（理工大学本科生毕业设计）高度超静定斜拉桥的非线性分析研究（中英文对照）

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GNARE,INGENERAL,NOTFULFILLEDSINCETHEBRIDGESPANISLARGEANDNOPRETENSIONFORCESEXISTININCLINEDCABLES,QUITELARGEDEFLECTIONSANDVERYLARGEBENDINGMOMENTSMAYAPPEARINTHEGIRDERSANDTOWERSANOTHERITERATIONTHENHASTOBECARRIEDOUTINORDERTOREDUCETHEDEFLECTIONANDTOSMOOTHTHEBENDINGMOMENTSINTHEGIRDERANDFINALLYTOFINDTHECORRECTINITIALSHAPESUCHANITERATIONPROCEDUREISNAMEDHERETHESHAPEITERATIONFORSHAPEITERATION,THEELEMENTAXIALFORCESDETERMINEDINTHEPREVIOUSSTEPWILLBETAKENASINITIALELEMENTFORCESFORTHENEXTITERATION,ANDANEWEQUILIBRIUMCONFIGURATIONUNDERTHEACTIONOFDEADLOADANDSUCHINITIALFORCESWILLBEDETERMINEDAGAINDURINGSHAPEITERATION,SEVERALCONTROLPOINTSNODESINTERSECTEDBYTHEGIRDERANDTHECABLEWILLBECHOSENFORCHECKINGTHECONVERGENCETOLERANCEINEACHSHAPEITERATIONTHERATIOOFTHEVERTICALDISPLACEMENTATCONTROLPOINTSTOTHEMAINSPANLENGTHWILLBECHECKED,IE,|SPANMIOITSCTRLDILEVERTCA|THESHAPEITERATIONWILLBEREPEATEDUNTILTHECONVERGENCETOLERANCE,SAY104,ISACHIEVEDWHENTHECONVERGENCETOLERANCEISREACHED,THECOMPUTATIONWILLSTOPANDTHEINITIALSHAPEOFTHECABLESTAYEDBRIDGESISFOUNDNUMERICALEXPERIMENTSSHOWTHATTHEITERATIONCONVERGESMONOTONOUSLYANDTHATALLTHREENONLINEARITIESHAVELESSINFLUENCEONTHEFINALGEOMETRYOFTHEINITIALSHAPEONLYTHECABLESAGEFFECTISSIGNIFICANTFORCABLEFORCESDETERMINEDINTHEINITIALSHAPEANALYSIS,ANDTHEBEAMCOLUMNANDLARGEDEFLECTIONEFFECTSBECOMEINSIGNIFICANTTHEINITIALANALYSISCANBEPERFORMEDINTWODIFFERENTWAYSALINEARANDANONLINEARCOMPUTATIONPROCEDURE1LINEARCOMPUTATIONPROCEDURETOFINDTHEEQUILIBRIUMCONFIGURATIONOFTHEBRIDGE,ALLNONLINEARITIESOFCABLESTAYEDBRIDGESARENEGLECTEDANDONLYTHELINEARELASTICCABLE,BEAMCOLUMNELEMENTSANDLINEARCONSTANTCOORDINATETRANSFORMATIONCOEFFICIENTSAREUSEDTHESHAPEITERATIONISCARRIEDOUTWITHOUTCONSIDERINGTHEEQUILIBRIUMITERATIONAREASONABLECONVERGENTINITIALSHAPEISFOUND,ANDALOTOFCOMPUTATIONEFFORTSCANBESAVED2NONLINEARCOMPUTATIONPROCEDUREALLNONLINEARITIESOFCABLESTAYEDBRIDGESARETAKENINTOCONSIDERATIONDURINGTHEWHOLECOMPUTATIONPROCESSTHENONLINEARCABLEELEMENTWITHSAGEFFECTANDTHEBEAMCOLUMNELEMENTINCLUDINGSTABILITYCOEFFICIENTSANDNONLINEARCOORDINATETRANSFORMATIONCOEFFICIENTSAREUSEDBOTHTHESHAPEITERATIONANDTHEEQUILIBRIUMITERATIONARECARRIEDOUTINTHENONLINEARCOMPUTATIONNEWTONRAPHSONMETHODISUTILIZEDHEREFOREQUILIBRIUMITERATION42STATICDEFLECTIONANALYSISBASEDONTHEDETERMINEDINITIALSHAPE,THENONLINEARSTATICDEFLECTIONANALYSISOFCABLESTAYEDBRIDGESUNDERLIVELOADCANBEPERFORMEDINCREMENTWISEORITERATIONWISEITISWELLKNOWNTHATTHELOADINCREMENTMETHODLEADSTOLARGENUMERICALERRORSTHEITERATIONMETHODWOULDBEPREFERREDFORTHENONLINEARCOMPUTATIONANDADESIREDCONVERGENCETOLERANCECANBEACHIEVEDNEWTONRAPHSONITERATIONPROCEDUREISEMPLOYEDFORNONLINEARANALYSISOFLARGEORCOMPLEXSTRUCTURALSYSTEMS,AFULLITERATIONPROCEDUREITERATIONPERFORMEDFORASINGLEFULLLOADSTEPWILLOFTENFAILANINCREMENTITERATIONPROCEDUREISHIGHLYRECOMMENDED,INWHICHTHELOADWILLBEINCREMENTED,ANDTHEITERATIONWILLBECARRIEDOUTINEACHLOADSTEPTHESTATICDEFLECTIONANALYSISOFTHECABLESTAYEDBRIDGEWILLSTARTFROMTHEINITIALSHAPEDETERMINEDBYTHESHAPEFINDINGPROCEDUREUSINGALINEARORNONLINEARCOMPUTATIONTHEALGORITHMOFTHESTATICDEFLECTIONANALYSISOFCABLESTAYEDBRIDGESISSUMMARIZEDINSECTION44243LINEARIZEDVIBRATIONANALYSISWHENASTRUCTURALSYSTEMISSTIFFENOUGHANDTHEEXTERNALEXCITATIONISNOTTOOINTENSIVE,THESYSTEMMAYVIBRATEWITHSMALLAMPLITUDEAROUNDACERTAINNONLINEARSTATICSTATE,WHERETHECHANGEOFTHENONLINEARSTATICSTATEINDUCEDBYTHEVIBRATIONISVERYSMALLANDNEGLIGIBLESUCHVIBRATIONWITHSMALLAMPLITUDEAROUNDACERTAINNONLINEARSTATICSTATEISTERMEDLINEARIZEDVIBRATIONTHELINEARIZEDVIBRATIONISDIFFERENTFROMTHELINEARVIBRATION,WHERETHESYSTEMVIBRATESWITHSMALLAMPLITUDEAROUNDALINEARSTATICSTATETHENONLINEARSTATICSTATEQACANBESTATICALLYDETERMINEDBYNONLINEARDEFLECTIONANALYSISAFTERDETERMININGQA,THESYSTEMMATRICESMAYBEESTABLISHEDWITHRESPECTTOSUCHANONLINEARSTATICSTATE,ANDTHELINEARIZEDSYSTEMEQUATIONHASTHEFORMASFOLLOWSMAQ”DAQ2KAQPTTAWHERETHESUPERSCRIPTADENOTESTHEQUANTITYCALCULATEDATTHENONLINEARSTATICSTATEQATHISEQUATIONREPRESENTSASETOFLINEARORDINARYDIFFERENTIALEQUATIONSOFSECONDORDERWITHCONSTANTCOEFFICIENTMATRICESMA,DAAND2KATHEEQUATIONCANBESOLVEDBYTHEMODALSUPERPOSITIONMETHOD,THEINTEGRALTRANSFORMATIONMETHODSORTHEDIRECTINTEGRATIONMETHODSWHENDAMPINGEFFECTANDLOADTERMSARENEGLECTED,THESYSTEMEQUATIONBECOMESMAQ”2KAQ0THISEQUATIONREPRESENTSTHENATURALVIBRATIONSOFANUNDAMPEDSYSTEMBASEDONTHENONLINEARSTATICSTATEQATHENATURALVIBRATIONFREQUENCIESANDMODESCANBEOBTAINEDFROMTHEABOVEEQUATIONBYUSINGEIGENSOLUTIONPROCEDURES,EG,SUBSPACEITERATIONMETHODSFORTHECABLESTAYEDBRIDGE,ITSINITIALSHAPEISTHENONLINEARSTATICSTATEQAWHENTHECABLESTAYEDBRIDGEVIBRATESWITHSMALLAMPLITUDEBASEDONTHEINITIALSHAPE,THENATURALFREQUENCIESANDMODESCANBEFOUNDBYSOLVINGTHEABOVEEQUATION44COMPUTATIONALGORITHMSOFCABLESTAYEDBRIDGEANALYSISTHEALGORITHMSFORSHAPEFINDINGCOMPUTATION,STATICDEFLECTIONANALYSISANDVIBRATIONANALYSISOFCABLESTAYEDBRIDGESAREBRIEFLYSUMMARIZEDINTHEFOLLOWING441INITIALSHAPEANALYSIS1INPUTOFTHEGEOMETRICANDPHYSICALDATAOFTHEBRIDGE2INPUTOFTHEDEADLOADOFGIRDERSANDTOWERSANDSUITABLYESTIMATEDINITIALFORCESINCABLESTAYS3FINDEQUILIBRIUMPOSITIONILINEARPROCEDURELINEARCABLEANDBEAMCOLUMNSTIFFNESSELEMENTSAREUSEDLINEARCONSTANTCOORDINATETRANSFORMATIONCOEFFICIENTSAJAREUSEDESTABLISHTHELINEARSYSTEMSTIFFNESSMATRIXKBYASSEMBLINGELEMENTSTIFFNESSMATRICESSOLVETHELINEARSYSTEMEQUATIONFORQEQUILIBRIUMPOSITIONNOEQUILIBRIUMITERATIONISCARRIEDOUTIINONLINEARPROCEDURENONLINEARCABLESWITHSAGEFFECTANDBEAMCOLUMNELEMENTSAREUSEDNONLINEARCOORDINATETRANSFORMATIONCOEFFICIENTSAJAJ，AREUSEDESTABLISHTHETANGENTSYSTEMSTIFFNESSMATRIX2KSOLVETHEINCREMENTALSYSTEMEQUATIONFORQEQUILIBRIUMITERATIONISPERFORMEDBYUSINGTHENEWTONRAPHSONMETHOD4SHAPEITERATION5OUTPUTOFTHEINITIALSHAPEINCLUDINGGEOMETRICSHAPEANDELEMENTFORCES6FORLINEARSTATICDEFLECTIONANALYSIS,ONLYLINEARSTIFFNESSELEMENTSANDTRANSFORMATIONCOEFFICIENTSAREUSEDANDNOEQUILIBRIUMITERATIONISCARRIEDOUT443VIBRATIONANALYSIS1INPUTOFTHEGEOMETRICANDPHYSICALDATAOFTHEBRIDGE2INPUTOFTHEINITIALSHAPEDATAINCLUDINGINITIALGEOMETRYANDINITIALELEMENTFORCES3SETUPTHELINEARIZEDSYSTEMEQUATIONOFFREEVIBRATIONSBASEDONTHEINITIALSHAPE4FINDVIBRATIONFREQUENCIESANDMODESBYSUBSPACEITERATIONMETHODS,SUCHASTHERUTISHAUSERMETHOD5ESTIMATIONOFTHETRIALINITIALCABLEFORCESINTHERECENTSTUDYOFWANGANDLIN,THESHAPEFINDINGOFSMALLCABLESTAYEDBRIDGESHASBEENPERFORMEDBYUSINGARBITRARYSMALLORLARGETRIALINITIALCABLEFORCESTHERETHEITERATIONCONVERGESMONOTONOUSLY,ANDTHECONVERGENTSOLUTIONSHAVESIMILARRESULTS,IFDIFFERENTTRIALVALUESOFINITIALCABLEFORCESAREUSEDHOWEVERFORLARGECABLESTAYEDBRIDGES,SHAPEFINDINGCOMPUTATIONSBECOMEMOREDIFFICULTTOCONVERGEINNONLINEARANALYSIS,THENEWTONTYPEITERATIVECOMPUTATIONCANCONVERGE,ONLYWHENTHEESTIMATEDVALUESOFTHESOLUTIONISLOCATEINTHENEIGHBORHOODOFTHETRUEVALUESDIFFICULTIESINCONVERGENCEMAYAPPEAR,WHENTHESHAPEFINDINGANALYSISOFCABLESTAYEDBRIDGESISSTARTEDBYUSEOFARBITRARYSMALLINITIALCABLEFORCESSUGGESTEDINTHEPAPERSOFWANGETALTHEREFORE,TOESTIMATEASUITABLETRIALINITIALCABLEFORCESINORDERTOGETACONVERGENTSOLUTIONBECOMESIMPORTANTFORTHESHAPEFINDINGANALYSISINTHEFOLLOWING,SEVERALMETHODSTOESTIMATETRIALINITIALCABLEFORCESWILLBEDISCUSSED51BALANCEOFVERTICALLOADS52ZEROMOMENTCONTROL53ZERODISPLACEMENTCONTROL54CONCEPTOFCABLEEQUIVALENTMODULUSRATIO55CONSIDERATIONOFTHEUNSYMMETRYIFTHEESTIMATEDINITIALCABLEFORCESAREDETERMINEDINDEPENDENTLYFOREACHCABLESTAYBYTHEMETHODSMENTIONEDABOVE,THEREMAYEXISTUNBALANCEDHORIZONTALFORCESONTHETOWERINUNSYMMETRICCABLESTAYEDBRIDGESFORSYMMETRICARRANGEMENTSOFTHECABLESTAYSONTHECENTRALMAINSPANANDTHESIDESPANWITHRESPECTTOTHETOWER,THERESULTANTOFTHEHORIZONTALCOMPONENTSOFTHECABLESTAYSACTINGONTHETOWERISZERO,IE,NOUNBALANCEDHORIZONTALFORCESEXISTONTHETOWERFORUNSYMMETRICCABLESTAYEDBRIDGES,INWHICHTHEARRANGEMENTOFCABLESTAYSONTHECENTRALMAINSPANANDTHESIDESPANISUNSYMMETRIC,ANDIFTHEFORCESOFCABLESTAYSONTHECENTRALSPANANDTHESIDESPANAREDETERMINEDINDEPENDENTLY,EVIDENTLYUNBALANCEDHORIZONTALFORCESWILLEXISTONTHETOWERANDWILLINDUCELARGEBENDINGMOMENTSANDDEFLECTIONSTHEREINTHEREFORE,FORUNSYMMETRICCABLESTAYEDBRIDGES,THISPROBLEMCANBEOVERCOMEASFOLLOWSTHEFORCEOFCABLESTAYSONTHECENTRALMAINSPANTIMCANBEDETERMINEDBYTHEMETHODSMENTIONEDABOVEINDEPENDENTLY,WHERETHESUPERSCRIPTMDENOTESTHEMAINSPAN,THESUBSCRIPTIDENOTESTHEITHCABLESTAYTHENTHEFORCEOFCABLESTAYSONTHESIDESPANISFOUNDBYTAKINGTHEEQUILIBRIUMOFHORIZONTALFORCECOMPONENTSATTHENODEONTHETOWERATTACHEDWITHTHECABLESTAYS,IE,TIMCOSITISCOSI,ANDTISTIMCOSI/COSI,WHEREIISTHEANGLEBETWEENTHEITHCABLESTAYANDTHEGIRDERONTHEMAINSPAN,ANDI,ANGLEBETWEENTHEITHCABLESTAYANDTHEGIRDERONTHESIDESPAN6EXAMPLESINTHISSTUDY,TWODIFFERENTTYPESOFSMALLCABLESTAYEDBRIDGESARETAKENFROMLITERATURE,ANDTHEIRINITIALSHAPESWILLBEDETERMINEDBYTHEPREVIOUSLYDESCRIBEDSHAPEFINDINGMETHODUSINGLINEARANDNONLINEARPROCEDURESFINALLY,AHIGHLYREDUNDANTSTIFFCABLESTAYEDBRIDGEWILLBEEXAMINEDACONVERGENCETOLERANCEE104ISUSEDFORBOTHTHEEQUILIBRIUMITERATIONANDTHESHAPEITERATIONTHEMAXIMUMNUMBEROFITERATIONCYCLESISSETAS20THECOMPUTATIONISCONSIDEREDASNOTCONVERGENT,IFTHENUMBEROFTHEITERATIONCYCLESEXCEEDS20THEINITIALSHAPESOFTHEFOLLOWINGTWOSMALLCABLESTAYEDBRIDGESINSECTIONS61AND62AREFIRSTDETERMINEDBYUSINGARBITRARYTRIALINITIALCABLEFORCESTHEITERATIONCONVERGESMONOTONOUSLYINTHESETWOEXAMPLESTHEIRCONVERGENTINITIALSHAPESCANBEOBTAINEDEASILYWITHOUTDIFFICULTIESTHEREAREONLYSMALLDIFFERENCESBETWEENTHEINITIALSHAPESDETERMINEDBYTHELINEARANDTHENONLINEARCOMPUTATIONCONVERGENTSOLUTIONSOFFERSIMILARRESULTS,ANDTHEYAREINDEPENDENTOFTHETRIALINITIALCABLEFORCES7CONCLUSIONTHETWOLOOPITERATIONWITHLINEARANDNONLINEARCOMPUTATIONISESTABLISHEDFORFINDINGTHEINITIALSHAPESOFCABLESTAYEDBRIDGESTHISMETHODCANACHIEVETHEARCHITECTURALLYDESIGNEDFORMHAVINGUNIFORMPRESTRESSDISTRIBUTION,ANDSATISFIESALLEQUILIBRIUMANDBOUNDARYCONDITIONSTHEDETERMINATIONOFTHEINITIALSHAPEISTHEMOSTIMPORTANTWORKINTHEANALYSISOFCABLESTAYEDBRIDGESONLYWITHACORRECTINITIALSHAPE,AMEANINGFULANDACCURATEDEFLECTIONAND/ORVIBRATIONANALYSISCANBEACHIEVEDBASEDONNUMERICALEXPERIMENTSINTHESTUDY,SOMECONCLUSIONSARESUMMARIZEDASFOLLOWS1NOGREATDIFFICULTIESAPPEARINCONVERGENCEOFTHESHAPEFINDINGOFSMALLCABLESTAYEDBRIDGES,WHEREARBITRARYINITIALTRIALCABLEFORCESCANBEUSEDTOSTARTTHECOMPUTATIONHOWEVERFORLARGESCALECABLESTAYEDBRIDGES,SERIOUSDIFFICULTIESOCCURREDINCONVERGENCEOFITERATIONS2DIFFICULTIESOFTENOCCURINCONVERGENCEOFTHESHAPEFINDINGCOMPUTATIONOFLARGECABLE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