高度超静定斜拉桥的非线性分析研究

高度超静定斜拉桥的非线性分析研究1.摘要一个拉索高度超静定的斜拉桥的非线性分析比较在研究中被实行。包括桥的几何学和预应力分配的初始形状是使用双重迭代的方法决定的,也就是,一个平衡迭代和一个形状迭代。对于开始的形状分析，一个线性和一个非线性计算程序被建立。以前斜拉桥所有非线性被忽视，而且形状迭代是不考虑平衡而实行的。后来桥的所有非线性被考虑到，而且平衡和形状的重复都实行了。基于收敛于一点的起始形状由不同的程序决定,自振频率和震动模态也被详细地研究。数字的结果表明收敛于一点的起始形状能由二个环的重复方法快速地得到,合理的起始形状能由线性的计算程序决定，而且那样许多计算工作将被节省。在由线性的和非线性计算程序决定的结果之间的几何学和预应力分配中只有很小的不同。然而，对于自振频率和震动模态的分析来说，基本的频率和震动模态将会有显著的不同，而且斜拉桥反应的非线性只出现在由非线性计算得到的初始形状的基础之上的模态中。2.序言在过去的三十年中斜拉桥分析和建筑中取得了飞速的进步。进步主要是由于计算机技术的领域发展,高强度的钢拉索,正交异性钢板和建筑技术产生的。既然第一座现代的斜拉桥1955年在瑞典被建造，他们的名声在全世界得到快速地增长。因为它的直立美学的外观，经济原因和便于直立，斜拉桥被认为是跨径范围从200m到大约1000m的最合适的建筑类型。世界上现在最长的斜拉桥是日本的横跨岛海、连接本州四国的多多罗桥。多多罗斜拉桥在1999年5月1日被开通，它有890m的一个中央跨径和1480m的总跨度。一座斜拉桥由三个主要的成分所组成，也就是主梁、索塔和斜拉索。主梁在沿纵向方向由拉索弹性支撑以使主梁能跨越一个更长的距离而不需要中间桥墩。主梁的永久荷载和车辆荷载通过拉索传递给索塔。很大的拉力存在于拉索中减小了索塔中大部分和梁的一部分压力。斜拉桥的非线性的来源主要地包括拉索下垂,梁柱的偏压和大的偏转效应。因为在未施加活载前拉索中存在高度预应力，斜拉桥的初始形状和预应力由每条拉索决定。他们不能够被独立地看成是传统的钢或者是高强混凝土桥。因此开始的形状必须被在桥的分析之前正确的决定。只有基于正确的起始形状才能得到一个正确的偏转和震动分析。这篇论文的目的要提供一个高度冗余的斜拉桥的非线性分析的比较，桥的开始形状将会由线性和非线性计算程序迭代来决定。基于开始的形状计算,桥的震动频率和模态被确定。3.系统方程3.1一般的系统方程当只有非线性在刚体中被考虑到，而且系统的衰减矩阵被认为是恒定的时候,在非线性动力学中结构的一个有限元模型才能从虚工作原则中得到，如下:Kjbαj-∑Sjajα=Mαβqβ”+Dαβqβ’3.2线性化系统方程为了要不断的解决更大的偏转问题,线性化系统的方程必需用到。通过泰勒的一般方程的扩展的最早的条目，对于一个小的时间（或荷载）间隔的线性化的方程便得到，如下：MαβΔqβ”+ΔDαβqβ’+2KαβΔqβ=Δpα-upα3.3在静力学中的线性化系统方程在非线性静力学中，线性化系统方程变成：2KαβΔqβ=Δpα-upα4．非线性分析4.1.起始形状分析斜拉桥的初始形状提供了几何学的结构和桥在主梁和索塔的恒载、斜拉索的拉力作用下的预应力分配。作用的平衡条件，指定的边界条件和建筑的设计需求应该被满足。因为计算的形状,主梁和索塔的永久荷载必须被考虑，拉索的自重被疏忽，而且拉索下垂的非线性应包括在内。形状的计算通过使用二重迭代的方法运行,也就是,平衡重复和形状重复循环。这能用拉索中的任意小的张力开始。基于参考结构(建筑设计形式)，没有歪斜和零的预应力在主梁和索塔中,斜拉桥平衡位置在恒载作用下是由迭代首先确定的(平衡迭代)。虽然首先决定结构的是使平衡情况和边界情况得到满足，但是建筑的设计需求大体上没有得到实现。因为桥的跨径是很大的而没有预应力存在斜拉索中,相当大的偏转和非常大的弯矩可能在主梁和索塔中出现。那么另外的一个迭代有必要执行来减少偏转和使主梁的弯矩平滑并最后找出正确的初始形状。如此的一个迭代程序在这里命名为‘形状迭代’。对于形状迭代,在先前步骤中确定的基本的轴线力将会被作为下个重复采取的初始基本力,这样一个新的平衡结构在恒载和这个初始力下再次被确定。在形状迭代的时候，一些控制点(主梁和拉索连接的点)将会被选择检验应力集中。在每次形状迭代过程中，主跨的控制点的垂直位移比率将会被检验。也就是,形状迭代将会重复直到应变可以达到所说的10-4。当应变达到的时候,计算将会停止而斜拉桥的初始形状就找到了。数字的实验表明重复收敛于一点是没什么作用的，并且所有的三个非线性对最后的几何初始形状有比较少的影响。只有拉索下垂作用在确定初始形状分析中有显著作用，而偏压柱和大的偏转效应变则无关重要。开始的分析能以二种不同的方式被实行:一个线性和一个非线性计算程序。(1)线性的计算程序:为了要找到桥的平衡结构,斜拉桥的所有非线性因素被疏忽，而只是线性的弹性拉索、梁单元、同等的线形的变形系数被使用。形状迭代是不考虑平衡迭代而实行的。合理的收敛于一点的起始形状被得到，而且许多计算的工作能被节省。(2)非线性计算程序:斜拉桥所有的非线性因素在整个的计算程序中被考虑。非线性拉索元素的下沉作用、主梁元素的稳定系数和非线性变形调整系数被应用。形状的迭代和平衡迭代都在非线性计算中实行。牛顿-瑞普生方法在这里被用于平衡迭代。4.2静态偏转分析基于确定的起始形状,斜拉桥在活载作用下的非线性静态偏转分析可通过模数或迭代运行。荷载模数方法导致很大的数字错误是广为人知的。迭代方法比较适于非线性计算，而且需要的应变应能被达到。牛顿-瑞普生的迭代程序将被使用。因为非线性分析较大或复杂的结构系统,一个‘完整’的迭代程序(重复为一个单一全部荷载运行步骤)将会时常失败。一个模数-迭代程序高度地被推荐,荷载将会被增加，而且重复将会在每个荷载步骤中实行。斜拉桥的静态偏转分析将会从使用线性或非线性计算程序决定的初始形状开始。斜拉桥静态的偏转分析的运算法则在第4.4.2节中被概述。4.3.线性振动分析当一个结构系统是足够稳固而且外部的刺激不是太强烈,系统可能以一个确定的非线性的静态系数作一个小振幅振动，由振动引起的非线性静态系数的变化是很小的和可以忽略的。这种以一个非线性静态系数以一个小振幅的振动被称作线性化振动。线性化振动不同于线性振动，系统用很小的振幅以一个线性静态系数振动。非线性静态系数qαa能由非线性偏转分析决定。在决定qαa之后，系统矩阵可能被建立有关于如此的一个非线性静态系数，线性化系统的等式如下所示：MαβAqβ”+DαβAqβ’+2KαβAqβ=pα(t)-TαA上面的上标字母‘A'代表在非线性静态系数qαa被计算的数量。这个等式用恒定系数矩算而不是线性的计算确定的正确的初始形状才能得到斜拉桥的振动频率和模态的正确分析。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-stayedbridgesissummarizedinSection4.4.2.4.3.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.Based