Thenonlinearstaticstateqαacanbestaticallydeterminedbynonlineardeflectionanalysis.Afterdeterminingqαa,thesystemmatricesmaybeestablishedwithrespecttosuchanonlinearstaticstate,andthelinearizedsystemequationhastheformasfollows:MαβAqβ”+DαβAqβ’+2KαβAqβ=pα(t)-TαAwherethesuperscript‘A’denotesthequantitycalculatedatthenonlinearstaticstateqαa.ThisequationrepresentsasetoflinearordinarydifferentialequationsofsecondorderwithconstantcoefficientmatricesMαβA,DαβAand2KαβA.Theequationcanbesolvedbythemodalsuperpositionmethod,theintegraltransformationmethodsorthedirectintegrationmethods.Whendampingeffectandloadtermsareneglected,thesystemequationbecomesMαβAqβ”+2KαβAqβ=0ThisequationrepresentsthenaturalvibrationsofanundampedsystembasedonthenonlinearstaticstateqαaThenaturalvibrationfrequenciesandmodescanbeobtainedfromtheaboveequationbyusingeigensolutionprocedures,e.g.,subspaceiterationmethods.Forthecable-stayedbridge,itsinitialshapeisthenonlinearstaticstateqαa.Whenthecable-stayedbridgevibrateswithsmallamplitudebasedontheinitialshape,thenaturalfrequenciesandmodescanbefoundbysolvingtheaboveequation.4.4.Computationalgorithmsofcable-stayedbridgeanalysisThealgorithmsforshapefindingcomputation,staticdeflectionanalysisandvibrationanalysisofcable-stayedbridgesarebrieflysummarizedinthefollowing.4.4.1.Initialshapeanalysis1.Inputofthegeometricandphysicaldataofthebridge.2.Inputofthedeadloadofgirdersandtowersandsuitablyestimatedinitialforcesincablestays.3.Findequilibriumposition(i)Linearprocedure?Linearcableandbeam-columnstiffnesselementsareused.?Linearconstantcoordinatetransformationcoefficientsajαareused.?EstablishthelinearsystemstiffnessmatrixKαβbyassemblingelementstiffnessmatrices.?Solvethelinearsystemequationforqα(equilibriumposition).?Noequilibriumiterationiscarriedout.(ii)Nonlinearprocedure?Nonlinearcableswithsageffectandbeam-columnelementsareused.?Nonlinearcoordinatetransformationcoeffi-cientsajα;ajα，βareused.?Establishthetangentsystemstiffnessmatrix2Kαβ.?Solvetheincrementalsystemequationfor△qα.?EquilibriumiterationisperformedbyusingtheNewton–Raphsonmethod.4.Shapeiteration5.Outputoftheinitialshapeincludinggeometricshapeandelementforces.6.Forlinearstaticdeflectionanalysis,onlylinearstiff-nesselementsandtransformationcoefficientsareusedandnoequilibriumiterationiscarriedout.4.4.3.Vibrationanalysis1.Inputofthegeometricandphysicaldataofthebridge.2.Inputoftheinitialshapedataincludinginitialgeometryandinitialelementforces.3.Setupthelinearizedsystemequationoffreevibrationsbasedontheinitialshape.4.Findvibrationfrequenciesandmodesbysub-spaceiterationmethods,suchastheRutishauserMethod.5.EstimationofthetrialinitialcableforcesIntherecentstudyofWangandLin,theshapefindingofsmallcable-stayedbridgeshasbeenperformedbyusingarbitrarysmallorlargetrialinitialcableforces.Theretheiterationconvergesmonotonously,andtheconvergentsolutionshavesimilarresults,ifdifferenttrialvaluesofinitialcableforcesareused.Howeverforlargecable-stayedbridges,shapefindingcomputationsbecomemoredifficulttoconverge.Innonlinearanalysis,theNewton-typeiterativecomputationcanconverge,onlywhentheestimatedvaluesofthesolutionislocateintheneighborhoodofthetruevalues.Difficultiesinconvergencemayappear,whentheshapefindinganalysisofcable-stayedbridgesisstartedbyuseofarbitrarysmallinitialcableforcessuggestedinthepapersofWangetal.Therefore,toestimateasuitabletrialinitialcableforcesinordertogetaconvergentsolutionbecomesimportantfortheshapefindinganalysis.Inthefollowing,severalmethodstoestimatetrialinitialcableforceswillbediscussed.5.1.Balanceofverticalloads5.2.Zeromomentcontrol5.3.Zerodisplacementcontrol5.4.Conceptofcableequivalentmodulusratio5.5.ConsiderationoftheunsymmetryIftheestimatedinitialcableforcesaredeterminedindependentlyforeachcablestaybythemethodsmentionedabove,theremayexistunbalancedhorizontalforcesonthetowerinunsymmetriccable-stayedbridges.Forsymmetricarrangementsofthecable-staysonthecentral(main)spanandthesidespanwithrespecttothetower,theresultantofthehorizontalcomponentsofthecable-staysactingonthetoweriszero,i.e.,nounbalancedhorizontalforcesexistonthetower.Forunsymmetriccable-stayedbridges,inwhichthearrangementofcable-staysonthecentral(main)spanandthesidespanisunsymmetric,andiftheforcesofcablestaysonthecentralspanandthesidespanaredeterminedindependently,evidentlyunbalancedhorizontalforceswillexistonthetowerandwillinducelargebendingmomentsanddeflectionstherein.Therefore,forunsymmetriccable-stayedbridges,thisproblemcanbeovercomeasfollows.Theforceofcablestaysonthecentral(main)spanTimcanbedeterminedbythemethodsmentionedaboveindependently,wherethesuperscriptmdenotesthemainspan,thesubscriptIdenotestheithcablestay.Thentheforceofcablestaysonthesidespanisfoundbytakingtheequilibriumofhorizontalforcecomponentsatthenodeonthetowerattachedwiththecablestays,i.e.,Timcosαi=Tiscosβi,andTis=Timcosαi/cosβi,whereαiistheanglebetweentheithcablestayandthegirderonthemainspan,andβi,anglebetweentheithcablestayandthegirderonthesidespan.6.ExamplesInthisstudy,twodifferenttypesofsmallcable-stayedbridgesaretakenfromliterature,andtheirinitialshapeswillbedeterminedbythepreviouslydescribedshapefindingmethodusinglinearandnonlinearprocedures.Finally,ahighlyredundantstiffcable-stayedbridgewillbeexamined.Aconvergencetolerancee=10-4isusedforboththeequilibriumiterationandtheshapeiteration.Themaximumnumberofiterationcyclesissetas20.Thecomputationisconsideredasnotconvergent,ifthenumberoftheiterationcyclesexceeds20.TheinitialshapesofthefollowingtwosmallcablestayedbridgesinSections6.1and6.2arefirstdeterminedbyusingarbitrarytrialinitialcableforces.Theiterationconvergesmonotonouslyinthesetwoexamples.Theirconvergentinitialshapescanbeobtainedeasilywithoutdifficulties.Thereareonlysmalldifferencesbetweentheinitialshapesdeterminedbythelinearandthenonlinearcomputation.Convergentsolutionsoffersimilarresults,andtheyareindependentofthetrialinitialcableforces.7.ConclusionThetwo-loopiterationwithlinearandnonlinearcomputationisestablishedforfindingtheinitialshapesofcable-stayedbridges.Thismethodcanachievethearchitecturallydesignedformhavinguniformprestressdistribution,andsatisfiesallequilibriumandboundaryconditions.Thedeterminationoftheinitialshapeisthemostimportantworkintheanalysisofcable-stayedbridges.Onlywithacorrectinitialshape,ameaningfulandaccuratedeflectionand/orvibrationanalysiscanbeachieved.Basedonnumericalexperimentsinthestudy,someconclusionsaresummarizedasfollows:(1).Nogreatdifficultiesappearinconvergenceoftheshapefindingofsmallcable-stayedbridges,wherearbitraryinitialtrialcableforcescanbeusedtostartthecomputation.Howeverforlargescalecable-stayedbridges,seriousdifficultiesoccurredinconvergenceofiterations.(2).Difficultiesoftenoccurinconvergenceoftheshapefindingcomputationoflargecable-stayedbridge,whentrialinitialcableforcesaregivenbythemethodsofbalanceofverticalloads,zeromomentcontrolandzerodisplacementcontrol.(3).Aconvergedinitialshapecanbefoundrapidlybythetwo-loopiterationmethod,ifthecablestresscorrespondingtoabout80%ofEeq=Evalueisusedforthetrialinitialforceofeachcablestayinthemainspan,andthetrialforceofthecablesinsidespansisdeterminedbytakinghorizontalequilibriumofthecableforcesactingonthetower.(4).Thereareonlysmalldifferencesingeometryandprestressdistributionforces.Theiterationconvergesmonotonouslyinthesetwoexamples.Theirconvergentinitialshapescanbeobtainedeasilywithoutdifficulties.Thereareonlysmalldifferencesbetweentheinitialshapesdeterminedbythelinearandthenonlinearcomputation.Convergentsolutionsoffersimilarresults,andtheyareindependentofthetrialinitialcableforces.7.ConclusionThetwo-loopiterationwithlinearandnonlinearcomputationisestablishedforfindingtheinitialshapesofcable-stayedbridges.Thismethodcanachievethearchitecturallydesignedformhavinguniformprestressdistribution,andsatisfiesallequilibriumandboundaryconditions.Thedeterminationoftheinitialshapeisthemostimportantworkintheanalysisofcable-stayedbridges.Onlywithacorrectinitialshape,ameaningfulandaccuratedeflectionand/orvibrationanalysiscanbeachieved.Basedonnumericalexperimentsinthestudy,someconclusionsaresummarizedasfollows:(1).Nogreatdifficultiesappearinconvergenceoftheshapefindingofsmallcable-stayedbridges,wherearbitraryinitialtrialcableforcescanbeusedtostartthecomputation.Howeverforlargescalecable-stayedbridges,seriousdifficultiesoccurredinconvergenceofiterations.(2).Difficultiesoftenoccurinconvergenceoftheshapefindingcomputationoflargecable-stayedbridge,whentrialinitialcableforcesaregivenbythemethodsofbalanceofverticalloads,zeromomentcontrolandzerodisplacementcontrol.(3).Aconvergedinitialshapecanbefoundrapidlybythetwo-loopiterationmethod,ifthecablestresscorrespondingtoabout80%ofEeq=Evalueisusedforthetrialinitialforceofeachcablestayinthemainspan,andthetrialforceofthecablesinsidespansisdeterminedbytakinghorizontalequilibriumofthecableforcesactingonthetower.(4).Thereareonlysmalldifferencesingeometryandprestressdistributionbetweentheresultsofinitialshapesdeterminedbylinearandnonlinearprocedures.(5).Theshapefindingusinglinearcomputationoffersareasonableinitialshapeandsavesalotofcomputationefforts,sothatitishighlyrecommendedfromthepointofviewofengineeringpractices.(6).Insmallcable-stayedbridges,thereareonlysmalldifferenceinthenaturalfrequenciesbasedoninitialshapesdeterminedbylinearandnonlinearcomputationprocedures,andthemodeshapesarethesameinbothcases.(7).Significantdifferencesinthefundamentalfrequencyandinthemodeshapesofhighlyredundantstiffcablestayedbridgesisshowninthestudy.Onlythevibrationmodesdeterminedbytheinitialshapebasedonnonlinearproceduresexhibitthenonlinearcablesagandbeam-columneffectsofcable-stayedbridges,e.g.,thefirstandthirdmodesofthebridgearedominatedbythetransversalmotionofthetower,notofthegirder.Thedifferenceofthefundamentalfrequencyinbothcasesisabout12%.Henceacorrectanalysisofvibrationfrequenciesandmodesofcable-stayedbridgescanbeobtainedonlywhenthe‘correct’initialshapeisdeterminedbynonlinearcomputation,notbythelinearcomputation.高度超靜定斜拉橋的非線性分析研究1.摘要一個(gè)拉索高度超靜定的斜拉橋的非線性分析比較在研究中被實(shí)行。包括橋的幾何學(xué)和預(yù)應(yīng)力分配的初始形狀是使用雙重迭代的方法決定的,也就是,一個(gè)平衡迭代和一個(gè)形狀迭代。對(duì)于開(kāi)始的形狀分析，一個(gè)線性和一個(gè)非線性計(jì)算程序被建立。以前斜拉橋所有非線性被忽視，而且形狀迭代是不考慮平衡而實(shí)行的。后來(lái)橋的所有非線性被考慮到，而且平衡和形狀的重復(fù)都實(shí)行了。基于收斂于一點(diǎn)的起始形狀由不同的程序決定,自振頻率和震動(dòng)模態(tài)也被詳細(xì)地研究。數(shù)字的結(jié)果表明收斂于一點(diǎn)的起始形狀能由二個(gè)環(huán)的重復(fù)方法快速地得到,合理的起始形狀能由線性的計(jì)算程序決定，而且那樣許多計(jì)算工作將被節(jié)省。在由線性的和非線性計(jì)算程序決定的結(jié)果之間的幾何學(xué)和預(yù)應(yīng)力分配中只有很小的不同。然而，對(duì)于自振頻率和震動(dòng)模態(tài)的分析來(lái)說(shuō)，基本的頻率和震動(dòng)模態(tài)將會(huì)有顯著的不同，而且斜拉橋反應(yīng)的非線性只出現(xiàn)在由非線性計(jì)算得到的初始形狀的基礎(chǔ)之上的模態(tài)中。2.序言在過(guò)去的三十年中斜拉橋分析和建筑中取得了飛速的進(jìn)步。進(jìn)步主要是由于計(jì)算機(jī)技術(shù)的領(lǐng)域發(fā)展,高強(qiáng)度的鋼拉索,正交異性鋼板和建筑技術(shù)產(chǎn)生的。既然第一座現(xiàn)代的斜拉橋1955年在瑞典被建造，他們的名聲在全世界得到快速地增長(zhǎng)。因?yàn)樗闹绷⒚缹W(xué)的外觀，經(jīng)濟(jì)原因和便于直立，斜拉橋被認(rèn)為是跨徑范圍從200m到大約1000m的最合適的建筑類(lèi)型。世界上現(xiàn)在最長(zhǎng)的斜拉橋是日本的橫跨島海、連接本州四國(guó)的多多羅橋。多多羅斜拉橋在1999年5月1日被開(kāi)通，它有890m的一個(gè)中央跨徑和1480m的總跨度。一座斜拉橋由三個(gè)主要的成分所組成，也就是主梁、索塔和斜拉索。主梁在沿縱向方向由拉索彈性支撐以使主梁能跨越一個(gè)更長(zhǎng)的距離而不需要中間橋墩。主梁的永久荷載和車(chē)輛荷載通過(guò)拉索傳遞給索塔。很大的拉力存在于拉索中減小了索塔中大部分和梁的一部分壓力。斜拉橋的非線性的來(lái)源主要地包括拉索下垂,梁柱的偏壓和大的偏轉(zhuǎn)效應(yīng)。因?yàn)樵谖词┘踊钶d前拉索中存在高度預(yù)應(yīng)力，斜拉橋的初始形狀和預(yù)應(yīng)力由每條拉索決定。他們不能夠被獨(dú)立地看成是傳統(tǒng)的鋼或者是高強(qiáng)混凝土橋。因此開(kāi)始的形狀必須被在橋的分析之前正確的決定。只有基于正確的起始形狀才能得到一個(gè)正確的偏轉(zhuǎn)和震動(dòng)分析。這篇論文的目的要提供一個(gè)高度冗余的斜拉橋的非線性分析的比較，橋的開(kāi)始形狀將會(huì)由線性和非線性計(jì)算程序迭代來(lái)決定。基于開(kāi)始的形狀計(jì)算,橋的震動(dòng)頻率和模態(tài)被確定。3.系統(tǒng)方程3.1一般的系統(tǒng)方程當(dāng)只有非線性在剛體中被考慮到，而且系統(tǒng)的衰減矩陣被認(rèn)為是恒定的時(shí)候,在非線性動(dòng)力學(xué)中結(jié)構(gòu)的一個(gè)有限元模型才能從虛工作原則中得到，如下:Kjbαj-∑Sjajα=Mαβqβ”+Dαβqβ’3.2線性化系統(tǒng)方程為了要不斷的解決更大的偏轉(zhuǎn)問(wèn)題,線性化系統(tǒng)的方程必需用到。通過(guò)泰勒的一般方程的擴(kuò)展的最早的條目，對(duì)于一個(gè)小的時(shí)間（或荷載）間隔的線性化的方程便得到，如下：MαβΔqβ”+ΔDαβqβ’+2KαβΔqβ=Δpα-upα3.3在靜力學(xué)中的線性化系統(tǒng)方程在非線性靜力學(xué)中，線性化系統(tǒng)方程變成：2KαβΔqβ=Δpα-upα4．非線性分析4.1.起始形狀分析斜拉橋的初始形狀提供了幾何學(xué)的結(jié)構(gòu)和橋在主梁和索塔的恒載、斜拉索的拉力作用下的預(yù)應(yīng)力分配。作用的平衡條件，指定的邊界條件和建筑的設(shè)計(jì)需求應(yīng)該被滿(mǎn)足。因?yàn)橛?jì)算的形狀,主梁和索塔的永久荷載必須被考慮，拉索的自重被疏忽，而且拉索下垂的非線性應(yīng)包括在內(nèi)。形狀的計(jì)算通過(guò)使用二重迭代的方法運(yùn)行,也就是,平衡重復(fù)和形狀重復(fù)循環(huán)。這能用拉索中的任意小的張力開(kāi)始。基于參考結(jié)構(gòu)(建筑設(shè)計(jì)形式)，沒(méi)有歪斜和零的預(yù)應(yīng)力在主梁和索塔中,斜拉橋平衡位置在恒載作用下是由迭代首先確定的(平衡迭代)。雖然首先決定結(jié)構(gòu)的是使平衡情況和邊界情況得到滿(mǎn)足，但是建筑的設(shè)計(jì)需求大體上沒(méi)有得到實(shí)現(xiàn)。因?yàn)闃虻目鐝绞呛艽蟮亩鴽](méi)有預(yù)應(yīng)力存在斜拉索中,相當(dāng)大的偏轉(zhuǎn)和非常大的彎矩可能在主梁和索塔中出現(xiàn)。那么另外的一個(gè)迭代有必要執(zhí)行來(lái)減少偏轉(zhuǎn)和使主梁的彎矩平滑并最后找出正確的初始形狀。如此的一個(gè)迭代程序在這里命名為‘形狀迭代’。對(duì)于形狀迭代,在先前步驟中確定的基本的軸線力將會(huì)被作為下個(gè)重復(fù)采取的初始基本力,這樣一個(gè)新的平衡結(jié)構(gòu)在恒載和這個(gè)初始力下再次被確定。在形狀迭代的時(shí)候，一些控制點(diǎn)(主梁和拉索連接的點(diǎn))將會(huì)被選擇檢驗(yàn)應(yīng)力集中。在每次形狀迭代過(guò)程中，主跨的控制點(diǎn)的垂直位移比率將會(huì)被檢驗(yàn)。也就是,形狀迭代將會(huì)重復(fù)直到應(yīng)變可以達(dá)到所說(shuō)的10-4。當(dāng)應(yīng)變達(dá)到的時(shí)候,計(jì)算將會(huì)停止而斜拉橋的初始形狀就找到了。數(shù)字的實(shí)驗(yàn)表明重復(fù)收斂于一點(diǎn)是沒(méi)什么作用的，并且所有的三個(gè)非線性對(duì)最后的幾何初始形狀有比較少的影響。只有拉索下垂作用在確定初始形狀分析中有顯著作用，而偏壓柱和大的偏轉(zhuǎn)效應(yīng)變則無(wú)關(guān)重要。開(kāi)始的分析能以二種不同的方式被實(shí)行:一個(gè)線性和一個(gè)非線性計(jì)算程序。(1)線性的計(jì)算程序:為了要找到橋的平衡結(jié)構(gòu),斜拉橋的所有非線性因素被疏忽，而只是線性的彈性拉索、梁?jiǎn)卧⑼鹊木€形的變形系數(shù)被使用。形狀迭代是不考慮平衡迭代而實(shí)行的。合理的收斂于一點(diǎn)的起始形狀被得到，而且許多計(jì)算的工作能被節(jié)省。(2)非線性計(jì)算程序:斜拉橋所有的非線性因素在整個(gè)的計(jì)算程序中被考慮。非線性拉索元素的下沉作用、主梁元素的穩(wěn)定系數(shù)和非線性變形調(diào)整系數(shù)被應(yīng)用。形狀的迭代和平衡迭代都在非線性計(jì)算中實(shí)行。牛頓-瑞普生方法在這里被用于平衡迭代。4.2靜態(tài)偏轉(zhuǎn)分析基于確定的起始形狀,斜拉橋在活載作用下的非線性靜態(tài)偏轉(zhuǎn)分析可通過(guò)模數(shù)或迭代運(yùn)行。荷載模數(shù)方法導(dǎo)致很大的數(shù)字錯(cuò)誤是廣為人知的。迭代方法比較適于非線性計(jì)算，而且需要的應(yīng)變應(yīng)能被達(dá)到。牛頓-瑞普生的迭代程序?qū)⒈皇褂谩Ｒ驗(yàn)榉蔷€性分析較大或復(fù)雜的結(jié)構(gòu)系統(tǒng),一個(gè)‘完整’的迭代程序(重復(fù)為一個(gè)單一全部荷載運(yùn)行步驟)將會(huì)時(shí)常失敗。一個(gè)模數(shù)-迭代程序高度地被推薦,荷載將會(huì)被增加，而且重復(fù)將會(huì)在每個(gè)荷載步驟中實(shí)行。斜拉橋的靜態(tài)偏轉(zhuǎn)分析將會(huì)從使用線性或非線性計(jì)算程序決定的初始形狀開(kāi)始。斜拉橋靜態(tài)的偏轉(zhuǎn)分析的運(yùn)算法則在第4.4.2節(jié)中被概述。4.3.線性振動(dòng)分析當(dāng)一個(gè)結(jié)構(gòu)系統(tǒng)是足夠穩(wěn)固而且外部的刺激不是太強(qiáng)烈,系統(tǒng)可能以一個(gè)確定的非線性的靜態(tài)系數(shù)作一個(gè)小振幅振動(dòng)，由振動(dòng)引起的非線性靜態(tài)系數(shù)的變化是很小的和可以忽略的。這種以一個(gè)非線性靜態(tài)系數(shù)以一個(gè)小振幅的振動(dòng)被稱(chēng)作線性化振動(dòng)。線性化振動(dòng)不同于線性振動(dòng)，系統(tǒng)用很小的振幅以一個(gè)線性靜態(tài)系數(shù)振動(dòng)。非線性靜態(tài)系數(shù)qαa能由非線性偏轉(zhuǎn)分析決定。在決定qαa之后，系統(tǒng)矩陣可能被建立有關(guān)于如此的一個(gè)非線性靜態(tài)系數(shù)，線性化系統(tǒng)的等式如下所示：MαβAqβ”+DαβAqβ’+2KαβAqβ=pα(t)-TαA上面的上標(biāo)字母‘A'代表在非線性靜態(tài)系數(shù)qαa被計(jì)算的數(shù)量。這個(gè)等式用恒定系數(shù)矩陣MαβA、DαβA、2KαβA表現(xiàn)第二的次序一組線性的一般差別的等式。這個(gè)等式能被模型的重疊方法，整體的變形方法或直接的整合方法解答。當(dāng)減幅效應(yīng)和荷載限制被忽略的時(shí)候,系統(tǒng)等式變成：MαβAqβ”+2KαβAqβ=0這個(gè)等式表現(xiàn)基于非線性靜態(tài)系數(shù)qαa的不減幅的系統(tǒng)天然振動(dòng)。天然振動(dòng)的頻率和模態(tài)可以從上面的等式運(yùn)用程序，舉例來(lái)說(shuō)子空間重復(fù)方法來(lái)得到。對(duì)于斜拉橋,它的起始形狀是非線性靜態(tài)系數(shù)qαa。斜拉橋由于以開(kāi)始的形狀為基礎(chǔ)小振幅振動(dòng)的時(shí)候,天然的頻率和模態(tài)能被找到來(lái)解決上述的等式。4.4.斜拉橋計(jì)算運(yùn)算法則分析斜拉橋的形狀的確定計(jì)算、靜態(tài)偏轉(zhuǎn)分析和振動(dòng)分析的計(jì)算法則簡(jiǎn)短的概述如下。4.4.1起始形狀分析1.橋的幾何和實(shí)際的數(shù)據(jù)輸入。2.主梁和索塔的恒載的輸入而且適當(dāng)?shù)毓烙?jì)了起始拉索中的受力。3.確定平衡位置(i)線性的程序o運(yùn)用線性的拉索和主梁剛性單元。o運(yùn)用線性的恒定變形調(diào)整系數(shù)ajα。o建立線性系統(tǒng)剛度矩陣Kαβ通過(guò)排列元素的剛度矩陣。o求解線性等式得到qα。(平衡位置)o沒(méi)有平衡的迭代被實(shí)行。(ii)非線性程序o非線性下垂效應(yīng)的拉索和主梁?jiǎn)卧皇褂?。o非線性變形調(diào)整系數(shù)ajα;ajα，β被使用。o建立接觸的系統(tǒng)剛度矩陣2Kαβ。o求解增量系統(tǒng)的等式以得到△qα。o平衡迭代使用牛頓-瑞普生方法運(yùn)行△qα。4.變形迭代。5.包括幾何形狀和基本力的初始的形狀輸出。6.對(duì)于線性偏轉(zhuǎn)分析，只有線性剛度單元和變形系數(shù)被采用且沒(méi)有平衡迭代的實(shí)行。4.4.3.振動(dòng)分析1.橋的幾何的和物理的數(shù)據(jù)輸入。2.包括開(kāi)始的幾何和開(kāi)始單元常受力的初始形狀數(shù)據(jù)的輸入。3.建立以初始形狀的自由振動(dòng)的線性化系統(tǒng)等式。4.運(yùn)用子空間重復(fù)方法得到振動(dòng)頻率和模態(tài),例如Rutishauser方法。5.初始拉索受力估算：在王教授和林教授的最近研究中,小型的斜拉橋的通過(guò)任意小或任意大試驗(yàn)初始拉索應(yīng)力來(lái)實(shí)現(xiàn)。如果不同的初始拉索受力試驗(yàn)評(píng)價(jià)被采用，在那里重復(fù)單調(diào)的迭代，而最后的結(jié)論有相似的結(jié)果。然而對(duì)于大型斜拉橋，確定形狀的計(jì)算變得更困難以達(dá)到一致。在非線性分析中，牛頓-瑞普生類(lèi)型迭代計(jì)算能收斂到一點(diǎn),只有當(dāng)解決的被估計(jì)的價(jià)值是在真正的價(jià)值附近時(shí)才能實(shí)現(xiàn)。當(dāng)斜拉橋分析的形狀在王的論文中建議用任意小的初始拉索應(yīng)力開(kāi)始時(shí)，收斂到一點(diǎn)的困難可能會(huì)出現(xiàn)。因此,估計(jì)適當(dāng)?shù)脑囼?yàn)開(kāi)始的拉索應(yīng)力來(lái)得到一致的結(jié)論對(duì)于形狀確定分析變得重要起來(lái)。接下來(lái),一些估計(jì)試驗(yàn)初始拉索應(yīng)力的方法將會(huì)被討論。5.1.垂直荷載的平衡5.2.零5.3.零5.4.拉索等價(jià)系數(shù)比的概念5.5.不對(duì)稱(chēng)的考慮如果估計(jì)的初始拉索應(yīng)力是用上面介紹的方法對(duì)每條拉索獨(dú)立地確定的，非對(duì)稱(chēng)的斜拉橋索塔中可能會(huì)存在不平衡的水平受力。因?yàn)橹虚g跨(主跨)和邊跨的對(duì)稱(chēng)的拉索布置對(duì)索塔的水平分力的合力為零，也就是沒(méi)有不平衡的水平力。而對(duì)于非對(duì)稱(chēng)的斜拉橋，拉索在中間跨和邊跨分布是不對(duì)稱(chēng)的，拉索在索塔上產(chǎn)生的分力分別獨(dú)立計(jì)算，很明顯的索塔中的不平衡的水平力將會(huì)引起很大的彎矩和偏移。因此，非對(duì)稱(chēng)的斜拉橋的這個(gè)問(wèn)題可以按如下解決。中間跨(主要部份)的拉索受力可以通過(guò)上面獨(dú)立介紹的方法確定，其中上標(biāo)m代表主跨，上標(biāo)i代表第i條拉索。然后邊跨上拉索受力通過(guò)與索塔連接的拉索的水平平衡方程確定，即Timcosαi=Tiscosβi,andTis=Timcosαi/cosβi,αi是指斜拉索與主跨梁的夾角，βi是指斜拉索與邊跨梁的夾角。6.例子在這項(xiàng)研究中，二座不同的類(lèi)型小型斜拉橋從文學(xué)中取得,而且他們的起始形狀將會(huì)用先前描述的形狀確定方法使用線性和非線性程序來(lái)確定。最后，一座高度冗余的斜拉橋?qū)?huì)被研究。對(duì)于平衡迭代和形狀迭代都采用應(yīng)變10-4。重復(fù)的最大周期定為20。如果重復(fù)的循環(huán)數(shù)超過(guò)20則計(jì)算被認(rèn)為是不收斂于一點(diǎn)的。接著的二個(gè)小型斜拉橋的開(kāi)始形狀在第6.1節(jié)和6.2首先采用任意小的試驗(yàn)拉索應(yīng)力決定的。在這二個(gè)例子中收斂于一點(diǎn)是重復(fù)單調(diào)的。他們的收斂于一點(diǎn)的起始形狀可以很容易地獲得。由線性和非線性計(jì)算決定的開(kāi)始形狀之間只有很小的不同。收斂于一點(diǎn)的結(jié)論顯示同樣的結(jié)果，而且他們與試驗(yàn)的拉索應(yīng)力無(wú)關(guān)。7.結(jié)論通過(guò)線性和非線性計(jì)算二重循環(huán)的建立而得到斜拉橋的初始形狀。這個(gè)方法能達(dá)到建筑的設(shè)計(jì)形式有統(tǒng)一的預(yù)應(yīng)力分配,而且使所有的平衡和邊界情況滿(mǎn)足。初始的形狀確定是斜拉橋分析中最重要的工作。只有一個(gè)正確的起始形狀，才能得到一個(gè)有意義的和正確的偏轉(zhuǎn)及震動(dòng)分析。基于研究的數(shù)字實(shí)驗(yàn),一些結(jié)論概述如下:(1).對(duì)于小型斜拉橋初始形狀的確定不會(huì)出現(xiàn)現(xiàn)在的困難，任意的初始試驗(yàn)拉索應(yīng)力都能用來(lái)計(jì)算。然而對(duì)于大跨度的斜拉橋，循環(huán)的收斂于一點(diǎn)會(huì)產(chǎn)生很大問(wèn)題。(2).在跨度的斜拉橋的形狀確定的收斂于一點(diǎn)通常會(huì)產(chǎn)生困難，當(dāng)拉索的試驗(yàn)應(yīng)力通過(guò)垂直荷載平衡、零彎矩控制、零位移控制的方法給出時(shí)。(3).如果主跨的每條拉索的預(yù)應(yīng)力符合到約Eeq的80%，通過(guò)兩重循環(huán)的方法可以很快的找到收斂于一點(diǎn)的初始形狀，而且邊跨拉索的初始應(yīng)力是由作用于索塔上的水平等式?jīng)Q定的。(4).由線性程序和非線性的程序得到的初始形狀的結(jié)果對(duì)于幾何和預(yù)應(yīng)力的分配的結(jié)果只有很小的不同。(5).使用線性的計(jì)算能提供一個(gè)合理的起始形狀而且節(jié)省很多的計(jì)算工作,所以在工程實(shí)踐中它高度的被推薦。(6).在小型的斜拉橋中,由線性和非線性計(jì)算程序確定的初始形狀為基礎(chǔ)得到的自振頻率只有很小的差別，而模態(tài)情形在兩種情況中是一樣的。(7).基本頻率和高度冗余剛性斜拉橋的顯著不同在研究中被展示。只有基于非線性程序的初始形狀確定的模態(tài)展示非線性拉索下沉和主梁的偏轉(zhuǎn)效應(yīng)。舉例來(lái)說(shuō)，橋的第一和第