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1318/JOUGEOMETRICNONLINEARANALYSISOFFRAMESTRUCTURESBYPSEUDODISTORTIONSByPrafullaV.Makode,1RossB.Corotis,2andMartinR.Ramirez3ABSTRACT:Inthispaper,thepreviouslydescribedpseudodistortionmethodisextendedforthecaseofgeo-metricnonlinearity.Thegeneraltheoryofstabilityisfollowedtoaccountforaxialforce-bendinginteractionandP2Deffects.Theresultingsecond-orderplastichingeanalysismaybesolvedbytheimpositionofpseudodistortionsonmembers,andthecompletebehaviorofthestructuremaybemodeledbyasingleelasticanalysisoftheoriginalframeandthedeterminationoftheappropriatepseudodistortions.Theadvancedtheoryofthepseudodistortionmethodisderivedandthenappliedtosampleframes.Resultsarecomparedtoacon-ventionalnonlinearanalysisinvolvingreformulationoftheglobalstiffnessmatrix.INTRODUCTIONThepseudodistortionmethod(PDM)hasbeendevelopedinacompanionpaper(Makodeetal.1999)forthereanalysisofframestructures.Themethodwasderivedasanextensionforatechniqueapplicabletopin-jointedstructures(Holnicki-SzulcandGierlinski1989;Holnicki-Szulc1991),andisanefficientapproachforoptimizationandreliabilitycomputa-tionsinwhicharelativelyfewmembersizesarechangedateachiteration(Makode1995).Eventhoughthemethodisbasedonsuperposition,thecompanionpaper(Makodeetal.1999)showedhowitcouldbeusedforplastichingeanalysis.Inthispaper,thePDMwillbeextendedtoincludetheeffectsofgeometricnonlinearity,thusenablingsecond-orderplasticanalysisofframestructures.SECOND-ORDERANALYSISInframestructures,axialforcesactthroughtransversede-flectioncausedbythebendingeffecttoproduceadditionaldeflection(andmoments)inthemember(P2Deffect).Thedeflectionsandmomentscausedbytheprimary-bendingeffectarereferredtoasprimary,andtheadditionalonescausedbyaxial-forceeffectassecondary(ChenandLui1987).Iftheaxialforcesarelarge,forexample,inmultistorybuild-ings,thesecondaryeffectbecomessignificantandshouldbeincludedintheanalysisofthestructure.Duetothepresenceofthesecondaryeffect,theelasticloaddeflectionbehaviorofthestructurebecomesnonlinearandwillbereferredtoassec-ond-orderanalysis.Anapproachbasedonpseudodistortionwillbedevelopedtoperformsecond-order(elastic)analysis.Itwillthenbein-tegratedwiththeplastichingeformulationtodevelopaveryefficientapproachforsecond-orderplastichingeanalysis.GENERALTHEORYOFSECOND-ORDERANALYSISDuetothenonlinearnatureofsecond-orderanalysis,theexternalloadsareincreasedproportionallyinincrements,i.e.,lisincreasedbytheincrementofDl.Toobtaintheincre-1Struct.Engr.,Ammann&WhitneyConsultingEngineers.,NewYork,NY10014.2Dean,Coll.ofEngrg.andAppl.Sci.,Univ.ofColorado,Boulder,CO80309-0422.3Dir.,CurriculumandLearningAssessment,IllinoisMathematicsandSci.Acad.,Aurora,IL60506.Note.AssociateEditor:ScottA.Burns.DiscussionopenuntilApril1,2000.Separatediscussionsshouldbesubmittedfortheindividualpapersinthissymposium.Toextendtheclosingdateonemonth,awrittenre-questmustbefiledwiththeASCEManagerofJournals.ThemanuscriptforthispaperwassubmittedforreviewandpossiblepublicationonSep-tember22,1998.ThispaperispartoftheJournalofStructuralEngi-neering,Vol.125,No.11,November,1999.qASCE,ISSN0733-9445/99/0011-1318–1327/$8.001$.50perpage.PaperNo.19293.RNALOFSTRUCTURALENGINEERING/NOVEMBER1999mentalresponseofthestructure,theglobalstiffnessmatrixisformedandsolved.ThegeneralmemberstiffnessmatrixforanindividualmemberisAEAE00200LL12EI6EI12EI6EI0ff02ffABAB3232LLLL6EI4EI6EI2EI0ff02ffBCBD22LLLLmK̄=AEAE20000LL12EI6EI12EI6EI02f2f0f2fABAB3232LLLL6EI2EI6EI4EI0ff02ffBDBC22LLLL(1)whereA=cross-sectionalarea;E=modulusofelasticity;L=elementlength;I=momentofinertia;andfA,fB,fC,andfD=stabilityfunctions.Thestabilityfunctionsrepresenttheeffectofaxialforceonthebendingstiffness.Theexpressionsforstabilityfunctionsforcompressiveaxialforcesare(ChenandLui1987)3(kL)sinkLf=(2)A12f2(kL)(12coskL)f=(3)B6f(kL)(sinkL2coskL)f=(4)C4f(kL)(kL2sinkL)f=(5)D2fwheref=222coskL2kLsinkL;k2=P/EI;andP=axialload.Thestiffnessmatrixin(1)issimplifiedbyusingaTaylorseriesexpansionforthefiandf.IfonlythefirsttwotermsoftheTaylorseriesareretained,theresultingstiffnessmatrixwillbe(ChenandLui1987)mmm¯¯K=K1K(6)Gwhere=first-order(linear)elasticstiffnessmatrix,ex-mKpressedasAEAE00200LL12EI6EI12EI6EI002LLLL32326EI4EI6EI2EI002LLLL22mK=AEAE20000LL12EI6EI12EI6EI02202LLLL32326EI2EI6EI4EI002LLLL22(7)and=geometricstiffnessmatrix,expressedasmK̄G0000006PP6PP0205L105L10P2PLPPL0202m10151030K̄=(8)G0000006PP6PP020225L105L10PPLP2PL002210301015Aftereveryloadincrementtheadditionalresponseandmemberforcesareadded.Forthenextloadincrement,thetotalaxialloadfromthepreviousstepisusedtoformthememberstiffnessmatrix[(1)–(8)].Theprocessisrepeatedun-tilthedesiredloadlevelisreached.Intheconventionalmethodforsecond-orderanalysis,theglobalstiffnessmatrixisreformedandsolvedaftereveryloadincrement.Toavoidtherepeatedformulationoftheglobalstiffnessmatrix,pseudodistortionscanbeusedtosimulatetheeffectofaxialforceonthebendingstiffness(i.e.,tosimulatethechangeinstabilityfunctions).Theorderofthematrixtobesolvedaftereveryincrementis2m,wheremisthenumberofmem-bersthathaveasignificantaxialforceeffectontheirbendingstiffness.Whenthefractionofsuchmembersisrelativelylow,thePDMwillprovideasignificantsavingsincomputationaleffort.SECOND-ORDERANALYSISBYPDMDuringtheloadingprocess,theincrementalresponseoftheframeisrequiredateachloadstep.Intheformulationthatfollows,theequationswillbeexpressedintermsofloadandresponse.Insecond-orderanalysis,theseequationsareappliedforeachloadincrement.Forelementswithhighaxialload,theelementstiffnessma-trixisgivenby(1).ThisstiffnessmatrixisbasedonFig.1.Theindividualrowsin(1)areobtainedbyusingslope-deflec-tionequations,givenhereformember1-2b6Kf1212cd¯¯M̄=4Kfu12Kfu1(ȳ2ȳ)(9)12121211212212L12b6Kf1212dc¯¯M̄=2Kfu14Kfu1(ȳ2ȳ)(10)21121211212212L12bba6Kf6Kf12Kf121212121212¯¯¯¯F=2F=u1u1(ȳ2ȳ)(11)122112122LLL121212FIG.2.TransverseShearinMember1-2FIG.1.Force-DisplacementRelationshipinMember1-2whereand=stabilityfunctionsfA,fB,fC,andabcdf,f,f,fijijijijfD,respectively,formemberi-j.ThetransverseshearforceF̄12(andhenceF̄21)isthesumoftwoparts:(1)Transverseshearforceduetorotationofthemember[Fig.2(a)];andF̄912(2)transverseshearforceduetoswayofthemember[Fig.F̄0122(b)].Thus¯¯¯F=F91F0(12)121212ThetransverseshearforceduetorotationsisgivenbyF̄912[Fig.2(a)]¯¯M1M1221F̄9=(13)12L12substituting(9)and(10)into(13)givesbbb6Kf6Kf12Kf121212121212¯¯F̄9=u1u1(ȳ2ȳ)(14)1212122LLL121212where21bcdf=f1f(15)12121233Thetransverseshearforceduetoswayisgivenby[Fig.F̄0122(b)]¯2PD2P(ȳ2ȳ)1212F̄0==(16)12LL1212Adding(14)and(16)gives(11),inwhichP̄L1212abf=f2(17)121212K12JOURNALOFSTRUCTURALENGINEERING/NOVEMBER1999/1319Using(14)and(16)intheelementstiffnessmatrix[(1)],weget(formember1-2)AEAE00200LL12K6K12K6K12121212bbbb0ff02ff1212121222LLLL121212126K6K1212bcbd0f4Kf02f2Kf121212121212LL1212mK̄=AEAE20000LL12K6K12K6K12121212bbbb02f2f0f2f1212121222LLLL121212126K6K1212bdbc0f2Kf02f4Kf121212121212LL1212000000¯¯PP121202000LL12120000001000000¯¯PP121200020LL1212000000(18)whichmaybeexpressedasmmm¯¯¯K=K1K(19)12Theequilibriumequationofmember1-2ismmm¯¯¯P=Kd(20)substituting(19)into(20)mmmm¯P̄=(K1K)d(21)12Transferringtotheleft-handsidemK̄2mmmm¯¯¯¯¯P1(K)d=(K)d(22)21substitutingforandwegetmm¯¯KK210¯¯P(ȳ2ȳ)P121212¯LF1212M̄0121P̄021¯¯F2(ȳ2ȳ)P211212¯LM12210AEAE00200LL12K6K12K6K12121212bbbb0ff02ff1212121222LLLL121212126K6K1212bcbd0f4Kf02f2Kf121212121212LL1212=AEAE20000LL12K6K12K6K12121212bbbb02f2f0f2f1212121222LLLL121212126K6K1212bdbc0f2Kf02f4Kf121212121212LL12121320/JOURNALOFSTRUCTURALENGINEERING/NOVEMBER1999FIG.3.EquivalentShearLoadFIG.4.ConversionoffAintoTransverseShearLoadū1ȳ1ū1?ū2ȳ2ū2(23)where=equivalenttransverseshearloadthatin-mm¯¯(2K)d2ducesthesameeffectasP̄12(Fig.3).Foragivenloadstep,andP̄12arenotknownin(ȳ2ȳ)12advance,andtherespectivevaluesfromthepreviousstepareused.Substitutingfor1ontheleft-handsideofmmmm¯¯¯¯P9P(K)d2(22)mmm¯¯¯P9=(K)d(24)1Thus,theequilibriumequations[(20)]aretransformedinto(24).Fig.4showsthetransformationof(20)[Fig.4(a)]into(24)[Fig.4(b)].Thestabilityfunctionisconvertedtoaf12equivalenttransverseshearloads[Fig.4(b)].sT̄ijAssemblingthememberstiffnessmatrixandequivalenttransverseshearload,wegettheequilibriumequationsforthewholestructure¯¯¯P9=Kd(25)1Thesolutionof(25)isobtainedbyusingoriginalelasticanalysisP̄9=Kd(26)andtheappropriatepseudodistortions(formulationispre-sentedinthenextsection).Thus,inthePDM:(1)theP2Deffectissimulatedbyimposingequivalenttransverseshearloads;and(2)theaxialloadeffectonthebendingstiffnessissimulatedbyimposingpseudodistortionsattheendoftheaffectedmembers.FORMULATIONOFSECONDARYEFFECTBYPSEUDODISTORTIONSOneMemberConsiderthatthesolutionofaframe[Fig.5(a)]foracertainloadincrementisdesired,i.e.,thesolutionoftheglobalstiff-FIG.5.SimulationofSecondaryEffectbyPseudodistortionsnessequilibriumequation[(25)]isrequired.Supposeonlymember1-2hasasignificantsecondaryeffect.Thiseffectwillbeconsideredasstructuralresponsemodificationtothemem-berintheoriginalelasticlinearstructureandissimulatedbyimposingthepseudodistortionsandontheoriginalstruc-00uu12ture[Fig.5(c)].Toobtaintheexpressionforunknownpseudodistortions0u1andanytwoofthethreeequilibriumconditionscanbe0u,2useddM̄=M1M(27)121212dM̄=M1M(28)212121dF̄=F1F(29)121212Inthedevelopmentshownhere,(27)and(28)willbeused.Thesameexpressionwouldbeobtainedbyusinganycom-binationoftwo.Forthememberwithasignificantsecondaryeffect,(9)and(10)mustbeused.Formember1-2withpseudodistortionsthesebecomeb16Kf1212c1d12¯¯M̄=4Kfu12Kfu1(ȳ2ȳ)(30)12121211212212L12b26Kf1212d12c2¯¯M̄=2Kfu14Kfu1(ȳ2ȳ)(31)21121211212212L12whereand=generalstructuralmodi-b1b2c1c2d12f,f,f,f,fijijijij12ficationfactors.Allofthestructuralmodificationscanbede-rivedbydefiningthesefactorsappropriately.Forstandardsec-ond-orderanalysis,====andb1b2c1c2cbfff,fff,121212121212=d12dff.1212Inthecompanionpaper(Makodeetal.1999)thefollowingequationwasdevised:6K1200¯¯M̄=4Ku12Ku1(ȳ2ȳ)24Ku22Ku12121122121211222L12(32)Similarly(31)leadstob13(f21)12d12c2¯¯(f21)u12(f21)u1(ȳ2ȳ)12112212L1200=2u22u12(35)Solvingthesetwoequationsforand00uu12d12c1c2d12¯¯(f24f13)u12(f2f)u1212112122b2b13(f22f11)121201(ȳ2ȳ)=3u121L12(36)c1d12d12c2¯¯2(f2f)u1(f24f13)u1212112122b1b23(f22f11)121201(ȳ2ȳ)=3u122L12(37)Tosimplifytheaboveequations,thefollowingtermsareintroduced:1d12c12d12c2a=f24f13;a=f24f13(38a,b)ijijijijijij1c2d122c1d12b=2(f2f);b=2(f2f)(38c,d)ijijijijijij1b2b12b1b2g=3(f22f11);g=3(f22f11)(38e,f)ijijijijijijSubstituting(38)into(36)and(37)gives1g12110¯¯au1bu1(ȳ2ȳ)=3u(39)121122121L122g12220¯¯bu1au1(ȳ2ȳ)=3u(40)121122122L12Forstandardsecond-orderanalysis===b1b2bc1fff,f12121212=and=c2cd12dff,ff.12121212Substitutingin(38)itcanbeseenthat==121aa,b121212and=212b,gg.121212Let====and==121212***aaa,bbb,ggg,121212121212121212andsubstitutethemin(39)and(40)*g120¯¯**au1bu1(ȳ2ȳ)=3u(41)121211212L12*g120¯¯**bu1au1(ȳ2ȳ)=3u(42)121221212L12Introducingexpressionsfor(41)and(42)fromthecom-ū,panionpaper(Makodeetal.1999),andrearranginginmatrixform,thesebecomeSubstituting(30)into(32)yieldsb16Kf1212c1d12¯¯4Kfu12Kfu1(ȳ2ȳ)121211212212L126K1200¯¯=4Ku12Ku1(ȳ2ȳ)24Ku22Ku121122121211222L12(33)Canceling2K12andrearrangingresultsinb13(f21)12c1d12¯¯2(f21)u1(f21)u1(ȳ2ȳ)12112212L1200=22u2u12(34)***ggg121212******aD1bD1(B2B)23aD1bD1(B2B)au1bu(y2y)112111211222122212121212121212120LLLu1212121=2(43)FG0FGFGu***ggg2121212******bD1aD1(B2B)bD1aD1(B2B)23bu1au(y2y)11211121122212221212121212121212LLL121212wheredccd**a=f24f13;b=2(f2f)(44a,b)121212121212b*g=3(12f)(44c)1212Pseudodistortionsandrequiredtosimulatetheeffectof00ff12axialloadonbendingstiffnessaregivenby(43).MultipleMembersDuringtheloadingprocessmanymemberscouldhavesig-nificantsecondaryeffects.Theformulationtosimulateasec-ondaryeffectinmultiplemembersispresentedinthissection.Considerthatatagivenloadstepmmembershaveasignifi-JOURNALOFSTRUCTURALENGINEERING/NOVEMBER1999/1321cantaxialforceeffectontheirbendingstiffness,anditisde-siredtoobtaintheresponseoftheframewiththesemmembershavingasecondaryeffectincluded.Responseofthisframeissimulatedbyimposing2mpseudodistortionsontheoriginalframe.Theexpressionfortheunknownpseudodistortionsto0u1isobtainedfromtheequilibriumconditionthatmember0u2mforcesanddisplacementsinthemodified(structurewithaxialforceeffectonthebendingstiffnessofmmembers)anddis-tortedstructuresareidentical.Thedisplacementexpressionsarethesameaswerederivedinthecompanionpaper(Makodeetal.1999),butwith2nreplacedby2m2m0ū=u1Du(45)jjjiiOi=12m0ȳ=y1Bu(46)jjjiiOi=12m0ū=u1Cu(47)jjjiiOi=1Toobtain2mpseudodistortionsweuse2mequilibriumcon-ditions.Foratypicalmemberl-kwithsignificantaxialforce,thisconditioncanbeanytwoof(27)–(29),withthesubscripts12replacedbyl-k.Inthedevelopmentshownhere,themomentequilibriumequationswillbeused.Thederivationproceedsaswithasin-glemember(withsubscripts12replacedbyl-k),leadingtothefollowingsetofequationsinplaceof(43):2m*glk0**aD1bD1(B2B)23dulikilikiliiOFlklkGLlki=1*glk**=2au1bu2(y2y)lklkFlklkGLlk(48)2mg*lk0**bD1aD1(B2B)23dulikilikikiiOFlklkGLlki=1*glk**=2bu1au2(y2y)lklkFlklkGLlk(49)Therearemsuchsetofequationsformmemberswithsig-nificantaxialforceeffect.Toget2mpseudodistortions,the2mequationsaresolvedsimultaneously.Thematrixtobesolvedisoforder2m32m.SolutionProcedureForsecond-orderanalysisbythePDM,theloadsarein-creasedproportionallyinincrements.Aninitialelasticanalysisforincrementalloadisperformed,andthedecomposedglobalstiffnessmatrixisstored.Aftereveryincrement,incrementalresponseandmemberforcesareaddedtototalresponseandtotalmemberforcesfromthepreviousincrement.Thetotalaxialforceforeachelementischecked.Ifthetotalaxialforcereachesorexceedsapredefinedlevel,thememberisconsid-eredtohavesignificantsecondaryeffect.Thestabilityfunctionvalueforthememberiscalculated.Toaccountforadditionalendmomentsduetoaxialforces,i.e.,P2Deffects,theequivalentshearloadforeachmemberiscalculated.Theincrementalcontributionofshearloadde-pendsontheresultsofthepreviousincrement.Theadditionalincrementalmoment(duetoaxialforces)(Fig.6)(GraffandEisenberger1991)issDM=P(Du2Du)1DP[(u2Du)2(u2Du)](50)lklklklkllkk1322/JOURNALOFSTRUCTURALENGINEERING/NOVEMBER1999FIG.6.P2DEffectandtheincrementalequivalenttransverseshearloadattheendsoftheelementaresDMlksDT=(51)lkLlkTheincrementalshearloadsareaddedtotheincrementalex-ternalloadstoevaluatetheresponseofthestructureinthenextincrement.Forthenextincrement,theoriginalstructureisreplacedbytheonewithmodifiedmembersthathavesignificantsecond-ordereffects.Supposethattherearemsuchmembersintheframe.Toobtaintheresponseofthisstructure,2mpseudo-distortionsareimposedontheoriginalstructureattheendsofthosemmembers.Theappropriatepseudodistortionsthatsim-ulatetheexacteffectofstabilityfunctionsarecalculatedbysolving2mequations.These2mequationsconsistof(48)and(49)foreverymemberwithsecond-ordereffects.Theoriginalresponseontheright-handsideoftheseequationsisduetoincrementalloadandtheequivalentshearloadfromthepre-viousincrement(thisoperationisasimplematrixmultiplica-tionoftheoriginaldecomposedstiffnessmatrixandtheloadvectorcorrespondingtotheloadincrement).Aftersolvingforpseudodistortionstothemodified00uu,12mincrementalrotationsandmodifiedincrementaldisplace-ūmentsandareevaluatedusing(45)–(47).Theincrementalȳū,memberforcesinanymemberi-jaregivenbymmm¯¯¯P=Kd(52)where=modifiedmemberforcevector;=modifiedmm¯¯PKmemberstiffnessmatrix;and=modifiedmemberdisplace-md̄mentvector.Theincrementalresponseisaddedtothetotalresponsefromthepreviousincrement.Thetotalaxialforcesintheotherele-mentsarecheckedagainsttheprescribedlevel.Moremembersaremodifiedifnecessary.Theprocessisrepeateduptothedesiredlevelortheprescribedlimitofanyparticularnodaldisplacement.Thus,thesingleanalysisoftheoriginalstructureandde-terminationofappropriatepseudodistortionsreplacesrepeti-tiveformulationandsolutionofthefull-sizeglobalstiffnessmatrix.Thematrixtobesolvedisoforder2m(m=numberofmemberswithsignificantsecondaryeffect).SECOND-ORDERPLASTICHINGEANALYSISInthecompanionpaper(Makodeetal.1999),thepseudo-distortion-basedformulationforfirst-orderplastichingeanal-ysis(materialnonlinearity)wasdeveloped.InthepreviousFIG.7.PossibleStructuralResponseModificationsinSec-ond-OrderPlasticHingeAnalysissections,theformulationforsecond-orderanalysis(geometricnonlinearity)waspresented.Inthissectiontheformulationforsecond-orderplastichingeanalysis(bothmaterialandgeo-metricnonlinearity)isdeveloped.Duringtheloadingprocessinsecond-orderplastichingeanalysis,amembermayundergodifferentstructuralresponsemodifications,i.e.,amembermayexhibitsignificantaxialforceeffectand/orplastichingesmayformatoneorbothendsofthemembers.Fig.7summarizesthedifferentpossiblestruc-turalresponsemodifications.Formemberswithnosignificantaxialforceeffect,thefor-mulationfordifferentcaseswasdevelopedinthecompanionpaper(Makodeetal.1999).Formemberswithsignificantaxialforceeffectandnohinges,theformulationwaspresentedintheprevioussections.Inthenexttwosectionstheformulationforhinge(s)inmemberswithsignificantaxialforceeffectisdeveloped.HingesatBothEndsinMemberswithSignificantAxialForceEffectConsiderthattherearempseudodistortionsimposedonthestructuretosimulatedifferentstructuralresponsemodifica-tions.Forthememberexhibitingsignificantaxialforceeffectandhingesatbothends,thechangeinthehingemomentofHlkandHklatendslandkisM̄=H(53)lklkM̄=H(54)klklTheseequationswereusedinthecompanionpaper(Makodeetal.1999).Therefore,thefinalexpressionsforpseudodistor-tionstosimulatehingesatbothendsofamemberwithsig-nificantaxialforceeffectarethesameasderivedthere,withupperlimitonsummationmreplacedby2masshownhere2m10D2d1(B2B)ulililikiiOFGLlki=11H1kl=2u1(y2y)2H2llklkFSDGL23Klklk(55)2m10D2d1(B2B)ukikilikiiOFGLlki=11H1lk=2u1(y2y)2H2klkklFSDGL23Klklk(56)Theseexpressionsforpseudodistortionsapplytoanymem-ber(withorwithoutsignificantaxialforceeffect)withhingesatbothends.HingeatOneEndinMemberswithSignificantAxialForceEffectConsiderthatthereare2mpseudodistortionsimposedonthestructuretosimulatedifferentstructuralresponsemodifica-Jtions.Toobtaintheexpressionforpseudodistortionsusedtosimulateahingeatoneendinatypicalmemberl-kwithsig-nificantaxialforceeffect,thetwomomentequilibriumcon-ditionswillbeused.IfthehingeisformedatendlandthechangeinhingemomentisHlk[(53)],theequationanalogousto(9)isb6Kflklkcd¯¯H=4Kfu12Kfu1(ȳ2ȳ)(57)lklklkllklkklkLlkThiscanbesolvedforūldbf3fHlklklk¯¯u=2u2(ȳ2ȳ)1(58)lklkccc2f2fL4KflklklklklkSubstitutingthisintotheequationanalogousto(10)andrearrangingd2d2(f)6K2(f)lklklkcc¯M̄=4Kf2u1f2kllklkklkFGFSDGcc4fL34flklklkdHflklk?(ȳ2ȳ)1lkc2flk(59)Formemberl-kwithasignificantaxialforceeffectandahingeatendl,theslopedeflectionequationsare(53)and(59).Similarly,ifthereisahingeatendkonly,andthehingemomentisHkl,theequationsbecomeM̄=H(60)klkld2d2(f)6K2(f)lklklkcc¯M̄=4Kf2u1f2(ȳ2ȳ)lklklkllklkFGFSDGcc4fL34flklklkdHfkllk1c2flk(61)Forthedevelopmentoftheformulation,ageneralmodifi-cationcaseisconsideredb1d6KfHflklkkllkc1d12¯¯M̄=4Kfu12Kfu1(ȳ2ȳ)1H1lklklkllklkklklkcL2flklk(62)b2d6KfHflklklklkd12c2¯¯M̄=2Kfu14Kfu1(ȳ2ȳ)11HkllklkllklkklkklcL2flklk(63)and=generalstructuralmodificationb1b2c1c2d12f,f,f,f,fijijijijijfactors.Allstructuralmodificationscanbederivedbydefiningthesefactorsappropriately.ForamemberwithsignificantaxialforceandahingeatlwithhingemomentHlkb1c1d12f=f=f=0;H=0(64a,b)lklklkkld2d22(f)(f)lklkb2cc2cf=f2;f=f2(64c,d)lklklklkFSDGFGcc34f4flklkUsing(64)in(62)and(63)produces(53)and(59).ForamemberwithsignificantaxialforceandahingeatkwithhingemomentHkld2d2(f)2(f)lklkc1cb1cf=f2;f=f2(65a,b)lklklklkFGFSDGcc4f34flklkb2c2d12f=f=f=0;H=0(65c,d)lklklklkUsing(65)in(62)and(63)produces(60)and(61).Considerthefollowingequationfromthecompanionpaper(Makodeetal.1999):OURNALOFSTRUCTURALENGINEERING/NOVEMBER1999/13236Klk00¯¯M̄=4Ku12Ku1(ȳ2ȳ)24Ku22Ku(66)lklkllkklklkllkkLlkSubstituting(62)into(66)givesb1d6KfHflklkkllkc1d12¯¯4Kfu12Kfu1(