ConstitutivemodelforfibrereinforcedconcretebasedontheBarcelonatest

AnaBlancoa*,PabloPujadasa,SergioCavalaroa,AlbertdelaFuentea,AntonioAguadoa

aDepartmentofConstructionEngineering,UniversitatPolitècnicadeCatalunya,UPC,JordiGirona1-3,08034Barcelona,Spain.

*Correspondingauthor.Tel.:+34-93-401-7347;fax:+34-93-401-1036;e-mail:ana.blanco@upc.eduAbstractSeveralconstitutivemodelsforfibrereinforcedconcrete(FRC)havebeenreportedinthepastyearsbasedontheflexuralperformanceobtained inabending test.TheBarcelona testwaspresentedasanalternative tocharacterizethetensilepropertiesofFRC;however,noconstitutivemodelwasderivedfromit.Inthisarticle,aformulationtopredictthetensilebehaviourofFRCisdevelopedbasedontheresultsoftheBarcelonatest.Theconstitutive model proposed is validated by simulating the results of an experimental program involvingdifferent types of fibres and fibre contents bymeans of finite element software. Moreover, the simplifiedformulationproposediscomparedwithconstitutivemodelsfromEuropeancodesandguidelines.Keywords:FibreReinforcedConcrete;ConstitutiveModel;BarcelonaTest;Modelling;Design1.IntroductionThedesigncodesandguidelinesforfibrereinforcedconcrete(FRC)proposedifferentconstitutivemodels that shareacommonapproach: theparameters thatdetermine the tensilebehaviourareretrievedfromtheresultsofflexuraltestsonbeams[1].Thesetupandloadconfigurationadoptedproduceagradualevolutionofenergyreleaseduringtheloadingprocedure,whichsimplifiesthecontrol and the assessment of the response of the material depending of the level of damage.Moreover,anisostaticconfigurationisused,meaningthattheinternalforcesmaybeeasilyderived.As an indirect consequence, the beam test has become the reference for the systematic qualitycontrolofFRC.Nevertheless, despite these advantages, themethod also shows drawbacks. On one hand, someauthors suggest that the shape and size of the specimen and its production process favours apreferential alignment of the fibres along the axis of the beam [2]. This tends to increases themechanicalefficiencyandtheoveralltensileresponseoftheFRC,whichmightnotoccurinthecaseoffull-scalestructureswithoutamarkedpreferentialorientation[3-6].Ontheotherhand,thesmallareaof thebeamsubjected tocracking reduces the totalamountofnon-elasticenergymobilisedandcontributes to increase thescatter in theresults [7,8]. Inaddition to that, theweightof thespecimenandthetypeofequipmentrequiredcomplicatethetestprocedureandlimitthenumberofelementscharacterizedperbatch.This leads toa seriouscontradiction in termsof thequalitycontrolofFRCsinceitisessentialtocharacterizeaminimumnumberofelementsinordertoobtainreliabletensilestrengthvalues,especiallyinamaterialsaffectedbyahighscatter.Alternativetestshavebeenproposedwiththeaimofreducingthisfavourableorientationandthescatter.ThisisthecaseoftheroundpanelandtheEFNARCpaneltests.Althoughbothofthemseemto overcome the issues mentioned, the size of the specimen required increases the setupcomplexity,thuslimitingevenmorethenumberofresultsobtainedperbatch.Inthiscontext,theBarcelona test [8, 9] according to UNE 83515:2010 [10] has been proposed as an intermediatealternativebetweenthebeamandthepaneltests.Eventhoughitmightshowsomedisadvantagesregardingthecontrolofcrackinitiationandtheestimationoftheinternalstressdistribution,itissimpler to perform, less-time demanding andmore sustainable than othermethods in terms ofvolumeofconcrteconsumed[8,11-12].Furthermore,sinceabiggercrackedsurfaceismobilized,ityieldsvaluesoftheresidualtensilestrengthandtoughnesswithanaveragecoefficientofvariationthatareusuallybelow thoseof thebeamtest [8].Suchevidencessuggest that theBarcelona test

mightbeanadequateoptionforthesystematicqualitycontrolofFRC.However,itsacceptanceinpracticeisstillhinderedbytheabsenceofsimplifiedformulationtoderivethetensileconstitutivemodelsfromthetestresults.TheobjectiveofthispaperistoproposeananalyticalformulationfortheestimationofthetensileconstitutivecurveoftheFRCdirectlyfromtheresultsoftheBarcelonatest.Forthat,ananalyticaldeduction is performed considering the changes in the resistantmechanism experienced by thespecimenduring the test procedure. Then, the formulation obtained is validated through a FEMbackanalysisusingresultsoftestsconductedinlaboratorywithconcretereinforcedwithsteelandplasticfibres.Finally,theconstitutiveequationsderivedfromtheBarcelonatestandfromthebeamtest for the same concrete are compared. The results obtained confirm the validity of theformulation proposed in this paper, thus representing a contribution towards the reliable andsimplecharacterisationofthematerial.2.DescriptionoftheBarcelonatestTheBarcelonatest isadoublepunchtest(DPT)performedonacylindricalFRCspecimenwithadiameter and a height of 150mm, according to the specifications in the Spanish standard UNE83515[10].Cylindricalsteelpuncheswithaheightof24mmandadiameterof37.5mmareplacedat the centre of the top and the bottom surfaces of the specimen. As shown in Figure 1a, anextensometerchainisplacedathalf-heightofthespecimentomeasuretheTotalCircumferentialOpeningDisplacement(TCOD)experienced.Aconstantrelativedisplacementrateof0.5±0.05mmisappliedbythepistonofthepress.TheforceandtheTCODarealsomeasured.

Figure1.a)Barcelonatestsetup,b)topviewofthespecimenafterthetest,c)frontalviewofthespecimenafter

crackingandd)topviewofthespecimenaftercracking.Recentstudieshaveshownthat itispossibletoestimatetheTCODwithouttheneedofusingtheextensometerchain.Carmonaetal. [12]proposedanexperimental correlationbetween theaxialdisplacementand theTCOD.Subsequently,Pujadasetal. [13]presentedananalytical correlationthatisvalidforthewholeextentofthecurveandforanytypeofFRC.Thesameauthorshavealso

Steelpunch

Steelpunch

a) b)

FP

dδp/2

𝝈𝝈

𝝈

dδL

c) d)

demonstratedthatthetestmaybeconductedwithcubicspecimenswith150mofsidewithouttheextensometerchain[14].During the test, theapplied loadproducesa tensile stress field inside the specimen. In this firststage, the concrete matrix is responsible for bearing the stresses. When the tensile strength ofconcrete is reached,a transition stage occurs: between2 to 4 radial cracks are abruptly formedperpendiculartothestressfield(seeFigure1b)andtwowedgesareformedunderthecylindricalpuncheswheretheloadisapplied.Thesewedgesmaybeidealizedasconeswiththesamediameterofthepunches[9,10,12,15,16].Inthismoment,partoftheelasticenergyisreleasedandachangein the resistant mechanism is observed since the fibres become active; being responsible forbearingthetensilestressesatasectionallevel(seeFigure1c).Oncethecrackshaveappeared,thespecimen enters a kinematic stage, inwhich the conical wedges slide into the specimenwith acertaindisplacement (δp) causing the lateraldisplacement (δL)of theconcretesegmentsand thecorrespondingcrackopening(seeFigures1cand1d).Furtherdetailonthefailuremechanismmaybefoundin[13].Differencesinthefractureofthespecimenshelptoexplaintheparticularities intheresultsfromtheBarcelonatestandtheflexuraltest.Noticethatthetotalcrackedsurfaceintheformerisupto2.2timesthatofthelatter.Consequently,thetotalnon-elasticenergymobilized in theBarcelonatestwillbehigher,favouringsmallerscatterinthepost-crackingresults.IntheBarcelonatest, thebiggestpartoftheelasticenergyisreleasedabruptlyatthemomentofcrackingandaremainingsmallpartisreleasedduringthepostcrackingstage.Onthecontrary,inthe flexural test, the releaseoccursatamuchslowerrateduringalmost thewholepost crackingstage since the crack depth increases gradually.On one hand, thismay bean advantage for theBarcelonatestgiventhattheresultsobtainedatasectionallevelwillreflectbetterthecontributionof the fibres. On the other hand, thismeans that it ismore likely to present instabilities in theprecisemomentthatcrackingoccurs.Itisimportanttoremark,however,thattheauthorshavenotexperiencedanyprobleminthecontrolofthetestduetothehigherdissipationofelasticenergyinthemomentofcracking.3.AnalyticalexpressionsforthetensilestrengthSeveral analytical expressionswere reported in the literature for determining the direct tensilestrength (fct) derived from the DPT [9, 11, 15-19]. In Table 1, a summary of the closed-formexpressions ispresented.Themainparametersare themaximum load (P), thediameterand theheightofthespecimen(dandh,respectively)andthediameteroftheplate(d').Theseexpressionsallowdeterminingthefctforplainconcreteand,inthecaseoftheexpressionbyMolinsetal.[9],theresidualstrengthsinthepost-crackingstage.However,thereisnoformulationthatprovidestheσ-εrelationandthatisvalidforboththelinear-elasticandpost-crackingstages.

Table1.Analyticalexpressionsforthetensilestrength.

Study Hypothesis/Approach Expression

Chen(1970)1 Limitanalysisofperfectelasto-plasticmaterial

𝑓$% =𝑃

𝜋)1.2𝑑2 ℎ − 0𝑑′2 2

34

ChenandYuan(1980)2Concreteasanelasto-plasticstrain-hardeningandfracturematerial(FEM

analysis)

𝑓$% =0.75𝑃

𝜋)1.2𝑑2 ℎ − 0𝑑′2 2

34

Bortolotti(1988)3 ModificationoftheCoulombfailurecriterionforconcrete

𝑓$% =𝑃

𝜋)𝑑2 ℎ − 8𝑑92 :

3cot𝛽4

Marti(1989)4 Non-linearfracturemechanics 𝑓$% = 0.4𝑃

4 8𝑑2:3 √01+

𝑑𝜆𝑑C

2

Molinsetal.(2007) Strut-and-tiemodel 𝑓$% =4𝑃

9𝜋𝑑9ℎ

1,2:BothexpressionsareproposedinChenandYuan(1980).3:where𝛽 = E

3− F

3isthefailureangleand𝜑istheinternalfrictionangle.

4:where𝑑C isthemaximumaggregatesizeand𝜆isanexperimentalparameterdependingonthematerial.4.FormulationtopredictthetensilebehaviourofFRC4.1Formulationtoestimatethestress(σ)Alldeductionsareperformedconsideringacylindricalcoordinatesystemwiththelongitudinalaxiscoinciding with the axis of the Barcelona test sample. Furthermore, only half of the sample isanalysedinordertosimplifydevisualizationofthephenomenon.Thecrackingsurfaceof theconicalwedge isdeterminedby the failureangleof thematerial (β),whichinturndependsoftheinterlockingeffectbetweentheaggregatesandthefibrescrossingthecontact surface. After the cracking, the force applied by themetallic plate (FP) generates at thesurfaceoftheconicalwedgeafrictionforce(Ffr)andanormalforce(FN),asshowninFigure2a.Tosimplifythe2Drepresentation,FfrandFNarepositionedatonesideeventhoughinrealitytheyaredistributedallaroundsurface.Figure2bshowsthebalanceofforcesacting.NoticethattheforceFPis completelybalancedby the vertical components ofFfr andFN. However,a horizontal resultingradialforcenamedFRappears.Suchforceisnotoutofbalancesinceitisdistributedallaroundthelateralsurfacepointingtowardsthecentreoftheconicalwedge,thuscancellingitself.

Figure2.Detailofa)interplayofforcesattheconicalwedge,b)balanceofforcesattheconicalwedge,c)

interplayofforcesatthespecimenandd)balanceofforcesatthespecimen.TheforceFfrmayberepresentedaccordingwithequation(Eq.1)astheproductofakineticfrictioncoefficient (μk) and the normal force (FN). It is important to remark that the kinetic frictioncoefficientshouldbeusedinthiscasesincetheconicalwedgeisconstantlymovingduringthetest.Byapplyingequation(Eq.1)andsimpletrigonometryinFigure3b,equations(Eq.2)and(Eq.3)may

β

Ff 𝑭𝑵

φ

𝑭𝑷a) 𝑭𝑹

𝑭𝑵

𝑭𝑷𝝁𝒌 ∙ 𝑭𝑵 = 𝑭𝒇𝒓 β

β

b)

β

Ffr 𝑭𝑵

𝑭𝑷

c) 𝑭𝑹𝑭𝑵

𝑭𝑷

𝝁𝒌 ∙ 𝑭𝑵 = 𝑭𝒇𝒓 β

βd)

be deduced to estimate FP and FR,respectively. After combining equations (Eq.2) and (Eq.3),equation(Eq.4)isobtainedtoestimate𝐹R dependingonthevalueofFP.

By the principle of action and reaction, the forces applied to the conical wedge should also beapplied with the same magnitude and opposite direction to the rest of the specimen. This isrepresentedinFigure2calongwiththebalanceofforces(seeFigure2d).Again,aresultingradialforceappearswithamagnitudethatcouldbeestimatedthroughequation(Eq.4).However,duetothepresenceofradialcracks,inthiscase𝐹R mustbegeneratedbythecontributionofthefibres.This contribution is estimated using the infinitesimal slice from Figure 3. This slice receives anormalstressthatbalancestheinfinitesimalradialforcedFr.Inordertosimplifythedeductionitisassumed that an average normal stress (σ) is uniformly distributed over thewhole area of thecracked radial surface (𝐴). Before themain cracks appear, the concrete behavesas amonolithicelementandtheresistantmechanismgeneratesanon-uniformstressdistribution.Oncethecracksareformed,thestressproducedinsidethespecimenisgovernedinthecrackedsectionsbythepull-outresponseofthefibres[20].

Figure3.Infinitesimalsliceofaconcretesegment.

It is observed that during the test an approximately constant opening appears throughout thecrackingplane.Inotherwords,thecrackwidthclosetothecentreofthespecimenisapproximatelythe same as at the perimeter. Consequently, the stress distribution generated by the pull-out offibresmaybeassumedasuniformlydistributedinthecrackingsurface.It isimportanttoremarkthat, in reality, a slightdifferentialopeningmightexist.However, theconsiderationofauniformstressisafareestimatethatisonthesafeside.Consideringthepreviousassumption,theresultingforceactingatthecracksmustequal𝜎 ∙ 𝐴(seeFigure3c). Inthiscontext,theequilibriumofforcesintheinfinitesimalslicemayberepresentedthroughequation(Eq.5),whichconsidersthatforsmallvaluessin𝑑𝜃 ≈ 𝑑𝜃.ThetotalvalueofFRincylindrical coordinates may be obtained by integrating equation (Eq.5) around the completecircumferentialangularsection,leadingtoequation(Eq.6).

𝐹Z[ = 𝜇] ∙ 𝐹 (Eq.1)

𝐹_ = 𝐹 ∙ 𝑠𝑒𝑛𝛽 + 𝐹Z[ ∙ 𝑐𝑜𝑠𝛽 = 𝐹 ∙ (𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽) (Eq.2)

𝐹R = 𝐹 ∙ 𝑐𝑜𝑠𝛽 + 𝐹Z[ ∙ 𝑠𝑒𝑛𝛽 = 𝐹 ∙ (𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽) (Eq.3)

𝐹R = 𝐹_ ∙𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

(Eq.4)

dθ𝝈𝑨𝒅𝜽𝟐Infinitesimal

section

R

𝒉𝟐

𝒅𝑭𝑹

c)

σA

σA

dθ

σ

𝒅𝜽𝟐

𝒅𝑭𝑹

𝝈𝑨𝒅𝜽𝟐

a) b)

𝝈𝑨𝒅𝜽𝟐

σA

σA

σ

Substituting equation (Eq.4) in (Eq.5) gives equation (Eq.7),which allows the estimation of thetensilestress(σ)resistedbytheFRCdependingoftheloadappliedbythepress(FP).Inthiscase,theareaofthecrackedradialsurface(𝐴)shouldbeobtainedinequation(Eq.8)astheareaofthesectionalcutofonequarterofthespecimensubtractedbythesectionalareaofhalfofoneconicalwedge,whichisnotpartofthecrackedsurface.

4.2Formulationtoestimatethestrain(ε)Duringthetest,thespecimenisdividedinanumberofsegmentsthatequalsthetotalnumbernofcracksformed.Inordertosimplifythedeductionitisassumedthatallconcretesegmentshavethesamesizewithaninternalangle2π/n.Whenthekinematicmechanismstarts,eachconicalwedgeslidedδP/2 into thespecimen leading to the lateraldisplacementof theconcretesegments (dδL),whichisdepictedinFigure4a.

Figure4.a)Lateraldisplacementofconcretesegmentandb)infinitesimalsectionofsegment.

Bothdisplacementsarerelatedwiththeangleβoftheconicalwedgeaccordingtoequation(Eq.9).Consideraninfinitesimalelementinasliceoftheconcretesegment(seeFigure4b)withalateralareadAthatequalsdR·dh.Thedifferentialwork(dτ)donebythetensilestressinthiselementmaybeexpressedas shown inequation (Eq.10).Notice that the lateraldisplacementoccurswith thesame direction and magnitude in all infinitesimal elements since it may be assumed that theconcretesegmentsmoveasindependentbodies.

Inthiscontext,theworkproducedateachsegmentmaybecalculatedthroughthesumof𝑑𝜏forallinfinitesimalelementsincludedinthesegment.Thisisequivalenttoatripleintegrationof𝑑𝜏intheinterval(-π/n,π/n),foraradiusrangingfrom0totheradiusofthespecimen(R)andforaheight

𝑑𝐹R = 2 ∙ 𝜎 ∙ 𝐴 ∙𝑑𝜃2 = 𝜎 ∙ 𝐴 ∙ 𝑑𝜃 (Eq.5)

𝐹R = m 𝑑𝐹R =3E

n2 ∙ 𝜋 ∙ 𝜎 ∙ 𝐴 (Eq.6)

𝜎 =𝐹_

2 ∙ 𝜋 ∙ 𝐴 ∙𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

(Eq.7)

𝐴 =𝑑 ∙ ℎ4 −

𝑑93

4 ∙ 𝑡𝑎𝑛𝛽 (Eq.8)

𝑑𝛿r =stu3· tan𝛽 (Eq.9)

𝑑𝜏 = 02𝜎𝑑𝐴𝑑𝜃22 · 𝑑𝛿r · cos𝜃 = 𝜎 · 𝑑ℎ · 𝑑𝑅 0

𝑑𝛿_2 tan𝛽2 · cos𝜃 · 𝑑𝜃 (Eq.10)

𝝈𝒅𝑨𝒅𝜽𝟐

dFH

σdA

σdA

𝒅𝜹𝑳 =𝒅𝜹𝑷 𝒕𝒂𝒏 𝜷

𝟐

θ d𝒅𝜽

𝟐

𝝈𝒅𝑨𝒅𝜽𝟐

d

a) b)

𝒅𝜹𝑳 =𝒅𝜹𝑷 𝒕𝒂𝒏 𝜷

𝟐

varyingfrom0tohalftheheightofthespecimen(h/2),asshowninequation(Eq.11).Thelattermaybemultipliedbythenumberofsegments(n)toobtainthetotalworkofthespecimen,whichisshowninequation(Eq.12).

Supposenowanequivalentspecimensubjectedtothesamekinematicmechanismbutthatwasnotallowedtohaveradialcracks.Inthiscase,thelateraldisplacementδLexperiencedwouldbecausedbyacircumferentialdeformationduetothetensilestresses(Figure5).

Figure5.Circumferentialdeformationofthespecimen.Thecircumferentialdeformationmaybewrittenaccordingwithequation(Eq.13).Thelatteristheusedinequation(Eq.14)toestimatetheworkdonebythetensilestressinaninfinitesimalelementwithalateralareadA.

Following thesameproceduredescribedpreviously, the totalworkof thespecimen isestimatedthroughthetripleintegrationofdτintheinterval(-π,π),foraradiusrangingfrom0totheradiusof thespecimen (R)and fora height varying from0 to half the height of the specimen (h/2), asshowninequation(Eq.15).

Assuming that the circumferential deformation of the specimen is equivalent to the lateraldisplacement of the concrete section, the increments of work in (Eq.11) and (Eq.12) may beconsideredequal,asshowninexpression(Eq.16)andin(Eq.17).

Finally, the strain in the specimen may be written as in expression (Eq.18) or in terms ofincrementsasinequation(Eq.19).

𝜏� = m m m 𝑑𝜏E�

�E�

R

n

�3

n= 𝜎 ·

ℎ2 · 𝑅 · 𝑑𝛿_ · tan𝛽 · sin8

𝜋𝑛:

(Eq.11)

𝜏 = 𝑛 ∙ 𝜏� = 𝑛 ∙ 𝜎 ∙ℎ2 ∙ 𝑅 ∙ 𝑑𝛿_ ∙ tan𝛽 ∙ sin 8

𝜋𝑛:

(Eq.12)

𝜀 = 𝑑𝑅 𝑅⁄ (Eq.13)

𝑑𝜏 = 2 ∙ 𝜎 ∙ 𝑑𝐴 ∙𝑑𝜃2 ∙ 𝑑𝑅 = 𝜎 ∙ 𝑑ℎ ∙ 𝑑𝑅 ∙ 𝑑𝜃 ∙ (𝑅 ∙ 𝜀) (Eq.14)

𝜏 = ∫ ∫ ∫ 𝑑E�E

Rn 𝜏

��n = 3E·�·�·�·R�

�= 𝜋 · 𝜎 · �

3· 𝜀 · 𝑅3 (Eq.15)

𝑛 · 𝜎 ·ℎ2 · 𝑅 · 𝑑𝛿_ · tan· 𝛽 · sin 8

𝜋𝑛: = 𝜋 · 𝜎

ℎ2 · 𝜀 · 𝑅

3 (Eq.16)

𝑛 · 𝑑𝛿_ · tan𝛽 · sin 8𝜋𝑛: = 𝜋 · 𝜀 · 𝑅 (Eq.17)

σdA

σdA𝝈𝒅𝑨

𝒅𝜽𝟐

dθ

𝝈𝒅𝑨𝒅𝜽𝟐

dR

R

R+dR

σdA

σdA

dR

d

Pf

Pi

4.3Valuesoffailureangle(β),frictioncoefficient(μ)andnumberofcracks(n)Intheformulationproposed,thevaluesofthefailureangle(β),thekineticfrictioncoefficient(μk)and thenumberof cracks (n)are required.Thevalueofβ isusually defined in the literaturebymeansoftheinternalfrictionangleofthematerial(φ)sinceitdeterminesthecrackingsurfaceoftheconicalwedge.Areviewonthedifferentvaluesforφproposedbyseveralauthorsduringthepast50yearsmaybefound in [12].Theactual lengthof theconicalwedgewasmeasured in theexperiencesconductedattheLaboratoryofStructureTechnologyLuisAgulló.Figure6ashowstheprocedure of opening the specimen, which was performed avoiding any damage of the conicalwedge.

Figure6.a)Openingofthespecimenandb)measurementsofthelengthoftheconicalwedge.

After the conicalwedgewas separated from the concrete segments, its lengthwasmeasured asobserved in Figure 6b. The measurements revealed that the length of the conical wedge (l) isapproximately40mm.Takingintoaccountthedimensionsofthesteelpunch(d’=37.5mm)andusingexpression(Eq.20),theinternalfrictionangleis65°.Hence,thevaluesareφ=65°andβ=25°.

Regarding the friction coefficient, the values presented in the literature usually cover the staticfriction coefficient of concrete (µs). Nevertheless, very limited information is available for thekinetic frictioncoefficient (µk). It is known that the value ofµk for the same surface tends to besmaller than µs. This should be especially true for two cracked concrete surfaces subjected toconsiderable relative displacement, such as observed in theBarcelona test. As the surfaces startmoving,theroughnessissmoothenedduetothecrackingoftheirirregularities,thusreducingthevaluesofthefrictioncoefficient.InFigure6badetailofthesurfaceoftheconicalwedgeafterthetestispresented.

In the absence of reliable values of µk, the μs proposed in the Model Code 2010 [21] andsummarizedinTable2isusedasareference.Consideringtheaspectoftheconicalwedgeafterthetest,itisassumedthattheμsshouldbeintherangeofroughsurfaces.However,thevalueshouldbereducedtotakeintoaccountthedifferencesbetweenµkandμsaswellasthemechanismobservedduring the test. Then, a µk equal to 0.7 is considered a reasonable initial approximation. It is

𝜀 =𝑛𝛿_𝜋𝑅 · tan𝛽 · sin8

𝜋𝑛:

(Eq.18)

𝛥𝜀 =𝑛𝛥𝛿_𝜋𝑅 · tan𝛽 · sin 8

𝜋𝑛:

(Eq.19)

𝑙 =𝑑′2 · 𝑡𝑎𝑛𝜑

(Eq.20)

Conicalwedge

Conicalwedge

65°

a) b)

importanttoremarkthatmorestudiesarerequiredtocharacterizeµkandthevariablesaffectingthisparameter.

Table2.FrictioncoefficientforplainconcreteaccordingtotheModelCode2010.

Thenumberofcracks(n) may be determinedexperimentally through the observation of the specimens after the test. The influence of thisparameteronthefinalstress-straincurvewiththeconstitutivemodelproposedhereispresentedin Figure 7. For that, the experimental result of the characterisation of one specimenwas usedconsideringnvaluesof2,3and4.Itisevidentthat,thechangeinndoesnotaffectsignificantlytheshapenortheabsolutevaluesofstressestimatedforacertainstrain.Therefore,ifnoinformationonthenumberofcracksisavailable,anumberofcracksequalto3maybeadoptedsinceitisthemostcommonoutcome.

Figure7.Influenceofthenumberofcracks(n)ontheconstitutivemodel.

5.ExperimentalprogramIn order to validate the formulation proposed and to compare the new formulation with theconstitutivemodelsfromEuropeancodesandguidelinesbasedonflexuraltests,theresultsofthreeexperimental studies involving Barcelona tests and 3-point bending tests are considered. TheseresultscorrespondtofourseriesofSFRC,twoofthemfrom[23]andtheothertwofrom[24].TwoseriesofPFRCfromthestudyof[6]werealsoanalysed.InTable3,thespecimensoftheBarcelonatest(C)andthebeamsbendingtest(B)arelistedaccordingtotheircorrespondingreference.Thesewereproducedaccordingto[10]and[22],respectively.

Table3.SeriesofFRCbeamsandcylinders.

Accordingtothenotationadopted,thelettersA,BandCidentifyadifferenttypeoffibreandthenumber following the letter corresponds to the fibre content (in kg/m3). The letters B and C,appendedattheend,differentiatethecylindricalspecimensfromthebeams,followedbyanumber

0.0

1.0

2.0

3.0

0 5 10 15 20

Stress[M

Pa]

Strain[‰]

n=2

Interfaceroughness Frictioncoefficientμ[-]Smoothinterface 0.5-0.7Roughinterface 0.7-1.0

Veryroughinterface 1.0-1.4

Series Reference Cylinders(C) Beams(B) NotationA40 Blanco2013 5 6 A40_C1orA40_B2A60 Blanco2013 2 1 A60_C2orA60_B3B30 Monsó2011 10 8 B30_C3orB30_B4B50 Monsó2011 10 10 B50_C4orB50_B5C5 Pujadas2014 6 3 C5_C5orC5_B6C7 Pujadas2014 5 3 C7_C6orC7_B1

identifyingthespecimensoftheseries.Thetypeoffibreusedsteelfibre(SF)orplasticfibre(PF)andadditionalfibrecharacteristicsarepresentedinTable4.

Table4.Characteristicsofthefibres(providedbythemanufacturer).Theaverageresultsat28daysofthecompressivestrength(fcm)andthemodulusofelasticity(Ecm)assessedaccordingtoUNE83507:2004[25]andUNE83316:1996[26],respectively,arepresentedinTable5.TheresultsoftheflexuraltestincludedinTable6correspondtotheaveragevaluesofthe limit of proportionality (fL) and the residual flexural tensile strengths fR1, fR2, fR3 and fR4associatedtocrackmouthopeningCMODof0.05mm,0.50mm,1.50mm,2.50mmand3.50mm,respectively,accordingtoEN14651:2005[22].

Table5.MainpropertiesoftheFRCseries.

It shouldberemarked that thestudyby[24]doesnotprovideanyresults regardingEcm and thevalueswereestimatedfrom fcmwiththeformulationintheSpanishcodeEHE-08[27].Noticethatduetothecontroldevicemalfunctioning;onlytheresultsofonebeamareavailableforA60.6.SimplifiedmodelFollowingtheapproachofthecurrentEuropeandesigncodesandguidelinesforFRC,whichincludesimplifiedconstitutivemodels,asimplifiedversionoftheformulationisproposedintheshapeofamultilinearσ-εdiagram,asindicatedinFigure8.Table6summarizestheparametersdefiningthesimplified σ-ε diagram derived from the Barcelona test. Notice that the values stresses may beexpressedasfunctionoftheloadassociatedtoacertaindisplacement,consideringtheconstant(ω).TheprocedurefollowedtoobtainthesimplifiedversionisdescribedinAppendixA.

Characteristics Unit A B CMaterial - SF SF PFLength(L) [mm] 50 50 48Diameter(d) [mm] 0.62 1.05 -Aspectratio(L/d) [-] 83 48 -Tensilestrength(fy) [MPa] 1270 1115 550Modulusofelasticity(E) [GPa] 210 210 10Numberoffibresperkg [-] 8100 2800 35000

Characteristics A40 A60 B30 B50 C5 C7

Modulusofelasticity EcmAvg.[MPa] 29029 31597 30200 30000 31150 31930CV[%] 0.96 1.10 - - 1.69 2.14

Compressivestrength fcmAvg.[MPa] 46.73 54.30 44.85 43.97 52.15 54.64CV[%] 0.77 1.51 1.80 4.80 1.52 0.82

Residualflexuralstrengths

fLAvg.[MPa] 5.29 3.72 13.14 13.76 4.61 5.14CV[%] 2.23 - 9.4 7.5 2.19 8.71

fR1 Avg.[MPa] 6.13 6.40 3.73 8.58 2.01 3.59CV[%] 13.71 - 40.3 21.9 22.30 9.20

fR2Avg.[MPa] 7.04 6.12 3.91 9.47 2.25 4.66CV[%] 15.77 - 45.3 28.5 28.53 7.05

fR3Avg.[MPa] 7.08 6.24 3.89 9.29 2.46 5.14CV[%] 15.05 - 50.0 32.0 26.84 6.50

fR4Avg.[MPa] 6.62 6.47 3.62 8.88 2.48 5.17CV[%] 12.08 - 48.7 32.3 23.47 6.43

Table6.Parametersofthesimplifiedmodel.

Figure8.Simplifiedσ-εdiagram.Figure9showsthecomparisonbetweenthesimplifiedandthecompleteσ-εmodelsdeductedinthisstudyforonespecimenofseriesA40,A60,B30,B50,C5andC7.Thecurvesrevealagoodfitandthevalidityforsteelandplasticfibresaswellasfordifferentfibrecontentsofthesimplifiedmodel,whichrepresentswithgoodaccuracytheshapeofthecurve.

Figure9.Completeandsimplifiedmodelsfora)A40_C6,b)A60_C2,c)B30_C5,d)B50_C2,e)C5_C1andf)C7_C2.7.Numericalvalidation7.1ModeltypeandgeometryThefiniteelementsoftwareDIANA9.4waschosentomodeltheBarcelonatestduetoitsextensivematerial libraryandanalysiscapabilities.Thesetupofthetestrequireda3Danalysis,usingsolidand interface elements to simulate the behaviour of the specimen during the test. Given thesymmetryofthetest,onlyhalfofthespecimenwasmodelledtofavourtheefficiencyoftheanalysisand amore refinedmesh discretization (see Figures 10a, 10b and 10c). Furthermore, the steelpunchusedintheBarcelonatesttotransmittheloadtothespecimenwasnotmodelled.Instead,theloadwasdirectlyappliedinanequivalentareaonthetopsurfaceoftheFRCspecimen.

0.0

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0 5 10 15 20

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Pa]

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σ-εmodelSimplifiedversion

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Pa]

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0 5 10 15 20Strain[‰]

σ-εmodelSimplifiedversion

Strain[‰] Stress[MPa]

𝜀� =𝜎�𝐸$�

𝜎� =𝐹_�C�2 ∙ 𝜋 ∙ 𝐴 ∙

𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

= 𝜔𝐹_�C�

𝜀3 = 𝜀� + 0.1 𝜎3 =𝐹_n.n3��2 ∙ 𝜋 ∙ 𝐴 ∙

𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

= 𝜔𝐹_n.n3��

𝜀� = 4.0 𝜎� =𝐹_n.����2 ∙ 𝜋 ∙ 𝐴 ∙

𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

= 𝜔𝐹_n.����

𝜀� = 20 𝜎� =𝐹_�.nn��2 ∙ 𝜋 ∙ 𝐴 ∙

𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

= 𝜔𝐹_�.nn��ε

σ2 σ3

ε3

σ1 σ

ε2 ε1 ε4

σ4

a) b) c)

d) e) f)

A40_C6

Figure10.a)SpecimenoftheBarcelonatest,b)halfofthespecimenmodelledwithFEM,c)definitionofthe

conicalwedgeinthemodelandd)detailoftheconicalwedge.The fracture process leads to a change of the resistant mechanism. This change is difficult toreproducewitha singlemesh for all the volume since it entails localized large displacements atcertain locations (particularly in the formation of the cone) that may cause divergences and adifferentresponsefromtherealone.Forthisreason,thespecimenwasnotaddressedasasinglevolume but as the summation of the cone and the rest of the specimen. This approach ofpredefiningthefailuremechanisminadoublepunchtestwasalreadyappliedinpreviousstudiesbydefiningthefailureplanes[28].Inaddition,toavoidconcentratedloadsinthevertex,theconewas approximated as an inverted truncated conewith a top diameter equal to that of the steelpunch,abottomdiameterof4mmandaheightof40mm(seeFigure10d).Aninterfaceelementwasdefinedbetweenthelateralsurfaceoftheconeandthespecimentoallowthedisplacementoftheconeandthecracking.Themeshesofeachofthe4partsofthespecimenwereradiallygeneratedsoastoreducetheinfluencethatthediscretizationadoptedmayhaveonthe mesh and the results. Symmetry conditions were imposed by restraining the verticaldisplacement of the bottom face of the half specimen. The load case consisted of a verticaldisplacement acting simultaneously at all nodes on the loading surface. Each load step wasequivalenttoaverticaldisplacementof0.025mm.7.2MaterialpropertiesA fracture energy based on total strain rotating crackmodel [29]was used for the FRC of thespecimens assuming the multilinear (σ-ε) curve in tension proposed in section 5. GeneralcharacteristicsandtheparametersofthismodelwereestablishedinaccordancewithEurocode2[30].Sincenocrackingisexpectedinthecone(asobservedexperimentally),forefficiencypurposesitwasdefinedasanelasticmaterialwithamodulusofelasticityequaltothatofthecorrespondingFRC.The interfacematerialbetween theconeand therestof thespecimen ischaracterizedbyalinearnormalandlineartangentialstiffnessandafrictionalbehavioursimulatedusingaCoulombfrictionmodelwithabrittlegappingcriterion.Table7includesasummaryofthemainpropertiesofthematerials.NoticethatforthevalidationoftheformulationsixtypesofFRCwereconsidered:fourofSFRC(A40,A60,B30andB50)andtwoofPFRC (C5 and C7). Notice that the same base mixture compositionwas used for A40 and A60,changingonlythefibrecontentfrom40kg/m3intheformerto60kg/m3inthelatter.Thesame

150mm 4mm

SpecimenoftheBarcelona

test

Symmetryplane Halfofthespecimen

modelledwithFEMConicalwedge

a) b)

d)c)

75mm37.5mm

40mm

Symmetryplane

holdstrueforthemixtureB30andB50withfibrecontentsrespectivelyof30kg/m3and60kg/m3and for C5 and C7, which present 5 kg/m3 and 7 kg/m3. To simplify the presentation of theinformation,themainpropertiesoftheFRCareclassifiedaccordingtoseriesA(forA40andA60),B(forB30andB50)andC(forC5andC7).Onlytwospecimenspereachseriesandfibrecontentarepresentedsincethetendenciesaresimilar.

Table7.MaterialpropertiesconsideredintheFEMmodel.

Taking into account the inspections of the specimens after the Barcelona test and theconsiderationsincludedinsection4.3,afailureangle(β)of0.438radandadynamicfrictionangle(μk)of0.7wereusedinallFEMandinthesimplifiedmodelproposedhere.Itwasalsoassumedthatthenumberofcracks(n)thatappearduringthetestwas3sincethisisthemostcommonoutcomeofthetest.

Modelpart Materialproperties Value Reference

Interfacematerial

Normalstiffness[MN/m3] 1.0·105 Tangentialstiffness[MN/m3] 1.0·105 Tangentoffrictionangle 0.7 Tangentofdilatancyangle 0.0

Conicalwedge Poissonratio[-] 0.2 Eurocode2Averagemodulusofelasticity[GPa] 2.9/3.2/3.0/3.0/3.1/3.2 A40/A60/B30/B50/C5/C7

SFRC(SeriesA)

Averagecompressivestrength[MPa] 46.7/54.3 A40/A60Averagemodulusofelasticity[GPa] 29.0/31.6 A40/A60Poissonratio[-] 0.2 Eurocode2Tensilestrengthσ1[MPa] 2.9/2.9/3.3/3.4 A40_C5/_C6/A60_C1/_C2Residualstrengthσ2[MPa] 2.4/2.2/3.2/3.2 A40_C5/_C6/A60_C1/_C2Residualstrengthσ3[MPa] 2.0/1.5/2.7/2.8 A40_C5/_C6/A60_C1/_C2Residualstrengthσ4[MPa] 1.3/1.0/1.6/1.7 A40_C5/_C6/A60_C1/_C2Strainε1[‰] 0.1 A40_C5/_C6/A60_C1/_C2Strainε2[‰] 0.2 A40_C5/_C6/A60_C1/_C2Strainε3[‰] 4.0 A40_C5/_C6/A60_C1/_C2Strainε4[‰] 20.0 A40_C5/_C6/A60_C1/_C2

SFRC(SeriesB)

Averagecompressivestrength[MPa] 44.9/44.0 B30/B50Averagemodulusofelasticity[GPa] 30.2/30.0 B30/B50Poissonratio[-] 0.2 Eurocode2Tensilestrengthσ1[MPa] 2.8/2.5/2.7/2.8 B30_C5/_C8/B50_C2/_C8Residualstrengthσ2[MPa] 2.4/2.0/2.5/2.3 B30_C5/_C8/B50_C2/_C8Residualstrengthσ3[MPa] 1.6/1.6/1.9/1.4 B30_C5/_C8/B50_C2/_C8Residualstrengthσ4[MPa] 0.7/0.9/1.0/0.9 B30_C5/_C8/B50_C2/_C8Strainε1[‰] 0.1 B30_C5/_C8/B50_C2/_C8Strainε2[‰] 0.2 B30_C5/_C8/B50_C2/_C8Strainε3[‰] 4.0 B30_C5/_C8/B50_C2/_C8Strainε4[‰] 20.0 B30_C5/_C8/B50_C2/_C8

PFRC

Averagecompressivestrength[MPa] 52.2/54.6 C5/C7Averagemodulusofelasticity[GPa] 31.3/32.1 C5/C7Poissonratio[-] 0.2 Eurocode2Tensilestrengthσ1[MPa] 3.3/3.1/3.5/3.3 C5_C2/_C5/C7_C2/_C5Residualstrengthσ2[MPa] 1.4/1.8/1.5/1.8 C5_C2/_C5/C7_C2/_C5Residualstrengthσ3[MPa] 0.8/1.0/1.0/0.9 C5_C2/_C5/C7_C2/_C5Residualstrengthσ4[MPa] 0.5/0.6/0.6/0.6 C5_C2/_C5/C7_C2/_C5Strainε1[‰] 0.1 C5_C2/_C5/C7_C2/_C5Strainε2[‰] 0.2 C5_C2/_C5/C7_C2/_C5Strainε3[‰] 4.0 C5_C2/_C5/C7_C2/_C5Strainε4[‰] 20.0 C5_C2/_C5/C7_C2/_C5

7.3ResultsInFigure11,theexperimentalandthepredictedcurvesarecomparedfortwospecimensofeachseries. It must be remarked that in the experimental results there is an accommodation effectbetween the surfaces of the piston and the specimen (due to irregularities of the surface of thespecimen)thatleadstolargerdisplacementsintheearlystages.Thiseffectwasnotobservedinthesimulationwiththefiniteelementsoftwaresincethecontact isperfectfromthebeginningofthetest.Forthisreason,themaximumloadoftheexperimentalcurvewasmovedtomatchthepeakofthesimulatedcurve.

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B30_C5Model

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B30_C8Model

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B50_C2Model

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B50_C8Model

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C5_C2Model

f)d) e)

i)g) h)

l)j) k)

c)a) b)

Figure11ExperimentalandnumericalP-δcurvesfor:a)A40_C5,b)A40_C6,c)A60_C1,d)A60_C2,e)B30_C5,f)B30_C8,g)B50_C2,h)B50_C8,i)C5_C2,j)C5_C5,k)C7_C2andl)C7_C5.

Ageneraloverviewofthecurvesrevealsthatthemodel isnotabletopredictthesuddendropofload after the cracking occurs. In general, the model provides a peak load close to theexperimentallyobtainedforseriesAandB,whereasthepredictedvalueforseriesCislowerthantheexperimentalvalue.Itshouldberemarkedthatthenoveltyoftheformulationproposedisthatitisvalidforthepost-crackingstageand,forthatreason;theaimofthefiniteelementanalysisistovalidatethecapacityofthemodeltoreproducethatstageoftheexperimentalcurves.Regardingthepost-crackingstage,theresultsindicatethatthepredictedcurvesexhibitatendencyvery similar to the one experimentally observed, especially from a displacement of 2.0 mmonwards.However,atthesametime,agrowingdifferencebetweentheexperimentalandpredictedresultsisdetectedasthedisplacementincreases.ThisisparticularlyevidentinseriesB30andB50fordisplacementsbiggerthan4.0mm.Inordertofurtheranalysethisbehaviour,thedifferencesinthevaluesofloadfordisplacementsof1.0mm,3.0mmand5.0mmareincludedinTable8.Noticethatapositivevalueindicatesthattheprediction overestimates the value of load, whereas a negative number corresponds to anunderestimationoftheexperimentalvalue.Table8.Differencesbetweenexperimentalandpredictedloadvaluesfordisplacementsof1.0,3.0and5.0mm.

From the results, it is found that the difference between both curves is highest for a value ofdisplacement of 1.0 mm. It is important to highlight the difficulty that the modelling of theBarcelonatestentailssince,aspreviouslyintroduced,afterthecrackingoftheconcretematrixthefailureisgovernedbyakinematicmechanism.Thischangeintheresistantmechanismisveryhardtoreproduceevenwithadvancedfiniteelementsoftware.Inthiscase,thedifferencesbetweentheresults provided by the model and the experimental data may be partially attributed to the

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C7_C5Model

Series SpecimenDifferencesbetweenexperimentaldataandmodel[%]

1.0mm 3.0mm 5.0mm

A(SFRC)

A40_C5 -15.7 0.1 -2.1A40_C6 -13.2 4.7 12.5A60_C1 -2.1 -7.8 -6.6A60_C2 -13.8 -3.7 0.0

B(SFRC)

B30_C5 -28.9 -1.7 27.2B30_C8 -23.6 7.0 26.5B50_C2 -26.5 0.6 23.2B50_C8 -12.8 9.4 14.9

C(PFRC)

C5_C2 -11.5 -1.0 -C5_C5 -14.4 -1.0 -C7_C2 -6.8 2.1 32.0C7_C5 -4.7 9.1 -

difficulty to reproduce such phenomenon. Nevertheless, as the response FRC stabilizes, theexperimentalresponseandthepredictionofthemodelbecomecloser(aroundadisplacementof2.0and3.0mm).In the last stretch of the post-cracking stage (around a displacement of 5.0mm), the differencebetweenthemodelandtheexperimentalresultsincreasesagain.Infact,theoverestimationat5.0mmofthemodelsrangesfrom15%to30%forseriesBandC.InthecaseofseriesA,thedifferenceremainsbelow20%.Atsuchadvancedstageofthetest, theinfluenceofthetypeof fibreandthefibrecontentmaybesignificant.NoticethattheaveragedifferenceforseriesB50islowerthanforseriesB30,whichhaslowerfibrecontentthantheformer.ThisbehaviourmayalsobeobservedforseriesC,sincetheaveragedifferenceislowerforC7thanforC5.

Despitethedifferences,themodelyieldsresultswithatendencyandanabsolutevaluesimilartothosefromtheexperimentaltest.TakingintoaccounttheparticularitiesintheresponseofFRCandthenumericaldifficultytosimulatetheBarcelonatestinafiniteelementmodel,theresultsindicatethatthesimplifiedconstitutiveequationfromthisworkprovidessatisfactoryresults.8.ComparativeanalysiswiththemodelsinEuropeancodesThe formulation based on the Barcelona test is compared in this section with the constitutivemodelsintheEuropeancodesandrecommendationsforthedesignofFRC,whicharebasedontheflexuralperformanceof smallbeams.For thatpurpose,the trilinear/multilinearmodels from theDBV[31],theRILEM[32],theEHE-08[27]andtheModelCode2010[21]wereselectedsincetheyreproducemoreaccuratelythecontributionofthefibresaftercracking.InthecaseoftheCNR-DT204 [33], the bilinear model was considered. Even though the most recent recommendationsalready include certain plastic fibres as structural fibres, it should be remarked that theseconstitutivemodelsarebasedontheexperienceswithsteelfibrereinforcement.ThevaluesoftheparametersdefiningeachoftheconstitutivemodelsforseriesA40,A60,B30,B50,C5andC7arepresentedinTable9.Noticethatthetensilestrengthisσ1forallmodelsexcepttheM2010, in which the tensile strength is represented by σ2 due to the shape of the diagram.Additionally,itshouldbepointedoutthatpartialsafetyfactorswerenotusedtoobtainanyvaluesofstressandstrain.ThenotationusedtoreferthemodelsinthecodesisDBV,RILEM,CNR-DT,EHEandMC(orbeammodelsasagroup);whilefortheformulationproposeditisBCN.

Table9.ParametersdefiningtheconstitutivemodelsofseriesA40,A60,B30,B50,C5andC7.

Series Modelsσ1 ε1 σ2 ε2 σ3 ε3 σ4 ε4

[MPa] [‰] [MPa] [‰] [MPa] [‰] [MPa] [‰]

A40

DBV 2.536 0.104 0.642 0.204 - - 0.522 10.000RILEM 5.907 0.203 2.811 0.303 - - 2.451 25.000CNR-DT 2.811 0.097 - - - - 1.916 20.000EHE 3.038 0.105 2.811 0.205 - - 2.274 20.000MC 3.089 0.106 3.432 0.150 2.806 0.193 2.274 20.000BCN 2.851 0.098 2.191 0.198 1.447 4.000 0.907 20.000

A60

DBV 3.395 0.107 1.200 0.207 - - 0.996 10.000RILEM 6.657 0.211 2.879 0.311 - - 2.394 25.000CNR-DT 2.879 0.091 - - - - 1.146 20.000EHE 3.423 0.091 2.879 0.208 - - 1.839 20.000MC 3.481 0.110 3.868 0.150 2.855 0.473 1.839 20.000BCN 3.368 0.107 3.199 0.207 2.784 4.000 1.704 20.000

B30 DBV 2.918 0.097 0.040 0.197 - - 0.000 6.300

Theresultsregardingthetensilestrength(σ1)revealthatthelowestvaluesalwayscorrespondtothe DBV and the highest to the RILEM. In this analysis, the value of the CNR-DT model is notconsideredsinceit isasimplifiedbilinearmodelwhichcannotbecomparedtoatrilinearmodel.ThevaluesprovidedbytheBCNmodelremainbetweenthelimitvaluesoftheDBVandtheRILEM.Infact,thevalueofσ1oftheDBVisonly11.0%,0.8%,14.8%,5.4%,1.2%and0.7%lowerthanthevalueoftheBCNforseriesA40,A60,B30,B50,C5andC7,respectively;whereasthevalueoftheRILEMis107.2%,97.6%,125.1%,106.6%,93.7%and94.8%higherthanthevalueofBCNforA40,A60,B30,B50,C5andC7,respectively.Ingeneral,thebeammodelthatpresentstheclosestvaluestotheBCNistheMC(notethatthetensilestrengthoftheMCcorrespondstoσ2).TheotherparametersinTable9,particularlythevaluesofstress,maybemoreeasilyidentifiedandanalysedifplotted.InFigure12,theconstitutivemodelsobtainedfromtheaverageresultsofthebeam tests are compared to the models determined by means of the average results of theBarcelonatestforeachseries.Ingeneral,thehighestvaluesofcorrespondtotheRILEM,theEHEortheMC.TheBCNmodelrangesbetweentheonespresentingthelowestandthehighestvaluesofresidual strengths (except for series A60). The curves of series A40 are an example of suchbehaviour.Figures12aand12bcorrespondtothesametypeofsteelfibrewithahighaspectratio(L/d=83)for fibre contents of 40 kg/m3 and60 kg/m3, respectively. An increase of 20 kg/m3 in the fibrecontent leads to a significant increment in the values of stress, which is observed if the curvesobtainedwiththeBCNmodelforA40andA60arecompared.Infact,thevaluesofσ2,σ3andσ4forA60are46.0%,92.3%and87.9%higherthanforA40.It isreasonablethatthelowest incrementcorresponds to σ2 since after the cracking the reinforcement capacity of the fibres is not fullydevelopeduntilhighervaluesofstrain,whichcorrespondtoσ3.Likewise,itcouldbeexpectedthatthe increment in the value of the ultimate stress (σ4) should be lower than for σ3 due to thedebonding-slippingmechanisms.

RILEM 5.722 0.189 1.677 0.289 - - 1.338 25.000CNR-DT 1.677 0.056 - - - - 0.884 20.000EHE 2.943 0.097 1.677 0.197 - - 1.201 20.000MC 2.993 0.099 3.325 0.150 1.671 0.266 1.201 20.000BCN 2.542 0.094 2.056 0.194 1.417 4.000 0.688 20.000

B50

DBV 2.871 0.096 0.332 0.196 - - 0.175 10.000RILEM 5.629 0.188 3.860 0.288 - - 3.284 25.000CNR-DT 3.860 0.129 - - - - 2.312 20.000EHE 2.895 0.109 3.860 0.196 - - 2.931 20.000MC 2.944 0.098 3.271 0.196 3.143 16.346 2.931 20.000BCN 2.724 0.091 2.439 0.191 1.629 4.000 0.803 20.000

C5

DBV 3.289 0.105 0.303 0.205 - - 0.244 10.000RILEM 6.449 0.206 0.818 0.306 - - 0.769 25.000CNR-DT 0.818 0.026 - - - - 0.608 20.000EHE 3.748 0.120 0.818 0.220 - - 0.692 20.000MC 3.391 0.108 3.768 0.150 0.848 0.189 0.692 20.000BCN 3.329 0.106 1.339 0.206 0.805 4.000 0.491 20.000

C7

DBV 3.409 0.106 0.594 0.206 - - 0.655 10.000RILEM 6.685 0.208 1.561 0.308 - - 1.931 25.000CNR-DT 1.561 0.049 - - - - 2.074 20.000EHE 3.438 0.121 1.561 0.207 - - 1.868 20.000MC 3.515 0.110 3.906 0.150 1.487 0.180 1.868 20.000BCN 3.432 0.107 1.487 0.207 0.910 4.000 0.610 20.000

However,suchincreaseinthevaluesofstressdoesnotoccurinthecaseofthebeammodels,whichmightbeattributedtothefactthattheconstitutivemodelsofseriesA60wereobtainedwiththeresults of only one beam (the other tests failed due to a problem with the test equipment).Therefore,conclusiveremarkscannotbemadefromcomparingthebeammodelsofseriesA40andA60;eventhoughasignificantincreaseintheresponseshouldbeexpectedifmorebeamshadbeentested.

Figure12.ComparisonoftheformulationbasedontheBarcelonatestwiththeconstitutivemodelsbasedonbeam

tests:a)A40,b)A60,c)B30,d)B50,e)C5andf)C7.Regarding12cand12d,correspondingtoseriesB30andB50,anoticeableincreaseintheresponseisobservedinthebeammodelswhentheamountoffibresincreasesfrom30kg/m3to50kg/m3.TheincreaseinthecaseoftheBCNmodel ismoresubtlethanforseriesA40andA60;beingthevaluesofσ2,σ3andσ4 forB50only18.6%,15.0%and16.8%higherthanforB30.ThereasonforthisdifferencebetweenseriesAandBmaybeattributed to the typeof fibre.Despitehaving thesamelength,theSFofseriesBhasasmalleraspectratiothantheSFofseriesAand,consequently,thenumberoffibresintheconcretematrixislower.Therefore,despiteincreasingthefibrecontentin 20 kg/m3 in both cases, the number of fibres is higher for series A, thus leading to a higherrestrictionofthecrackopeningandhighervaluesofstress.

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 5 10 15 20 25

Stress[M

Pa]

Strain[‰]

DBVRILEMCNR-DTEHEMC

0.0

1.0

2.0

3.0

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5.0

6.0

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0 5 10 15 20 25Stress[M

Pa]

Strain[‰]

DBVRILEMCNR-DTEHEMC

0.0

1.0

2.0

3.0

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Stress[M

Pa]

Strain[‰]

DBVRILEMCNR-DTEHEMCBCN

0.0

1.0

2.0

3.0

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5.0

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0 5 10 15 20 25

Stress[M

Pa]

Strain[‰]

DBVRILEMCNR-DTEHEMC

0.0

1.0

2.0

3.0

4.0

5.0

6.0

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0 5 10 15 20 25

Stress[M

Pa]

Strain[‰]

DBVRILEMCNR-DTEHEMCBCN

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0 5 10 15 20 25

Stress[M

Pa]

Strain[‰]

DBVRILEMCNR-DTEHEMCBCN

a) b)

c) d)

e) f)

FromthebehaviourobservedinFigures12cand12daquestionregardingthedifferencesbetweenthebeammodelsandtheBCNtestmayberaised.Whytheincreaseinfibrecontentleadstobiggerincrementsinthestressesforthebeammodels?Thisisprobablyrelatedtotheamountofenergydissipatedatthemomentthemaincracksappearandduringthewholetest.ThestudybyGuàrdia[34]revealedthattheenergydissipatedintheBarcelonatestisaround4to5timesbiggerthaninthebeamtest. Suchdifference in theamountofenergyreleasedmaybeattributed to thebiggercrack surface in the case of theBarcelona test.Hence, an equivalent increment in the values ofstressintheBCNmodelwouldrequireabiggerincreaseofthefibrecontent.Inthecaseoftheplasticfibres(seeFigures12eand12f),anincreaseinthefibrecontentof2kg/m3(from0.55%to0.77%involume)doesnotresultinasignificantincreaseofthevaluesofstressforneitherthebeammodelsnortheBCNmodel.However,asignificantdifferenceintheresponseofthe beam models is observed between series C5 and C7 since the latter exhibits hardeningbehaviour.ThisphenomenondoesnotappearinthecaseoftheBCNmodelandthereasonmaybefoundinthefailuremechanismoftheBarcelonatest. FailureintheBarcelonatestoccurswhenthestressinthespecimensreachesthetensilestrengthofconcrete.Atthispoint,twoscenariosarepossible:asofteningbehaviourorahardeningbehaviour.Thesofteningbehaviouristhemostcommonresponseandusuallytakesplaceforlowormoderatefibre contents since the restriction to the opening of the radial cracks is small. In such case, theconical wedgewould slide inside the specimen leading to a sudden reduction of the load (andenergy dissipation) until themaincracks openand the fibres start their bridgingmechanism. If,however,theamountoffibresisbigenough,ahigherrestrictiontothecrackopeningwouldoccurasthefibresrestricttheslideoftheconicalwedgeintothespecimen.This isparticularlytrueforshort fibres since theirbridgingcapacity is activated for smallercrack openings. In this case,anincreaseintheloadvalueandahardeningbehaviourofthematerialareexpected.Thesameargumentcouldbevalidforthebeamtest.Howeverwhyisitthatforthesameamountandtypeoffibresthehardeningbehaviouronlyoccursinthebeammodels?Thisis,again,relatedtotheamountofenergydissipatedwhenthemaincracksappearandduringthewholetest.GiventhattheenergydissipatedinthebeamtestissmallerandlessabruptthanintheBarcelonatest,itispossiblethatforthesamefibrecontentthisishighenoughtoresultinahardeningbehaviourinthebendingtestbutstillnotenoughtogeneratesuchresponseintheBarcelonatest.Infact,hardeningintheBarcelonatestwouldonlyoccuriftheamountoffibreswasveryhighandaftertheformationoftheconicalwedgethefibresdidnotbreak.9.ConclusionsThispaperpresentsanewanalyticalformulationtopredictthepost-crackingtensilebehaviourofFRCfromtheresultsoftheBarcelonatest.ItalsorepresentsastepforwardontheknowledgeofthefailuremechanismoftheBarcelonatest.Theanalyticalformulationproposedwassimplifiedintheform of a multilinear σ-ε diagram in which the values of strain are predefined. This simplifiedversion of the model may be easily implemented in the current design tools and adopted byprofessionals.

ThefractureprocessoftheBarcelonatestleadstoachangeoftheresistantmechanismthatisveryhard to simulate, evenwith advanced FEM programs. Indeed, an approach inwhich the conicalwedge formed during the test was geometrically predefined in the finite element analysis wasrequired to avoid divergences. The results provided by the model were used to validate theformulation by comparing them to the experimental results of several FRC with different fibretypesandcontents.Despitethedifferencesbetweentheexperimentalandthepredictedbehaviour,

themodelprovidedsatisfactoryresultswithasimilartrendtothatoftheexperimentalresults.Thenumericalresultsrevealedthattheformulationisvalidtoestimatethepost-crackingresponseofconcretereinforcedwitheithersteelorplasticfibres.

The development of the analytical formulation and the numerical study allowed identifying theneedforreliablevaluesofthekineticfrictioncoefficientforaconcretetoconcretecontact.Giventheabsenceofthisinformationintheliterature,thevaluesofthestaticfrictioncoefficientprovidedinthedesignrecommendationswereconsideredasareference.The comparison of the model proposed with the constitutive models in European codes andguidelinesrevealedthattheformerremainsintherangeofstressesofthelatter.Onfuturestudies,the constitutive model presented may be used to predict the response of certain structuralapplicationstodeterminewhetheritspredictionismoreaccuratethanthebeammodels.Acknowledgements

Theauthorsof thisdocumentwish to show their gratitude for theeconomic support received through theResearch Project BIA2010-17478: Construction processes by means of fibre reinforced concretes. Theexperimental program was conducted in collaboration with ESCOFET. The first and second authorsacknowledge the grant FI provided by the Comissionat per a Universitats del DIUE de la Generalitat deCatalunyaidelFonsSocialEuropeuandthesupportoftheDepartamentd’Universitats,RecercaiSocietatdelaInformaciódelaGeneralitatdeCatalunya.ThefirstalsoacknowledgesthesupportoftheCol·legid’EnginyersdeCamins,CanalsiPortsdeCatalunya.References

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[17] W.F.Chen, R.L. Yuan, Tensile strength of concrete: double punch test, Journal of Structural Divisions.106(8)(1980)1673-1693.[18]P.Marti,SizeEffectinDouble-PunchTestsonConcreteCylinders,ACIMat.J.86(6)(1989)597-601.[19]X.X.Wei,K.T.Chau,Finitesolidcircularcylinderssubjectedtoarbitrarysurfaceload.PartII—Applicationtodouble-punchtest.Int.J.SolidsStruct.37(40)(2000)5733-5744.[20]F.Laranjeira,C.Molins,A.Aguado,Predictingthepulloutresponseofinclinedhookedsteelfibers,Cem.Concr.Res.40(10)(2010)1471-1487.[21]fib.ModelCode2010,InternationalFederationforStructuralConcrete,Lausanne,2010.[22]CEN.EN14651:2005Testmethodformetallicfibreredconcrete-Measuringtheflexuraltensilestrength(limitofproportionality(LOP),residual),EuropeanCommitteeforStandardization,Brussels,2005.[23]A.Blanco,CharacterizationandmodellingofSFRCelements,DoctoralThesis,UniversitatPolitècnicadeCatalunya-BarcelonaTech,Barcelona,2013.http://www.tdx.cat/handle/10803/116195.[24]A.Monsó,AnálisisdelcomportamientodelhormigónreforzadoconfibrasparaelensayoBarcelonaydeflexotracción, Bachelor Thesis, Universitat Politècnica de Catalunya-BarcelonaTech, Barcelona, 2011.http://upcommons.upc.edu/pfc/handle/2099.1/14264?locale=en.[25] AENOR. UNE 83507:2004. Hormigones con fibras. Rotura por compresión., Asociación Española deNormalizaciónyCertificación,Madrid,2004.[26] AENOR. UNE 83316:1996. Ensayos de hormigón. Determinación del módulo de elasticidad encompresión.,AsociaciónEspañoladeNormalizaciónyCertificación,Madrid,1996.[27]CPH.InstruccióndelHormigónEstructuralEHE-08,MinisteriodeFomento,Madrid,2008.[28]A.Pros,P.Díez,C.Molins,Numericalmodelingof thedoublepunchtest forplainconcrete, Int. J.SolidsStruct.48(2011)1229-1238.[29]R.Burguers,J.C.Walraven,G.A.Plizzari,G.Tiberti,StructuralBehaviorofSFRCTunnelSegmentsduringTBMOperations, inProceedingsof theUndergroundSpace- the4thDimensionofMetropolises:WorldTunnelCongress2007and33rdITA/AITESAnnualGeneralAssembly,Prague,CzechRepublic,2007,Barták.,Hrdina.,Romancov.Zlámal.Eds.Taylor&FrancisGroup,1461-1467.[30] CEN. ENV 1992-1-1 Eurocode 2: Design of Concrete Structures - Part 1: General rules and rules forbuildings,EuropeanCommitteforStandardization,Brussels,1992.[31] DBV. Guide to Good Practice "Steel Fibre Concrete", German Society for Concrete and ConstructionTechnology,Berlin,2001.[32]L.Vandewalleetal.Testanddesignmethodsforsteelfibrereinforcedconcrete-σ-εDesignMethod-FinalRecommendation,Mater.Struct.36(8)(2003)560-567.[33]CNR.CNR-DT204/2006GuidefortheDesignandConstructionofFiber-ReinforcedConcreteStructures,ItalianNationalResearchCouncil,Rome,2006.[34]J.Guàrdia,Caracteritzaciódelcomportamentatracciódeformigód'altatreballabilitatreforçatambfibresd'acermitjançantl'assaigBarcelona,BachelorThesis,UniversitatPolitècnicadeCatalunya,Barcelona,2008.AppendixAIn this appendix, the procedure followed to propose a simplified version of the constitutive models isdeveloped.Thefirststageofthecurveoftheconstitutivemodelcorrespondstothelinear-elasticstagepriortocracking,wheretheconcretematrixbearsthetensilestresses.Thetensilestrength(σ1)anditscorrespondingstrain(ε1)canbecalculatedasindicatedinexpressions(Eq.A1)and(Eq.A2).

Thevalueofσ1canbeeasilydeterminedbyknowingthemaximumloadregisteredduringthetest(FPmax)andthevaluesofthefailureangle(β)andthekineticfrictioncoefficient(μk).Whenthevalueofσ1isknown,thestrainε1maybeobtainedusingthemodulusofelasticity(Ecm).

Oncethetensilestrengthoftheconcretematrixisreached,crackingoccursandthereisadropofstressuntilthefibresstarttheirbridgingcapacity,characterizedbyσ2andε2.Thevalueofε2isestablishedfollowingthephilosophyinmostEuropeancodesandguidelineswhichdefinethisparameterasinequation(Eq.A3).

Inordertodeterminethevalueofσ2, the loadassociatedwiththevalueofε2mustbeknown.Forthat, thedisplacement equivalent to the strain ε2 must be calculated isolating𝛥𝛿�in equation (Eq.19), which gives

𝜎� =𝐹_�C�2 ∙ 𝜋 ∙ 𝐴 ∙

𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

(Eq.A1)

𝜀� =𝜎�𝐸$�

(Eq.A2)

𝜀3 = 𝜀� + 0.1‰ (Eq.A3)

equation(Eq.A4).Ifanumberofcracksequal3,aβequalto0.438radanda𝛥𝜀equalto0.1‰(𝜀3 − 𝜀�fromequation (Eq.19) are assumed, the value 0.02mm of displacement equivalent to the strain ε2 is obtained.Noticethatthe0.02mmismeasuredfromthedisplacementcorrespondingtoFPmaxonwards.

Thestressofσ2maybedeterminedasshownin(Eq.A5)forthevalueofloadcorrespondingtoadisplacementof0.02mm(𝐹_n.n3��).

Thesameprocedurewasusedtoestimatethevaluesof𝜎�relatedwiththestrainε3.Afteranalysingseveralexperimental results, a ε3 equal to 4.0‰ was chosen in order to obtain a good fit with the completeconstitutivecurve.ThiscorrespondstoaΔεof3.9‰(ε3-ε1=4.0‰-0.1‰),assumingthatfortypicalconcreteε1 shouldbe close to0.1‰.Substituting this value inequation (Eq.A4)givesarelativedisplacement𝛥𝛿�ofapproximately0.8mm.Therefore,thevaluestressσ3maybecalculatedasindicatedinexpression(26).

The value of ε4 is assumed as 20‰, following the tendency observed in several instruction and codesregardingthemaximumstrainfortheconstitutivecurvesofFRC.Consideringthesameassumptionsasforε3,thedisplacementassociatedto20‰estimatedinequation(Eq.19)isapproximately3.9mmand, therefore,thevalueofσ4maybedeterminedasindicatedinexpression(Eq.A7).

𝛥𝛿_ =𝛥𝜀𝜋𝑅

𝑛𝑡𝑎𝑛𝛽sin(𝜋𝑛)=

0.0001 · 𝜋 · 75𝑚𝑚3 · tan(0.436) · sin(𝜋3)

= 0.02𝑚𝑚 (Eq.A4)

𝜎3 =𝐹_n.n3��2 ∙ 𝜋 ∙ 𝐴 ∙

𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

(Eq.A5)

𝜎� =𝐹_n.����2 ∙ 𝜋 ∙ 𝐴 ∙

𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

(Eq.A6)

𝜎� =𝐹_�.nn��2 ∙ 𝜋 ∙ 𝐴 ∙

𝑐𝑜𝑠𝛽 − 𝜇] ∙ 𝑠𝑒𝑛𝛽𝑠𝑒𝑛𝛽 + 𝜇] ∙ 𝑐𝑜𝑠𝛽

(Eq.A7)