Modelling Unreinforced, TRM‐strengthened and FRP‐strengthened brick masonry walls subjected to eccentric compressive load

ModellingUnreinforced,TRM‐strengthenedandFRP‐strengthenedbrick masonrywallssubjectedtoeccentriccompressiveload Bernat‐Maso,Ernest1;Gil,Lluís2;Roca,PereRoca3 ABSTRACT Themixedbending/bucklingcollapsemodeofbrickmasonrywallsmightbepreventedbyinstalling strengtheningsystemswhichprovidetensilestrength.Amongothers,theTextileReinforcedMortar (TRM)andFibreReinforcedPolymers(FRP)laminatesarethemostcommonsolutions.Theirstructural responseundersecondorderbendingeffectsisstudiedbymeansofimplementinganumerical simulationadaptableforunreinforcedandstrengthenedwalls.Theproposedmodelisbasedonabidimensionalsimplificationandtheuseofsimplifiedmicromodellingtechniques.Theuseofcontact elementshasprovedtobeessentialatsimulatingthistypeofstructures.Theobtainedresultsare comparedwiththeexperimentalevidenceson44real‐scaletests,concludingthattheimplemented numericalmodelisaccurateatpredictingtheload‐bearingcapacityofmasonrywallssubjectedto eccentriccompressiveloads. Keywords:Simplifiedmicromodel;Masonrywalls;FibreReinforcedPolymer;TextileReinforcedMortar; CohesiveZoneModel. 1. INTRODUCTION Thestructuralresponseandfailuremodesofload‐bearingbrickmasonrywallsischaracterisedbythe almostnegligibletensilestrengthofthismaterial.Amongthepossiblecollapsemechanisms,the buckling‐bendingmixedfailureissignificantbecauseofitssuddendevelopment.UsingFibreReinforced Polymer(FRP)orTextileReinforcedMortar(TRM)havebeenprovedtobeeffectivestrengthening solutionstoprovidetensilestrengthandavoidthisfailuremodeinpreviousresearches[1,2]. StrengtheningmasonrystructureswithFRPlaminatesisacommonactuationamongtheconservation andmaintenancetaskscarriedoutonmasonrybuildingswhichrequirestructuralupgradingor rehabilitation.TheFRPisspeciallyindicatedforthoseinterventionswhichrequirelimitingthelateral deflectionofthewallsandlookforalessflexibleout‐of‐planeresponseofthestructurebecauseofits outstandingstiffness.However,thedrawbacksofthisstrengtheningsystem(watervapour impermeabilityandinsufficientmechanicalcompatibilitywiththemasonry),whichhavebeenpointed outbyseveralauthors(see,forexampletheworkbyBarattaetal.[3],orthearticlesbyPapanicolaou 1 Departmentofstrengthofmaterialsandengineeringstructures.PolytechnicUniversityofCatalonia(SPAIN). [email protected](Correspondingauthor) 2 Departmentofstrengthofmaterialsandengineeringstructures.PolytechnicUniversityofCatalonia(SPAIN). [email protected] 3 DepartmentofConstructionEngineering.PolytechnicUniversityofCatalonia(SPAIN). [email protected] 403 Modelling Unreinforced, TRM-strengthened and FRP-strengthened brick masonry walls subjected to eccentric compressive load Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. etal.[4,5]),mightmakeothertechniquesmoresuitableforgeneralstructuralinterventionson masonry. ThisisthecaseoftheTRM(alsocalledFabricReinforcedCementitiousMatrix,FRCM),whichisa compositematerialconsistingofatextilegridembeddedintoaninorganicmatrix.Thistextilegrid shouldbemadeofahighstrengthmateriallikeglass,basaltorcarbonfibres,whereastheinorganic matrixistypicallyacementorlimemortar.Itisthephysicalpropertiesoftheseinorganicmatrixeswhat allowsbetterphysicalcompatibilitywithmasonrysubstratesthanotheradhesivesused,forexample, intheFRPsolutions.Inthisstudyfield,theexperimentalstudiesabouttheuseofTRMatstrengthening masonrywallshavebeenfocusedonanalysingthestructuralresponseagainstin‐planeloads[6,7]or out‐of‐planecyclicloads[8].However,thesecondorderbendingeffectsonTRMstrengthenedmasonry walls,whichmightbecrucialinnon‐seismicareas,hasdeservedlessattention,andafewcontributions, like[9],areavailable. Numericalmodellingtechniquesprovideaninterestingandinexpensivetoolfordesigningandassessing structures.Nevertheless,littlebibliographyonthenumericalsimulationofunreinforced,TRM‐ strengthenedorFRP‐strengthenedbrickmasonrywallsisavailableforthecaseofeccentric compressionloadingconfiguration.Aboutthenumericalmodellingofunreinforcedmasonrywalls, differentfiniteelementapproacheshavebeenproposedfromsimplifiedone‐dimensional(1D)beam descriptionsapproaches(see[10])tothree‐dimensional(3D)models,liketheoneproposedbyMartini [11].Betweenthesetwoapproaches,thesophisticatedmicro‐modelspresentedbyLourenço[12]have tobehighlighted.Asacompromisebetweencomputationalcostandresult’saccuracy,thereisa growingtendencytoworkinhomogenised2Dmodelstorepresentmasonrystructures.Therequired homogenisationtechniqueshavebeenstudiedbymanyresearchers([13,14]). RegardingthenumericalmodellingofTRM‐strengthenedmasonrywalls,thereareonlyafew publicationsaboutthesimulationofthesetypeofstructures,likethepreviouswork[15].Similarly, researchesdealingwiththenumericalsimulationofthestructuralresponseoftheTRMasasole materialhavebeenrecentlypublished[16],whereasotherresearchershavestudiedhowtomodelthe TRM‐masonryinteractionusingmasonryprismsasreferencesamples[17].Tosomeextent,the modellingofTRMreinforcedconcretecanbeconsideredasareferencecaseandmayprovidesome insightatmodellingTRMreinforcedmasonrystructures.Inthisline,someauthorshavedealtwiththe numericalmodellingTRM‐strengthenedreinforcedconcretebeamssubjectedtoflexuralloading conditions[18],otherhavebeenfocusedontheshearstrengthening[19]. Finally,therearenotknownresearchesaboutthenumericalsimulationofFRP‐strengthenedmasonry wallssubjectedtoeccentriccompressiveloadswhichtakeintoaccountthesecondorderbending effects.However,otherstructuralconfigurationshavebeenmodelled,likethein‐planeresponse[20] ortheout‐of‐planebehaviour[21]. Tosumup,thereislittleresearchaboutthenumericalmodellingofmasonrywallssubjectedto eccentriccompressiveloads.Inaddition,thereareevenlessevidencesiftheconsideredwallsareTRM‐ strengthenedandfinally,therearenotknownstudiesaboutthefiniteelementcalculationofFRP‐ strengthenedmasonrywallssubjectedtoeccentriccompressiveloads.Thus,theaimofthecurrent 404 Ernest Bernat-Maso1, Lluís Gil2 and Pere Roca3 researchistomodelthementionedcasesandtocomparetheobtainedresultstoenhancethe understandingabouttheeffectofstrengtheningmasonrywalls. 2. METHODOLOGY Themethodologyconsistedofimplementinganumericalmodelforpredictingtheload‐bearingcapacity ofunreinforcedbrickmasonrywallssubjectedtoeccentriccompressiveloads(see[22]).Then,this modelwasadaptedtotakeintoaccounttheeffectoftheTRM(see[15]),whichiscontinuously distributed,andthecontributionoftheFRP,whichisappliedasdiscretelaminatesfollowingdifferent orientations.Itisthislastadaptationprocedure,theonewhichisanalysedwithmoredetailinthis paper.Thethreeresultingnumericalmodelsaretobeunderstoodasthesameapproachthatevolved soitissignificanttodescribe,compareanddiscusstheadaptabilityofthemodeltothedifferent structuralconfigurations.Comparisonwithexperimentalresultsisalsoprovidedtoassesstheaccuracy ofthemodel. 2.1. Samples Forty‐fourfullscalebrickmasonrywallsweretesteduptofailureduetoaneccentricallyapplied compressiveload.Allwallsweretestedinapinned‐pinnedconfigurationwiththeaimofclearly identifyingtheboundaryconditionsandtoeasethecomparisonwiththenumericalmodels.Twentyof thesewallswereunreinforced(W#1‐W#20),ninewereTRM‐strengthened(W#21‐W#29)andfifteen wereFRP‐strengthened(W#30‐W#44).Inthecorrespondingcases,thestrengtheningsystem(TRMor FRP)wasplacedonthesideofthewallsubjectedtotensilestresses.Themaingeometriccharacteristics ofthesestructuresandthecorrespondingtypologyofstrengtheningaresummarisedinTables1,2and 3. Thefirstfivecolumnsofthesetablescorrespondtothenameofthewall,thewidthofthewall,b, theeffectiveheightofthewall,Hef(measuredinverticaldirectionbetweenthetwohingesusedtoset thepinned‐pinnedconfiguration),thethicknessofthewall,t,andtheeccentricityoftheloadatmidheightatthebeginningofthetest,e0.Thelastcolumnpresentsthedimensionlessmeasurementofthe load‐bearingcapacityofthewall,ɸ,calculatedastheexperimentalmaximumappliedforceoutofthe maximumcompressiveforcethatacrosssectionofthewallcouldbearunderuniformstress distributionhypothesis.Fortheunreinforcedcases(Table1),thecompressivestrengthofthemasonry, fc,anditsflexuralstrength,fx,arepresentedinthesixthandseventhcolumnsrespectively.TheYoung’s modulusofthemasonrywas780MPaforunreinforcedandTRM‐strengthenedwallsandtheaverage densityoftheusedmasonrywas1732kg/m3forallcases.FortheTRM‐strengthenedcasesthe compressivestrengthofthemasonrywas10.8MPa,itsflexuralstrengthwas0.36MPa(see[2]).Thelast fivecolumnsinTable2arededicatedtocharacterisetheTRMandtheyincludethetypeof strengtheningmortar(MforthePortlandbasedmortar,Rforthelime‐basedmortarandXforthe pozzolanicmortaraspresentedin[2]),thetypeoffibregrid(GforglassandCforcarbon),thepresence ofconnectors,thenumberoffibregridsinstalledintothemortarlayerandtherealthicknessofthe TRMlayer. Finally,thestrengtheningpatternsoftheFRP‐strengthenedwallsaredetailedinTable3.Columnssixth toninthpresentthenumberofverticallaminatesuniformlydistributedonthewidthofthe strengthenedface,#v,FRP,thenumberofhorizontallaminatesuniformlydistributedontheheightof thestrengthenedface,#h,FRP,andthenumberofinclinedlaminatesplacedatthesamesection– constantheight–,#i,FRP,consideringtwodirectionsatanangleof±α(°). 405 Modelling Unreinforced, TRM-strengthened and FRP-strengthened brick masonry walls subjected to eccentric compressive load Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. Table1.Geometryoftheunreinforcedbrickmasonrywalls Wallb(mm)Hef(mm) t(mm) e0(mm) f c(MPa) f x(MPa)Ф(%) W#19002,947132 5.6 18.2 0.23 7.8 W#29002,927132 19.6 12.9 0.36 4.3 W#38702,922132 1.2 8.7 W#48701,980132 7.0 37.8 W#59002,917132 33.7 15.6 W#68752,857132 13.1 13.7 1.8 W#78852,872132 ‐7.6 8.3 W#88792,942132 21.5 8.0 W#98882,892132 0.2 6.7 W#108881,865132 13.9 10.8 33.0 W#118701,892132 1.6 28.9 W#128821,841132 31.8 36.7 W#139001,861132 21.2 6.5 W#149001,861132 29.0 40.4 W#158951,845132 19.5 18.4 W#168941,860132 88.0 31.8 W#178941,905270 20.8 38.3 W#188711,197132 30.3 62.6 W#198711,220132 33.0 53.5 W#208681,865132 5.6 11.9 Table2.GeometryoftheTRM‐strengthenedbrickmasonrywalls Wallb (mm) Hef (mm) t (mm) e0 (mm) TRM mortar Fibre grid Connectors #fibre grids tTRM (mm) Ф (%) W#218651,83213230.3 M G 0 113.024.3 W#228721,82713233.2 M G 0 18.026.4 W#238681,82213233.9 R G 0 19.521.9 W#248671,84013225.4 R G 0 19.023.1 W#258681,82813232.7 R G 0 27.533.5 W#268691,82313229.7 M G 0 28.031.5 W#278731,82213230.4 X C 0 18.027.8 W#288711,82813223.7 X C 6 19.025.2 W#298681,82713231.4 X C 9 111.026.7 Threefailuremodeswereexperimentallyobserved:themixedbending/bucklingfailureofmostofthe unreinforcedbrickmasonrywallsduetothemechanismformationcollapse(amasonryjointopened), thetensilefailureoftheTRMassociatedwithamechanismformationprocess(onlyforW#21and W#23)andtheshear/compressivefailureneartheendingsofthewallfortherestofthestrengthened walls.Therealgeometricimperfectionsweremeasuredandintroducedintothenumericalmodel throughthegeometrydefinitionlateron.Themeasurementprocessconsistedondeterminingthe distancebetweenaverticallyalignedprofileandthewallwhenitwasplacedinthetestingposition.A lasersensorwasusedforthispurposeandmeasurementswerecarriedouteverytwomasonryrows atthetwolargestedgesofoneofthetwobiggestfacesofthewalls.Thesemeasurementsallowed determiningtheshapeofthewall. 406 Ernest Bernat-Maso1, Lluís Gil2 and Pere Roca3 Table3.GeometryoftheFRP‐strengthenedbrickmasonrywalls Wallb(mm)Hef(mm)t(mm) e0(mm) #v,FRP #h,FRP #i,FRPα(˚) Ф (%) W#308371,798126.5 29.5 200 ‐‐‐ 27.3 W#318401,815125.9 31.9 28.6 W#328381,798126.1 29.8 28.9 W#338291,811126.4 29.7 300 ‐‐‐ 31.5 W#348331,814126.3 31.2 31.5 W#358371,824126.3 29.4 39.2 W#368431,788126.1 29.4 220 ‐‐‐ 31.3 W#378351,792124.4 32.7 31.9 W#388271,797124.7 30.1 33.1 W#398331,777123.8 28.7 350 ‐‐‐ 29.1 W#408301,784124.7 27.2 27.9 W#418311,761124.5 27.6 24.1 W#428261,830124.3 26.8 00461 34.5 W#438311,831124.6 28.8 24.0 W#448271,812123.8 27.4 30.7 2.2. Numericalmodelforunreinforcedcases Theimplementedmodelsassumethehypothesisofplanestrainappliedtosimplifythethreedimensional(3D)realcaseintoabi‐dimensional(2D)model.Inaddition,simplifiedmicromodellinghas beenusedbecausethisapproachdoesnotrequiresomuchinformationofthematerialpropertiesas detailedmicromodellingbutperformsbetterthanmacromodels.Theproposedmethodmodelseach bricktogetherwiththesurroundingmortarasahomogenisedmaterial,calledmasonry,andusesthe correspondingexperimentalproperties.Finally,thenumericalmodelconsiderslargedeformationsto accuratelyrepresentthebucklingphenomenaoriginatedbythesecondorderbendingeffects.The definitionofmaterials,contactsandgeometryhavebeensetasgeneralaspossibletoassurethatthe modelcanbereproducedingeneralpurposeFiniteElementAnalysis(FEA)packages,liketheusedone: ANSYS®. 2.2.1. Geometry Thegeometricmodelisdefinedasaseriesofpiledrectangularpartscorrespondingtothedifferent masonryrowsexistingintheanalysedwall.Twoextrabodies,whicharetwotriangles,oneateachend ofthewall(seeFigure1),aredefinedtoeasilyrepresentthehingesofthepinned‐pinnedconfiguration consideredforallcases.Thesetrianglesallowplacingtheloadapplicationpointaccuratelyandsetting therotationaxes.Inaddition,therealshapeofeachwallisusedtocreatethecorrespondingmodel, takingintoaccountitsinitialout‐of‐planeimperfections,whichmightaffectthedevelopmentofsecond orderbendingeffects. 2.2.2. Materialsandcontacts Themechanicalresponseofthebrickmasonryisorthotropicbecauseofthepatternofthejoints.In addition,dependingonthepropertiesofthecomponents(units,mortarandinterfaces)ofthis compositematerial,themasonryshowsdifferentfailuremodes.Thiscomplexresponseisnot consideredtoavoidusingadvancedsimulationtools,whichwouldbeagainstthefirstaimofthe implementedmodels:keepingthemassimpleaspossible. 407 Modelling Unreinforced, TRM-strengthened and FRP-strengthened brick masonry walls subjected to eccentric compressive load Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. Figure1.Geometricdefinitionofageneralwal l Thus,masonryismodelledasahomogeneousmaterial,whosecompressiveresponseisdefinedwith anisotropiclinearelasticwithperfectplasticityconstitutivemodel.Thus,onlythecompressivestrength andtheYoung’smodulusarerequiredtocharacterisethecompressiveresponseinthemodel.Onthe otherhand,thetensionalbehaviouriscontrolledbycontactelements,whichhaveneitherthickness norelasticproperties.ThecurrentsteeldefinitionincludedintheFEAsoftwareisusedtomodelthe materialofthepartscorrespondingwiththehinges. Apartfromtheexperimentallydeterminedpropertiesofthemasonrypresentedbefore(seesection 2.1),thePoisson’scoefficientissetto0.35.Thisassumptionisbasedonbibliographicevidenceslikethe workbyBosiljkovetal.[23],whoproposedvaluesofPoisson’sratioupto0.4whenusingmortarwith lowcontentofcement.Infact,accordingtotheseauthors,thehighestPoisson’svaluescorresponded withthelowestvaluesofYoung’smodulus. Thetensileresponseofthemasonryismodelledusingcontacts,whichareplacedbetweenmasonry rows.Thesecontactsare“bonded”atthefirstcalculationstep,meaningthatthecorrespondingnodes oftheadjacentpartsaresettomoveequallyandtogether.Thestressincreaseoneverycontact,which isassociatedtothestepbysteploadingprocessdescribedinsection2.2.4,mightcausetheopening (debonding)ofthesecontactsaccordinglywithabilinearconstitutivelaw.Thisresponseisimplemented usingacohesivezonemodel(CZM)thatwasoriginallyproposedbyAlfanoetal.[24].TheCZMsetsan elasticbehaviouruptothebrittlefailureofthecontact.Whenthetensilestrengthisreached,the contactopensandthecorrespondingstressdecreaseslinearlywithaslopethatisdefinedbythe correspondingfractureenergy.Thus,onlytwoparameters(tensilestrengthandthecorresponding 408 Ernest Bernat-Maso1, Lluís Gil2 and Pere Roca3 fractureenergy)arerequiredtouseCZManditisbelievedthatthisisthesimplestandmostgeneral waytorepresentthetensileresponseofthemasonry. Theflexuraltensilestrength,fx(summarisedinTable1),whichwasexperimentallydeterminedcarrying outbond‐wrenchtests,isusedinsteadofthedirecttensilestrengthintheCZMdefinition.Itisbecause theflexuraltensilestrengthisconsideredtobemorerepresentativeoftheresponseofthemasonry underthesecondorderbendingphenomena.Infact,thecontactsbetweenmasonryrowscanbe bendedorcompressedbutnotsubjectedtoatensileaxialeffortundertheconsideredloading configuration.Inaddition,thefractureenergy,GfI,isestimatedusingEq.(1).Thisexpressionhasbeen adjustedusingtheexperimentalresultspresentedin[25]andwaspreviouslyusedin[26,27]. 󰇛 ⁄󰇜   36.65   󰇛󰇜 (1) Theslidingfailurebetweenmasonryrowsisnotconsideredbecauseitwasnotexperimentally observed.Finally,aslongasthetestedwallsandthesteelhingeswerealwaysincontactduringthe experiments,thedebondingprocessisnotconsideredinthesecontacts. 2.2.3. Mesh Figure2showsatypicalmesh.Masonryismeshedwith8‐nodesuniformstructuredquadrilateral elementswithquadraticintegrationofthedisplacements.Theaveragesizeoftheelementsis5mmfor themasonryofthewallswithaneffectiveheightbelow2m,and20mmforthetallerwalls.Theparts representingthesteelhinges(triangles)aremeshedwithanunstructureduniformmeshcomposedby 6‐nodestriangularelementsand8‐nodesquadrilateralelementswithquadraticintegrationofthe displacementsandanaveragesizeof5mm. Figure2.FiniteElementmeshandpositionofthecontactsforstrengthenedwalls 2.2.4. Boundaryconditionsandloadingprocess Theboundaryconditionsofthemodelconsistedoffixingthedisplacementofthelowerhingeby restrainingthehorizontalandverticalmovementsofthelowestvertexofthecorrespondingtriangular. Additionally,thehorizontaldisplacementofthetopvertexofthetriangularpartrepresentingtheupper 409 Modelling Unreinforced, TRM-strengthened and FRP-strengthened brick masonry walls subjected to eccentric compressive load Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. hingeisfixed.Theseconstrainsallowthestructuretorotatearoundtherestrainednodeswhile impedinganypossiblelateraldisplacementatthewall’sends. Theloadisindirectlyappliedasaverticaldescendingdisplacementofthetopvertexofthemodel.This movementisappliedataconstantrate(stepbystepcalculation)andtheforcereactionatthe applicationpointistakenasthemeasurementoftheappliedloadforcomparingwiththeexperimental values.Thisdisplacement‐controlledsimulatedloadingprocessmakesitpossibletocalculatesome stepsafterreachingthemaximumload,clarifyingtheinterpretationofthefailurepattern.The calculationisperformedstepbystepsoeachnewdisplacementincreaseisappliedonthedeformed configurationofthewallduetothepreviousloadingsteps. 2.3. AdaptationofthenumericalmodelfortheTRM‐strengthenedcases Thepreviouslydescribednumericalmodelforunreinforcedbrickmasonrywallswasupdatedtoinclude thepossibilityofcalculatingTRM‐strengthenedwallssubjectedtoeccentriccompressiveloads.This adaptationrequiredmodifyingthegeometry,thematerialsandcontactdescriptionsandthemesh definition.Onlythemodificationsaredescribedbelowsotherestofthedefinitionisthesamethanfor unreinforcedmasonrywalls. 2.3.1. Geometry TwomainchangesinthegeometrywerenecessarytorepresenttheTRM‐strengthenedwalls:a) consideringadditionalpartstomodeltheTRMlayerandb)dividingthemasonrypartsneartheendsof thewallsfollowinganinclinedlinetoallowthecompressive/shearfailureofthemasonryobservedin theexperimentaltests. TheTRMlayerwasmodelledusingrectangularbodiesplacednexttotheonesrepresentingthe masonry.TherewereasmanyTRMpartsasmasonryrowsandthecontactsbetweenthemweredefined toallowthetensilefailureoftheTRM,whichwasobservedintwoexperimentalcases(see[2]).The interfacesbetweenmasonrypartsandTRMpartswerealsodescribedbymeansofcontactelements. ThethicknessoftheTRMpartswasconsideredconstantalongthewall’sheightandequaltorealTRM thicknessatmid‐heightforeverywall. Inaddition,fictitiouscontactswereplacedinaninclinedlinecrossinganddividingthemasonrybodies neartheextremesofthewalls(seeFigure1)tomodelthecompressive/out‐of‐planeshearcollapse mode,whichwasexperimentallyobservedinmostoftheTRMstrengthenedwalls.Thisfailuremode wascharacterisedbytheopeningofonediagonalcrack,whichalwaysstartedfromoneextremeofthe strengthenedsideofthewallandgrewtothecompressedsideofthestructure.Thechosenorientation forthesedividinglineswassettobeconstantandcorrespondedtotheaverageinclinationofthe experimentalcollapses. 2.3.2. Materialsandcontacts ThecompressivebehaviouroftheTRMisgovernedbythemortarmatrix,whereasthetensileresponse isinfluencedbya)themortarmatrix(atlowloads,beforecrackingandmobilisingthefibremesh),b) 410 Ernest Bernat-Maso1, Lluís Gil2 and Pere Roca3 thefibregridandc)theadherencebetweenthemortarmatrixandthefibregrid(atlargerloads,after cracking).Thus,itisacomplexresponsewhichdependsondifferentparameters.However,asimplified materialmodelwasusedtorepresenttheTRMtoassureanefficientcalculationoftheloadbearing capacityoftheTRMstrengthenedwalls.Hence,TRMwasmodelledasanisotropicandhomogeneous materialincompression.Itscompressiveresponsewassettobelinearelasticwithperfectplasticity, whereasthetensileresponsewascontrolledbythecontactelementsdefinedusingtheCZM. ThecompressivebehaviouroftheTRMisdefinedbytheYoung’smodulus(Em)andcompressive strengththestrengtheningmortar(fcm),whosevaluesaresummarisedinTable4.Thefirstvariablewas experimentallydeterminedwhereastheYoung’smoduluswasprovidedbytheproducerofthemortar. Table4.PropertiesofthestrengtheningmortarsfortheTRMsolutions MortarFlexuralstrength, fxm(MPa) Compressivestrength, fcm(MPa) Bondingstrength, fbm(MPa) Modulusofelasticity, Em(MPa) M8.1(0.18) 42.2(0.27) 2*11000* R6.6(0.03) 14.5(0.08) 0.8*8000* X9.4(0.10) 34.5(0.08) 0.8*15000* Dependingonthelocationofthecontacts(seeFigure2),thesemightbeclassifiedintorealcontacts andfictitiouscontacts.Thefirstonescorrespondedwiththejointsofthemasonry(describedinsection 2.2.2)andtheinterfacebetweentheTRMandthemasonry.Incontrast,thesecondonesweresetin ordertoallowthedevelopmentoftheexperimentallyobservedfailuremodes.AllcontactsforTRM‐ strengthenedwallsaresummarisedinTable5. Table5.PropertiesofthecontactsfortheTRMwalls ContactWallsContactFailure f xt (MPa) GfI (N/m) (MPa) GfII (N/m) Real.Masonry‐TRM AllTRMNone ‐‐‐ ‐‐‐ ‐‐‐ ‐‐‐ Fictitious.Masonry‐Masonry (inclined)AllTRMDebondingand Sliding2.8*1000.5620 Fictitious.TRM‐TRM W#21 W#22 Debonding8.1295 ‐‐‐ ‐‐‐ W#23 W#24Debonding6.6240 W#25Debonding 12.0 440 W#26Debonding 11.3 412 W#27Debonding 20.0 733 W#28Debonding 17.8 652 W#29Debonding 14.5 533 *Thisvaluecorrespondstothedirecttensilestrength,nottotheflexuraltensilestrength. TherealcontactsbetweenmasonryandTRMpartsweredefinedas“bonded”,soneitheropeningnor slidingprocessesareallowed,whichcorrespondedtotheexperimentalobservations.Thefictitious contactsbetweenpairsofTRMpartsweredefinedtoallowtheopeningbutnottheslidingbetween TRMparts.Thus,theymaketheTRMtensilefailure(experimentallyobservedinwallsW#21andW#23) possible.Thistensileresponsefollowsabilinearconstitutivelaw,whichischaracterisedbythetensile strengthandthecorrespondingfractureenergy.Thefirstparameter,fxt,iscalculatedasthemaximum 411 Modelling Unreinforced, TRM-strengthened and FRP-strengthened brick masonry walls subjected to eccentric compressive load Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. toapplicabilitylimitations,combinedwiththepossibleunderestimationoftheYoung’smodulusofthe masonry.Inthisline,placingathickerlayerofadhesivemighthavecontributedtolimittheexperimental lateraldeformationofthetestedwallsbecauseitmeansintroducinganadditionalmaterialstifferthan masonry.Furthermore,theerrorofthenumericalmodelmightberelatedwiththedefinitionofthe materials.Inparticular,thecompressivebehaviourofthemasonryandthefractureenergiesof masonry,FRPandinterfacesbetweenthemmightbedefinedinamorerealisticwaytomodelthe experimentalresponseofthestudiedstructureswithmoreaccuracy.Regardingthecompressive responseofthemasonry,twoimprovementsarepossible:implementingaparabolicstress‐strain compressiveresponseandusingamorerepresentativevalueoftheYoung’smodule.However,thefirst improvementwillbeagainsttheaimofkeepingthemodeldefinitionassimpleaspossible.Thepossible underestimationoftheYoung’smodulusofthemasonryusedinthenumericalmodelmightexplainthe errorinthelateraldeformation.Inaddition,highervaluesoftheYoung’smodulusofthemasonrywould resultinlowerlateraldeformationsandabetteradjustmentoftheexperimentaldata.Incontrast, increasingtheYoung’smodulusofthemasonrywillderiveinalargerload‐bearingcapacity, accentuatingthecurrentoverestimationtrendofthemodel,soothervariablesseemtobenecessarily changed.Aboutthefractureenergies,itseemsnecessarytodevelopanalyticalapproachesto determineeachfractureenergyasfunctionofthecorrespondingstrengthamongotherpossible parameters.Thisimprovementwillprovideascientificallyjustifiedalternativetothefirstapproachof usingofthesamerelationshipforallfracturephenomena,whichhasbeenadoptedhereintoavoid providingarbitraryvalues.Infact,theuseddefinitionofthefractureenergiesmightexplainagreatdeal oftheobtainederrorandimprovingitwouldhelptoreachmoreaccurateresults.Anothersimplification whichmighthavecontributedtotheerrorofthenumericalmodelistheuniformlineardistributionof themechanicalpropertiesofthediscretestrengtheningsystem(laminates)alongthewidthofthewall toallowthebidimensionalhypothesis.Thisfactmightberelatedwiththeoverestimationoftheloadbearingcapacityofthewallsbecausetheusedsimplificationassumesauniformstressdistributionon theinterfacesalthoughitisknownthat,forexample,thetangentialstressdistributionisnotuniform andtherearesectionswithminorshearstrengththantheassumedone. Inaddition,fortheFRP‐strengthenedcases,thepredictedfailuremodesshowsagreementwiththe experimentalobservations.Inparticular,reachingthetangentialstrengthofthefictitiousinclined contactinthemasonryneartheendsofthewallispointedoutasthecollapsecauseofthestudiedFRP‐ strengthenedwallsinagreementwiththenumericalresults.Inaddition,ithastoberemarkedthatthe definitionofthecontactsorientedtorepresenttheeffectofthelaminatescrossingoverotherscauses aparticularstressdistributioninthemasonry‐FRPcontact(seeFigure6,right).Infact,thereisaslight discontinuityinthetangentialstressdistributionduetothebondedrestrainimposedatthepositions wherethelaminateswerecrossed(1/3and2/3ofthewall’sheightfortheW#38caserepresentedin Figure6right).Thisresultindicatesthattherestrainingeffectofthehorizontallaminatesmightleadto increasingthetangentialstressofthemasonry‐FRPcontactinpunctualareas,whichmightcause adherenceproblemsinthesimulatedmodel.Thus,installinghorizontalcrossinglaminatestoavoidthe debondingphenomenamightbecounter‐productivebecausethetangentialstressdistributionofthe verticalonesismodifiedincreasingthepossibilityofapunctualslidingfailurethatstartsaglobalfailure ofthelaminateadherence.However,thisultimateeffecthasnotbeenexperimentallyobservednor numericallypredicted. 418 Ernest Bernat-Maso1, Lluís Gil2 and Pere Roca3 Table7.ResultsofthenumericalmodelatcalculatingFRP‐strengthenedwalls Wallvmax/Hef(∙10‐3)hmax/t(%) ɸmax(%)  (kN) Abs.rel.err.(%)Av.err.(% ) W#305.230.033.6763.445.8 26.9 W#31 5.029.731.1707.028.9 W#32 3.116.025.9587.66.0 W#33 4.826.134.0761.827.3 18.8 W#34 4.624.733.1745.724.1 W#35 4.323.231.5713.45.0 W#36 3.317.226.8610.31.0 8.1 W#37 3.620.324.5545.09.3 W#38 4.828.431.9703.913.9 W#39 4.624.932.7722.132.4 36.5 W#40 3.618.530.7681.730.0 W#41 3.719.430.0665.646.9 W#42 4.222.133.8744.015.6 35.4 W#43 4.322.832.6722.059.8 W#44 4.423.234.0746.031.0 Comparingtheresultsofthefiniteelementanalysisamongthem,itisnoticedthatrestrainingthe masonry‐FRPcontactstorepresentthepresenceofhorizontallaminatescausesaload‐bearingcapacity decrease(comparingW#30‐32withW#36‐38andW#33‐35withW#39‐41).Thisisinlinewiththe previouscomment.Thus,accordingwiththemodel,andfortheproposedstructuralconfigurationand loadingpattern,includinghorizontalFRPlaminatesisnoteffective.Incontrast,themodelreproduces theeffectofinstallingthreeverticallaminates(W#33‐35)incomparisonwiththecaseswithtwo laminates(W#30‐32).Aminorload‐bearingcapacityincreaseispredictedwhenaddingvertical laminatesasexperimentallyobserved.Finally,ithastoberemarkedthattheaverageloadbearing capacitypredictedbythemodelforthecasesW#42‐44andW#33‐35arepracticallythesame.Thisfact supportsthehypothesisofassumingareductionfactor(¾)fortheFRPpropertiesassociatedwith inclinedlaminatesinthedescribedconfiguration. 4. CONCLUSIONS Theobjectiveofimplementinganumericalmodel,whichwascapableofpredictingtheload‐bearing capacityofUnreinforced,TRM‐strengthenedandFRP‐strengthenedwallssubjectedtoeccentric compressiveloadatalowcomputationalcost,hasbeenachieved.Theparticularadaptationforeach casehavebringspecificconclusions.Thus,forthesimulationofunreinforcedwalls,itmightbe concludedthatthenumericalmodelismoreaccurateforthecaseswithhigheccentricityandmore slenderness.Inaddition,theobtainedresultsemphasisethatarealisticdescriptionofthejoint behaviourisessentialtoaccuratelypredicttheload‐bearingcapacityofunreinforcedmasonrywalls, especiallywhenthefailuremodeisprimarilysecond‐orderbending. RegardingTRM‐strengthenedwalls,itisimportanttoremarkthattheload‐bearingcapacitypredicted bythemodelcorrectlyadjuststheexperimentalresults.Theaverageerroriswithinthetypical experimentalscatteringofdestructivetestsonmasonrywalls,around30%,anditisbetterthanthe errorobtainedfortheunreinforcedwalls.Thisfactagreeswiththegreateruniformityofthestructural 419 Modelling Unreinforced, TRM-strengthened and FRP-strengthened brick masonry walls subjected to eccentric compressive load Third International Conference on Mechanical Models in Structural Engineering University of Seville. 24-26 june 2015. responseshowedbytheTRM‐strengthenedwallsincomparisonwiththebehaviourofunreinforced masonrywalls.Inaddition,itcanbeconcludedthattheinfluenceofthecompressivestrengthofthe masonryanditsYoung’smodulusisverysignificantinthosecaseswithsmallloadeccentricities,which failedbyshear/compression.Incontrast,theresponseofthewallsthatfailduetobending/buckling phenomena,whicharetheoneswiththelargerloadeccentricitiesand/orslenderness,islessinfluenced bytheseparameters.Moreover,theload‐bearingcapacityofthewallssubjectedtolargerload eccentricitytendstobeunderestimatedbytheproposednumericalmodel.Aconservativedefinitionof themechanicalvariableswhichdirectlydescribethebendingresponseofthewallsmightexplainthis underestimationtendency. AlsoinrelationwiththenumericalmodellingofTRM‐strengthenedwalls,itisworthnoticingthatthe numericalmodelonlyoverestimatestheexperimentalcasesofthewallsstrengthenedwithcarbonfibre grids.Thisresultmightpointoutthattheadherencebetweentheinorganicmatrixesandthecarbon fibregridsisnotreliable.Finally,ithasbeenobservedthatthecompressive/shearfailuremode happens,inthenumericalsimulations,whentheshearstrengthisreachedinmorethanahalfofthe considereddividinglineonthemasonry. TheanalysisoftheresultsofthenumericalsimulationofFRP‐strengthenedwalls,allowsconcluding thattheproposedmodelaccuratelypredictstheexperimentallyobservedfailuremodeandthesmall influenceofthestrengtheningpatternontheresistanceoftheanalysedcases.Inaddition,itmightbe concludedthatasignificantpartoftheerrorofthenumericalmodelmightbeduetotheapproximate andmaybecontroversialdefinitionofthefractureenergies. Asgeneralconclusions,itisimportanttohighlightthatconsideringamorerealisticformulationto representthecompressiveresponseofthemasonrymighthelpimprovingtheobtainedresults.In addition,arealisticYoung’smodulusmustbeusedbecauseitdeterminesthedeformationleadingto thegeometricinstabilityassociatedwithbucklingfailure.Thus,futureworksmightincludeaccurately obtaintheYoung’smodulusofmasonryelements,performingparametricanalyseordevelopingthreedimensionalmodelsincludingtheout‐of‐planeshearinthestudy. ACKNOWLEDGEMENTS 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