Ley tensión-deformación en compresión para el análisis no lineal de estructuras de hormigón reforzado con fibras de acero

ScienceDirect

www.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

,

Ángel

de

la

Rosa

,

Sébastien

Wolf

,

Elisa

Poveda

b_{ArcelorMittalFibres,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:[email protected](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=1.05εcfE_ff

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−σ_R∗ ε∗−12 (5)

whereσ_R∗isthefollowingfunctionoftheparametersthat char-acterizethefiber:

σ_R∗ =0.8279+0.3888∗_f 35.03φf −1

< 1 (6) Actually,σ_R∗ isthenon-dimensionalstresscorrespondingto

ε∗=3,asrepresentedinFig.1,whichimpliesthatσ_R∗ <1as statedinEq.(6).Thissecondstretchinterceptsthex-axisat:

ε∗_u=1+ 2

1−σ_R∗ (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

(σ_R∗,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 (σ_R∗, 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-ingσ_R∗ (Eq.(6))isderivedasfollows.Theareabelowtheσ–ε

curvesbetweenthepeakand3εcf iscorrelatedwiththe same

areaof themodelexpressedasafunctionofσ_R∗.The correla-tionprocesstoobtainσ_R∗ 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-eredinthecorrelationforcalculatingσ_R∗ becausetheparabola shouldnotincrease(thatisthereasonweimposeσ_R∗ <1).In thecasethatweseektoaccountforhardeningincompression therecouldberecoursetothelimitcaseofahorizontalstraight line(σ_R∗ =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 (Eo_f =Ef/E0, f_cfo =f_cf/f_c0 andεo_cf =ε_cf/ε_c0) 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:

f_cfo =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:

f_cfo =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(thesignificanceofλforEo_f 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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