MECHANICAi BEHAVIOUR OF MASONRY PANEiS UNDER IN-PiANE ioads
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1 Structural Analysis of Historical Constructions Jerzy Jasieńko (ed) 2012 DWE, Wrocław, Poland, ISSN , ISBN MECHANICAi BEHAVIOUR OF MASONRY PANEiS UNDER IN-PiANE ioads Barsotti ok N I Bennati pk O ABSTRACT qhisworkexaminesthemechanicalbehaviourandbearingcapacityofmasonrypanelssubjectedto verticalandhorizontalloads. tewillshowthatbyadoptingthejohrjcoulombfailurecriterionandapplyingsomesimple considerationsdrawnfromclassicallimitanalysisithevalueofthehorizontalloadcapableofleading topanelcollapsecanbepredicteditogetherwithsomeotheressentialparameterscharacterisingthe failuremodeitself. rpperboundestimatesofthecollapsemultipliervaluecanbeeasilyobtainedbyexaminingthe possiblemechanismsofpanelcollapsebyslidingorrotation.iowerboundestimatescaninsteadbe determinedbybuildingsuitableinonjuniformstaticallyadmissiblestressfields. qheestimatedcollapseloadvaluesobtainedforthesinglepanelareusedtomakesomepredictions abouttheloadcapacityofamasonrywallsubjectedtobothverticalandhorizontalloads. heywords: Masonry panelsi fnjplane loadsi iimit analysis N. INTRODUCTION aeterminingthebearingcapacityofamasonrywallsubjectedtoloadsactinginitsmidplanehasbeen thesubjectofagreatdealofexperimentalandtheoreticalresearchstudiesconductedonarelatively systematicbasissincethenvtmseseeiforexampleixniozf.aespitesuchwidejrangingeffortsithetrue mechanismsunderlyingtheresistanceofmasonrystilldonotseemtohavebeencompletely elucidateditherebymakingthisissueoneofthemostfundamentalunresolvedquestionsinthefieldof structuralmechanicseventoday. fntheworkpresentedhereinweintendtoshowthatispossibletoevaluatethehorizontallimitloadof amasonrypanelandthecollapsemechanismsbyusingonesinglesimplefailurecriterionforthe masonryandthetypicaltoolsoflimitanalysis.aftersomeintroductoryconsiderationsithefollowing sectionsaddresstheproblemofdeterminingtheupperandlowerboundsofthelimitloadofapaneli respectivelyobtainedbyexaminingsuitabletypesofkineticallycompatiblecollapsemechanismsand nonjuniformstaticallyadmissiblestressfields.corthesakeofsimplicityiwelimitthestudytothe particularcaseinwhichthemasonryisinfinitelyresistanttocompressionandentirelyunableto transmittensions.tewilllastlyshowhowtheresultsobtainedforasinglepanelcanbeusedto determineanapproximatelimitvalueforthehorizontalloadofamasonrywalliusingtheexampleof asimpleverticalrectangularwallwithanopeningatitsbase. O. THE MASONRY PANEi qhestatedproblemistodeterminethehorizontallimitloadofamasonrypanelofrectangularshapei fixedatitsbaseandsubjectedatitstoptoadistributionofverticalloadshavingaresultantofassigned magnitudeimiactingparalleltothepanelverticalaxisatadistanceeiandadistributionofhorizontal actionsstaticallyequivalenttoaforceilmiwherel representsthehorizontalforcemultiplieresee cig.nf.qhepanel sownweightisneglectedinthecalculations. N Ass.mrof.Iaept.ofCivilbngineeringIrniversityofmisaIr.barsotti]ing.unipi.it O cullmrof.iaept.ofcivilbngineeringirniversityofmisais.bennati]ing.unipi.it 4NS
2 corourpurposeshereithemasonrypanelisschematizedasatwojdimensionalbodyiinastateof planestressicomposedofarigidjplasticmaterial.qhechosenfailurecriterionisthejohrjcoulomb criterionidefinedbytherelationw t c - ks I ENF betweenthenormalandtangentialcomponentsis andtirespectivelyiofthegenericstressvector. Fig. Njasonrypanelsubjectedtoloadsactingatitstop qhecohesioncandthetangentofinternalfrictionangleikiarerelatedtothemasonry stensileand compressivestrengthsis t ands c IEbothassumedtobepositiveFaccordingtothewellJknownrelationsW c s cs t s c -s t I k tanf. O 4 EOF bquationseofshowthatiforfinitevaluesofthetensileandcompressivestrengthsicandkaretwo limitedquantitiesiandarebothgreaterthanzeroifs c [s t.fntheparticularcaseinwhichthe compressionresistanceisinsteadinfinitei k and c IandENFreducesto s s. EPF t fnotherwordsiinthislimitcasethejohrjcoulombcriterionendsupcoincidingwithdalileo s criterion. fnthefollowingwelimittheanalysistotheparticularcasewhere s and s M.qhecollapse valueofthehorizontalforcemultiplieril, isestimatedbydeterminingtheupperandlowerbounds throughapplicationofthebasictheoremsoflimitanalysis.corthesakeofsimplicityiweassumethat themasonrycanbemodelledasastandardrigid perfectlyplasticmaterial.fnthisregardiitshould berecalledthatoadenkovic stheoremsenableextendingthestaticandkinematictheoremsoflimit analysisalsotothecaseofnonjstandardmaterialsxpz. c t P. UPPER BOUNDS OF THE iimit ioad qhesearchfortheupperboundsofthehorizontalforcecollapsemultiplierisconductedbymeansof thekinematictheoremoflimitanalysis.oadenkovic sfirsttheoremguaranteesthatievenfornonj standardmaterialsithetruevalueofthecollapsemultiplierisinanyeventsmallerthanthose 4NT
3 correspondingtocollapsemechanismswithassociatedplasticflowiwhichshallthereforebethose takenupinthefollowing. Fig. Odenericcollapsemechanism ConsistentwiththechoiceoftakingthejohrJCoulombcriterionasthatformasonryfailureIwe assumethatthegenericcollapsemechanismischaracterizedbyaplanesurfaceiorthogonaltothe panelmidplaneialongwhichthedisplacementfieldexhibitsadiscontinuity.qhissurfaceisthus identifiedassoonasvalueshavebeenassignediforinstanceitotheslopeia,ofsegmentabwith respecttothehorizontalianddistancedofpointbfromthepanel supperbaseecig.of.qhemotion thatthepanel supperpartundergoeswithrespecttotheloweroneisshowninthesamefigure.apart fromslidingoneovertheotherithetwopartsofthepanelalsomoveapartinsuchawaythatthe vectorrepresentativeofthevelocityofanygivenpointofthepanel supperportionisalwaysinclined withrespecttoabbythematerial sinternalfrictionangle. AlongthelimitsurfaceIexpressionEPFreducestosimply s M andtheequilibriumequationsforthe panelupperportionimmediatelyenablesobtainingtherelationsw b + Oe l I æ N ö a arctan ç. E4F Od è l ø corapaneloffixeddimensionsithestrictupperboundofthecollapsemultiplieristhereforew N+ Ox l I ERF c Oh where h h L b istheslendernessofthepaneliand x e L b istheratiooftheeccentricityofthe resultantoftheverticalcompressiveactionstothepanelwidth. 4. iower BOUNDS OF THE iimit ioad AccordingtothestatictheoremoflimitanalysisIthesearchforthelowerboundsoftheloadmultiplier canbecarriedoutbyexaminingasuitablecollectionofstressfieldsthatareinequilibriumwiththe appliedloadsandatthesametimecompatiblewiththeboundsimposedonthematerial sstrength. teassumethatthepanelisdividedintoastressjfreenonjreactingpartandareactiveportioninastate ofmonoaxialcompression. 4NU
4 pinceibytheassumedhypothesesinobodyforcesareactingonthepaneliitcanimmediatelybe concludedthattheisostaticlinesinthepanel sreactiveportionarestraightlinesegmentsx4z. Fig. Pdenericisostaticlineofcompressioninthepanel sreactiveparteaf;plotoftheisostaticsforl l s EbF tedenoteby b b ExF theinclinationwithrespecttothehorizontaloftheisostaticlineof compressionpassingthroughthepointsituatedontheabscissaxofthepanel supperbaseecig.paf. qhehorizontalandverticalcomponentsqandpofthesurfaceforcesactingonthissamepointare relatedbytheequationw p q I ESF tan b andtheresultantofthehorizontalactionscanbeexpressedasw b pexf l m ò dx. ETF tan bexf M bquationetfshowsthatforanassigneddistributedverticalloadipexfi multiplierl increaseswith decreasingvaluesofangleb.fnthisregardibearinginmindthatalltheisostaticshavebeenassumed tointersectthepanellowerbaseiitfollowsthatateverypointontheupperbasethefollowingbound mustberespectedw h tan b ³ I EUF x whichindicatesthatangleb cannotfallbelowaminimumvalue. qhusiifweset tan b hlxinetfirecallingthatthestraightlineofactionoftheresultantofthe verticalactionsisatdistancee fromthepanelverticalaxisiitfollowsimmediatelythattheexpression forthemaximumstaticallyadmissiblevalueil s Iofthehorizontalforcemultiplierforapaneloffixed dimensionscanbeexpressedasw b N xpexf N æ b ö N+ Ox l ò dx mç + e I EVF s m M h mh è O ø Oh whereiasusuali h h L b isthepanelslendernessand x e L b. 4NV
5 qhesituationcorrespondingtothismultipliervalueisrepresentedgraphicallyinfigurepb.qhe isostaticsallpassthroughthelowerbasevertexaandthecompressionsmoreoverdivergeataxoz. qhevaluesofthehorizontalforcemultiplierfurnishedbyevfarethereforestaticallyadmissibleonlyin thelimitcaseofmasonrywithinfinitecompressivestrength.corfinitevaluesofcompressive resistanceiinsteadithevaluesofmultiplierl s thatarecompatiblewithpanelequilibriumandatthe sametimewiththemasonry sstrengthwillbeconsistentlysmallerthan l s. qhe lvaluesfurnishedbyevfcoincidewiththekineticallyadmissibleonesobtainedviaerf. qhereforeiwithsoleregardtothelimitcaseconsiderede s c and s t MFI N+ Ox l s l c I ENMF Oh representstheactualvalueil p Iofthecollapsemultiplier.bquationENMFshowsthatregardlessofthe eccentricityoftheresultantoftheverticalactionsithevalueofl p willineverycasefallwithinthe limitvalues M l p NLh. fnthecasethatthepanel stopisfreetorotateianestimateofthevalueofthehorizontalforce multipliercanbeobtainedbysetting x M inenmfiwhence l p NL Oh. N.RM l N.OR N.MM valuesdrawnfromxrz valuesdrawnfromxsz M.TR M.RM M.OR hhlb Fig. 4Comparisonbetweenl valuesfurnishedbyenmf andsomeexperimentalresultsinthecaseofacentredverticalload fnthisregardiafirstcomparisonmadebetweenthevaluesofthehorizontalforcemultiplierfurnished bybq.enmfforthecaseof x Mandsomeexperimentalresultsobtainedbysubjectingmasonry panelstoverticalloadsofrathermodestintensitywithrespecttothosethatwouldcausefailureby crushingedrawnfromxrzandxszfihighlightsthemorethanacceptableagreementbetweenthetwo setsofvaluesecig.4f. thenthepanel stopispreventedfromrotatingitheeccentricityofforcemcanbedifferentfromzero. fnthelimitcaseinwhichmisappliedincorrespondencetotheuppervertexaeandhencei e b L O FI l NLh andthecompressionsarenilatallpointsofthepanelwiththeexceptionofthosebelonging p tothesegmentaaiwhere s. M M.R M.S M.T M.U M.V N N.N N.O c R. AN EXAMPiE APPiICATIONW iimit HORIZONTAi ioad OF A MASONRY WAii fnordertoexemplifyanapplicationoftheresultsdescribedintheforegoingwithreferencetoasingle masonrypaneliletusconsiderawallwithanopeninginitsbaseiasillustratedinfigurer.fndicatingm astheweightofthewall supperportioniabcaiweaimtoestimatethemaximumvalueofthe 4OM
6 horizontalactioncmcompatiblewithboththeequilibriumofthedifferentpartsofthewallandthe strengthofitsconstituentmasonry. Fig. RtallwithopeningatitsbaseEleftF;internalactionsexchangedbetweenthelowerpanelspositioned atthewallbaseandtheupperportionerightf aeterminingtheexactsolutiontothisproblemisanythingbutsimple.bvengiventhesimplifying hypothesesadoptedregardingthemasonry sresistancees t MFItheproblemwouldnotadmitany solutionintheeventthatiforinstanceithemasonry sspecificweightweretakentobeuniformandthe horizontalactionsuniformlydistributedthroughoutthewall. eereiwiththeaimofarrivingatanapproximatesolutioniweforsakeanyattempttoverifythe equilibriumconditionsateachpointofthewallandinsteadchoosetoworkthroughasimpleschemei thatisibyimposingglobalequilibriumofeachofthethreeportionsintowhichwehavedividedthe wallwthetwopanelsflankingtheopeningoneachsideandtheupperrectangularportioniabcai overlyingthetwopanels. BywayofhypothesisIweassumethatthetwolowerpanelsareinalimitcondition.joreoverIifwe neglectthepanel sownweightswithrespecttotheweightoftheoverlyingpartibyvirtueofbq.enmfi wecansetw N+ Ox N N+ Ox ln I O lo I ENNF Oh N Oh O where x N en L Ob I x O e O L b I h N h L Ob and h O h L b. ByinsertingENNFintotheequilibriumequationsofthewallportionoverlyingthetwopanelsflankingthe openingiweobtaintheexpressionsfortheverticalcompressiveactionsexertedonthetwobasepanelsw æ N- Ox ö ç O æ mn m I O + 4x ö è P+ 4xN - Ox ç N m O m. ENRF O ø è P+ 4xN - Ox O ø joreoverifrom l Nm N + lomo cmitfollowsw O + 4xN b c. ENSF P+ 4xN - Ox O h 4ON
7 oelationsenrfandensfexpressthevariationsinthevaluesoftheresultantoftheverticalactions exertedonthetwobasepanelsim 1 andm 2 IaswellasinthevalueofthehorizontalforcemultipliercI withvaryingvaluesofthetwoeccentricities x N en L Ob and x O e O L b. qhegoalnowistodeterminethemaximumvalueofmultiplierc undertheconditionsw m N > M e m O > M. ENTF ConditionsENTFalwaysholdfor - NL O x N NL O and -NL O x O NL O.Byobservingthat c b L h Iasit isalsoconfirmedbytheplotofthefunction c c E x NI x O F showninfiguresaiitcanbeconcludedthat creachesitsmaximumvalueincorrespondenceto x O NL O Ewithany x N whateverfiwhereitis b c. ENUF max h qhereforeiwhenthehorizontalforceisdirecttowardstherightiasshowninfigurerithepaneltothe leftoftheopeningisunloadede m N M FIwhiletherightJsidepanelissubjectedtotheactionofaforce m O m actingincorrespondencetotheleftvertexoftheupperbase.fftheupperpartofthewallis assumedtobeinfinitelyresistantithissituationcorrespondstotheactualcollapsemechanismforthe masonrywall. chlb chlb a) x O x N x N b) Fig. SPaplotofthevaluesofthehorizontalforcemultiplier forrightwardeafandleftwardebfhorizontalactions x O Fig. Tpchematicillustrationoftheisostaticlinesofcompression inthereactivezonesofthebasepanels ffweinsteadconsideranhorizontalactionpointinginthedirectionoppositethatshowninfigureri calculationsanalogoustotheforegoingenableconcludingthat c max b L h Ecig.SbFI m m N and 4OO
8 m M.fnthiscaseIhoweverIitcanbeenseenthatpositivevaluesof O x N areunacceptableinthatthey wouldinvolveverticaltensileactionsononeofthetwopanels.joreoveriintheleftpaneliwhich sustainstheweightoftheentireupperpartitheresultantoftheverticalactionsturnsouttohavean eccentricity x N M.fnotherwordsIthecompressedpanelmanagestodeveloponlyhalfthatwhich wouldbeitsmaximumbearingcapacityforhorizontalactions.lnceagainiiftheupperpartofthe wallisassumedtobeinfinitelyresistantithissituationcorrespondstotheactualcollapsemechanism forthemasonrywall. S. CONCiUSIONS qhemechanicalbehaviourandbearingcapacityofmasonrypanelssubjectedtoverticalandhorizontal loadshavebeeninvestigatedbyadoptingthejohrjcoulombfailurecriterionandapplyingsome simpleconsiderationsdrawnfromclassicallimitanalysis.rnderthesehypothesesithevalueofthe horizontalloadcapableofleadingtopanelcollapsehavebeenpredicteditogetherwithsomeother essentialparameterscharacterisingthefailuremodeitself. rpperboundestimatesofthecollapsemultipliervaluehavebeenobtainedbyexaminingthepossible mechanismsofpanelcollapse byslidingorrotation.iowerboundestimatesareinsteadarrivedatby buildingsuitableinonjuniformstaticallyadmissiblestressfields. qheresultsobtainedwithreferencetoasinglemasonrypanelarethenusedtoestimatethevalueofthe limitloadofasimplemasonrywallsubjectedtobothverticalandhorizontalloads.qhisexample applicationshowsthattheproposedmethodcanalsobeextendedtocasesofwallswithmore complicatedshapesiforexampleihavingoneormoreopenings. REFERENCES xnz jagenesd.icalvid.j.envvtf.fnjplaneseismicresponseofbrickmasonrywalls.barthquake bngineering and ptructural aynamicsiosinmvnjnnno. xoz oocam.eommsf.assessmentofmasonryshearjwallsbysimpleequilibriummodels.construction and Building MaterialsIOMIOOVJOPU. xpz iublinerg.envvufmlasticity theoryk jacmillanmubco. x4z jansfieldb.e.envsuf.qensioncieldqheory.fnwmrock 12th fntk Congress on Applied MechanicsIPMRJPOMIppringerIkewvork. xrz xsz iourencom.b.illiveiraa.s.ioocam.ilrdunaa.eommrf.aryjointstonemasonrywalls subjectedtoinplanecombinedload.gournal of ptructural bngineeringinpnennfinssrjnsto. 4OP
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