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ABSTRACTPARTIIIOFTHISTHREEPARTSERIESOFPAPERSDESCRIBESTHESYNTHESISOFROLLERANDSLIDINGHYDRAULICSTEELGATESTRUCTURESPERFORMEDBYTHEMIXEDINTEGERNONLINEARPROGRAMMINGMINLPAPPROACHTHEMINLPAPPROACHENABLESTHEDETERMINATIONOFTHEOPTIMALNUMBEROFGATESTRUCTURALELEMENTSGIRDERS,PLATES,OPTIMALGATEGEOMETRY,OPTIMALINTERMEDIATEDISTANCESBETWEENSTRUCTURALELEMENTSANDALLCONTINUOUSANDSTANDARDCROSSECTIONALSIZESFORTHISPURPOSE,SPECIALLOGICALCONSTRAINTSFORTOPOLOGYALTERATIONSANDINTERCONNECTIONRELATIONSBETWEENTHEALTERNATIVEANDFIXEDSTRUCTURALELEMENTSAREFORMULATEDTHEYHAVEBEENEMBEDDEDINTOAMATHEMATICALOPTIMIZATIONMODELFORROLLERANDSLIDINGSTEELGATESTRUCTURESGATOPGATOPHASBEENDEVELOPEDACCORDINGTOASPECIALMINLPMODELFORMULATIONFORMECHANICALSUPERSTRUCTURESMINLPMS,INTRODUCEDINPARTSIANDIITHEMODELCONTAINSANECONOMICOBJECTIVEFUNCTIONOFSELFMANUFACTURINGANDTRANSPORTATIONCOSTSOFTHEGATEASTHEGATOPMODELISNONCONVEXANDHIGHLYNONLINEAR,ITISSOLVEDBYMEANSOFTHEMODIFIEDOA/ERALGORITHMACCOMPANIEDBYTHELINKEDTWOPHASEMINLPSTRATEGY,BOTHIMPLEMENTEDINTHETOPCOMPUTERCODEANEXAMPLEOFTHESYNTHESISISPRESENTEDASACOMPARATIVEDESIGNRESEARCHWORKOFTHEALREADYERECTEDROLLERGATE,THESOCALLEDINTAKEGATEINASWANIIINEGYPTTHEOPTIMALRESULTYIELDS294PERCENTOFNETSAVINGSWHENCOMPAREDTOTHEACTUALCOSTSOFTHEERECTEDGATE1998JOHNWILEYOPTIMIZATIONTOPOLOGYOPTIMIZATIONDISCRETEVARIABLEOPTIMIZATIONMIXEDINTEGERNONLINEARPROGRAMMINGMINLPTHEMODIFIEDOA/ERALGORITHMMINLPSTRATEGYHYDRAULICGATESLIDINGGATEROLLERGATEASWAN1INTRODUCTIONTHISPAPERDESCRIBESTHEMIXEDINTEGERNONLINEARPROGRAMMINGMINLPAPPROACHTOTHESYNTHESISOFROLLERANDSLIDINGGATESTRUCTURES,IETHESIMPLESTTYPESAMONGVERTICALLIFTHYDRAULICSTEELGATES,SEEFIGURE1ROLLERANDSLIDINGGATESAREALSOREGARDEDASTHEMOSTFREQUENTLYMANUFACTUREDTYPESOFHYDRAULICSTEELGATESFORHEADWATERCONTROLTHEYAREUSEDTOREGULATETHEWATERSTREAMONHYDROELECTRICPLANTS,DAMSORSPILLWAYSASHYDRAULICSTEELGATESAREVERYSPECIALSTRUCTURES,ONLYAFEWAUTHORSHAVEDISCUSSEDTHEIROPTIMIZATION,EGKRAVANJAETAL,1DJONGELINGANDKOLKMANASWELLASALMQUISTETALPARTICULARINTERESTWASSHOWNINTHEOPTIMIZATIONNOTOFTHESEROLLERANDSLIDINGGATESBUTOFSIMILARSTRUCTURESINSUCHINVESTIGATIONS,VANDERPLAATSANDWEISSHAARASWELLASGURDALETALOPTIMIZEDSTIFFENEDLAMINATEDCOMPOSITEPANELS,BUTLERANDRINGERTZOPTIMIZEDSTIFFENEDPANELS,FARKASANDJARMAI1OPTIMIZEDWELDEDRECTANGULARCELLULARPLATES,FINCKENORETAL1TREATEDSKINSTRINGERCYLINDERSANDGENDYETAL1STIFFENEDPLATESALMOSTALLAUTHORSUSEDNONLINEARPROGRAMMINGNLPTECHNIQUESGURDALETALPROPOSEDTHEGENETICALGORITHM,WHILEKRAVANJAETAL1DINTRODUCEDMINLPALGORITHMSANDSTRATEGIESTOTHESIMULTANEOUSTOPOLOGYANDPARAMETEROPTIMIZATIONOFTHEGATEINPARTSIOFTHISTHREEPARTSERIESOFPAPERS,AGENERALVIEWOFTHEMINLPAPPROACHTOTHESIMULTANEOUSTOPOLOGYANDPARAMETEROPTIMIZATIONOFSTRUCTURESISPRESENTEDPARTIIDESCRIBESTHEEXTENSIONTOTHESIMULTANEOUSSTANDARDDIMENSIONOPTIMIZATIONBASEDONTHESUPERSTRUCTUREAPPROACHDEFINEDINPARTSIANDII,THEMAINOBJECTIVEOFTHISPAPERPARTIIIISTHEMINLPSYNTHESISOFROLLERANDSLIDINGHYDRAULICSTEELGATESTRUCTURES,OBTAINEDATMINIMALGATECOSTSANDSUBJECTEDTODEFINEDDESIGN,MATERIAL,STRESS,DEFLECTIONANDSTABILITYCONSTRAINTSASTHEMINLPAPPROACHENABLESSIMULTANEOUSTOPOLOGY,PARAMETERANDSTANDARDDIMENSIONOPTIMIZATION,ANUMBEROFGATESTRUCTURALELEMENTSGIRDERSANDPLATES,THEGATEGLOBALGEOMETRY,INTERMEDIATEDISTANCESBETWEENSTRUCTURALELEMENTSANDALLCONTINUOUSANDSTANDARDDIMENSIONSAREOBTAINEDSIMULTANEOUSLYTHISLASTPARTOFTHETHREEPARTSERIESOFPAPERSISDIVIDEDINTOTHREEMAINSECTIONS1SECTION2DESCRIBESHOWDIFFERENTTOPOLOGYANDSTANDARDDIMENSIONALTERNATIVESAREPOSTULATEDANDHOWTHEIRINTERCONNECTIONRELATIONSAREFORMULATEDBYMEANSOFEXPLICITLOGICALCONSTRAINTSINORDERTOPERFORMTOPOLOGYANDSTANDARDDIMENSIONALTERATIONSWITHINTHEOPTIMIZATIONPROCEDURE2SECTION3REPRESENTSAGENERALMINLPOPTIMIZATIONMODELFORROLLERANDSLIDINGGATESTRUCTURESGATOP3FINALLY,INSECTION4,THEPROPOSEDSUPERSTRUCTUREMINLPAPPROACHISAPPLIEDTOTHESYNTHESISOFANALREADYERECTEDROLLERGATE,THESOCALLEDINTAKEGATEINASWANIIINEGYPT2SUPERSTRUCTUREALTERNATIVESANDTHEMODELLINGOFTHEIRDISCRETEDECISIONS21POSTULATIONOFTOPOLOGYANDSTANDARDDIMENSIONALTERNATIVESTHEFIRSTSTEPINTHESYNTHESISOFTHEGATEISTHEGENERATIONOFANMINLPSUPERSTRUCTUREINWHICHDIFFERENTTOPOLOGY/STRUCTUREANDSTANDARDDIMENSIONALTERNATIVESAREEMBEDDEDTOBESELECTEDASTHEOPTIMALRESULTTHEGATESUPERSTRUCTUREALSOCONTAINSTHEREPRESENTATIONOFSTRUCTURALELEMENTSWHICHMAYCONSTRUCTEACHPOSSIBLESTRUCTUREALTERNATIVEASWELLASDIFFERENTSETSOFDISCRETEVALUES,DEFINEDFOREACHSTANDARDDIMENSIONALTERNATIVEASBOTHTHEROLLERANDSLIDINGGATESHAVEALMOSTTHESAMESTRUCTURE,ITWASREASONABLETOPROPOSEASUPERSTRUCTURE,WHICHCOULDWELLBEUSEFULFORBOTHOFTHEM211TOPOLOGYALTERNATIVESTHEGATESUPERSTRUCTURETYPICALLYINCLUDESAREPRESENTATIONOFMAINGATEELEMENTS,WHEREEACHGATEELEMENTISCOMPOSEDOFVARIOUSSTRUCTURALELEMENTS,SUCHASHORIZONTALGIRDERS,VERTICALGIRDERS,STIFFENERSANDPLATEELEMENTSOFTHESKINPLATE,SEEFIGURE2THESUPERSTRUCTURECOMPRISESNMAINGATEELEMENTS,N3N,EACHCONTAININGMHORIZONTALGIRDERS,M3M,THE32VNUMBEROFVERTICALGIRDERSTHROUGHTHEENTIREGATE,V3,ANDTHECORRESPONDINGM122VNUMBEROFSKINPLATEELEMENTSTOEACHMTHHORIZONTALGIRDEROFTHENTHMAINGATEELEMENTANEXTRABINARYVARIABLEYN,MISASSIGNEDTHENUMBEROFHORIZONTALGIRDERSANDCORRESPONDINGPLATEELEMENTSOFTHESKINPLATE,DISTRIBUTEDOVERTHENTHMAINGATEELEMENT,CANTHEREFOREBEDETERMINEDSIMPLYBYNOTE,MNYTHATTHEPROPOSEDMINIMALNUMBEROFIDENTICALVERTICALGIRDERSIS3ANDTHATTHEYCANTAKEONLYODDNUMBERSIFABINARYVARIABLEYVISASSIGNEDTOEACHV,VV,THENUMBEROFVERTICALGIRDERSCANBEOBTAINEDBY32VYVINTHESAMEWAYANEVENNUMBER22VYVOFEQUALHORIZONTALPARTITIONSOFTHEENTIREGATEISPROPOSEDINTHECASEOFVERTICALGIRDERS,WECANSEETHATTHESTRUCTURALELEMENTSCANALSOBEDETERMINEDBYSUITABLELINEARCOMBINATIONSOFBINARYVARIABLESAMONGTHEMAXIMALNUMBERMMAXOFHORIZONTALGIRDERSPEREACHMAINGATEELEMENT,THEUPPERANDLOWERGIRDERSTOGETHERWITHTHEMINIMALNUMBERMMINOFINTERMEDIATEHORIZONTALGIRDERSANDTHEADJOININGSKINPLATEELEMENTSARETREATEDASFIXEDSTRUCTURALELEMENTS,WHICHAREALWAYSPRESENTINTHEOPTIMIZATIONALLOTHERMMAXMMININTERMEDIATEHORIZONTALGIRDERSWITHTHECORRESPONDINGNUMBEROFADJOINEDSKINPLATEELEMENTSARETHENREGARDEDASALTERNATIVESTRUCTURALELEMENTS,WHICHMAYEITHERDISAPPEARORBESELECTEDSINCEONLYALTERNATIVESTRUCTURALELEMENTSPARTICIPATEINTHEDISCRETEOPTIMIZATION,THESIZEOFTHEDISCRETEDECISIONSISSIGNIFICANTLYREDUCEDEACHPOSSIBLECOMBINATIONBETWEENSELECTEDALTERNATIVESTRUCTURALELEMENTSANDFIXEDSTRUCTURALELEMENTSFORMSANEXTRAGATETOPOLOGY/STRUCTUREALTERNATIVE212STANDARDDIMENSIONALTERNATIVESFOURSTANDARDDIMENSIONSAREADDITIONALLYDEFINEDTOREPRESENTTHESTANDARDTHICKNESSESOFSHEETIRONPLATESTHETHICKNESSOFTHESKINPLATETSNFOREACHNTHMAINGATEELEMENT,THEFLANGETHICKNESSOFTHEHORIZONTALGIRDERTFN,THEWEBTHICKNESSOFTHEOUTERHORIZONTALGIRDERM1OR,OUTWNMMM,ANDTHEWEBTHICKNESSOFTHEINNERHORIZONTALGIRDER,1MMIT,SINCETHETHICKNESSTSNHASACOMMONVALUEFORTHEENTIRESKINPLATEOFTHENTHMAINGATEELEMENTANDTHETARETHESAMEFORALLHORIZONTALGIRDERSOFTHENTHMAINGATEELEMENT,IETHEYCORRESPONDTOTHECOMMONSTANDARDDESIGNVARIABLESFORTHESUPERSTRUCTUREORITSPARTFROMTHESPECIAL,STCOMDMINLPMSMODELFORMULATIONINPARTIISIMILARLY,THEWEBTHICKNESSES,WHICHTAKEACOMMONVALUEFORBOTHOUTERHORIZONTALGIRDERSOFTHEOUTWNMNTHMAINGATEELEMENT,AND,WHICHARETHESAMEFORALLTHEINNERINWMTHORIZONTALGIRDERS,CORRESPONDTOTHECOMMONSTANDARDDESIGNVARIABLESOFTHEALTERNATIVESTRUCTURALELEMENTSANEXTRASETOFDISCRETEVALUES,STCOMADOFSTANDARDDIMENSIONALTERNATIVESANDASPECIALSETOFTHESAMESIZEOFBINARYVARIABLESAREINTRODUCEDFOREACHMENTIONEDSTANDARDDIMENSIONEACHSTANDARDDIMENSIONTSNSHALLBEEXPRESSEDWITHINTHEGIVENISTANDARDDIMENSIONALTERNATIVES,II,STANDARDDIMENSIONTFNBYKALTERNATIVES,KK,STANDARDDIMENSIONBYPALTERNATIVES,PP,ANDSTANDARD,OUTWNMDIMENSIONBYRALTERNATIVES,RRTHEVECTOROFIBINARYVARIABLES,INWMTYN,IANDTHEVECTOROFIDISCRETEVALUESQN,IAREASSIGNEDTOTHEVARIABLETSN,THEVECTORSOFKELEMENTSYN,KANDQN,KTOTHEVARIABLETFN,THEVECTORSOFPELEMENTSYN,PANDQN,KTOTHEVARIABLEANDTHEVECTORSOFRELEMENTS,INWMTYN,VANDQN,VTOTHEVARIABLECONSEQUENTLY,THEOVERALLVECTOROFBINARY,INWMTVARIABLESASSIGNEDTOTHEGATESUPERSTRUCTUREISYYN,M,Y,V,YN,I,YN,P,YN,V22MODELINGOFDISCRETEDECISIONSTHEPOSTULATEDGATESUPERSTRUCTUREOFTOPOLOGYANDSTANDARDDIMENSIONALTERNATIVESCANBEFORMULATEDASANMINLPPROBLEMUSINGASPECIALMINLPMODELFORMULATIONMINLPMSFORSIMULTANEOUSTOPOLOGY,PARAMETERANDSTANDARDDIMENSIONOPTIMIZATIONOFMECHANICALSUPERSTRUCTURES,DESCRIBEDINPARTIIASCANBESEENFROMTHEMINLPMS,THEOBJECTIVEFUNCTIONISTYPICALLYSUBJECTEDTOSTRUCTURALANALYSISANDLOGICALCONSTRAINTSWHILESTRUCTURALANALYSISCONSTRAINTSREPRESENTAMATHEMATICALMODELOFASYNTHESIZEDSTRUCTURE,LOGICALCONSTRAINTSAREUSEDFORTHEEXPLICITMODELINGOFLOGICALDECISIONSMODELINGOFDISCRETEDECISIONSTODETERMINETOPOLOGYALTERNATIVESISANOBJECTIVEOFTHEHIGHESTIMPORTANCEFORTHESYNTHESISINORDERTOPERFORMTHESEDECISIONSWITHINTHEMINLPOPTIMIZATION,INTERCONNECTIONLOGICALCONSTRAINTSDYRD,PRAREPROPOSEDWHILEVARIABLES,THEIRBOUNDSANDMOSTOFTHECONSTRAINTSOFTHEMINLPMSMODELFORMULATIONAREREPRESENTEDINPARTII,INTERCONNECTIONLOGICALCONSTRAINTSANDTHECONSTRAINTSDEFININGTOPOLOGYALTERATIONSAREDESCRIBEDINTHISSECTIONTHELATTERONESAREDERIVEDFROMTHEFOLLOWINGBASICINTEGERORMIXEDINTEGERLOGICALCONSTRAINTSAMULTIPLECHOICECONSTRAINTSFORSELECTINGAMONGASETOFUNITSISELECTEXACTLYMUNITS1IIYSELECTMUNITSATTHEMOST2IIMSELECTATLEASTMUNITS3IIYBIFTHENCONDITIONSIFUNITKISSELECTEDTHENUNITIMUSTBESELECTEDYKYI04CACTIVATIONORDEACTIVATIONOFCONTINUOUSVARIABLES,INEQUALITIESOREQUATIONS1EXAMPLETORELATECONTINUOUSVARIABLEXTOTHESCALARVALUEUXUY5IF1,0YXUIFYX2ANOPPOSITERELATIONXU（1Y）610,YXIFYXU3EXAMPLEFORTHEBOUNDSOFCONTINUOUSVARIABLEXX1,0YXXUPY7IF1,0,0UPYXIFYXDINTEGERCUTSCONSTRAINTELIMINATESUNNECESSARYINTEGERCOMBINATIONSYK0,1MEGTHOSEFOUNDATPREVIOUSMINLPITERATIONS,1,I8K|BIKKIIIBINIYWHERE,I|KI|Y0INORDERTODESCRIBETHEMODELINGOFDISCRETEDECISIONS,AGENERALGATESUPERSTRUCTUREFROMFIGURE2ISADDRESSEDINWHICHTHEDEFINEDSTRUCTURALELEMENTSARETYPICALLYHORIZONTALANDVERTICALGIRDERSASTHEMODELINGOFVERTICALGIRDERSISSIMPLIFIEDANDNEEDSNOSPECIALINTERCONNECTIONLOGICALCONSTRAINTS,THEMODELINGOFDISCRETEDECISIONSREGARDINGHORIZONTALGIRDERSPROVEDTOBEMORESOPHISTICATED221MODELINGOFTOPOLOGYALTERATIONSLETUSCONSIDERTHEVERTICALCROSSSECTIONOFTHEGATEELEMENTSUPERSTRUCTUREWITHFIXEDANDALTERNATIVEHORIZONTALGIRDERS,SEEFIGURE3ATHENUMBEROFFIXEDANDALTERNATIVEGIRDERSANDTHEIRLOCATIONSINTHESUPERSTRUCTURECANBEDESCRIBEDBYTHEFOLLOWINGLOGICALCONSTRAINTS9MINMAXMY101,23,111MAXYLOGICALCONSTRAINT9DEFINESTHEMINIMALMMINANDMAXIMALMMAXNUMBEROFSTRUCTURALELEMENTSGIRDERSWHILENUMBERMMINREPRESENTSTHENUMBEROFFIXEDSTRUCTURALELEMENTS,THEDIFFERENCEBETWEENTHEMAXIMALANDMINIMALNUMBEROFELEMENTSMMAXMMINGIVESTHENUMBEROFALTERNATIVESTRUCTURALELEMENTSCONSTRAINT10DEFINESTHEDIRECTIONOFTHEREMOVALOFALTERNATIVEELEMENTSFROMTHETOPDOWNTHESUPERSTRUCTUREFROMFIGURE3ISEVIDENTTHATTHEMOSTUPPERELEMENTISTHEFIXEDONE,WHICHISSETBYTHECONSTRAINT11ITTHENFURTHERFOLLOWSFROMCONSTRAINT10THATALLTHERESTFIXEDELEMENTSARELOCATEDATTHEBOTTOMOFTHESUPERSTRUCTUREHENCE,CONSTRAINTS911REPRESENTTHEEXPLICITMODELFORTOPOLOGYALTERATIONSOFHORIZONTALGIRDERS222MODELINGOFINTERCONNECTIONRELATIONSINTERCONNECTIONRELATIONSBETWEENALTERNATIVEANDFIXEDSTRUCTURALELEMENTSWITHINTHESUPERSTRUCTUREREQUIRESPECIALATTENTIONPAIDTOTHESTRUCTURALSYNTHESISPERFORMEDBYTHEMINLPAPPROACHINTERCONNECTIONRELATIONSEITHERRESTORETHECONNECTIONSBETWEENCURRENTLYSELECTEDEXISTINGSTRUCTURALELEMENTSORCANCELTHERELATIONSBETWEENCURRENTLYREJECTEDDISAPPEARINGSTRUCTURALELEMENTSSINCEMINLPMETHODSOPTIMIZETHETOPOLOGYANDPARAMETERSSIMULTANEOUSLY,ITISNECESSARYTODEFINETHESEINTERCONNECTIONRELATIONSINANEXPLICITEEQUATIONALFORM,SOTHATTHEYCANENABLEINTERCONNECTIONSANDDISCONNECTIONSBETWEENTHEELEMENTSDURINGTHEOPTIMIZATIONPROCESSSPECIALINTERCONNECTIONLOGICALCONSTRAINTSFORINTERCONNECTIONRELATIONSBETWEENTHEALTERNATIVEANDFIXEDSTRUCTURALELEMENTSHAVEBEENPROPOSEDTHEYWILLBEEMBEDDEDINTOTHEMINLPOPTIMIZATIONMODELOFTHEGATESTRUCTURE,ENABLINGTHELATTERTOTHUSBECOMESELFSUFFICIENTFORAUTOMATICTOPOLOGYANDPARAMETEROPTIMIZATIONTHEMODELINGOFINTERCONNECTIONLOGICALCONSTRAINTS,HOWEVER,REQUIRESADDITIONALEFFORT,SINCEMOSTELEMENTCONSTRAINTSINCLUDEFUNCTIONSNOTONLYOFTHEIROWNVARIABLESBUTALSOOFTHEVARIABLESBELONGINGTOTHEIRADJOININGSTRUCTURALELEMENTSSUCHANEXAMPLEIS,EGTHECONSTRAINTOFTHEMOMENTOFINERTIAIN,MOFTHEMTHHORIZONTALGIRDEROFTHENTHMAINGATEELEMENTSEEEQUATION23INTHEFOLLOWINGSECTION,WHICHINCLUDESTHESUBSTITUTEDEXPRESSIONS6OFTHESKINPLATEEFFECTIVEWIDTHWITHTHE,NMSBHEIGHTSBETWEENGIRDERSANDTHESILLHM1ANDHM1OFBOTHADJOININGGIRDERSTHECONSTRAINTSOFTHEMTHINTERMEDIATEHORIZONTALGIRDERARETYPICALLYFUNCTIONSOFTHREEHEIGHTSHM1,HMANDHM1,ANDTWOVERTICALDISTANCESBETWEENHORIZONTALGIRDERS,ANDTHEDISTANCEIS1D1DMHDSIMPLYDEFINEDBYTHECONSTRAINTM2,3,M1121MHDTHEPROBLEMARISESIFHM1ISNOTDEFINEDWHENTHEADJOININGUPPERALTERNATIVEGIRDERTOTHEMTHHORIZONTALGIRDERDOESNOTEXISTFOREXAMPLE,LETUSCONSIDERTHETHIRDGIRDERINFIGURE3AWHICHISTHEUPPERMOSTEXISTINGINTERMEDIATEELEMENTINORDERTODEFINEH4SOASTOFULFILTHECONSTRAINTSOFGIRDER3,H4SHOULDTEMPORARILYBECOMEEQUALTOTHEHEIGHTOFTHEUPPERMOSTFIXEDGIRDERH6HMFIGURE3BTHEMAINIDEAISTOSETALLHEIGHTSOFNONEXISTINGINTERMEDIATEGIRDERSGIRDERS4AND5INFIGURE3ATOTHEVALUEOFH6BYMEANSOFTHELOGICALCONSTRAINTS,M2,3,M113MUPHDY,M2,3,M114,LOWEXNOTE,THATCONSTRAINTS13AND14RESTORETHEUPPERANDLOWERMUPHD,MLOWEXHDBOUNDSOFTHEDISTANCEWHENTHECORRESPONDINGGIRDEREXISTSYM1MHDANDSETITTOZERO,OTHERWISEWHENTHEDISTANCEISZERO,ITFOLLOWSFROMCONSTRAINT12THATHMBECOMESEQUALTOHINTHISWAYALLDISTANCESANDHEIGHTSAREDEFINEDFORANYGIRDERTHATBECOMESTHEUPPERMOSTSELECTEDINTERMEDIATEONEANDREESTABLISHESITSCONNECTIONTOTHEUPPERMOSTFIXEDGIRDERASTHEUPPERMOSTSELECTEDINTERMEDIATEGIRDERISCONNECTEDTOTHEUPPERMOSTFIXEDGIRDEREGGIRDER3TOGIRDER6INFIGURE3A,THELATTERSHOULDALSO,INTHESIMILARWAY,BECONNECTEDTOTHEFORMERONEGIRDER6TOGIRDER3INFIGURE3ACONSTRAINTSFORTHEUPPERMOSTFIXEDGIRDERARETHENJUSTFUNCTIONSOFTWOHEIGHTSHMANDHM1ANDADISTANCETHE1MHDPROBLEMARISESIFSOMEINTERMEDIATEGIRDERSDONOTEXIST,EGGIRDERS4AND5INFIGURE3AINSUCHCASES,HM1SHOULDNOTBECONSIDEREDINSTEAD,THEHEIGHTHSH3INFIGURE3AOFTHEUPPERSELECTEDINTERMEDIATEGIRDERSHOULDBEDEFINEDANDSUBSTITUTEDFORHM1THEVERTICALDISTANCEDHSOFTHEUPPERMOSTSELECTEDINTERMEDIATEGIRDERISTHENDEFINEDBYTHECONSTRAINT15HSMSDTHESELECTIONOFTHEHEIGHTHSAMONGALLHMCANBEPERFORMEDBYTHEFOLLOWINGLOGICALCONSTRAINTS,M2,3,M21611UPUPSMSMSHY,M2,3,M217SSSH1811UPSMSMY19SSHCONSTRAINTS16AND17SETHSTOTHEHEIGHTHMOFTHATMTHEXISTINGHORIZONTALGIRDERYM1,WHICHHASTHEEXISTINGADJOININGLOWERGIRDERYM11ANDTHEDISAPPEARINGADJOININGUPPERGIRDERYM10HOWEVER,FORM“M1THENEXTUPPERGIRDERALWAYSEXISTS,SINCEITISFIXED,IEYM1INTHISCASEWENEEDADDITIONALCONSTRAINTS,IE18AND19,WHICHSETHSTOTHEHEIGHTHM13THEMINLPOPTIMIZATIONMODELFORROLLERANDSLIDINGHYDRAULICSTEELG
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