ScienceDirect
www.sciencedirect.comwww.e-ache.com HormigónyAcero2018;69(S1):75–80 www.elsevierciencia.com/hya
Model
for
the
compressive
stress–strain
relationship
of
steel
fiber-reinforced
concrete
for
non-linear
structural
analysis
Ley
tensión-deformación
en
compresión
para
el
análisis
no
lineal
de
estructuras
de
hormigón
reforzado
con
fibras
de
acero
Gonzalo
Ruiz
a,∗,
Ángel
de
la
Rosa
a,
Sébastien
Wolf
b,
Elisa
Poveda
a aETSICaminos,C.yP.,UniversidaddeCastilla-LaMancha,Avda.CamiloJoséCelas/n,13071CiudadReal,SpainbArcelorMittalFibres,RoutedeFinsterthal,L-7769Bissen,Luxembourg
Received17October2018;accepted22October2018 Availableonline3December2018
Abstract
Amodelfornon-linearcalculationsofsteelfiber-reinforcedconcrete(SFRC)elementsincompressionisproposedanddescribedtechnologically. ThesamecurveproducedbyEurocode2(EC2)isuseduntilcompressivestrength,althoughnon-dimensionalvariablesrefertofiberconcrete parameters.Beyondthepeak,weuseaparabolathatisderivedsothattheaverageenergyconsumptionofthematerialequalsthissamemean energyobtainedfromadatabasecomprisedof197tests.EstimatesforthecompressivestrengthofSFRCandthecorrespondingcriticalstrainare alsoprovidedbycorrelationwiththedatabaseusingthesurfaceresponsemethodology.ThetotalenergyconsumptionofSFRCincompressionis approximatelyfourtimeshigheronaveragethanthecorrespondingenergyofthebaseplainconcrete,whichisincludedinthemodel.Therefore, itcanconsiderablyimprovethemodelingofductilityinSFRCstructures.
©2018Asociaci´onEspa˜noladeIngenier´ıaEstructural(ACHE).PublishedbyElsevierEspa˜na,S.L.U.Allrightsreserved.
Keywords:Steelfiber-reinforcedconcrete;Compressivestress-strainrelationship;Non-linearstructuralanalysis;Response-surfacemethodology
Resumen
Seproponeysedescribe,enformatotecnológico,unmodeloparaelcálculono-linealdeelementosdehormigónreforzadoconfibrasdeacero (HRFA)encompresión.Hastaelpicodecarga,seusalacurvadadaporelEurocódigo2(EC2),aunquelasvariablesadimensionalesserefierena losparámetrosdelhormigónreforzadoconfibras.Despuésdelpico,usamosunaparábolaquesecalculademodoquelaenergíaconsumidapor elmaterialseaigualalvalordeesamismaenergíadadaporunabasededatoscompuestapor197ensayos.Tambiénseproporcionanestimaciones delaresistenciaacompresióndelHRFAydeladeformacióncríticacorrespondiente,lascualesseobtienenporcorrelacionesconlabasede datosusandolametodologíadelassuperficiesderespuesta.LaenergíaconsumidaporelHRFAencompresiónes,comomedia,unascuatroveces mayorquelaquetendríasumatrizsinreforzar,locualesreproducidoporelmodelo.Así,éstepuedemejorarconsiderablementeelmodeladode laductilidadenestructurasdeHRFA.
©2018Asociaci´onEspa˜noladeIngenier´ıaEstructural(ACHE).PublicadoporElsevierEspa˜na,S.L.U.Todoslosderechosreservados.
Palabrasclave: Hormigónreforzadoconfibrasdeacero;Leytensión-deformaciónencompresión;Análisisno-lineal;Metodologíadelassuperficiesderespuesta
∗Correspondingauthor.
E-mailaddress:Gonzalo.Ruiz@uclm.es(G.Ruiz).
1. Introduction
Thistechnicalnotedescribesamodel for the compressive stress–strain(σ–ε)behaviorof steelfiber-reinforcedconcrete. Themodelisbasedonfunctionsobtainedfromcorrelationswith
https://doi.org/10.1016/j.hya.2018.10.001
Figure1.Schematicrepresentationofthestress-strainrelationshipinSFRCfor structuralanalysis(solidcurve),comparedwiththatofthecorrespondingbase concrete(brokenlinecurve).
anextensivedatabasecomprisedof197well-documentedSFRC compressivetests[1–20].
The following section describes the σ–ε model. Subse-quently,wejustifytheassumptionsmadetoformulatethemodel andprovideabriefdescriptionofthedatabasethatsupportsafew oftheexpressionsofthemodel,togetherwithashortexplanation oftheresponse-surfacemethodologyandtheprocessfollowed inordertoobtaintheresponses.
2. σ–εrelationshipfornon-linearstructuralanalysisof SFRC
Therelationshipbetweenσandε,showninFig.1,maybe usedtomodeltheresponseofSFRCtoshorttermuniaxial com-pression.Ithastwodistinctcurves.Thefirstrunsfromtheorigin of the axesto themaximumstress (curve1 inFig. 1) andis describedbythefollowingequation:
σ∗= αε
∗−ε∗2
1+(α−2)ε∗ (1)
where:
σ∗=fσ
cf Non-dimensionalstress
fcf CompressivestrengthofSFRC α=1.05εcfEff
cf Non-dimensionalcoefficient
εcf Criticalstrain,i.e.strainthatcorrespondstofcf Ef ElasticmodulusofSFRC
ε∗= εε
cf Non-dimensionalstrain
Thecompressivestrengthandcorrespondingstrainplusthe elasticmodulusofSFRCcanbeeasilyobtainedbytesting.These valuescanalsobeestimatedbyusingthefollowingequations:
fcf =fc0
1+4.174∗fφf
(2)
εcf =εc0
1+0.4823 λ
φf−0.002606∗f
(3)
Ef =E0 (4)
where:
E0 elasticmodulusofthebaseconcrete
Thesecondcurve(labeledas2inFig.1)isasofteningbranch thatrunsfrompeakstresstozero.Thefollowingparaboladefines thecurve:
σ∗=1−1 4
1−σR∗ ε∗−12 (5)
whereσR∗isthefollowingfunctionoftheparametersthat char-acterizethefiber:
σR∗ =0.8279+0.3888∗f 35.03φf −1
< 1 (6) Actually,σR∗ isthenon-dimensionalstresscorrespondingto
ε∗=3,asrepresentedinFig.1,whichimpliesthatσR∗ <1as statedinEq.(6).Thissecondstretchinterceptsthex-axisat:
ε∗u=1+ 2
1−σR∗ (7)
Note thatEqs.(1) and(5)andrelatedparametersconsider thatstressesandstrainsarepositiveincompression.Likewise, the softeningpartof thecurve,Eqs.(5)–(7),isonlyvalidfor SFRCwithhook-endedfibers.
3. Justificationofthemodel
Themodeldescribedabovedependsbasicallyontwopoints. Thefirstisthepeakofthestress–straincurve,thatis,thepoint whichrepresentsthestrengthoftheSFRC,fcf,andits
corre-spondingstrain,εcf(criticalstrain).Thesecondisthestressfor
astrainthreetimesthecriticalstrain,i.e.σR−3εcf.Weadopt
anon-dimensionalrepresentationwherestressesaredividedby
fcf andstrainsbyεcf.Therefore,thesepointsarejust(1,1)and
(σR∗,3)inthenon-dimensionalcurve,σ∗−ε∗.
Thefirstpartofthecurvefromtheorigintothepeakstressis modeledbyusingthesameσ–εcurvethatisalreadyusedinthe EC2forplainconcrete.Itshouldbenotedthoughthatthenew curvereferstothestrengthandcriticalstrainoftheSFRC,rather thantothoseofthebaseconcrete.Theelasticmodulus ofthe material,Ef,determinestheslopeattheorigin,whichisαinthe
non-dimensional curve.Asstatedabove,fcf,εcf,andEf are
easytoobtainbytesting.Inthecasethatanestimateofthese val-uesisneededandtestingcannotbeperformed,themodeloffers Eqs.(2)and(3)inordertoobtainfcfandεcfbasedonthe
param-eterscharacterizingthetypeof fiber(thefiberlength,f,and
thefiberaspectratio,λ)andthefibercontent(φf).These
Figure2.StrengthandmechanicalcharacteristicsofconcreteaccordingtoTable5.1oftheEurocode2[21](fc0inthepapercorrespondstofcminTable5.1).
samples.Thederivationprocessisexplainedinthe following section.ThereisnoequationfortheelasticmodulusoftheSFRC becausethecorrelationwiththedatabaseindicatesthatthefiber parametersarenotsufficientlysignificantinordertodetermine theEf / E0ratio.
The second part of the curve is an inverted vertical-axis parabola the vertex of which is point (1, 1) and that passes through point (σR∗, 3). This point was chosen as a reference because all the tests in the database with a σ–ε curve at least reached it. In other words, the shortest σ–ε tail in the database reached the abscissa 3εcf. The formula for
obtain-ingσR∗ (Eq.(6))isderivedasfollows.Theareabelowtheσ–ε
curvesbetweenthepeakand3εcf iscorrelatedwiththe same
areaof themodelexpressedasafunctionofσR∗.The correla-tionprocesstoobtainσR∗ issimilartothoseforfcf/ fc0 and
εcf/ εc0,and is explainedin the followingsection. The last
validpointofthemodelisε∗u,whichissimplyderivedasthe intercept of theparabola withthe x-axis.Note that modeling theSFRCsofteningthrough thisparabola errs on theside of caution,sincemostof the softeningbranchesinthe database areactuallylongerandconsumemoreenergythantheproposed curve.
Itshouldbenotedthatveryfewspecimensinthedatabase showedhardening(onlythree; SFRCsinthe database donot exceedafibercontentof3%involume).Theywerenot consid-eredinthecorrelationforcalculatingσR∗ becausetheparabola shouldnotincrease(thatisthereasonweimposeσR∗ <1).In thecasethatweseektoaccountforhardeningincompression therecouldberecoursetothelimitcaseofahorizontalstraight line(σR∗ =1)andarestrictedvalueofεu.
InregardtothetotalenergyconsumptionofSFRCin com-pression,the proposed model reflects that it is several times higherthanthecorrespondingenergyofthebaseplainconcrete, as suggestedinFig.1 (thearea belowthe dimensionalcurve representstheenergyconsumptionperunitvolume).Infact,the averageenergy consumptionof the SFRCduringthe ascend-ing/softeningbranchinthedatabaseis1.5/2.8timestheenergy of the corresponding base concrete in the ascending branch. Thus,thetotalenergyconsumedinSFRCisaroundfourtimes largeronaveragethanthatofthecorrespondingbaseconcrete (andthisonlyconsiderstheenergyupto3εcf).Consequently,
theuseoftheproposedσ–εcurvecanconsiderablyimprovethe modelingofductilityinSFRCstructures.
4. Databaseandresponse-surfacemethodology
Thedatabasegeneratedforthisstudycontains197 compres-sivetestsonSFRC150×300mm2cylinders[1–20].Allhave information concerning the baseconcrete. Likewise,all con-creteswerereinforcedexclusivelywithsteelhook-endedfibers withonlyonebend.80ofthetestsreportedthecompleteσ–ε
curve.Someoftherelevantparametersofthedatabasefallwithin thefollowingranges:
• CompressivestrengthofSFRC(fcf):29.4–93.5MPa
• Maximumaggregatesize(dm):10–25mm
• Volumetricfiberratio(φf):0.24–3.00%
• Fiberlength(f):10–80mm
• Fiberdiameter(df):0.2–1.2mm
• Aspectratio(λ=f / df):20–107
Thetimeatwhichthecompressivetestswereperformedwas 28days,withafewexceptions.
Theresponse-surfacemethodology[22]isacorrelation pro-cedure to determine a continuous function f for n of the parametersthatarethoughttobesignificantforitsresponse:
y=f(x1,x2,...,xn)+ξ (8)
where ξ is the error in the response as compared to the database.In general,fisunknownandthus,itisnecessaryto experimentwithconventionalpolynomialfunctionssuchas:
n
i=1
βixi+ n
i=1
βiix2i + n−1
i=1
n
j>i
βijxixj (9)
where βi are the coefficients for the linear terms, βii the
coefficients for the quadratic terms and βij with j>i the
coefficientsforthecombinationofvariables.
In this particular application, we look for the response of therelativevaluesoftheelasticmodulusEf,thecompressive
strengthfcf andthecriticalstrainεcf comparedwiththe
cor-respondingvaluesofthebaseconcrete,plustheresponseofσR
(σ for3εcf)relatedtothecompressive strengthoftheSFRC.
Wewanttoexpresstheserelativevaluesasfunctionsofbasic parametersofthefiberreinforcement,namelythefiberlength
f,thefiberaspectratioλandthefibervolumefractionφf.
Figure3.Coefficientsforfunctionstofitthedatabase(I.T.referstoanindependentterm).
datareferringtothecorrespondingvaluesofthebaseconcrete (Eof =Ef/E0, fcfo =fcf/fc0 andεocf =εcf/εc0) or thefiber
concrete(σ∗R=σR/fcf).Regardingthefiber parameters,only
thelengthhadtobeconvertedinordertobenon-dimensional byusing∗f =f / 0(notethatthechoicefor0isarbitrary
sinceitsactualvalueonlyaffectstheresultingnon-dimensional coefficients so that the final product is the same). The most completecorrelationinthisworkincludeslinear,combinedand quadraticterms.
Thecoefficientsobtainedforeachdesiredparameterareset forthinFig.3.Forexample,thefullresponse(linear,combined andquadratic)forthecompressivestrengthis:
fcfo =1+0.243∗f −0.00373λ−13.50φf +0.00356∗fλ
+7.81∗fφf+0.129λφf −0.1868∗f2−0.000009λ2−26φf2
(10)
where onlythe terms corresponding to ∗fφf and to ∗f2 are
actually significant (this is the reason that their coefficients arewritteninboldtypeinFig.3).Likewise,wealsoobtained coefficientsforthelinearplusthecombinedresponse,seeFig.3. Notethatthistimewedeterminethat∗fandλaresignificant.In ordertoderivesimplerequationswhereonlysignificantterms are present,we resorttothe correlationagain inlinearmode andonlywithtermsthathadalreadybeenfoundtobe signifi-cant.Theoutcomeoftenrevealsthatsometermswerenotreally significantand,inthesecases,weeliminatethemandresortto thecorrelationagain.Thisishowwereachthesimplest equa-tionfortheparameterunderstudy,inthiscase,thecompressive strength:
fcfo =1+4.174∗fφf (11)
Notethatwe roundedtheindependentterm to1, sincethe methodologydoesnotperformanytypeofasymptoticstudybut justsimplecorrelation.Theproceduretoobtaintheremaining responses is similar to the one previously described. Fig. 4 plotsEq.(11)superimposedoverthevaluesinthedatabase.It isclearthatthereisaremarkabledispersion,sincefcf mostly
correlates withthecompressive strengthof the baseconcrete
Figure4.Non-dimensionalcompressivestrengthfcf0=(fcf/fc0)asafunctionof
parameter∗fφf (=fφf /0)comparedwiththeexperimentalvaluesinthe
database.
fc0,whichisthevaluechosentogetnon-dimensionalfcf.This
is the reason for which we recommend that fcf and εcf are
actuallymeasured,sinceEq.(11)isonlyaroughapproximation oftheaveragebehavioroftestsinthedatabase.Similarly,Fig.5 plotsthedatabasevaluesforεcf andσRasfunctionsofsomeof
theinfluentialnon-dimensionalparametersinEqs.(3)and(6), namelyλφf and∗fφf.Italsoplotstheresponsesurfacesthat
correspondtotheseequations.Notethatthedatabasevaluesfor
εcf andσRshowlessscatterthanforfcf.
Despitehavingtriedtosimplifytheequationsbyincreasing the average significance of the terms,all the response levels couldbeused.Theresponseismoreaccurateindeedwhenthe morecomplexequationsareconsidered.Ontheotherhand,as alreadycommentedabove,wedidnotfindagoodcorrelation fortherelativeelasticmodulus(thesignificanceofλforEof is very weak,seetheTableinFig.3).The methodologyreveals thatfibercontentinlowdosagesdoesnotinfluencetheSFRC compressiveresponseatsmallstrains.Wedidnotproposeatype of phase-rule, i.e.such as Ef =E0(1−φf)+Esφf because
Figure5.Databasevaluesand3Dplotsoftheselectedfunctionsforthenon-dimensional(a)criticalstrainε0
cf(=εcf/εc0);and(b)stresscorrespondingto
ε=3εcf,σR∗(=σR /fcf).
5. Conclusions
This technical note proposes a σ–ε curve for steel fiber-reinforced concrete (SFRC) in compression, intended for non-linearcalculations.Themodelisdescribedtechnologically, similarlytothat used instructuralcodes.The model hastwo distinctstretches.Thefirstdescribestheσ–εbehavioruptothe maximumload,followingthesamecurveproposedbyEurocode 2[21]forplainconcrete,butusingnon-dimensionalvariables referringtofiberreinforcedconcrete.
Thesecondstretchisdefinedasaverticalparabolathe ver-texof whichcorresponds tocompressive strength. It ismust passthrough point σR−3εcf, whichin turn is estimated by
correlationtoanextensivedatabase ofactualσ–εcurves.The correlationtogetσRismadesothattheaverageenergybelow
theσ–εcurvebetweenεcf and3εcf inthedatabaseequalsthat
of the parabola for the samestretch.Therefore,this assump-tionforthesecondpartoftheσ–εcurveyieldsthesamemean energyconsumptionastestsinthedatabaseupto3εcf.Beyond
thispointwestillusetheparabola,whichisrelativelysafesince mostofthetestsinthedatabaseexhibitlongtails.Thetechnical notealsoprovidesestimatesforthecompressivestrengthofthe steelfiber-reinforcedconcreteandforthecorrespondingcritical strain.Alltheformulasarederivedbyusingtheresponse-surface methodologyandadatabaseformedby197testsonSFRC(only standardcylindersofSFRCwithhook-endedfiberswithonly onebend).
RegardingthetotalenergyconsumptionofSFRCin compres-sion,theproposedmodelreflectsthatitisapproximatelyfour times higheronaveragethanthecorresponding energyof the baseplainconcrete(consideringonlytheenergyupto3 εcf).
Therefore,theproposedσ−εcurvecanconsiderablyimprove themodelingofductilityinSFRCstructures.
6. Dedication
ThisworkisdedicatedtoDr.CarmenAndradebothforher extensiveanddeepcontributiontoresearchinthefieldof con-structiontechnologyandforherleadingpersonalexampleand friendship.
Acknowledgements
Funding from the Ministerio de Economía y Competitivi-dadthroughprojectBIA2015-68678-C2-1-RandInternational CampusofExcellenceCYTEMAisgratefullyacknowledged. ÁngeldelaRosaexpresseshisgratitudeforthefellowshipFPI BES-2016-077458.
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