three‐level performance‐based optimization method of steel ... · three‐level...

LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07

Three‐levelperformance‐basedoptimizationmethodofsteelframes

AbstractThispaperdescribesathree‐levelperformance‐basedoptimizationmodeland an estimatemethod of residual top displacement for steel frames atthreeearthquakelevels.Thesteelframesaresupposedtobeelasticatfre‐quent earthquake, inelastic and hardening at occasional and rare earth‐quakes,respectively.Theestimateformulaisderivedandestimateproce‐dureisgivenindetail.Theestimatemethodonlyneedstouse only onepushoveranalysisuntilsteelframesyield.Theyieldpointisobtainedauto‐maticallyintheproposedmethod.Theestimatemethodisabletomakeop‐timizationprocessuninterrupted.Optimaldesignof a3‐story2‐bay steelframeisdemonstratedtovalidatetheproposedmethod.

KeywordsEarthquake resistant structures, Earthquake engineering, Seismic design,Steelframes,Optimization,Optimizationmodels.

1INTRODUCTION

Performance‐baseddesignreferstothemethodologyinwhichstructuraldesigncriteriaareexpressedintermsofachievingasetofperformanceobjectives Ghobarah,2001 .Thestructuralseismicdesignneedstobebasedonthedefinedmultipleperformanceobjectivesandearthquakehazardlevels.Theperformanceobjectivescanbede‐finedasthreelevels i.e.,serviceability,lifesafety,andcollapseprevention associatedwiththreeearthquakehaz‐ardlevels i.e.,frequent,occasional,andrareearthquakes .ThestructureshouldhasnodamageatserviceabilitylevelwhenitmeetsthefrequencyearthquakeinEurocode8 EuropeanCommitteeforStandardization,2004 .Thestructureisallowedtohavemoderateandseveredamageatlifesafetyandcollapsepreventionlevelswhenitmeetsoccasionalandrareearthquakes,respectivelyinEurocode8 EuropeanCommitteeforStandardization,2004 .

Therefore,atlifesafetyandcollapsepreventionlevels,itisnecessarytoexplicitlyconsidertheinelasticbehav‐iorofthestructures.Thenonlineartime‐historyanalysisisbelievedtobethemostrigorousproceduretoevaluatetheinelasticbehaviorofstructures.However,thenonlineartime‐historyanalysismethodsarebelievednottobepracticalforeverydaydesignbecausetheyinvolvecomputationalandmodelingeffort,convergenceproblemandcomplexity Liuetal.,2010;GencturkandElnashai,2008 .Thesimplifiednonlinearanalysismethodsareprefera‐bletoevaluatetheinelasticbehaviorofstructuresincivilengineeringpractice Fajfar,1999,2000 .Thesimplifiednonlinearanalysismethodsarebasedonpushoveranalysistodeterminestructuralcapacitydiagramandondesignresponsespectratorepresentdemanddiagram.TheN2method Fajfar,1999,2000,2002;KreslinandFajfar,2012 andcapacityspectrummethod Freeman,1998;ZouandChan,2005;GencturkandElnashai,2008 are typicalsimplifiednonlinearanalysismethods.TheN2methodhasbeenimplementedinEurocode8 EuropeanCommitteefor Standardization,2004 .The capacity spectrummethodshavebeenapplied inATC40 AppliedTechnologyCouncil,1996 andFEMA356 FederalEmergencyManagementAgency,2000 indifferentiterativeprocedures.TheN2methodcandetermineperformancepointwithnoneed for iteration.Thesimplifiednonlinearanalysismethodshavebeenbrieflyreviewedinliterature Fajfar,2002 .

Thesimplifiednonlinearanalysismethodscanobtainseismicdemands topdisplacement,inter‐storydrifts,etc. atdifferenthazardlevels.Theseismicdemandsarecomparedwithperformancetargets specifiedlimitsontopdisplacement,inter‐storydriftsetc. fortherelevantperformancelevels.Iftheseismicdemandsareequaltoorlessthanperformancetargets, itmeansthestructuresperformwellatdifferenthazardlevels.Iftheseismicde‐mandsaregreaterthanperformancetargets,thestructuralperformanceisunsatisfiedatdifferenthazardlevels.Thenewstructuresmustbemodifiedandtheexistingstructuresmustbestrengtheneduntilthestructuresperformwellatdifferenthazardlevels.Therefore, it isstill tedioustodesignanewstructurewhichcanperformwellat

Original Article

QimaoLiua,b*

aABBCorporateResearchCenter,Vasteras,72226,SwedenbDepartmentofCivilEngineering,AaltoUniver‐sity,Espoo,02150,Finland

* qimao.liu@se.abb.com

http://dx.doi.org/10.1590/1679-78252954

Received:April01,2016InRevisedForm:December15,2017Accepted:February09,2018Availableonline:March21,2018

QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes

LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 2/18

differenthazardlevelsusingthesimplifiednonlinearanalysismethods,althoughitmayberelativelysimpletoeval‐uateperformanceoftheexistingstructuresatdifferenthazardlevels.

Theauthoreverproposedanoptimizationprocedureforseismicdesignofsteelframesformulti‐performanceandmulti‐hazard levels LiuandPaavola2015 .However, the inter‐storydriftsaredirectly treated in thecon‐straints,forabuildingwithmanystories,itisnotconvenienttoimplement.Inthispaper,anovelmethodtoestimateresidualtopdisplacementsofsteelframesatdifferenthazardlevelsisdeveloped.Athree‐levelperformance‐basedoptimizationmodelisproposed.Thepaperisarrangedasfollows.Insection2,threeperformancelevelsandasso‐ciatedthreehazardlevelsaredefined.Insection3,elasticandinelasticdemandspectrainADformatareobtainedbasedonelasticaccelerationresponsespectruminEC8.Insection4,aproposedmethodtoobtaincapacityspec‐trumdiagramisdevelopedbasedontheN2method.ComparedwiththeN2method, theproposedmethoduseonly onepushoveranalysisuntilsteelframesyield.Italsodoesn’tneedanyengineeringjudgementstogettheyieldpoint.Insection5,reductionfactorandductilityfactoraredetermined.Insection6,anestimatemethodofresidualtopdisplacementisdevelopedforthree‐levelperformance‐baseddesign.Insection7,computationalpro‐cedureofresidualtopdisplacementisgivenindetail.Insection8,athree‐levelperformance‐basedoptimizationmodelisformulatedbasedontheresidualtopdisplacements.Finally,astheillustrationofthedevelopedapproach,optimaldesignofa3‐story2‐baysteelframeisdemonstratedusingANSYSsoftware.

2DEFINITIONOFPERFORMANCEANDHAZARDLEVELS

Theperformancetargetsarespecifiedlimitsonanyresponseparameters,i.e.,topdisplacement,inter‐storydrifts,stresses,strains,etc.,atrelevantperformancelevels.Theperformanceobjectivesrequirethatthestructuralseismicdemandsareequaltoorlessthanperformancetargetsatrelevantperformancelevels.Thetopdisplace‐mentoftheMDOFsystemisagoodindicatoroftheglobaldeformationofthebuildingssubjectedtoearthquakeloading.Thetopdisplacementisoftentakenastheglobalseismicdemandinperformancebaseddesign KreslinandFajfar,2012;Fajfar,1999,2000,2002;Ghobarah,2001;ZouandChan,2005;GencturkandElnashai,2008 .However,thetopdisplacementcannotrevealtheglobaldamagedegreeofthebuildingssubjectedtoseveregroundmotions.Theresidualtopdisplacementisnotonlyagoodindicatoroftheglobaldeformationofthebuildingssub‐jectedtoearthquakeloading,butalsocanreveal theglobaldamagedegreeof thebuildingssubjectedtoseveregroundmotions.Inmulti‐levelperformancebaseddesign,theresidualdeformationofthebuildingsispresentatthelifesafetyandcollapsepreventionlevels.Inthispaper,theseismicdemandofsteelframeistheresidualtopdisplacement.Thelimitsonresidualtopdisplacementareperformancetargets.Theperformanceobjectivesrequirethattheresidualtopdisplacementofsteelframeisequaltoorlessthanthelimitsonresidualtopdisplacementatrelevantperformancelevels.Thethreeperformancelevels,correspondingdamagestatesandlimitsonresidualtopdisplacementaredefinedinTable1.Theperformancelevelsareassociatedwithearthquakehazardanddesignlevels.ThreeearthquakehazardlevelsassociatedwiththethreeperformancelevelsareproposedinTable2.

3SEISMICDEMANDDIAGRAM

Accordingtotheprecedingdefinition,theseismicdemanddiagramistheelasticaccelerationresponsespec‐trumatserviceabilitylevel,andinelasticaccelerationresponsespectraatlife‐safetyprotectionandcollapsepre‐ventionlevels.

Table1:Threeperformancelevels,correspondingdamagestatesandlimitsonresidualdisplacement.

Performancelevels Damagestates Limitsonresidualtopdisplacement

Serviceability Nodamage 1 0tpD

Lifesafety Moderatedamageandrepairable 2 5tpD

Collapseprevention Severedamage 3 15tpD

Table2:Proposedearthquakehazardlevels.

Earthquakefrequency

Returnperiodforyears

Probabilityofexceedance

Peakgroundaccelerationsinthispaper

Frequent 43 50%in30years 1ga 0.15g

Occasional 72 50%in50years 2ga 0.40g

Rare 475 10%in50years 3ga 0.60g

3.1HorizontalelasticaccelerationresponsespectruminADformat

Theelasticresponseaccelerationspectrum aeS T forthehorizontalcomponentsofseismicactionisdefined

inEurocode8 EuropeanCommitteeforStandardization,2004 as

0 : 1 2.5 1

4 : 2.5

B ae gB

B C ae g

CC D ae g

C DD ae g

TT T S T a S

T T T S T a S

TT T T S T a S

T T s S T a ST

where aeS T is the elastic response spectrum;T is the vibrationperiod of a linear single‐degree‐of‐freedom

system; ga isthedesigngroundaccelerationontypeAground;BT isthelowerlimitoftheperiodoftheconstant

spectralaccelerationbranch;CT istheupperlimitoftheperiodoftheconstantspectralaccelerationbranch;

isthevaluedefiningthebeginningoftheconstantdisplacementresponserangeofthespectrum; S isthesoilfactorand isthedampingcorrectionfactorwithareferencevalueof 1 for5%viscousdampingratio.Forground

typeA, 1.0S , 0 .1 5BT s, 0 .4CT s, 2.0DT s.

ForanelasticSDOFsystem,thedisplacementresponsespectrumis

24de ae

TS T S T

where deS T is the value in the displacement spectrum corresponding to the period T and a fixed viscous

dampingratio.ElasticresponsespectruminADformatisobtainedbyEqs. 1 and 2 .Forthehorizontalelasticacceleration

responsespectrumdefinedinEC8,thecut‐offperiodisobviously2sinADformat.

3.2HorizontalinelasticaccelerationresponsespectruminADformat

Therelationshipbetweeninelasticresponsespectrumandelasticresponsespectrum Vidicetal.,1994 is

S TS T

d deS T S TR

where dS T and aS T are the values in the displacement and acceleration spectra, respectively,

corresponding to the periodT and a fixed viscous damping ratio 5% . R is the reduction factor and is

ductilityfactor.BysubstitutingEqs. 3 and 4 intoEq. 2 ,weobtaintheinelasticresponsespectrumfunctioninADformat,

TS T S T

Forabilinearspectrum,

( 1) 1C

T CT T 6

R CT T 7

ThereductionfactorR andtheductilityfactor arerelatedtoboththestructuralresponseandtheelastic

accelerationresponsespectrum.

4STRUCTURALCAPACITYDIAGRAM

4.1Baseshear‐Topdisplacementdiagram MDOFsystem

Thenonlinearstaticanalysis alsocalledpushoveranalysis isusedtoobtainthebaseshearforce V ‐topdisplacement

tD diagramforMDOFsystem.Amonotonicallyincreasingpatternof lateralforcesisappliedto

structuresinpushoveranalysis.AplanarsteelframeshowninFigure1isassumedwhere n isthenumberofthestory,theheightofthe i thstoryis

ih andthemassofthe i thstoryis im .Theinvertedtriangularloadpatternwith

maximumloadingattopandzeroloadingatthegroundlevelisemployedinthispaper.Itisassumedthatthelateralforceatthe i thstoryshowninFigure1isproportionaltothecomponentoftheassumeddisplacementshape

weightedbythestorymassim FajfarandGaspersic,1996 ,

i i iP pm 8

wherethecomponentoftheassumeddisplacementshapei is

where p controls themagnitude of the lateral loads of steel frames. Thedisplacement shape follows the firstvibrationmodeofthebuilding.

Therefore,thebaseshearforceV is

i i ii i

V P p m pm

where1

istheequivalentmassoftheSDOFsystem.

Figure1:n‐storysteelframe.

Theequalloadingstepsareappliedtosteelframeinpushoveranalysis.TheloadingstepsizeiP atthe i th

storyis

i i i i i

pP m pm

where pp

c ,and c isaconstantinteger.

Theincrementofbaseshearforceis

Inthebaseshearforce V ‐topdisplacementtD diagramshowninFigure2, jV and tjD aretheincre‐

mentofbaseshearforceandincrementoftopdisplacementatthe jthloadstep,respectively.

Figure2: - tV D diagram.

4.2TheF ‐Ddiagram,yieldpointandelasticperiod equivalentSDOFsystem

ThedisplacementoftheequivalentSDOFsystemis

ThebaseshearforceoftheequivalentSDOFsystemis

where isaconstantandcalculatedas

i iin n

i i i ii i

Providedthatbothshearforce V andtopdisplacementtD aredividedby ,theforce‐displacement

relationshipdeterminedfortheMDOFsystem,i.e.,V ‐tD diagram,becomestheshearforceanddisplacementre‐

lationshipfortheequivalentSDOFsystem,i.e.,theshearforceF anddisplacementDdiagramshowninFigure3.

Figure3:F ‐Ddiagram.

InFigure3, jF and jD areshearforceincrementanddisplacementincrementoftheequivalentSDOFsys‐

tematthe jthloadstep,respectively.

Inthispaper,thesteelframeisassumedtoyieldifthefollowinginequalitiesaretrueatthe y thloadstep,

1yD D 19

whereisaconstantandgreaterthan1.Forstructuresteel,themodulusofelasticityismuchgreaterthanthetangentmodulus,inthispaper, 2 orbiggercanbeenoughtojudgethattheyieldinghappens.

Therefore,thedisplacementattheyieldpointinF ‐Ddiagramis

TheshearforceattheyieldpointinF ‐Ddiagramis

Inthispaper,thepost‐yieldstiffnessisidealizedtobezero AschheimandBlack2000 .TheidealizedF ‐D

diagramisshowninFigure4.Theelasticperiodoftheidealizedbilinearsystemcanbecalculatedas

Figure4:IdealizedF ‐Ddiagram thickline .

4.3CapacitydiagraminADformat

ThecapacitydiagraminADformatshowninFigure5isobtainedbydividingtheforcesintheforce‐defor‐

mationdiagram,i.e.,theidealizedF ‐Ddiagram,bytheequivalentmassm .

Inthispaper,thepushoveranalysisisuseduntilthesteelframesyieldandthecapacitydiagramisobtained.Itisrelativelyeasiertosucceedinperformingpushoveranalysisuntilasteelframeyieldsthanuntiltheframecol‐lapses.Itdoesn’tneedanyengineeringjudgementstogettheyieldpoint.Therefore,itcanmaketheoptimizationprocessuninterrupted.

Figure5:CapacitydiagraminADformat.

5DETERMINATIONOFREDUCTIONFACTORANDDUCTILITYFACTOR

ElasticresponsespectruminADformat,inelasticspectruminADformatandcapacitydiagraminADformat

areplottedinthesamepictureshowninFigure6 CT T orFigure7 CT T .ThereductionfactoruR isde‐

finedastheratiobetweentheaccelerationscorrespondingtotheelasticandinelasticsystems,

( )aeu

where ( )aeS Tis the acceleration value of the elastic spectrum diagram at the periodT and ayS is the yield

accelerationshowninFigure6orFigure7.

If ( )ay aeS S T ,i.e., 1uR ,thesteelframeresponseiselastic.Therefore,theductilityfactor is

1 when 1uR 25

If ( )ay aeS S T ,i.e., 1uR ,thesteelframeresponseisinelastic.Theductilityfactor canbecalculatedas

follows:

If CT T showninFigure6 ,

( ) ( )d de aeu

y y ay

S S T S TR

where ( )deS TisthedisplacementvalueoftheelasticspectrumattheperiodT and

dS isdisplacementvalueat

theintersectionoftheinelasticspectrumdiagramandcapacitydiagramshowninFigure6andFigure7.

If CT T showninFigure7 ,

1 1CTR

Figure6:ElasticspectrumADformat,inelasticspectrumADformatandcapacitydiagram CT T .

Figure7:ElasticspectrumADformat,inelasticspectrumADformatandcapacitydiagram CT T .

6ESTIMATEOFRESIDUALDEFORMATION

For thecaseof threeperformance levels i.e., serviceability, lifesafety,andcollapseprevention , the threecorrespondingstructuralcharacteristics i.e.,stiffness,strengthanddeformationcapacity dominatetheperfor‐mancesasshowninFigure8.ThetypicalperformancecurveofthesteelframesshowninFigure8indicatesthatnoresidualtopdisplacementispresentatserviceability level,moderateandtolerableresidualtopdisplacementispresentatlife‐safetylevel,andlargeresidualtopdisplacementispresentatcollapsepreventionlevel.Althoughthepermanentdeformationexistsatlifesafetyandcollapsepreventionlevels,thesteelframeisinstronghardeningphaseatserviceabilityandinweakhardeningphaseatcollapseprevention,respectively.

Figure8:Typicalperformancecurveofsteelframes.

ThedisplacementdemandoftheequivalentSDOFsystemcanbedeterminedfromthedefinitionofductilityas

*d yS D 28

ThedisplacementdemandoftheequivalentSDOFsystemistransformedbacktothetopdisplacementoftheMDOFsystem,

*t d yD S D 29

Ifthesteelframeisatelasticphase theductilityfactor 1 ,Noresidualtopdisplacementispresentafter

earthquake,i.e., 0tpD .

Ifthesteelframeisatinelasticandhardeningphase theductilityfactor 1 asshowninFigure9,accord‐

ingtotheidealizedF ‐Ddiagram,theresidualdisplacementdemandoftheequivalentSDOFsystemcanbeesti‐matedas

1p y e y y

DD D D D F

where eDistheelasticdisplacement.

Figure9:RealandidealizedF ‐Ddiagram.

IfthesteelframeisstillathardeningphaseasshowninFigure9,theestimatedvalueoftheresidualtopdis‐

placementofthesteel frame pDisgreaterthantherealresidualtopdisplacement hD

.Therefore, theestimated

valueoftheresidualtopdisplacementisconservativeinEq. 30 .TheresidualdisplacementoftheequivalentSDOFsystemistransformedbacktothetopresidualdisplacement

oftheMDOFsystem,

1tp p y y

DD D D F

AsshowninFigure8,thesteelframeisinelasticphaseatserviceability,instronghardeningphaseandinweakhardeningphaseatlifesafetyandcollapsepreventionlevels,respectively.Therefore,theresidualtopdisplacement

ofsteelframeatserviceability,lifesafetyandcollapsepreventionlevels,canbeestimatedas:if 1 , 0tpD ,if

1 ,theresidualtopdisplacementiscalculatedbyusingEq. 31 .

7COMPUTATIONALPROCEDUREFORRESIDUALTOPDISPLACEMENT

TheEq. 31 isusedtoestimatetheresidualtopdisplacementoftheMDOFsystemwhenthesteelframere‐sponseisatelasticphaseorinelasticandhardeningphase.Thedetailofcomputationalprocedureis:

Step1PerformpushoveranalysiswiththeequalloadingstepsizecontrolledbyEq. 11 untilthesteelframeyields InequalitiesEq. 18 orEq. 19 istrue .

Step2Obtaintheyieldpoint ,y yD F usingEq. 20 andEq. 21 .

Step3CalculatetheelasticperiodT oftheequivalentSDOFsystemusingEq. 22 .Step4Calculatethereductionfactor

uR withEq. 24 .

Step5If 1uR ,then 1 ,theresidualtopdisplacementis 0tpD .If 1uR ,calculatethereductionfactor

withEq. 26 if CT T orEq. 27 if CT T .Estimatetheresidualtopdisplacement tpD usingEq. 31 .

8THREE‐LEVELPERFORMANCEOPTIMIZATIONMODEL

The typeAgroundmotion isdefinedwith theelasticaccelerationresponsespectrumaccording toEq. 1 ,

whichhasbeennormalizedtopeakgroundacceleration ga equalto 1ga , 2

ga and 3ga atfrequent,occasionaland

rareearthquake,respectively.Thedesigngroundaccelerationsforthethreelevelperformancesdependondiffer‐entdesigncodesandcanbechosenbystructuralengineers.Theresidualtopdisplacementsofsteelframescanbe

estimatedbythecomputationalprocedureinsection7.Theresidualtopdisplacementsaredenotedas 1tpD , 2

and 3tpD at frequent,occasionalandrareearthquake,respectively. 1

, 2tpD

arethe limitson

residualtopdisplacementsrelatedtoserviceability,lifesafetyandcollapsepreventionlevels,respectively.Thethree‐levelperformanceoptimizationmodelisproposedas

1, 2, ,

S T D D

d d d i N

wheredand M d arethevectorofdesignvariablesandmassofsteelframe,respectively.id ,

id ,and id are

the i thdesignvariable, its lower andupperboundary, respectively. N is thenumberof designvariables.The

residualtopdisplacements 1tpD , 2

tpD ,and 3tpD aredependentonthevectorofdesignvariables d .

Inthispaper,thefirst‐orderoptimizationmethodofANSYSisemployedtosolvetheoptimizationmodelofEq.32 .Inthefirst‐orderoptimizationmethod,thesensitivityanalysistechniqueisusedtoconstructoptimizationalgorithmanddeterminethesearchingdirection.ANSYusesthefinite‐differencemethodstoperformsensitivityanalysis.Thefirst‐ordermethodconvertstheconstrainedoptimizationproblemintoaseriesofunconstrainedop‐timizationproblemsbyusingpenaltyfunctionmethods.AccordingtothelateralloadpatternofEq. 8 ,therelation

betweentheresidualtopdisplacementsjtpD 1,2,3j andresidualinter‐storydrifts

id r 1,2, ,i n is

Assumedthatthekthstoryofthestructureisweak/softstory,theresidualinter‐storydriftofthekthstoryisthedominantcomponentinthecontributiontotheresidualtopdisplacement.

jtp kD dr 34

Thedesignvariableofcolumnsatthekthstoryiskd .DifferentiatingEq. 34 withrespecttodesignvariable

kd ,wehave

Sincethekthstoryofthestructureisweak/softstory,thefirstordersensitivity k

willbeverylarge.In

Eq. 35 ,thefirstordersensitivityofresidualtopdisplacement,i.e.,j

,willbecomeverylarge.Thedesignwith

weak/softstoryisimpossibletobechosenastheoptimumdesign.Therefore,althoughtherearenoexplicitconstraintsoninter‐storydriftsinthree‐levelperformanceoptimi‐

zationmodelofEq. 32 ,theinter‐storydrifts localseismicdemand areindirectlyconstrainedbyusingresidualtopdisplacementconstraints,lateralloadpattern invertedtriangle andsensitivityanalysistechnique.

9EXAMPLE

9.1Optimaldesignofa3‐story2‐baysteelframe

Athree‐storytwo‐baysteelframeshowninFigure10isfixedattheground.Theheightofthefirst,secondandthirdstoriesare

1 5 .4h m ,2 3 .6h m and

3 3 .6h m ,respectively.Thesteelframeconsistsof4groupsinclud‐

ingB1,C1,C2,andC3showninFigure10.AllthemembersareH‐shapesectionshowninFigure10.ThedesignvariablesarethesizeofflangesandwebsalsoshowninFigure10.Thedesignvariablevectoris

1 1 1 1 2 2 2 2 3 3 3 3, , , , , , , , , , , , , , ,bf bf bw bw cf cf cw cw cf cf cw cw cf cf cw cwh t h t h t h t h t h t h t h t d

TheBeam189elementisusedtoanalyzethesteelframe.Beam189isa3‐D3‐nodeelement.Thiselementiswell‐suitedforlinear,largerotation,andlargestrainnonlinearanalysis.TheBeamelementisdevelopedbySimoandVu‐Quoc 1986 ,andIbrahimbegovic 1995 .ThepushoveranalysisiscarriedoutusingANSYSsoftware.Theproperties ofmaterial are defined as:Modulus of elasticity 200 GPaE , Poisson’s ratio 0.3 , Yield stress

235 MPay andSecantmodulusofplasticity 1.035 M P asE ,Density 37 8 5 7 k g /m .Themassofevery

floorisassumedtobe1000kg.Thedesignspace,i.e.,lowerandupperlimits,isshowninTable3.

Table3:Designspace,initialandoptimumdesigns.

Initialdesign

Optimumdesign

Designspace

Lowerlimits

Upperlim‐its

BeamB1 mm

Flangewidth 150 130 100 180Flangethickness 17 12 8 22Webdepth 400 319 205 600

Webthickness 14 7 6 20

ColumnC1 mm

ColumnC2 mm

ColumnC3 mm

Webthickness 14 11 4 20Structuralmass kg 6218 4014

PeriodofSDOF,T , s 0.164 0.216

YielddisplacementofSDOF, yD, cm 9.13 11.10

Ductilityfactor,

1stlevel 1.000 1.000 2ndlevel 1.000 1.039 3rdlevel 1.176 1.985

Topdisplace‐ment,

tD , cm

1stlevel 3.10 4.42Note: t yD D 2ndlevel 8.25 14.59

3rdlevel 13.57 27.88

Inthisexample,thepeakgroundaccelerationsareassumedas 1 0.15 gga , 2 0.4 gga and 3 0.6 gga

atfrequent,occasionalandrareearthquake,respectively.Thelimitsonresidualtopdisplacementsareassumedas 1 0tpD

, 2 5tpD cm,and 3 15tpD

Thefirst‐orderoptimizationmethodofANSYSisemployedtosolvetheoptimizationmodelofEq. 32 .Thefirst‐ordermethodwillperformamaximumof30iterationsuponexecution,usingalinesearchstepequalto100%ofthemaximumpossiblevalue,anda0.200%differenceappliedtothedesignvariablestoobtainthefirst‐ordersensitivity.Thetoleranceoftheobjectivefunctionis10kg,i.e.,thealgorithmconvergesifthedifferencebetweenthetwoadjacentobjectivevalues,achievedbytheadjacenttwolinearsearching,islessthan10kg.StartingwiththeinitialdesignshowninTable3,thefirst‐orderoptimizationmethodconvergesatthesixthiterationtoobtaintheoptimumdesignshowninTable3.Intheoptimizationprocess,theframemass,ductilityfactor,andresidualtopdisplacementsattheserviceability,lifesafetyandcollapsepreventionlevelsareshowninFigures11,12,and13,respectively.

Figure10:Three‐storysteelframewithH‐shapesectionsofmembers.

9.2Observationofoptimumdesign

Table3showsthattheresidualtopdisplacementoftheoptimumdesignatcollapsepreventionlevelis14.68cmandtheupperlimitofthedesignspaceis15cm,andtheresidualtopdisplacementoftheoptimumdesignatlifesafetylevelis1.40cmandtheupperlimitofthedesignspaceis5cm.Therefore,theconstraintoftheresidualtopdisplacement isactiveatcollapsepreventionlevel,notactiveat lifesafety level. If theconstraintupper limitofresidualtopdisplacementatcollapsepreventionlevelinthismathematicalmodelischanged,thedifferentoptimumdesignwillbeachieved.TheoptimumdesigniscomparedwiththeinitialdesignshowninTable3,andinFigures11,12and13.Themassofoptimumdesigndecreases.TheperiodandyielddisplacementoftheequivalentSDOFincrease,theductilityfactorincreases,andthetopdisplacementsofthesteelframeatserviceability,lifesafety,andcollapsepreventionlevelsincrease.

Table4:Inter‐storydriftsofoptimumdesign units:cm .

1st‐story 2nd‐story 3rd‐story

Serviceabilitylevel 1.79 1.56 1.07

Lifesafetylevel 6.18 5.08 3.33

Collapsepreventionlevel 13.53 9.14 5.25

Table5:Inter‐storydriftratioofoptimumdesign.

1st‐story 2nd‐story 3rd‐story

Serviceabilitylevel 0.003 0.004 0.003

Lifesafetylevel 0.011 0.014 0.009

Collapsepreventionlevel 0.025 0.025 0.015

Figure11:Objectivefunctionvariationinoptimizationprocess.

Inthedesigncodes,theinter‐storydriftsmustbeverifiedtosatisfythelimitations.Therefore,thepushoveranalysesoftheoptimumdesignshouldbeusedonceagain.AccordingtothetopdisplacementsshowninTable3,thepushoveranalysisstopsifthetopdisplacementisequalorgreaterthan27.88cmatcollapsepreventionlevel,equalorgreaterthan14.59cmatlifesafetylevel,equalorgreaterthan4.42cmatserviceabilitylevel.Theresultsshowthatthepushoveranalysisstopsat27.92cm,14.595cm,and4.424cm,atcollapseprevention,lifesafety,andserviceabilitylevels,respectively.Theinter‐storydriftsoftheoptimumdesignatcollapseprevention,lifesafety,andserviceabilitylevelsarelistedinTable4.ThepushoveranalysisprocessatcollapsepreventionlevelisshowninFigure14.Thediagramsoftopdisplacement,inter‐storydriftsshowninFigure14indicatethatthesteelframeisatthehardeningphaseatcollapsepreventionlevel.Therefore,theestimatedmethodoftheresidualtopdisplace‐mentproposedinthispaperisreasonable.Theinter‐storydriftandtopdisplacementdiagramsoftheoptimumdesignatserviceability,lifesafety,andcollapsepreventionlevelsareshowninFigure15.Theinter‐storydriftratiosoftheoptimumdesigninTable5indicatethatthedistributionofinter‐storydriftratiosarerelativelyuniformatserviceability,lifesafety,andcollapsepreventionlevels,respectively.

Figure12:Ductilityfactorvariationinoptimizationprocess.

0 1 2 3 4 5 6 7

Iteration number

Figure13:Residualtopdisplacementvariationinoptimizationprocess.

Figure14:Pushoveranalysistotopdisplacement27.92cm 27.88cm foroptimumdesign.

Figure15:Inter‐storydriftsandtopdisplacementof1st,2ndand3rdperformancelevels

9.3Discussions

Itisgenerallyacceptedthatthedamageofstructureisstrainanddisplacementrelated.Theresidualtopdis‐placementcandirectlymeasuretheglobaldamageofstructuressubjectedtothedifferentearthquakehazardlevels.

-10123456789

10111213141516

1900ral 1900ral 1900ral 1900ral 1900ral 1900ral 1900ral 1900ral

Iteration number

Serviceabilitylevel

1900ral

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

Drift (cm)

1st performance level, inter-story drift1st performance level, topdisplacement2nd performance level, inter-story drift2nd performance level, topdisplacement3rd performance level, inter-story drift3rd performance level, topdisplacement

Theestimatemethodofresidual topdisplacementassumesthat thesteel frameis inhardeningphasebefore itcollapses.Inthispaper,theestimatemethodofresidualtopdisplacementonlyneedsthedisplacementandbaseshearforceatyieldpoint.Therefore,thepushoveranalysisisuseduntilsteelframeyields InequalitiesEq. 18 orEq. 19 istrue .Thecomputationaltimetogettheyieldpointisfarlessthanthattousepushoveranalysisuntilthestructurecollapses.Ontheotherhand,itisnoteasytosucceedindemonstratingpushoveranalysisuntilstruc‐turescollapsebecausethedifferentstructureswillbeproducedintheoptimizationprocess.However,itiseasytosucceedinimplementingpushoveranalysisuntilstructuresyield.Therefore,theoptimizationprocesswillnotbeinterrupted.Theestimatemethodofresidualtopdisplacementcanbeusedinoptimizationdesignofsteelframesatserviceability,lifesafety,andcollapsepreventionlevels.WiththeaidofEq. 30 andFigure9,itcanbeobservedthattheestimatedvalueofresidualtopdisplacementisgreaterthantherealresidualtopdisplacement.Therefore,theestimatemethodofresidualtopdisplacementisconservative.TheresidualtopdisplacementcanbecalculatedusingthecomputationalprocedurediscussedinSection7.

Theinputpeakgroundaccelerations,i.e., 1ga serviceability , 2

ga life‐safety ,and 3ga collapseprevention

areoftengiveninthedesigncodes.Therefore,itisonlynecessarytodeterminethelimitsonresidualtopdisplace‐ments,i.e., 1

, 2tpD

,intheoptimizationmodelEq. 32 .Thentheoptimizationdesignofsteel

framescanbeimplemented.Theexampleinthispaperjustshowshowtoimplementathree‐levelperformance‐basedoptimizationmethodofsteelframes.Structuralengineersshouldsetupstructuralparametersandthelimitsonresidualtopdisplacementsinrealpracticeaccordingtorealstructureandlocaldesigncode,respectively.

Theinvertedtriangularloadpatternisinprincipleinaccurateforstructureswherehighermodeeffectsaresignificant.Itisaccurateenoughforstructureswherethefirstmodeisdominant.Becauseofthefactthatthenonlinearanalysis pushoveranalysis isbasedonatime‐independentdisplacementshape,itmaynotdetectthestruc‐turalweaknesseswhichmaybe generatedwhen the structures’ dynamics characteristics change after the for‐mationofthefirstlocalplasticmechanism.

Manyresearches Fajfar,1999,2000,2002;KreslinandFajfar,2012 indicate that theresultsobtainedbyusingtheN2methodarereasonablyaccurateprovidedthatthestructureoscillatespredominantlyinthefirstmode.Therefore,theN2methodhasbeenimplementedinEurocode8 EuropeanCommitteeforStandardization,2004 .Theestimatemethodof residual topdisplacementproposed in thispaper isdirectdevelopedbasedon theN2method.Formostbuildingstructures,thefirstmodealwaysdominatesthevibration.Theresidualtopdisplacementobtainedbyusingtheestimatemethodproposedinthispaperisreasonablyaccurateforperformancebaseddesign.TherestrictionoftheproposedmethodinthispaperisthesameastheN2method Fajfar2000 .

10CONCLUSIONS

Inthispaper,anestimatemethodofresidualtopdisplacementhasbeendevelopedforsteelframessubjectedtothreeearthquakehazardlevels.Athree‐levelperformance‐basedoptimizationmodelisproposedbasedontheestimatemethodofresidualtopdisplacement.Themainconclusionsare(1) The estimate method of residual top displacement needs to use pushover analysis until steel frames yield. The residual top displacement

is estimated by using the yield displacement value and yield loading value at the yield point. The estimated value of the residual top displacement obtained by the proposed method is greater than the real residual top displacement. The estimated value of residual top displacement is conservative.

(2) The computational time to get the yield point is far less than to get the collapse point. It is relatively easier to succeed in implementing pushover analysis until steel frame yields than until the frame collapses. It also doesn’t need any engineering judgements to get the yield point and the optimization process will not be interrupted in the proposed method.

(3) A three-level performance-based optimization model is proposed in this paper. In the optimization model, the peak ground accelerations related to three earthquake hazard levels and the limits on residual top displacement related to the three performance levels are required to start optimization design. The peak ground accelerations are often given in design codes. However, the limits on residual top displacement related to life safety and collapse prevention levels should be further investigated in the future.

(4) The pushover analyses of the optimum design indicate that the steel frame is at elastic phase at serviceability level, at inelastic and hardening phases at life safety and collapse prevention levels. It means that the assumption of residual top displacement is reliable when it is used in three-level performance-based optimization model.

References

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