Análisis numérico de vigas de hormigón armado reforzadas a cortante externamente mediante laminados de polímeros reforzados con fibras (PRF)

ScienceDirect

www.e-ache.com HormigónyAcero2018;69(285):113–120 www.elsevierciencia.com/hya

Numerical

analysis

of

reinforced

concrete

beams

strengthened

in

shear

by

externally

bonded

(EB)

fibre

reinforced

polymer

(FRP)

sheets

Análisis

numérico

de

vigas

de

hormigón

armado

reforzadas

a

cortante

externamente

mediante

laminados

de

polímeros

reforzados

con

fibras

(PRF)

Eva

Oller

Ibars

,

Denise

Ferreira

,

Antonio

Marí

Bernat

,

Jesús

Miguel

Bairán

García

a_{Ph.D.inCivilEngineering,Ass.Professor,TechnicalUniversityofCatalonia,UPC,Spain} b_{Ph.D.inCivilEngineering,Consultant,DIANAFEA,Netherlands}

c_{Ph.D.inCivilEngineering,Professor,TechnicalUniversityofCatalonia,UPC,Spain}

Received6April2017;accepted18April2017 Availableonline20June2017

Abstract

Inthispaper,afibrebeammodelpreviouslydevelopedbytheauthorsforthenonlinearanalysisofstrengthenedelements,includingtheeffects ofshear,isusedtopredicttheresponseofreinforcedconcrete(RC)beamsstrengthenedinshearwithfibrereinforcedpolymers(FRP)sheets. Thismodelhasbeenextendednotonlyforwrappedconfigurationsbutalsofordebondingfailureinordertoallowforitsapplicationtobeams strengthenedwithU-shapedandside-bondedconfigurations.Whensimulatingexistingexperimentaltests,themodelreproduces,withreasonable accuracythebehaviorofthebeams,beingthenausefultoolforpracticalengineering.

©2017Asociaci´onCient´ıfico-T´ecnicadelHormig´onEstructural(ACHE).PublishedbyElsevierEspa˜na,S.L.U.Allrightsreserved.

Keywords:Strengthening;Shear;FRPsheets;Debonding;Fibrebeammodel

Resumen

Enesteartículosepresentalaextensióndeunmodelodefilamentosybarrasparaelanálisisnolinealdeelementosreforzadosexternamentea cortanteconlaminadosdepolímerosreforzadosconfibras,teniendoencuentalosefectosdelcortante.Estemodelosehaextendidonosolopara configuracionesqueenvuelvendeformacompletalasecciónconellaminadodepolímerosreforzadosconfibras,sinotambiénparaloscasosen quesepuedeproducirdesprendimientoprematurodellaminado,parapermitirsuaplicaciónenvigasreforzadasenUoconlaminadosadheridos enlasalmas.Alsimularensayosexperimentalesexistentes,elmodeloreproduce,conunaexactitudrazonable,elcomportamientodelasvigas reforzadasy,portanto,esunaherramientaútilparalaingenieríapráctica.

©2017Asociaci´onCient´ıfico-T´ecnicadelHormig´onEstructural(ACHE).PublicadoporElsevierEspa˜na,S.L.U.Todoslosderechosreservados.

Palabrasclave: Refuerzo;Cortante;LáminasdePRF;Desprendimientoprematuro;Modelodefilamentosybarras

1. Introduction

TheshearstrengtheningbymeansofFRPsheetsorlaminates canbe performed indifferent configurations: (a)sheetsfully

∗_{Correspondingauthor.}

E-mailaddresses:[email protected](E.OllerIbars),

[email protected](D.Ferreira),[email protected]

(A.MaríBernat),[email protected](J.M.BairánGarcía).

wrapping thecross-section(wrapped);(b) sheetsorL-shaped laminates bonded onthe lateral sidesandthe bottom surface ofthebeam(U-shaped);and(c)sheetsorlaminatesbondedin thelateralsidesofthecross-section(side-bonded).Thesheets andlaminatescanbebondedinacontinuousordiscontinuous manner.

In the caseof wrapped configurations, the shearfailure is accompaniedbyFRPruptureortheFRPsystemcanfaildueto therupture ofthefibresatthe roundcorner ofthesection.In

https://doi.org/10.1016/j.hya.2017.04.022

U-shapedorside-bondedconfigurations,theFRPmaydebond beforereachingitsultimatecapacity.

Afibrebeammodel developedbythe authorsforthe non-linearanalysisofRCandstrengthenedelementsincludingthe effectsofshear[1–4]hasbeenimprovedtoaccountfor strength-eningbymeansofFRPsheets.Themodelhasbeenvalidated in[5,6]throughtheanalysisofRCbeamsstrengthenedinshear withFRPsheetsinvolvingdifferentconfigurationsbymeansof themodelization orthe experimentalcampaign ofAlzate[7], KhalifaandNanni[8],andMatthys[9].Themodelreproduces, withgood accuracy,the experimentalfailure loads, the load-deflection behavior and the strains instirrups and FRP with increasing load until failure. It also reflects the load-sharing between inner transversal steel reinforcement and EB FRP beforeandafterprematuredebondingfailure.Thisachievement isimportant duetoitssimplicity andcomputationalspeedto beappliedattruescalestructuralanalysis,makingitan attrac-tivetoolforpractical engineering.Thispaperbrieflyexplains themodelandshowssomeoftheresultsoftheirapplicationto simulatesomeexistingexperimentalresults.

2. Fibrebeammodel

2.1. Fundamentalsofthemodel

Thismodelisbasedonaflexuralfibrebeammodel[10]which considerstheinteractionaxialforce–shear–bendingmoment (N-V-M) and uses a displacement-based FE formulation for thenonlinearphasedanalysisofconcreteframestructures.The detailedformulationandvalidationofthe1Dmodelwithshear criticalbenchmarkswaspresentedelsewhere[1–4].Onlyabrief descriptionofthefundamentalsofthemodelispresentedhere.

Fig.1presentsthegeneralcharacteristics ofthe modelfor different levels of analysis: element, section, fibre and mate-rial. Regarding the element level, the model is based on the Timoshenkobeamtheorywiththecross-sectiondiscretizedinto fibres, the longitudinal reinforcement simulated by means of filamentsandtransversalreinforcementconsideredsmearedin concrete.Atthesectionallevel,ashear-sensitivemodelaccounts forthenonlinearforceinteraction(N-V-M).Theplane-section theory,thatallowsdeterminingthelongitudinalstrainsateach fibre as a function of the generalized strains of the section, is coupled with a constant shear stress constraint along the cross-section.Filamentsoflongitudinalreinforcementareonly submittedtoaxialstrainsandstresses,followingtheplane sec-tion theory. Transverse reinforcement (internal steel stirrups and/orEBFRP)isaccountedthroughitsvolumetricratioρstand

issubmittedtoaxial stressesσzst.Compatibilityrequirements

imposethattheverticalstrainεzinconcreteisequaltothestrain

inthetransversereinforcement.Thecomputedshearstressesτxz

mustequatetheimposedshearstressesgivenbythefixedstress constraintτ*ofthesectionalhypothesis.Byguaranteeingthese tworequirements,theverticalaxialstrainεzandshearstrainγxz

ofeachfibreareoutputted.Thisdeterminationisnotlinearand aniterativeprocedurewithinthefibrelevelisneeded.

Pertainingtothematerialsimulation,asmearedandrotating crack approachis consideredfor concrete. Then,the cracked

concreteissimulatedasacontinuousmaterialwithorthotropic nonlinear uniaxial equivalent characteristics considering full rotation of cracks.The Hognestad parabola isconsidered for concreteincompression. Lateraleffectsofsoftening[11]and strength enhancement [12] factors are included. When FRP strengtheningisplacedbymeansofawrappedconfiguration,the incrementofbothpeakstrengthandultimatestrainofconcrete duetotheconfinementactionisconsideredthroughthemodel ofSpoelstraandMonti[13].Alinearresponseisassumedfor uncracked concrete intension and a tension stiffening curve

[14] is considered for the remaining stresses in the cracked stage. Longitudinal and transverse reinforcements (steel and FRP)areunder1Dstress–strainstatesdeterminedthrough lin-earuniaxialconstitutiveequations,withkinematichardeningfor steel.Perfectbondisassumedbetweentheconcreteandthesteel reinforcement.

2.2. Debondingfailure

As experimentally observed in existing experimental pro-grammes, U-shaped and side-bonded configurations of FRP usually failduetodebonding afterthe formationof acritical shearcrack.Forshearstrengthening,thedebondingfailure ini-tiates oncethe shearcritical crackopens. Then, the laminate debondsiftheFRPbondedlengthfromtheshearcracktothe laminateendisnotenoughtoanchorortransferthetensileforce actingontheFRP.Intheside-bondedcase,debondingcanbe observedatbothsidesofthecriticalshearcrack.IntheU-shaped case,debondingoccursintheuppersideoftheshearcrack.

Thedebondingfailureapproachimplementedinthepresent model is that proposed by Olleret al. [15]. This model was originally developedtopredict debondingfailureatthe lami-nateendinbeamsexternallystrengthenedbyFRPinbending. Thisformulationcanalsobeappliedwhenpredictingdebonding forFRPshearstrengthening.ForU-shapedconfigurations,the bondedlengthLbof eachstripisthebondedlengthabovethe

criticalshearcrack.Forside-bondedconfigurations,thebonded lengthofeachstripistheminimumlengthofthelaminateabove orbelowthecriticalshearcrack.

ThemaximumtransferredforceFmaxalongthebondedlength Lb,canbeexpressedas[15]:

Fmax,Lb=bf2GfEftf

⎧ ⎪ ⎨ ⎪ ⎩ sin _π 2 Lb Llim

Lb≤Llim

1 Lb>Llim

(1)

Llim=

π

2

2GfEftf

τLM

(2)

wherebfistheFRPwidth;tfistheFRPthickness;EfistheFRP

modulus ofelasticity;τLM isthemaximumshearstressatthe

interfacegivenbyEq. (3);Gfisthefractureenergyor energy byunitareanecessarytoseparatethelaminatefromthesupport givenbyEq.(4).UnitsareinNandmm.

τLM =CτLM

Figure1.Shear-sensitivefibrebeammodelforFRPshearstrengthenedelements.

Gf =Cτ2LMCFfctm (4)

wherefcmisthemeanvalueofconcretecompressivestrength; fctm is the meanvalue of concretetensilestrength; CτLM is a

constantequalto0.87;andCFisaconstantequalto0.35. Fig.2summarizestheinputhypothesisconsideredinthe sec-tionalmodel,theoutputresultsandthecriteriaforcheckingFRP debondingfailure.Thegradientsofverticalstressesbetweenthe borderandtheshearcriticalfibrearecomputedtobecompared withthemaximumtransferforce.Theshearcriticalfibreis con-sideredtobe located at3/4·h being hthe height of the cross section.Thestressatthisfibreisconsideredthecriticaltensile

stressσzFRP (z=3/4·h).Thiscriterionisaconsequenceof the

basichypothesisofthemodel,resultingintohighershearstrains andhigherverticalstrainsinthemorecrackedareas[4].Since theverticalstressintheborderisnull,thegradientisequalto the tensilestressinthecriticalfibre. Thecritical stressinthe FRP,σzFRP(z=3/4·h),iscomparedwiththemaximumvertical

stressthatcanbetransferredtotheFRP,σmax,deb,givenbyEq. (5),thatcorrespondstothemaximumtransferredforceFmax,Lb.

WhenthestressesσzFRPintheFRPlaminateinthecriticalfibre

reach the maximumallowedstressthat canbe transferredby bondingmechanism,theareaoftheFRPreinforcementofthat cross-sectionissettozero,andtheanalysismaycontinuewith

redistributionofforcesintheremainingsteelstirrupsandFRP sheetsinothercrosssections.

σmax,deb=

Fmax,Lb

bftf

(5)

3. Numericalanalysisoftheexperimentalprogramme ofAlzate[7]

AnexperimentalprogrammeonFRP-shearstrengthenedRC beamswascarriedoutbyAlzate[7]withthepurposeofstudying thecontributionofFRPtotheshearresistanceofRCelements. Thebeamsweresimplysupported,4.5mlongandwitha rect-angularcrosssectionof0.42mheightand0.25mwidth.ARC beamcriticaltoshear(controlbeam)wasstrengthenedwith dif-ferentsolutionsofFRPintermsofconfigurationandquantity andtesteduntilfailure.Fromthesetofbeamstestedinthe exper-imentalcampaign,thispaperpresentsthenumericalsimulation ofsomeofthebeamsstrengthenedinshearwithverticalFRP strips.Aconcentratedloadwasappliedatadistanceof3times the totaldepth (a=3h=1.26m) from the support.Geometry, internalreinforcementandstrengtheningconfigurationsofthe specimensarerepresentedinFig.3.

BeamswerereinforcedwithFRPsheetsof300mmwidth pre-sentingtwodifferentthicknesses–S530representunidirectional fibres (530g/m2) withdryfibre thicknessesof 0.293mm and S330representunidirectionalfibres (300g/m2)withdryfibre thicknessesof 0.176mm– andtwodifferentconfigurations– wrappedandU-shaped.Thenamesofthetestedspecimensmean thefollowing:W90S530isthebeamwithwrappedS530FRP; U90S530isthebeamwithU-shapedS530FRP;W90S300isthe beamwithwrappedS300FRP;andU90S300isthebeamwith U-shapedS300FRP.ThefibresoftheFRPsheetsformedanangle of90◦

withrespecttothelongitudinalaxisandthesheetswere spacedat200mmfromedgetoedge.Thebeamswithwrapped

FRPstrengtheningpresentaductileshear-bendingrelated fail-ure withFRPrupture andcrushing of concretenear theload applicationpoint;incontrast,thebeamswithU-shaped config-uration presented abrittleshearfailuremechanism afterFRP debonding.Experimentaldataavailablein[7]includesvertical displacementsatmid-spanmeasuredandverticalstrainsin stir-rupsandintheFRPsheetsmonitoredbymeansofbondedstrain gages.Thelocationofthesensorsconsideredinthevalidation isrepresentedinFig.3.

Inthenumericalsimulationbeamelementswith0.1mlength, cross-section discretized into fibres of 0.005mheight, longi-tudinalreinforcementsimulatedwithsteelfilaments,boththe transversal steel and FRP reinforcement considered smeared withtheirrespectivequantitiesandmaterialproperties. Differ-entspecimensofeachtypeweretested(identifiedwith–a,–b

or–c)andalsosimulated;theonlydifferencebetweenthemis thecompressionstrengthofconcretefcm.

The materialpropertiesofconcreteandFRPconsideredin the model are listedinTable 1.Forthe beam withU-shaped configuration (U90), the parametersrelated tothe debonding failure criteria (τLM, Gf and tf) were determined as function

of fcm as shown inTable1.Forsteel reinforcement,

longitu-dinalandtransversal,thefollowingpropertieswereconsidered:

Es=200GPa,fsy=500MPa,fsu=580MPa,εsu=0.10.Loadwas

appliedincrementallyuntilfailure.

Theexperimentalandnumericalshearforcevs.deflectionat midspanarecomparedinFig.4forthetwobeamswithdifferent FRPconfigurations,wrapped(W90)andU-shaped(U90)andfor thetwoseries(S5meansseriesS530andS3meansseriesS300). Agoodagreementwasobservedbetweentheexperimentaland numericalresults.

Table2givesthenumericalresultswhichshowagood agree-mentwiththeexperimentalresponseintermsofultimateload andalongthenonlinearpathwithincreasingload.Themodelis abletopredictacorrectfailureloadofthebeamswithU-shaped

ø20 ø10

250

42

0

FRP

tfX300mm//500

2250

300 200

200 300

W90 (S530 & S300)

ø8//380

8-10 3-7

1-5

8-10 3-7

1-5

50

210

ø20 ø10

250

420

FRP

tfX300mm//500

2250

300 200

200 300

U90 (S530 & S300

ø8//380

8-10 3-7

1-5

50

210

8-10 3-7

1-5

350 380

530

350 380

530

Table1

Materialpropertiesemployedintheanalysis.

Tests Concreteproperties Bondproperties FRPstrengtheningproperties

fcm,MPa Ec,GPa fctm,MPa εc,u,%o Gf,MPa.mm τLM,MPa tf,mm ρf Ef,GPa εf,u,%o ff,u,MPa

U90S5-a 36.95 32.56 3.33 3.5 0.717 2.24 0.293 0.0088 240 15.0 4000 U90S5-b 28.01 29.97 2.77 3.5 0.596 1.85 0.293 0.0088 240 15.0 4000 U90S3-a 20.50 27.29 2.25 3.5 0.484 1.49 0.176 0.0053 240 15.5 3800 U90S3-b 22.58 28.09 2.40 3.5 0.516 1.59 0.176 0.0053 240 15.5 3800 U90S3-c 28.01 29.97 2.77 3.5 0.596 1.85 0.176 0.0053 240 15.5 3800 W90S5 49.90 34.98 3.90 18.0 Nodebondingcheck 0.293 0.0088 240 15.0 4000 W90S3-ab 37.00 32.58 3.33 13.0 Nodebondingcheck 0.176 0.0053 240 15.5 3800 W90S3-b 37.00 32.58 3.33 13.0 Nodebondingcheck 0.176 0.0053 240 15.5 3800

0 50 100 150 200 250 300 350

60 50 40 30 20 10 0

Shear

(kN

)

Displacement (mm)

Exp W90S3-a Exp W90S3-b Num W90S3-ab Exp U90S3-a Num U90S3-a Exp U90S3-c Num U90S3-c Debonding FRP

0 50 100 150 200 250 300

40 30

20 10

0

Shear

(k

N)

Displacement (mm)

Exp W90S5 Num W90S5 Exp U90S5-a Num U90S5-a Exp U90S5-b Num U90S5-b DebondingFRP

a

b

Figure4.Shearforcevs.displacementatmid-span:(a)SeriesS530;(b)SeriesS300.

Table2

Summaryofexperimentalandnumericalresultsatfailure.

Tests Experimentaldata Numericalresults

Failure Debondfailurerelatedresults

Pu(kN) Vu(kN) Failuremode Pu(kN) Vu(kN) Failuremode Pu,num/Pu,exp Vu,deb(kN) Vu,deb/Vu ␴_zFRP(MPa) ␴_max,deb(MPa)

U90S5-a 341 247 DS 341 241 DS 1.00 240 0.99 1109 1084 U90S5-b 326 236 DS 315 223 DS 0.97 222 0.99 991 988 U90S3-a 285 207 DS 263 186 DS 0.92 186 1.00 1151 1149 U90S3-c 320 232 DS 311 219 DS 0.97 219 1.00 1332 1275

W90S5 383 276 BS 402 284 BS 1.05 NP – – –

Figure5.Shearforcevs.straininthetransversereinforcement:(a)wrapped;(b)U-shaped.

FRPconfigurationsbecauseitaccountsfordebondingfailure. Laminate debonding failure in the U-shaped beam occurs before FRPreaches itsmaximumstrength (3800–4000MPa); ascanbeseeninthevaluesofσzFRPforthedebondinginstantin Table2.Whenthisvalueexceedsthemaximumstressallowed tobe transferred, σmax,deb,the debondingmechanism occurs,

setting the FRP area of the cross section to zero. From this pointforward,thisFRPelementceasesitscontributiontothe structuralresponse.ForalltheU-shapedstrengthened beams, themodel predictsfailurerightafterdebonding occurs,being not able toredistribute the forces;this isconsistent withthe experimental observations [7]. The beams with the wrapped configurations fail when FRP reaches the ultimate capacity; hencepresentinghigherultimateloadcarryingcapacities,which iscorrectlycapturedbythemodel.Theseresultshighlightthe importance of accounting for the bond failureof FRPinthe analysis of shear strengthened elements with U-shaped and side-bondedconfigurations;disregarding debondingmaylead tounsafepredictionsof ultimateloadcapacityas thistypeof bondisthecriticalfailuremode.

Thecomputedstrainsinthetransversalreinforcement(inner steel stirrups and EB FRP) with increasing shear force are

comparedwiththe experimentalmeasurements for thebeams withdifferentFRPstrengtheningconfigurations(Wrappedand U-shaped).Fig.5presentstheresultsofseriesS530forlocation of the sensorssee Fig.3.Only onespecimenof each typeis represented;theotherspecimenspresentedsimilarfittings.

Despitethedifficultyofthiscomparison,duetothediscrete formof therealcracksandtheassumptionofsmeared crack-ing bythemodel,agoodconsistencybetweennumericaland experimentalresultscanbeobserved.Theloadlevelforwhich thestirrupsandtheFRPreinforcementstarttocarryloadiswell capturedbythemodel.Thisloadlevelcorrespondstothe out-setofdiagonalcracking.Sensors1and5locatedinthebottom ofthebeamcanbemoreinfluencedbybendingcracking,and hence,ofmoredifficultcomparison.However,ingeneral,itcan beobservedthatthemodelisabletocapturetheoverallresponse ofthetransversereinforcement.

0 50 100 150 200 250 300 1200 1000 800 600 400 200 0 –200 Sh ea r (k N)

Stresses in transversal reinforcement (MPa) Stirrups FRP U90 S530 (L)

Debond 0 50 100 150 200 250 300 a b 1400 1200 1000 800 600 400 200 0 –200 Shear ( kN )

Stresses in transversal reinforcement (MPa) Stirrups FRP W90 S530 (L)

Yielding _Failure

0 50 100 150 200 250 300 2000 1500 1000 500 0 –500 Shear (k N)

Stresses in transversal reinforcement (MPa) Stirrups FRP W90 S300 (L)

Yielding Failure 0 50 100 150 200 250 1200 1000 800 600 400 200 0 –200 Sh ea r (k N)

Stresses in transversal reinforcement (MPa) Stirrups FRP U90 S300 (L)

Debond Yielding

Figure6.Stressesinthetransversereinforcement(steelandFRP):(a)wrapped;(b)U-shaped.

loadlevels. After the occurrence of debonding failure, shear stressesaretransferredforthesteelstirrupsthatwherealready yieldedatthisstageleadingtoitsfailureandconsequentfailure ofthebeaminshear.Forthewrappedconfiguration(Fig.4a), theFRPsheetscontinuetocarryloaduntilfailureoftheFRP; i.e.,theloadcarryingcapacityoftheFRPisnotlimitedbythe lossofbond.Inthesegraphs,theyieldingofstirrupsinstantis alsomarked;itcanbenoticedthat,beforethispoint,FRPand steel stressesaresimilar;after yielding,steel cannotincrease theloadcarryingcapacityandhence,theFRPincreasestheir stressessignificantly.Thisisobservedinbothcases,inwrapped andU-shapedconfigurations.

4. Conclusions

Thispaperdescribestheextensionofafibrebeammodelto simulateEB FRPshearstrengthening systems, considering a possibleFRPdebonding failureinside-bondedandU-shaped configurations.Experimentaltestsavailable inliteraturewere numericallysimulated.Fromtheseanalysesthefollowing con-clusionsaredrawn:

- Themodelisabletocorrectlycapturetheload-displacement response of the strengthened beams with wrapped and U-shapedconfigurations;

- Themodelcapturedtheoverallresponseofthetransverse rein-forcement (innersteel stirrupsandEB FRP), capturing the debonding of FRPandsubsequent failureof thebeams for theU-shapedconfigurations;

- WhendebondingfailureoccursandFRPceasesits contribu-tiontotheshearresistance,stirrupswerealreadyextensively yieldedandwerenolongerabletoabsorbtheredistribution offorces,andfailureoccurredrightafter.

- Thecomputationalandmodellingsimplicitymakesitsuitable torealscalepracticalapplications

Acknowledgments

Theauthorswanttoacknowledgethefinancialsupport pro-videdbytheSpanishMinistryofEconomyandCompetitiveness (MINECO)andtheEuropeanFundsforRegionalDevelopment (FEDER), through Research projects: BIA2015-64672-C4-1-RandBIA2015-64672-C4-3-R.Theauthorsacknowledge the supportofAlbertAlzate,AngelArteaga,DanielCisnerosand AnadeDiegofromtheInstitutodeCienciasdelaConstrucción EduardoTorrojaofSpain,ontheprovideddatarelatedtotheir experimentalprogramme.

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