Three‐level performance‐based optimization method of steel ... · Three‐level...
Transcript of 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
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
cm
Collapseprevention Severedamage 3 15tpD
cm
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
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 3/18
3.1HorizontalelasticaccelerationresponsespectruminADformat
Theelasticresponseaccelerationspectrum aeS T forthehorizontalcomponentsofseismicactionisdefined
inEurocode8 EuropeanCommitteeforStandardization,2004 as
2
0 : 1 2.5 1
: 2.5
: 2.5
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 T S T a S
TT T T S T a S
TT T
T T s S T a ST
1
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;
DT
isthevaluedefiningthebeginningoftheconstantdisplacementresponserangeofthespectrum; S isthesoilfactorand isthedampingcorrectionfactorwithareferencevalueof 1 for5%viscousdampingratio.Forground
typeA, 1.0S , 0 .1 5BT s, 0 .4CT s, 2.0DT s.
ForanelasticSDOFsystem,thedisplacementresponsespectrumis
2
24de ae
TS T S T
2
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
aea
S TS T
R
3
d deS T S TR
4
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,
2
24d a
TS T S T
5
Forabilinearspectrum,
( 1) 1C
TR
T CT T 6
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 4/18
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
i
weightedbythestorymassim FajfarandGaspersic,1996 ,
i i iP pm 8
wherethecomponentoftheassumeddisplacementshapei is
1
1
i
kk
i n
kk
h
h
9
where p controls themagnitude of the lateral loads of steel frames. Thedisplacement shape follows the firstvibrationmodeofthebuilding.
Therefore,thebaseshearforceV is
1 1
n n
i i ii i
V P p m pm
10
where1
n
i ii
m m
istheequivalentmassoftheSDOFsystem.
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 5/18
Figure1:n‐storysteelframe.
Theequalloadingstepsareappliedtosteelframeinpushoveranalysis.TheloadingstepsizeiP atthe i th
storyis
i i i i i
pP m pm
c 11
where pp
c ,and c isaconstantinteger.
Theincrementofbaseshearforceis
1
n
ii
V P
12
Inthebaseshearforce V ‐topdisplacementtD diagramshowninFigure2, jV and tjD aretheincre‐
mentofbaseshearforceandincrementoftopdisplacementatthe jthloadstep,respectively.
Figure2: - tV D diagram.
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 6/18
4.2TheF ‐Ddiagram,yieldpointandelasticperiod equivalentSDOFsystem
ThedisplacementoftheequivalentSDOFsystemis
tDD
13
ThebaseshearforceoftheequivalentSDOFsystemis
VF
14
where isaconstantandcalculatedas
1
2 2
1 1
n
i iin n
i i i ii i
mm
m m
15
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.
jj
VF
16
tjj
DD
17
Inthispaper,thesteelframeisassumedtoyieldifthefollowinginequalitiesaretrueatthe y thloadstep,
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 7/18
1
1
y
y
FF
D D
18
OR
1yD D 19
whereisaconstantandgreaterthan1.Forstructuresteel,themodulusofelasticityismuchgreaterthanthetangentmodulus,inthispaper, 2 orbiggercanbeenoughtojudgethattheyieldinghappens.
Therefore,thedisplacementattheyieldpointinF ‐Ddiagramis
1
y
y kk
D D
20
TheshearforceattheyieldpointinF ‐Ddiagramis
1
y
y kk
F F
21
Inthispaper,thepost‐yieldstiffnessisidealizedtobezero AschheimandBlack2000 .TheidealizedF ‐D
diagramisshowninFigure4.Theelasticperiodoftheidealizedbilinearsystemcanbecalculatedas
2 y
y
m DT
F
22
Figure4:IdealizedF ‐Ddiagram thickline .
4.3CapacitydiagraminADformat
ThecapacitydiagraminADformatshowninFigure5isobtainedbydividingtheforcesintheforce‐defor‐
mationdiagram,i.e.,theidealizedF ‐Ddiagram,bytheequivalentmassm .
a
FS
m
23
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 8/18
Inthispaper,thepushoveranalysisisuseduntilthesteelframesyieldandthecapacitydiagramisobtained.Itisrelativelyeasiertosucceedinperformingpushoveranalysisuntilasteelframeyieldsthanuntiltheframecol‐lapses.Itdoesn’tneedanyengineeringjudgementstogettheyieldpoint.Therefore,itcanmaketheoptimizationprocessuninterrupted.
Figure5:CapacitydiagraminADformat.
5DETERMINATIONOFREDUCTIONFACTORANDDUCTILITYFACTOR
ElasticresponsespectruminADformat,inelasticspectruminADformatandcapacitydiagraminADformat
areplottedinthesamepictureshowninFigure6 CT T orFigure7 CT T .ThereductionfactoruR isde‐
finedastheratiobetweentheaccelerationscorrespondingtotheelasticandinelasticsystems,
( )aeu
ay
S TR
S
24
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
D D S
26
where ( )deS TisthedisplacementvalueoftheelasticspectrumattheperiodT and
dS isdisplacementvalueat
theintersectionoftheinelasticspectrumdiagramandcapacitydiagramshowninFigure6andFigure7.
If CT T showninFigure7 ,
1 1CTR
T 27
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 9/18
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.
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 10/18
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
1
1p y e y y
DD D D D F
F
30
where eDistheelasticdisplacement.
Figure9:RealandidealizedF ‐Ddiagram.
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 11/18
IfthesteelframeisstillathardeningphaseasshowninFigure9,theestimatedvalueoftheresidualtopdis‐
placementofthesteel frame pDisgreaterthantherealresidualtopdisplacement hD
.Therefore, theestimated
valueoftheresidualtopdisplacementisconservativeinEq. 30 .TheresidualdisplacementoftheequivalentSDOFsystemistransformedbacktothetopresidualdisplacement
oftheMDOFsystem,
1
1tp p y y
DD D D F
F
31
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
tpD
and 3tpD at frequent,occasionalandrareearthquake,respectively. 1
tpD
, 2tpD
and 3
tpD
arethe limitson
residualtopdisplacementsrelatedtoserviceability,lifesafetyandcollapsepreventionlevels,respectively.Thethree‐levelperformanceoptimizationmodelisproposedas
1 1
2 2
3 3
Find
Min
. .
1, 2, ,
tp tp
tp tp
tp tp
i i i
M
S T D D
D D
D D
d d d i N
d
d
d
d
d
32
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 12/18
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
1
nj
tp ii
D dr
33
Assumedthatthekthstoryofthestructureisweak/softstory,theresidualinter‐storydriftofthekthstoryisthedominantcomponentinthecontributiontotheresidualtopdisplacement.
jtp kD dr 34
Thedesignvariableofcolumnsatthekthstoryiskd .DifferentiatingEq. 34 withrespecttodesignvariable
kd ,wehave
jtp k
k k
D dr
d d
35
Sincethekthstoryofthestructureisweak/softstory,thefirstordersensitivity k
k
dr
d
willbeverylarge.In
Eq. 35 ,thefirstordersensitivityofresidualtopdisplacement,i.e.,j
tp
k
D
d
,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.
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 13/18
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
Flangewidth 150 147 100 180Flangethickness 17 16 5 22Webdepth 400 431 250 600
Webthickness 14 11 4 20
ColumnC2 mm
Flangewidth 150 143 100 180Flangethickness 17 14 5 22Webdepth 400 355 250 600
Webthickness 14 11 4 20
ColumnC3 mm
Flangewidth 150 143 100 180Flangethickness 17 14 5 22Webdepth 400 349 250 600
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
cm.
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.
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 14/18
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
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 15/18
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.
3500
4000
4500
5000
5500
6000
6500
0 1 2 3 4 5 6 7
Str
uct
ura
l m
ass
(kg
)
Iteration number
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 16/18
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
Res
idu
al d
isp
lace
men
t (c
m)
Iteration number
Serviceabilitylevel
1900ral
1900ral
1900ral
1900ral
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
Sto
ry
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
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 17/18
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
tpD
, 2tpD
and 3
tpD
,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
AppliedTechnologyCouncil 1996 Seismicevaluationandretrofitofconcretebuildings,ReportATC‐40.
AschheimMandBlackEF 2000 YieldPointSpectraforSeismicDesignandRehabilitation.EarthquakeSpectra16:317‐336.
QimaoLiuThree‐levelperformance‐basedoptimizationmethodofstteelframes
LatinAmericanJournalofSolidsandStructures,2018,15 1 ,e07 18/18
EuropeanCommitteeforStandardization 2004 Eurocode8:Designofstructuresforearthquakeresistance‐Part1:generalrules,seismicactionsandrulesforbuildings.EuropeanstandardEN1998‐1 EnglishEdition .
FajfarPandGaspersicP 1996 TheN2methodfortheseismicdamageanalysisofregularbuildings.EarthquakeEngineeringandStructuralDynamics25:23‐67.
Fajfar P 1999 Capacity spectrummethod based on inelastic demand spectrum. Earthquake Engineering andStructuralDynamics28:979‐993.
FajfarP 2000 Anonlinearanalysismethodforperformancebasedseismicdesign.EarthquakeSpectra16:573‐592.
FajfarP 2002 Structuralanalysisinearthquakeengineering‐abreakthroughofsimplifiednon‐linearmethods.PaperReference843,Proceedingsof12thEuropeanConferenceonEarthquakeEngineering,London.
FreemanSA 1998 Thecapacityspectrummethodasatoolforseismicdesign.Proceedingsofthe11thEuropeanconferenceonearthquakeengineering,Paris.
GencturkBandElnashaiAS 2008 Developmentandapplicationofanadvancedcapacityspectrummethod.Engi‐neeringStructures30:3345‐3354.
GhobarahA 2001 Performance‐baseddesigninearthquakeengineering:stateofdevelopment.EngineeringStruc‐tures23:878‐884.
IbrahimbegovicA 1995 On finiteelement implementationofgeometricallynonlinearReissner’sbeamtheory:three‐dimensionalcurvedbeamelements.ComputerMethodinAppliedMechanicsandEngineering26:11‐26.
KreslinMandFajfarP 2012 TheextendedN2methodconsideringhighermodeeffectsinbothplanandelevation.BulletinofEarthquakeEngineering10:695‐715.
LiuQ,ZhangJandYanL 2010 AnoptimalmethodforseismicdriftdesignofconcretebuildingsusinggradientandHessianmatrixcalculations.ArchiveofAppliedMechanics80:1225‐1242.
LiuQandPaavolaJ 2015 AnOptimizationProcedureforSeismicDesignofSteelFramesforMulti‐PerformanceandMulti‐HazardLevels.AdvancesinStructuralEngineering18 1 59‐74.
SimoJCandVu‐QuocL 1986 Athreedimensionalfinitestrainrodmodel.PartII:computationalaspects.ComputerMethodinAppliedMechanicsandEngineering58:79‐116.
VidicT,FajfarPandFischingerM 1994 Constantinelasticdesignspectra:strengthanddisplacement,EarthquakeEngineeringandStructuralDynamics23 1994 502‐521.
Washington DC :FederalEmergencyManagementAgency 2000 Prestandardandcommentaryfortheseismicrehabilitationofbuildings,ReportFEMA‐356.
ZouXKandChanCM 2005 Optimalseismicperformance‐baseddesignof reinforcedconcretebuildingsusingnonlinearpushoveranalysis.EngineeringStructures27:1289‐1302.