on the nonlinear transient analysis of planar steel frames ... · on the nonlinear transient...

Original Article

LatinAmericanJournalofSolidsandStructures,2018,15 3 ,e28

OntheNonlinearTransientAnalysisofPlanarSteelFrameswithSemi‐RigidConnections:FromFundamentalsto

AlgorithmsandNumericalStudies

AbstractThispaperpresentsthefundamentalsforpredictionofamorerealisticbe‐haviorofplanar steel frameswith semi‐rigid connectionsunderdynamicloading.Themajorityoftheresearchinthisareaconcentratesonthenon‐linear static analysis of frames with semi‐rigid connections. Indeed, fewstudieshavecontributedtothenonlineardynamicandvibrationanalysesofframes.Therefore,thisarticlefirstdescribestheframes’semi‐rigidcon‐nectionbehaviorundermonotonicandcyclicloads,andpresentstheinde‐pendenthardeningtechniqueadoptedtosimulatethejointbehaviorundercyclicexcitation.Inafiniteelementcontext,thispaperpresentsanefficientnumericalmethodologythatisproposedinalgorithmicformtoobtainthenonlineartransientresponseofthestructuralsystem.Thepaperalsopre‐sents, inalgorithmic form,acompletedescriptionof theadoptedconnec‐tionhystereticmodel. Satisfying theequilibriumandcompatibility condi‐tions, and assuming only the connection’s rotational deformation due tobendingasvariable,thisworkobtainsthetangentstiffnessandmassma‐tricesofthebeam‐columnelementwithsemi‐rigidconnectionsattheends.The study concludes by verifying and validating the proposed numericalapproachusingfourstructuralsteelsystems:aL‐frame,atwo‐storyframe,asix‐storyframe,andafour‐bayfive‐storyframe.Theanalysesshowthatthe hysteresis of the semi‐rigid connection has a strong effect on theframes’responsesandisanimportantsourceofdampingduringthestruc‐turalvibration.

KeywordsSteelframes,Nonlineartransientanalysis,Geometricnonlinearities,Semi‐rigidconnections,Connectionstiffness,Connectionhystereticbehavior.

1INTRODUCTION

In the structural design and construction of reticulated steel structures involving one ormore floors, thebeam‐columnandcolumn‐baseconnectionsplayasignificantroleintheirstructuralresponse.Theyalsorepre‐sentasignificantfractionofthetotalcostofthestructure.Thus,thecharacteristicsoftheseconnectionsare, insteelstructures,economicallyandstructurallysignificant.Therefore,theengineermusthaveagoodunderstand‐ingoftheconnectionbehavior.Inmostcasesitisadvisabletousesemi‐rigidconnections.Infact,modernstand‐ards ABNT‐NBR8800,2008;AISC,2010;Eurocode3,2005 alreadyincludeprocedurestodefinethestiffnessaswellasstrengthofsemi‐rigidconnectionsforcomputationalanalysisprogramsandprescriptionsforthede‐signengineer.

Variousstudiesonthestaticanalysisofsteel frames King,1994;Lietal.,1995;Krugeretal.,1995;Chen,2000;ChanandChui,2000;SekulovicandNefovska‐Danilovic,2004;CabreroandBayo,2005;Ihaddoudèneetal.,2009;ValipourandBradford,2012 havealreadyshownthatthedeformationandbendingmomenttransmissioncapacityof theconnectionscansignificantlyalter the loadcarryingcapacity, the internal forcedistributionandglobalandlocalinstabilityofthesestructures.Inaddition,withinthecivil,naval,oceanic,andaeronauticalindus‐tries,moreresistantmaterialsandnewdesignandconstructiontechniqueshaveledtolighterandslenderstruc‐tures,whichare,however,usuallysusceptibletoexcessivevibrationproblems.Consequently,anessentialpartof

AndréaR.D.Silvaa*EvertonA.P.BateloaRicardoA.M.SilveiraaFranciscoA.NevesaPauloB.Gonçalvesb

aDepartamentodeEngenhariaCivil,EscoladeMinas,UniversidadeFederaldeOuroPreto,MinasGerais,Brasil.E‐mail:andreadiassil‐[email protected],[email protected],[email protected],[email protected],PontifíciaUniversidadeCatólicadoRiodeJaneiro,RiodeJaneiro,RJ,Brasil.E‐mail:paulo@puc‐rio.br

*Correspondingauthor

http://dx.doi.org/10.1590/1679-78254087

Received:June05,2017InRevisedForm:October22,2017Accepted:February27,2018Availableonline:March20,2018

AndréaR.D.Silvaetal.OntheNonlinearTransientAnalysisofPlanarSteelFrameswithSemi‐RigidConnections:FromFundamentalstoAlgorithmsandNumericalStudies

LatinAmericanJournalofSolidsandStructures,2018,15 3 ,e28 2/28

a structuraldesign should includeananalysisof theirdynamicbehaviorundervarious loading conditions.Re‐searchalongtheselines,accountingfortheeffectofconnectionflexibility,isstilllimited,particularlycomparedtothe amount of research on static analysis. Numerical investigations into the behavior of steel structures withsemi‐rigid connections submitted todynamic situationshavebeen carriedoutby, amongothers, the followingresearchers:ValipourandBradford 2012 ;ChanandHo 1994 ;LuiandLopes 1997 ;XuandZhang 2001 ;Sophianopoulos 2003 ;Silvaetal. 2008 ;Galvãoetal. 2010 ;Vimonsatitetal. 2012 ;NguyenandKim 2013 AttarnejadandPirmoz 2014 ;andAristizabal‐Ochoa 2015 .

Researchershavealsoexperimentallyevaluatedthecyclicresponsesofsemi‐rigidconnections.Bernuzzietal. 1996 ,Calado 2003 ,andShietal. 2007 carriedouttestsusingspecifictypesofconnectionsandproposedmathematicalmodelstorepresenttheobservedbehavior.

ThisarticlecanbeconsideredasanexpansionofanearlierworkbyGalvãoetal. 2010 .Inthiscontext,thecurrent paper evaluates, using the CS‐ASA Program Computational System for Advanced Structural Analysis;Silva,2009 ,thenonlineardynamicresponseofplanarsteelframe,consideringgeometricnonlineareffectsandthesemi‐rigidcharacteristicsof theconnectionsbetweenthestructuralelements.Whenthesemi‐rigidconnec‐tionsaresubmittedtoalternatingrepetitiveactions,theymaydevelopaninelasticbehaviorthatgivesrisetohys‐tereticdamping,whichisinmostcasesquitebeneficial.

CS‐ASAisaFEM‐basedcomputersystemimplementedinFortran90/95,originallydesignedforsteelstruc‐tures.Thisprogramcanperformadvancedstaticanddynamicnumericalanalysisconsideringsourcesofnonline‐aritiessuchassecondordereffectsandsemi‐rigidconnections usingpseudo‐springsatthefiniteelementends .Therefore,thispaperbringsthenumericalresultsobtainedfromanalteratedCS‐ASAversion,whichallowrepre‐sentthesemi‐rigidconnectionhystereticbehavior,inordertoallowabettermodelingofthenonlineartransientresponseofplanarsteelframeswithflexiblejoints.Actually,thisisanextensionoftheCS‐ASAinitialmodulethatcontemplatedthestaticnonlinearanalysisandsomecasesfordynamicnonlinearanalysisandvibration Galvãoetal.,2010;Silva,2009 .Asmaincontribution,inafiniteelementcontext,thispaperpresentsanefficientnumeri‐calmethodologytoobtainthenonlineartransientresponseofstructuralsystemswithsemi‐rigidjoints.Thepa‐peralsopresents,inalgorithmform,acompletedescriptionoftheadoptedconnectionhystereticbehavior.

The next section, Section 2, describes the connection behaviorwhen subjected tomonotonic and cyclicalloads,andpresents theadoptedtechniquetosimulatethisbehavior.Section3details themethodologyusedtoobtainthenonlineartransientresponseofthestructuralsystem.Section4presentstheformulationofthefiniteelementusedinthestructuralsystem’smodeling.Finally,tovalidatetheproposedanalysismethodology,Section5evaluatesthedynamicresponseoffourstructuralsystems.Theresultsdemonstratethatthestructuraldynamicbehaviorcanbegreatlyinfluencedbytheeffectsofthesemi‐rigidconnectionsandthatacarefuldynamicconnec‐tiondescriptionanddynamicanalysisisessentialforasafeandyetcost‐effectivedesign.

2SEMI‐RIGIDCONNECTIONS

Asnotedabove,connectionsplayakeyrole in theassembly,performance,andcostofasteelstructure. Inmostcases,theseconnectionscannotbemodeledasperfectlyrigidorideallyhinged.Rigidconnectionspreventanyalterationoftheanglebetweentheinterlinkedelements,transferringthefullmomentfromoneextremitytotheother.Inthecaseofhingedconnections,nomomentistransmittedbetweentheelements.Thesehypothesessimplifytheanalysisbutfailtorepresentthetruebehaviorofmoststructures.TheexperimentalinvestigationsofJonesetal. 1980;1983 andNethercotetal. 1998 demonstratedthatagreaterpartoftheconnectionsusedincurrentpracticesdisplaysemi‐rigidbehavior.Thiscansubstantiallyinfluencetheoverallstabilityofthestructur‐alsystemaswellasthedistributionoftheforcesactingonitselements.Indeed,geometricandmechanicaldiscon‐tinuitiescan,accordingtoColson 1991 ,introducelocalizedeffectsandimperfectionsintheconnectionsusedinsteelstructures,whichinterferewiththeoverallbehaviorofthestructure,providingjustificationforamorerig‐orousinvestigationintotheseconnectionbehavior.Suchstudiesshouldtakeintoaccountnotonlythemanufac‐turingandassemblypointofviewbutalsotheconnections’influenceonthestructuralresponse.

Notonlytheaxialandshearforcesbutalsothebendingandtorsionmomentsaretransmittedthroughacon‐nection.Foragreatmajorityofsteelstructures,however,theeffectsoftheseaxialandshearforcesonthedefor‐mationoftheconnectionaresmallcomparedtothosecausedbythebendingmoment Chenetal.,1996 .Forthisreason,andknowingthatintheanalysisofplanarstructuresthedeformationcausedbytorsionisnegligible,thisstudyconsidersonlytherotationaldeformationoftheconnectioncausedbybending.

Connectionbehaviorisgenerallydescribedbymoment‐rotationcurves,whichcanbeobtainedinthreeways:fromexperimentaltests,numericalsimulationinfiniteelements,orbytheoreticalmodels.Theselatteraredevel‐opedbyapplying curve‐adjustment techniques to the resultsobtainedby thenumerical simulationand/or the



experimentaltests.Díazetal. 2011 presentedthemainfeaturesofdifferentconnectionmodels,highlightingtheadvantagesanddisadvantagesofeachone.Otherstudieshavebeenperformedtoevaluatethebehaviorofcon‐nectionsundermonotonicandcyclic loads, including AckroydandGerstle,1982;Azizinaminietal.,1987;YeeandMelchers, 1986; Kishi and Chen, 1987a; Kishi and Chen, 1987b; Korol et al., 1990; Tsai and Popov, 1990;Abolmaalietal.,2003;Abolmaalietal.,2009 .Thefollowingtwosubsectionsdescribethesebehaviorsaswellasthenumericalmodelsadoptedhere.

2.1BehaviorunderMonotonicLoadingandMathematicalRepresentation

Figure1illustratesthetypicalbehaviorofthreetypesofsemi‐rigidconnections:theflushendplate,topandseatangle,andsinglewebangle.Inthefiguretherelativerotationbetweentheinterconnectedelements, cf ,due

toanappliedmomentisalsoshown.Theseconnections,whensubmittedtobending,displayasofteningnonlinearbehaviorwithdecreasingstiffness.Themoment‐rotationcurve is initiallypractically linear,but,as the load in‐creases,thenonlinearityincreases.Thischangeinbehaviorisinfluencedbytheexistenceofstressconcentration,geometricimperfectionsanddiscontinuitiesandtheplasticbehaviorofitscomponents.Atacertainloadinglevel,thesefactorsbegintointerfereintherotationalcapacityoftheconnection.Inthefinalloadingstage,themoment‐rotationcurvetendstowardanasymptoticalvalueknownastheultimatemomentcapacityofconnection.

Figure1:Moment‐rotationcurvesofthreeusualtypesofconnectionsundermonotonicloading.

Themost important characteristics thatdefine theconnectionbehaviorare its initial stiffnessand itsulti‐matemomentcapacity Bjorhovdeetal.,1990 .Theconnectionstiffness,denotedas cS ,isdeterminedbyexperi‐

mental tests.Mathematically, it represents the inclinationof the tangent to themoment‐rotationcurve Fig.1 .Thestiffnessvaluedoesnotdeterminewhetheraconnectionwillbehaveinarigidorflexiblemanner.Thisbehav‐iorcanonlybeestablishedbyusingaparameter that relates theconnection’s stiffness to thebendingstiffnessEI oftheelementtowhichitisconnected.Cunningham 1990 proposed,toprovideabetterunderstandingof

therotationalbehaviorof theconnection,a fixed factor, 1(1 3 / ( ))cEI S Lg -= + , inwhichLrepresents theele‐



ment’slength,where g variesbetween0and1.Foraperfectlyhingedconnection 0g = andforanideallyrigid

connection, 1g = ,since cS ¥ .Theconnectionstiffnesscanbecalculatedby:

cc

dMS

df= 1

whereM isthebendingmomentactingontheconnectionand cf isitsrotationaldeformation.

Themoment‐rotationcurvescangenerallybewrittenbyoneofthefollowingexpressions:

( )

( )c

c

M f

g M

ff

==

2

which relate themoment to the rotationwithmathematical expressions f and g,whichusually depend on thegeometryanddispositionoftheconnectingcomponentsandthecurveadjustmentparameters.

According to Chan and Chui 2000 , themathematicalmodels should provide a smoothmoment‐rotationcurvewiththefirstderivativepositiveandbeabletorepresentvarioustypesofconnections.Foralinearconnec‐tionmodelonlyoneparameterdefiningthejointstiffnessisnecessary.Inthiscase,themoment‐rotationrelation‐shipisgivenby:

cini cM S f= 3

where ciniS representstheinitialstiffness.Thisisthesimplestmodelandhasbeenwidelyusedintheanalysisof

semi‐rigidconnections Arbabi,1982;KawashimaandFujimoto,1984;Chan,1994;LuiandChen,1986 .Forlargedisplacementanalyses,however,itmayleadtoerroneousconclusions.

Unlike the linear model, the nonlinear models simulate the stiffness degradation as the load increases.Amongthevariousnonlinearmodelsfoundinliterature,onlythoseemployedinthisstudyarediscussed.

Thefirstisanexponentialmodel LuiandChen,1986;ChenandLui,1991 ,whosemathematicalexpressionforthemoment‐rotationcurveisgivenby:

01

1 exp 2

nc

j p cm

M M C Rm

ff

a=

é æ öù- ÷çê ú÷ç= + - +÷çê ú÷÷çè øê úë ûå 4

whereM isthemomentintheconnection, 0M istheinitialmoment, pR isthestrain‐hardeningstiffnessofthe

connection,a isanscalingfactor,and mC — 1,2,...m n= —isthecurve‐fittingcoefficient.

Alsousedhereisafour‐parametermodelproposedbyRichardandAbbott 1975 .Themodeldescribestheconnectionbehavioraccordingtotherelationship:

( )

( )1

0

+

1 /

cini p cp c

n ncini p c

S RM R

S R M

ff

f-

-=

é ù+ -ê úë û

5

in which pR is the strain‐hardening stiffness when cf tends to infinity; n is a parameter that defines the

sharpnessofthecurve,and 0M isareferencemoment.AstudymadebyKishietal. 2004 demonstratedthatthis

modeliseffectiveinestablishingthebehaviorofanendplateconnection.However,itcanbeappliedtoanytypeof connection. To do this, it is sufficient to either experimentally or numerically evaluate the four necessaryparameters.Fromthismodel,threeotherscanbeobtained:linear p ciniR S» ,bilinear n ¥ ,andignoring

the effect of the strain‐hardening stiffness, the three‐parameter potentialmodel presented by Kishi and Chen1986 . The stiffness of the connection is obtained using Eq. 1 , with M described by one of the nonlinearpresentedmodels.



2.2BehaviorunderCyclicLoadingandMathematicalRepresentation

Whenarealisticnumericalsimulationofagivenstructuralsystemunderdynamicloadsisdesired, it ises‐sentialtoconsiderthebehavioroftheconnectionundercyclicload.Dependingontheloadlevelandtimehistory,semi‐rigidconnectionsmaydevelopaninelasticbehavior.AtypicalconnectionresponseisillustratedinFig.2.

Figure2:Connectionbehaviorundercyclicloading.

InFigure2the0Acurvedescribestheconnection’sbehaviorduringtheinitialloadingprocessofthestruc‐ture,whichiscompatiblewiththedifferenttypesofconnectionshowninFig.1.ThefollowingABandBCpathscorrespondtotheunloadingprocesscharacterizedbyadecreasingintensityofthebendingmomentthatactsontheconnectionandthereversalofthedirectionoftheexternalforcing,asshownbythearrows.ThesubsequentCDandDEpathscorrespondtothenextunloadingandloadingcycles.Afterthefirstunloadingstageiscomplete,only a part of the total deformation caf is recovered and the connection sustains a permanent residual defor‐

mation cbf .Thisalsooccursaftereachofthefollowingloadingandunloadingcycles.Theloopsresultingfromthe

non‐coincidentmoment‐rotationcurvescharacterizetheconnectionhystereticbehavior,asshown,throughcyclicloadingtests,byTsaiandPopov 1990 ,Koroletal. 1990 andAbolmaalietal. 2003 .

Thenonlinearbehaviorofthesemi‐rigidconnectionswhensubmittedtocyclicloadscanbeefficientlysimu‐lated by the independent hardening technique, employed previously by, among others, Ackroyd and Gerstle1982 ,Azizinaminietal. 1987 ,Chenetal. 1996 ,ChanandChui 2000 andSekulovicandNefovska‐Danilovic2008 .Animportantadvantageofthisapproachisthatonecanadoptanymathematicalfunctionrepresentingthemoment‐rotationrelationshipundermonotonicloads.

Consideratypicaltrajectoryunderincreasingload trajectoriesindicatedbythecontinuouslinesinFig.3a beginningatapointwithcoordinate pf , the lastpermanent residual rotation at0A, the initial loadingphase,

0pf = ,thebendingmomentM atanypositionalongthetrajectoryDEcanbecalculatedasfollows:

( )c pM f f f= - 6

where f is defined in accordance with a mathematical model that describes the nonlinear moment‐rotationbehavioroftheconnectionwhensubmittedtoamonotonicload.Thus,theconnectionstiffnessisgivenby:



( )c c p

cc

dMS

d f f ff = -

= 7

Theloadingprocessbetweenatimestep t prior and t t+ D actual leadstoanincreaseinthebendingmoment, MD ,whichhasthesamesignofthebendingmoment,M .Therefore,theloadingconditionisverifiedif

0M M⋅ D > .Intheunloadingprocess,basedonexperimentalresults,alinearmoment‐rotationrelationwithaninclina‐

tionequaltotheinitialstiffnessoftheconnectionisconsidered,asillustratedbythesolidlinesinFigure3b.Con‐sider,forexample,thesegmentABofthehystereticcycle.Thestraightlinestartingatpoint ( ; )c aA Mf theendof

theprecedingloadingcycle withaninclination ciniS isdefinedas:

( )a cini ca cM M S f f= - - 8

where aM is the reversemoment at which unloading starts, which can be obtained from Eq. 6 considering

c caf f= .

Duringtheunloadingprocess,therelation 0M M⋅ D < isobeyed.Insomecase,however,additionalclarifi‐cationsarenecessary.ConsideragainFig.2.IftheconnectionisbeingunloadedfromAtoBandtheloadingpro‐cessbeginsbeforetheunloadinghasbeencompleted,sayatF,themoment‐rotationbehaviorofthiselementwillfollowtheFAtrajectoryuntilitreachesthelastreversemomentencountered.Atthispoint,theloadingprocessfollowstheoriginalmoment‐rotationcurveobtained,consideringthelastpermanentrotation pf .

Figure3:Independenthardeningmodel:loadingandunloadingportions.

Incomputingthe independenthardeningmodel, therotation pf shouldbestoredaftereachcompleteun‐

loadingcycle.Thereversemoment, aM ,shouldalsobestoredeverytime MD changessign.Inaddition,theinter‐

valsbetweenthetwosuccessivetime instantsshouldbesmallsothat thesevariablescanbedefinedwithade‐quateprecision.

3METHODOLOGYFORNONLINEARDYNAMICANALYSIS

Theequationsofmotionthatgovernsthedynamicresponseofastructuralsystemcanbeobtainedusingthevirtualworkprincipleorthevirtualdisplacementprinciple.Consideringtheinternal,inertialanddissipativeforc‐es,theequilibriumattime t t+ D ,canbeexpressedas ZienkiewiczandTaylor,1991 :

t t t

Tij ij k k k k k ek

V V V

dV d d dV d d dV d ft de + r d + m d = dò ò ò 9



where ijt represents the Cauchy tensor in equilibriumwith external excitation ekf ; ije represents the virtual

Green‐Lagrangedeformationcomponentscorrespondingtoarbitrarydisplacements kdd ,whicharecinematically

compatiblewith theboundary conditions; r is themassdensityof thematerial and m is theviscousdampingcoefficient. To determine the equilibrium configuration of the structure at t t+ D , the updated Lagrangianreferentialisused.Inthiscase,theconfigurationatinstanttisusedasthereferenceforanalysis.

TheadvantageofusingtheupdatedLagrangianreferentialdependsonthefiniteelementformulationadopt‐ed Bathe,1996 .WongandTin‐Loi 1990 andAlves 1993 showedthattheresultsobtainedusingtheinitialLagrangianreferentialleadstoaccumulatederrorsiflinearinterpolationfunctions Silveira,1995 areadoptedintheFEformulationduetopossiblerigidbodymotionsduringtheincrementalprocess.WithanupdatedLagrangi‐anreferential, thesesimplified interpolation functionscanbeapplied,as thereferential isupdatedatevery in‐crementandrigidbodyrotationsaredividedintosmallerparts.

Accordingtotheusualfiniteelementprocedures,andusingEq. 9 ,thefollowingmatrixequationintermsofthenodaldisplacementsisobtained:

( ) ( )i rtl+ + =MU CU F U F 10

whereM andC arethemassanddampingmatrices,respectively; iF istheinternalforcevector; U ,U andU ,

represent the displacement, velocity and acceleration vectors, of the structural system, respectively; rF is the

vectorthatdefinestheexternalexcitation;andl istheloadintensityatainstantt.Thestiffnessmatrixisafunctionofthreevariables—thenodaldisplacements,theinternalforcesofeachel‐

ement,andconnectionstiffnessandmustbecontinuouslyupdatedusinganincremental‐iterativesolverstrategytocapturethesecond‐ordereffects P -DandP d- andconnectionbehavior.Tothisend,anumericalproce‐dure that combines theNewmark Chopra,1995 andNewton‐Raphson BurdenandFaires,2004 methods isadopted.Thenecessary computational stepsaredetailed inAppendixA.Notice that the connection stiffness isupdatedattheendofeachiterativecycle.AppendixBsummarizes,inalgorithmicform,theconnectionhystereticmodelimplementedhereanddescribedinSection2.

4THESEMI‐RIGIDFINITEELEMENT

Inmoststructuralanalysisprogramsspringsareusedtomodeltheconnections.Figure4showstheplanarbeam‐columnelement,withnodalpointsiandj,andtwospringsfixedattheextremities.Theypermittheconnec‐tions’degreeoffreedomtobeincorporatedintothetangentstiffnessofthebeam‐columnelement.Onlythecon‐nection’srotationaldeformationduetobendingisconsidered.

Figure4:Beam‐columnelementwithsemi‐rigidconnections.

Figure5showsadeformedconfigurationof the finiteelement, the internal forcesanddeformations in thesprings,where ciS and cjS denotethestiffnessofthespringelements.Theconnectionscanbecharacterizedby

differentmoment‐rotationcurves Eq.6 .Theconnection’srelativerotation, cf , isdefinedasthedifferencebe‐



tweentherotationalanglesofthesideconnectedwiththeglobalnodeoftheelement, cq ,andtheoneconnected

tothebeam‐columnelement, bq .UsingEq. 6 andthemomentequilibriumofthespringelements,thefollowing

relationsareobtained:

θ

θci ci ci ci

bi ci ci bi

M S S

M S S

ì ü é ù ì üï ï ï ïD - Dï ï ï ïï ï ï ïê ú= =í ý í ýê úï ï ï ïD - Dê úï ï ï ïï ï ï ïî þ ë û î þ 11

θ

θcj cj cj cj

bj cj cj bj

M S S

M S S

ì ü é ù ì üï ï ï ïD - Dï ï ï ïê ú= =í ý í ýê úï ï ï ïD - Dê úï ï ï ïî þ ë û î þ 12

where cM and bM are the bendingmoments acting on the spring elements andon the beam‐columnelement,

respectively,andthesubscriptsiandjrefertotheelement’sextremities.Theeffectinducedbyconnectionflexibilityisaccountedforbymodifyingtheelementstiffnessmatrix.Here‐

in,thespringelementssimulatingtheconnection’sflexibilityhavenulllengthsandthemoment‐rotationequilib‐riumrelationisgivenby:

(3,3) (3,6)

(3,3) (3,3)

4 2 4 2 2

15 302 2 4 2 4

30 15

bi bi bi

bj bj bj

EI PL PL EI PL PIM k k L AL L AL

k kM EI PL PI EI PL PL

L AL L AL

q qq q

é ùê ú- + - +ì ü ì ü ì üé ùD D Dï ï ï ï ï ïê úï ï ï ï ï ïê ú= =í ý í ý í ýê úê úï ï ï ï ï ïD D Dê úê úï ï ï ï ï ïë ûî þ î þ î þ- + - +ê úë û

13

whereE is theYoung’smodulus; I andA are, respectively, the element cross section inertia and area; L is theelement’s length; andP is theaxial force.The termsof thematrix are the coefficientsof a conventionalbeam‐column element stiffnessmatrixwith perfectly rigid connections, determined using the nonlinear geometricalformulation proposed by Yang and Kuo 1994 , assuming large displacements and rotations, but smalldeformations.

CombiningEqs. 11 ‐ 13 ,resultsin:

θ

θ

θ

θ

(3,3) (3,6)

(6,3) (6,6)

0 0

0

0

0 0

ci cici ci

ci cibi bi

cj cjbj bj

cj cjcj cj

S SM

S S k kM

k S k SM

M S S

é ùì ü ì üï ï ï ï-D Dï ï ï ïê úï ï ï ïê úï ï ï ï- +D Dï ï ï ïê úï ï ï ï= =í ý í ýê úï ï ï ï+ -D Dê úï ï ï ïï ï ï ïê úï ï ï ïD D-ï ï ï ïê úï ï ï ïî þ î þë û

14

Assumingthattheloadsareonlyappliedattheglobalnodesoftheelement seeFig.5 ,theinternalmoments

biM and bjM areequaltozero,andthefollowingmoment‐rotationequilibriumrelationsisobtained:

θ

θ(6,6) (3,6)

(6,3) (3,3)

0 0 010 0 0

cjci ci ci ci ci

cj cj cj ci cj cj

S k kM S S S

M S S k S k Sb

æ öì ìé ùü é ù é ù é ù üï ïï ï+D D÷çï ïï ïï ï ï ïê úê ú ê ú ê ú÷ç= - ÷í ý í ýç ê úê ú ê ú ê ú÷çï ï ï ïD - + D÷÷ç ê úê ú ê ú ê úï ï ï ïè øï ïþ ë û ë û ë û þë ûï ïî î 15

inwhich ( )( )(3,3) (6,6) (6,3) (3,6)ci cjS k S k k kb = + + - .

Figures4and5showthat i ciM M= and j cjM M= .Thus,therelationshipbetweentheshearingforcesand

thebendingmoments,obtainedby theequilibriumforceandmoment,canbewritten inan incrementalmatrixformasfollows:

1/ 1/

1 0

1/ 1/

0 1

i

cii

cjj

j

L LQ

MM

L L MQ

M

é ùì üï ïDï ï ê úï ï ê ú ìï ï üï ïDDï ï ï ïê úï ï ï ï= =í ý í ýê úï ï ï ï- - DD ê úï ï ï ïïþïîï ï ê úï ïDï ï ê úï ïî þ ë û

16

with iMD and jMD beingtheincrementalnodalmoments, iQD and jQD beingtheincrementalshearingforces

intheequilibriumconfigurationt.



Figure5:Deformedconfigurationandspringelementdegreeoffreedom.

ObservingFig.6,itispossibletoestablishtherelations:

1/ 1 1/ 0

1/ 0 1/ 1

i

ci i

cj j

j

v

L L

L L v

q qq

q

ì üï ïDï ïï ïì ï ïü é ùï ïD - Dï ïï ïï ï ï ïê ú=í ý í ýê úï ï ï ïD - Dê úï ï ï ïïþ ë ûïî ï ïï ïDï ïï ïî þ

17

Figure6:Nodaldisplacementsofthesemi‐rigidbeam‐columnelement.

Bysubstituting 15 into 16 ,using 17 ,andperformingall thenecessaryoperations, thesemi‐rigidele‐ment stiffnessmatrix is then obtained consideringboth the connection flexibility and the geometric nonlineareffects.Usingthesenewtermsinthecomplete6x6ordermatrixoftheconventionalbeam‐columnelement,theelementstiffnessmatrix is finallyobtained in the localcoordinatessystem. In this formulation, therelationbe‐tweentheaxialforces iP and jP ,andthenodaldisplacementssuffernoalterationswhenconsideringtheconnec‐

tionflexibilityeffects.Thetransformationoftheinternalforcevectorandthestiffnessmatrixtotheglobalcoor‐dinatessystemiscarriedoutfollowingtheusualprocedures.



4.1Semi‐RigidFiniteElementMassMatrix

Theconsistentmassmatrixforthebeam‐columnelementisdefinedas KishiandChen,1986 :

Te

tV

dVr= òM H H 18

whereHistheinterpolationfunctionsmatrix.AccordingtoChanandChui 2000 ,thetransversaldisplacementvofthebeamelementcanbewritten,inan

incrementalform,usingthefollowingmatrixexpression:

( ) ( )2 21 2 2 1 1 1 2 2( ) 3 2 3 2bi i

bj j

vv x H H L H H L H H H H

v

qq

ì ü ì üD Dï ï ï ïï ï ï ïé ù é ùD = - + - -í ý í ýê ú ë ûë û ï ï ï ïD Dï ï ï ïî þ î þ 19

where ivD and jvD are the incremental vertical displacements of the nodes i and j, and biqD and bjqD are the

rotationsofthesesamenodes,asshowninFig.6.Inaddition, 1 1 /H x L= - and 2 /H x L= .

To introducetheconnection’s flexibilityeffect intothepreviousexpressionandtherebyobtainthedesiredmassmatrix,thebeam‐columnnodalrotationsarerelatedtotherotationsattheendsoftheelementwithsemi‐rigidconnectionsas:

14 2

0

2 4 0

cibi ci ci

bj cj cjcj

EI EIS SL L

EI EI SS

L L

q qq q

-é ùê ú+ì ü é ù ì üD Dï ï ï ïê úï ï ï ïê ú=í ý í ýê ú ê úï ï ï ïD Dê ú ê úï ï ï ïî þ ë û î þ+ê úë û

20

SubstitutingEq. 17 intoEq. 20 andusingEq. 19 , thefunctionthatdescribesthetransversaldisplace‐mentfieldofabeamwithsemi‐rigidconnectionsisobtained,i.e.:

{ }*( ) T

i i j jv x v vq qD = D D D DH 21

inwhich,

( ) ( )

1

2 21 2 2 1

1 1 2 2

4 20 1 / 1 1 / 0

( )2 4 0 1 / 0 1 / 1

3 2 0 3 2 0

ci ci

cjcj

EI EIS S L LL Lv x H H L H H L

EI EI S L LS

L L

H H H H

-é ùê ú+ é ù é ù-ê úé ù ê ú ê úD = - +ê úê ú ê ú ê úë û -ê ú ê ú ê úë û ë û+ê úë û

é ù- -ë û

22

Themassmatrixcomponents influencedbytheconnection’sstiffnessareobtainedusingtheH*matrix, in‐steadof theH inEq. 18 .Theother termsare identical to thosedeveloped for theconventionalbeam‐columnelementwithrigidconnections Silva,2009 .

5NUMERICALAPPLICATIONS

Inthissection,weusetheproposedmethodologytoobtainthenonlineardynamicresponseof fourplanarstructuralsteelsystems:aL‐frame,atwo‐storyframe,asix‐storyframe,andafour‐bayfive‐storyframe.Intheseexamples,viscousdamping isnotconsideredwhenthedissipativeeffectof theconnectionnonlinearhystereticbehavioronthedynamicresponseisevaluated.

5.1L‐Frame

ThefirstproblemtobeanalyzedistheL‐shapedframewithbuilt‐inextremitiesillustratedinFig.7,wheretherelevantdataisalsopresented.Theconnectionbetweenthesetwomembersiseitherperfectlyrigidorsemi‐rigid.Inthesemi‐rigidcase,anendplateconnectionswithaninitialstiffnessScini 137.3Nm/radisconsidered.Kawashima andFujimoto 1984 experimentallymeasured the frame’s natural frequencies considering a rigidbeam‐columnconnection.Theyalsonumericallydeterminedthesefrequenciesconsideringasemi‐rigidconnec‐



tion.Inadditiontotheseauthors,ShiandAtluri 1989 andChanandChui 2000 usedthisstructuretotesttheirnumericalformulationsfordynamicanalysesofplaneframesapplyinganimpactload 19.6N inthemiddleoftheframecolumn.Theapplicationofaloadwithlowvalueisjustifiedbythelowresistanceofthesemi‐rigidcon‐nectionofthebeam‐columnelements.Table1comparesthepresentresultswiththenaturalfrequenciescorre‐spondingtothefirsttwovibrationmodesobtainedbytheaforementionedauthors.Theresultsagreewellwiththosefoundinliterature.

Figure7:L‐shapedframe:geometryandloading.

Toevaluatetheinfluenceoftheconnection’snonlinearbehavioronthedynamicresponse,ashort‐durationimpactloadP t withaconstantintensityof19.6Nisappliedatthecenterofthecolumn,asshowninFig.7.Toverify the importance of the adopted semi‐rigid connection’smoment‐rotation function in this representation,twomodelsare considered: theexponential 59 and the four‐parametermodel ChenandLui,1991 .Thepa‐rametersfortheexponentialmodel,describedbyEq. 4 ,areasfollows:m 6,C1 1.078915,C2 18.148410,C352.890917,C4 ‐34.035435,C5 ‐37.194613,C6 43.896458,α 0.000384,Rp 95.55031kgfcm/radand

Mo 0.Forthefour‐parametermodel Eq.5 ,inadditiontothepreviousinitialstiffness,thefollowingvaluesareused:Rp 8.826Nm/rad,Mo 0.883Nmandn 1.7.

Table1:ThetwolowestnaturalfrequenciesoftheL‐frame rad/s .

TypesofConnections

Presentwork

KawashimaandFujimoto1984 ShiandAt‐

luri 1989ChanandChui

2000 Numerical

Experi‐mental

Rigidω1 15.81 ω1 16.30 ω1 15.50 ω1 15.87 ω2 34.74 ω2 35.90 ω2 30.80 ω2 35.30

Semi‐rigidω1 14.90 ω1 14.90 ω1 14.75 ω1 15.22ω2 32.77 ω2 33.00 ω2 32.06 ω2 33.52

Inthenumericalanalysisaconstanttimeincrement∆t 10‐4sisused.Figure8ashowstheinitialtimere‐

sponseofthehorizontaldisplacementuattheloadapplicationpointconsideringalinerconnectionbehavior theconnectionstiffnessScisconstant andcomparetheresultswiththoseobtainedbyChanandChui 2000 andShiandAtluri 1989 .TheresultsobtainedbyusingtheconnectionexponentialmodelarecomparedinFig.8bwiththosebyChanandChui 2000 andShiandAtluri 1989 ,whoadoptedtheRamberg‐Osgoodmodeltodescribetheconnection’snonlinearbehavior.ComparingFigs.8aand8b,itispossibletoverifythedifferenceinresponseswhenconsideringthesemi‐rigidconnectionwith linearandnonlinearmodel.Seethat the frameresponsewiththenonlinearconnectionmodelhasthedisplacementamplitudesreducedduetothehystereticdampingprovid‐edbythejointnonlinearbehavior.



Figure8:L‐shapedframe:geometryandloading.

Thecomparisonbetweentheconnection’s linearandnonlinearresponses isshowninFig.9.Thevibrationperiodispracticallythesameinbothcases,sincethesameinitialstiffnessoftheconnectionisconsidered.How‐ever,thenonlinearityoftheconnectionleadstoastrongdecreaseinthevibrationamplitude,duetoenergydissi‐pationduringeachhystereticcycle,asillustratedinFig.9b,wheretheconnection’smoment‐rotationresponseisdisplayed.

Toinvestigatetheinfluenceofthechosenmathematicalmodelfortheconnection’smoment‐rotationbehav‐ior,thesameanalysisiscarriedout,nowusingthefour‐parametermodel.Figure10shows,forboththeexponen‐tialandpotentialmodels,thehorizontaldisplacementattheloadpositionandtheconnection’shystereticbehav‐ior.Asmalldifferenceoccursintheconnection’shystereticbehavior Fig.10b butthetimeresponsesarecoinci‐dent Fig.10a .Thus,inthepresentcase,theconnection’sdescriptionhasnegligibleinfluenceontheframere‐sponse.



Figure9:DynamicbehavioroftheL‐framewithsemi‐rigidconnection.

Figure10:DifferentconnectionmodelsusedintheL‐frameanalysis.

5.2Two‐StoryFrame

Thenextstructuretobeinvestigatedisthesingle‐bay,two‐storyframesubjectedtohorizontalloadsappliedat thetopofeachpavement,as illustrated inFig.11.Forthebeams,aW360x72profile isadopted,andfor thecolumns, aW310x143profile. Young’smodulus is equal to205GPa.Two concentratedmassesof 3730kg areconsideredattheextremitiesofeachbeamandinthemiddle,amassof7460kg.Thestructureisthenanalyzedconsideringaperiodicloadandarectangularpulse, seeFig.11 .

Figure11:Single‐bay,two‐storyportalframe:geometryandloads.

Extendedflushendplateconnections,withaninitialstiffnessof34804kNm/rad,areconsideredbetweenthebeamsandthecolumns. Itsnonlinearmoment‐rotationbehavior isrepresentedbytheexponentialmodel,withparameters: m 6, C1 ‐0.25038x10‐3, C2 0.50736x106, C3 ‐0.30396x105, C4 0.75338x105, C5 ‐0.82873x105,C6 0.33927x105,α 0.31783x10‐3,Rp 0.96415x103kip/radandMo 0.



Figure12aillustratesthetimeresponseobtainedforthehorizontaldisplacementatthetopofthestructurewhensubmittedtoaperiodicload,wherethepresentresultsarecomparedwiththosebyChanandChui 2000 .Inthetransientregimethedisplacementreachesamaximumofaround 7cm.Afterapproximately6s,thestruc‐turereachesthepermanentregimewithanamplitudeof 5cm,whentheenergyintroducedintothestructureby the external loads and the energydissipated through the semi‐rigid connections are balanced. The connec‐tion’smoment‐rotationbehaviorduringthetotalresponseisshowninFig.12b.

ToinvestigatetheeffectofgravitationalloadsandtheP‐δeffectonthenonlineardynamicresponse,gravita‐tionalstatic loadsof36.6kNattheextremitiesandof73.2kNinthemiddleofthebeamareconsidered.Theseloadsinducesanincreaseoftheaxialforcesinthecolumnsandadditionalbendingmoments,reducingthestiff‐nessofthesemembersand,consequently,ofthestructuralsystem.Theanalysisisperformedconsideringarec‐tangularpulseof7.5kNactingduring1s.Threetypesofbeam‐columnconnectionsareconsidered:rigid, linearsemi‐rigidandnonlinearsemi‐rigid.Figures13a‐bexhibitsthetimehistoryofthehorizontaldisplacementatthetopofthestructurerespectivelywithandwithoutthepresenceofthegravitationalforces.Inbothcasesthesemi‐rigidconnectionleadstoastrongincreaseinthedisplacementsand,consequentlyonthestressesandstrains.Themaximumdisplacementforthenonlinearsemi‐rigidconnectionismorethanthreetimesthanthatoftheframewithrigidconnections.Thisunderlinestheimportanceofadetaileddynamicanalysisofportalframeswithsemi‐rigidconnectionsandexplainstheusuallylargevibrationamplitudesofthesestructures.

Figure12:Transientresponseofthesingle‐bay,two‐storyportalframeunderperiodicload.

For thisexample, inparticular, evenconsidering the influenceof the secondordereffects, thepresenceofgravitational forces induceda small increase in thedisplacements. In the rigid connections case, this influencewassmaller.While thevibrationamplitudesremainedconstant for thestructurewithrigidor linearsemi‐rigidconnections,itdecreasedinthenonlinearsemi‐rigidcaseduetothehystereticdamping.Ifviscousdampingwasalsoconsideredallresponsesconvergedtotherespectivestaticresponse.



Figure13:Transientresponseofthesingle‐bay,two‐storyportalframeunderarectangularpulse.

5.3TheSix‐StoryVogelFrame

Now,theinfluenceofbeam‐column’ssemi‐rigidconnectionsonthedynamicresponseofatwo‐bay,six‐storysteelframe,showninFig.14,isinvestigated.Theanalysisofthisstructurewithbeam‐columnrigidconnectionsisaclassicproblem,proposedbyVogel 1985 ,tocalibrateinelasticformulations.Thebeamsaresubjectedtouni‐formlydistributedstaticloadswithintensitiesof31.7kN/m topfloor and49.1kN/m otherfloors ,whicharetransformedintoequivalentnodalloadsandaddedasadditionalmassestotheself‐weightofthestructuralmem‐bers.Atthetopofthecolumns,ineachofthesixpavements,concentratedlateralharmonicloadsareappliedhav‐ingthefollowingintensities:F1 10.23sin ωct kNandF2 20.44sin ωct kN.Thebeamsandcolumns’steelpro‐filesarealsoshown in the figure.Young’smodulus isassumedtobeequal to205GPaandan initialgeometricimperfectionofΔ0 1/450forthecolumnsisadopted.Again,threetypesofbeam‐columnconnectionsarecon‐sidered:rigid, linearsemi‐rigidandnonlinearsemi‐rigid.Thesamesemi‐rigidconnectionsusedinthepreviousproblemareadoptedhere flushendplate .



Figure14:Two‐bay,six‐storyVogelframe:geometryandloading.

Thefundamentalnaturalfrequencyoftheframewithrigidconnectionsis2.41rad/s;whenadoptinglinearsemi‐rigidconnections,thevaluereducesto1.66rad/s,increasingtheeffectoflateralloadssuchaswindloadsonthestructuralresponse.Theframewithnonlinearsemi‐rigidconnectionshasthesamenaturalfrequency,sincetheinitialstiffnessisthesameasinthelinearcase.Figs.15a‐dillustratethetransientresponsesofthestructure’stop lateraldisplacement for increasingvaluesof the forcing frequencyωc.Thedisplacementsof the framewithnonlinearbeam‐columnconnectionscanbeamplifiedordampened,dependingontheforcingfrequency.Forlowforcing frequencies, see Fig. 15a ωc 1 rad/s , the horizontal displacement,when considering the nonlinearbehaviorof the connections Sc variable , is rather larger than theother two cases. InFig. 15b the forcing fre‐quencyisequaltothenaturalfrequencyoftheframewithlinearsemi‐rigidconnections,ωc 1.66rad/s,andthecorrespondingvibration amplitudes increases linearly.Therefore, itwas expected that the structurewith rigid



connections,whensubmittedtotheactionoftheloadwhosefrequencyisequaltothenaturalfrequency,itwouldshowaresonantresponse withvibrationamplitudesincreasinglinearly .Thiswasalsoexpectedfortheframewithsemi‐rigidconnections,however,when the frame isanalyzedwithsemi‐rigidconnectionswithnon‐linearbehavior,itsamplitudeofvibrationalsoincreases,butthisislimitedbythehystereticdampingwhich,asshownhere,hasabeneficialeffectintheresonanceregion.Forωc 2.41rad/s,theframewithrigidconnectionsdisplayaresonantresponse,asillustratedinFig.15c.Forωc 3.33rad/s,Fig.15d,outsidethemainresonanceregion,asharp decrease in the vibration amplitudes are observed in all cases. Shown in Figs. 16a‐b are the moment‐rotationhystereticcycles forthebeam‐columnconnectionidentifiedinsetFig.16bconsideringωc 1.66rad/sandωc 2.41rad/s,respectively.Comparingthehystereticcycles,itisobservedthattheamountofenergydissi‐patedwhenωc 1.66rad/sisgreaterthanthatforωc 2.41rad/s,whichisinagreementwiththetimerespons‐esinFigs.15b‐c,highlightingtheinfluenceofthenonlinearmoment‐rotationbehaviorofthesemi‐rigidconnec‐tionsonthedynamicsofsteelstructures.

Figure15:TransientresponseofthesixstoryVogelframeunderlateralharmonicforcing.



Figure16:Connectionhystereticbehavior.

5.4Four‐BayFive‐StoryFrame

Thislastexampleinvestigatesafour‐bay,five‐storysteelframelocatedinShanghai,China,underseismicex‐citation.Theresultsshowhowsemi‐rigidconnectionscansignificantlyalterthedynamicsofsteelstructuresun‐derbaseexcitationtoensureadequatestrengthandpreventcollapse.ItsgeometricconfigurationisshowninFig.17a.Theflushendplatebeam‐columnconnectionsareadoptedandtheirbehaviorisrepresentedbytheRichard‐Abbottmodelwiththeparameters:Scini 12336,86kNm/rad,Rp 112,97kNm/rad,M0 96,03kNmandn 1,6Nguyen,2010 .Gravitationalloadsarealsoincludedasadditionallumpedmassesatthebeams’nodes.Theeffectofameshrefinementinthebeamsisalsoconsidered.Theframeisassumedtobesubjectedtothefirsttensec‐ondsof the1940ElCentroN‐S earthquake component, shown inFig. 17b,with apeakgroundaccelerationof0.31g.

First, the free vibration analysis on the structure, considering the beam‐column rigid connections, is per‐formed.TheresultsarepresentedinTable2forthefirsttwonaturalperiodsofvibration.Table3presentstheresults considering semi‐rigid connections. In both cases an excellent agreement with the results by Nguyen2010 isobserved.Also,theinfluenceofthebeamdiscretizationisnegligible.Figure18showsthevariationofthefirst fivenatural frequenciesoftheframe,parametrizedbytherespectivevalueconsideringarigidconnec‐tion,asafunctionofthebeam‐columnconnectionsflexibility,γ.Allnaturalfrequencyratiosincreasewithγandconvergestooneatγ 1.Theconnectionflexibilityhasasignificantinfluenceonthevariationofthenaturalfre‐quencies,particularlyon the lowest frequencies.As thenatural frequenciesdecrease increases the influenceofenvironmentalloads,whosespectralcontentislocatedusuallyinthelowfrequenciesrange.Thisfactcanbeveryimportantforframesunderseismicloads,asthelowermodesmayhavestronginfluenceonabuilding’sseismicresponse.

Thehystereticmoment‐rotationbehaviorofthesemi‐rigidjointsAandC seeinsetinFig.19a ,areillustrat‐edinFig.19aand19b,respectivelyfortheframeundertheEl‐Centroearthquake.Figure20showsthetimere‐sponseofthehorizontaldisplacementatthestructure’srooflevel.



Figure17:Transientresponseofthesingle‐bay,two‐storyportalframeunderarectangularpulse.

Table2:Firsttwonaturalperiodsofvibration s fortheframewithrigidconnections.

ModesNguyen2010

PresentWorkOneele‐ment

Twoele‐ments

Fiveele‐ments

1 1.2282 1.1818 1.1820 1.18242 0.4196 0.4196 0.4201 0.4201



Thefirst‐andsecond‐orderelastictransientresponsesofthebuildingwithrigidconnectionsarepresentedinFig.20aandcomparedwiththenonlinearsecondorderresponseobtainedbyNguyen 2010 .

Table3:Firsttwonaturalperiodsofvibration s fortheframewithflexibleconnections.

ModesNguyen2010

PresentWorkOneele‐ment

Twoele‐ments

Fiveele‐ments

1 2.700 2.623 2.623 2.6232 0.777 0.733 0.773 0.770

Figure18:Influenceofbeam‐columnconnectionflexibilityonnaturalfrequencies.

Figure19:Beam‐columnconnectionhystereticbehavior.

Figure20bshowsthetransientresponsesforthesteelframewithflexibleconnections,butconsideringonlytheirlinearbehavior,inwhichcaseitisassumedthatSc Scini 12336,86kNm/rad.Finally,Figure20cpresentsthetransientsecond‐orderresponseofthesteelframewithflexibleconnections,butconsideringtheirnonlinear



hystereticbehavior RichardAbbottmodel andanincreasingnumberoffiniteelementsinthebeams’discretiza‐tionprocess.Again,agoodagreementwiththeresults foundin Nguyen,2010 isobserved.Comparingthere‐sults, themarked influenceof thestructure’sgeometricnonlinearityon thedynamicresponse isobserved.Theconsideration of the connections’ flexibility significantly amplifies the displacements compare Figure 20a andFigure20b .

Figure20:Timeresponseofthehorizontaldisplacementatthestructure’srooflevel.



Theconnectionsnonlinearity,Figure20b,decreases thedisplacements incomparisonwith the linearcase,buttheresultsarehigherthanthosewithrigidconnections.Thesecond‐ordereffectsleadtoaslightincreaseinthevibrationamplitudes.Finally,thebeamsdiscretizationhasnodetectableeffectontheresults.

6FINALREMARKS

Thisarticlepresentsthefundamentalsforpredictionofamorerealisticbehaviorofframeswithsemi‐rigidconnectionsunderdynamic loading. In thestructuralbehaviorevaluation, theeffectsofgeometricnonlinearityandconnectionflexibilityareconsidered.Differenttypesofconnectionsareconsideredandcompared.Numericalstudiesarecarriedoutforfourstructureswithdifferentgeometriesandloadconditions.Theobtainedresultsarecomparedwithexperimentsandsolutionsavailable inthe literature.Thecoincidenceornotwiththeresultsofothersauthorscanbestrongly influencedbythegeometricnonlinearfiniteelementformulationadopted,sincethestructuresstudiedareslender.ButGoodagreementisfoundinallcases,confirmingtheefficiencyofthefor‐mulation and numerical solution strategy adopted herein. Thus the numerical approach proposed here canbeefficientlyused to evaluate thenonlinear transient responseof planar steel structureswith semi‐rigid connec‐tions.

The connections flexibilitymaydecrease significantly thenatural frequencies, increasing consequently theeffectofenvironmentalloadssuchaswindandearthquakesonthedynamicresponseofthestructure.Thus,theseloadsmayinduceresonanceandthereforelargeamplitudeoscillationsbeyondtheacceptablemaximumvalues,whichdramaticallyreducethecomfort sea‐sickness andthereforelimittheuseofthebuilding,maycauselowcyclefatigueorevenimpairthesafetyofthestructure,producingunacceptableeconomiclosses.Forexample,theASCEStandard ASCE7‐05,2006 classifiesastructureasdynamicallysensitive,or“flexible”,ifthelowestnaturalfrequency 1Hz.Whenconsidering thehystereticbehaviorof the connection, a gradual reductionof thedis‐placement amplitudes is observed. The damping introduced by the connection’s nonlinear hysteretic behaviordecreasesthevibrationamplitudes,increasingthesafetyofthestructure.Thisdampingismeasuredbythecon‐nection’scapacitytodissipateenergyineachcycle,whichcanbeevaluatedbycalculatingtheenclosedareaofthecurvethatrelatestheforceactingontheconnectiontoitsdeformation.Viscousdampingwasnotconsideredinthepresentanalysistodrawattentiontotheinfluenceoftheconnection’sdissipativeproperties.Althoughbenefi‐cial,viscousdampingcoefficientsofslendersteelframesareusuallyverylow.Theconnectionbehaviorisshowntohaveastronginfluenceontheresults.Asthespringmodelisusedtorepresentthesemi‐rigidjointbehavior,thenumberofspringsdependsonthenumberofstructureconnections.Thustheconsiderationofthetruecon‐nectionbehavior isextremely important fora realistic forecastingof thedynamicresponseandresistanceofastructuralsteelframes.Finally,secondordereffectsareshowntoincreasethevibrationamplitudes,reducingthecomfortandsafetyoftallbuildingframes.

Acknowledgments

TheauthorswishtoexpresstheirgratitudetoCAPESandCNPq,FAPEMIGandUFOPforthesupportreceivedinthedevelopmentofthisresearch.SpecialthanksgotoProf.HarrietReisandProf.JohnWhitefortheireditorialreviews.

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Appendix A

Algorithmfornonlineartransientanalysis1: Definetheinputdata:geometric,materialandloadingpropertiesofthestructuralsystem2: Obtainthereferencenodalloadvector,Fr loadingdirection 3: 0t = 4:

1t t=

5: Considerthedisplacement,velocityandaccelerationvectorsprescribed:tU , tU ,and tU

6: SelectthetimeincrementΔt7: Definetheconstants:

0 1 2 3 4 52

1 1 1; ; ; 1 ; 1; 2

2 2

ta a a a a a

t tt

æ ö æ ög g D g÷ ÷ç ç÷ ÷ç ç= = = = - = - = -÷ ÷ç ç÷ ÷÷ ÷ç çbD bD b b bè ø è øbD

( )6 0 7 2 8 3 9 101a a ; a a ; a a ; a t and a tD g gD= =- =- = - =

whereβandγaretheNewmarkparameters8: for eachtimestepdo9:

1t t= Previoustimestep10:

1t t tD= + Previoustimestep11: Assemblethematrices:stiffness,K ;mass,M ;anddamping,C 12: Formtheeffectivestiffnessmatrix: 0 1

ˆ a a= + +K K M C

13: Formtheeffectiveloadvector: ( ) ( )12 3 4 5

ˆ t t t t t tr ia a a a= l + + + + -F F M U U C U U F

14: Solveforthedisplacementincrementvector ˆ ˆ:D D =U K U F 15: fork←1,maximumnumberofiterations do ITERATIVEPROCESS16: Evaluatetheacceleration,velocityanddisplacementvectorsapproximationsattime

1t :

+1 1 10 2 3 1 4 5 ; and t t tk k t t k k t t k t ka a a a a a= D - - = D - - = DU U U U U U U U U U U

17: Updatethestructuregeometry nodalcoordinates18: Evaluatetheinternalforcesvector: 1t k t k

i i F F K U

19: Evaluatethegradientvector: ( ) ( )1 1 1 1 11t t t t tk k k kr i

+ = l - + +R F M U C U F

20: Solvefortheresidualdisplacementvector: 11 1ˆ tk k K U R 21: Updatetheincrementaldisplacementvector: 1 1k k k U U U+ 22:

if ( ) ( )11 1 tk k tolerance factor+ +D £U U then

23: Exittheiterativeprocessandgotoline2624: endif25: endfor26: Updatetheacceleration,velocityanddisplacementvectorsattime

1t : 1 1 11 1 1 1 1 1

0 2 3 1 4 5; and +t t tk k t t k k t t k t ka a a a a a U U U U U U U U U U U

27: Updatetheinternalforcevectorattime1t :

28: Updatetheconnectionsstiffness SeeSection2andAppendixB29: endfor



Appendix B

Algorithmforhystereticbehaviorofsemi‐rigidconnection1: for eachfiniteelementdo2: Identifythenodalpointswithsemi‐rigidconnection3: for eachofthesenodalpointsdo 4: M = actualmomentinthisnode 5:

oldM = momentinthisnodeintheinstantt

6:oldM M MD = -

7: if 0oldM M⋅ < then

8:p c:oldf f= SeeFigures3 andAppendixA

9: 0Mai = and 0aM = SeeFigures3 andAppendixA

10: endif11: if 0M MD⋅ > then LOADINGPROCESS12:

c:old cf f=

13: Updatec :f c c cM / Sf f D= +

14:c c pDf f f= +

15: if 1Mai = then Unloadingprocessincomplete

16: if aM M< then

17:c ciniS S=

18: else19: Definethestiffnessconnection

cS forcDf SeeEquations7,and4or5

20: 0Mai = and 0aM =

21: endif23: elseif 0Mai = then

24: DefinethestiffnessconnectioncS for

cDf SeeEquations7,and4or5

25: endif26: elseif 0M MD⋅ < then UNLOADINGPROCESS27: if 0oldM M⋅ > and 0Mai = then

28:ca c:oldf f=

29:aM M MD- and 1Mai =

30: endif31:

c:old cf f=

32: Updatec :f c c cM / Sf f D= +

33:c ciniS S=

34: endif35: endfor36: endfor

on the nonlinear transient analysis of planar steel frames ... · on the nonlinear transient...

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