Effect of partial replacement of coarse aggregate by polystyrene balls on the shear behaviour of deep beams with web openings

Case

study

Effect

of

partial

replacement

of

coarse

aggregate

by

polystyrene

balls

on

the

shear

behaviour

of

deep

beams

with

web

openings

Ibrahim

G.

Shaaban

a,

*

,

Amr

H.

Zaher

b

,

Mohamed

Said

c

,

Wael

Montaser

d

,

Mohamed

Ramadan

d

,

Ghada

N.

Abd

Elhameed

d

a

CivilEngineeringandBuiltEnvironment,UniversityofWestLondon,UK b

StructuralEng.Dept.,FacultyofEng.,AinShamsUniversity,Cairo,Egypt

c_{CivilEngineeringDepartment,FacultyofEngineering,Shoubra,BenhaUniversity,Egypt} d_{StructuralEng.Dept.,FacultyofEng.,October6University,Giza,Egypt}

ARTICLE INFO

Articlehistory:

Received11September2019

Receivedinrevisedform6December2019 Accepted16December2019

Keywords: Coarseaggregate Polystyreneballs Deepbeams Shearstrength Webopenings Finiteelementmodel Strut-and-Tiemodel

ABSTRACT

Thirteenspecimenswereexperimentallytestedundersinglemidspanconcentratedloads tostudytheshearbehavioroflightweightconcrete(LWC)andnormalweightconcrete (NWC)deepbeamswithwebopenings.Inthisresearch,thetermLWCreferstotheconcrete obtained bypartially replacingaggregate by polystyrenefoam ballsnot the concrete containinglightweightaggregate.ThisresultedinaweightreductionofLWCbeamsinthis researchbyapproximately30%comparedtoNWCcompartments.Thestudiedvariables werethedimensionsandlocationofopenings,transversereinforcementratio,andshear spantodepthratio(a/d).Itwasfoundthattheoverallshearbehaviorandfailuremodefor LWCdeepbeamsarecomparabletothoseoftheNWCspecimens.Thisisverypromising andencouragingtobuildlighterdeepbeamsofsimilarstructuralbehaviourasthatofNWC deepbeams.Dimensionsoftheopeningshaveasignificanteffectonthebehaviourof failureandshearstrengthofLWCandNWCdeepbeams.Itwasfoundthatincreasingthe depthoftheopeningfrom20%to40%ofthebeamdepthledtoareductionintheultimate loadbyupto46.4%.Finiteelementmodellingofthetestbeamswascarriedouttoverify numerical resultsversus experimental workand bothwere very well correlated. In addition,aparametricstudywasconductedtoassesstheeffectofinternalstiffeningaround openings in deep beams. The maximum enhancement in the shear capacity was approximately30%forbeams,internallystrengthenedbyadditionalreinforcement on theperimeterofopeningscomparedtothebeamswithoutanyreinforcementaroundthe openings.Strut-and-Tiemodel(STM)wascarriedoutasarationalapproachtopredictthe shearbehaviourofstudiedbeams.ItwasfoundthatSTMunderestimatestheshearofthe studied beamscompared toexperimental resultsfor different testedbeams but the agreementbetweenbothofthemwasacceptable.Itisrecommendedthatthedepthof openingshouldnotexceed20%ofthedepthofthedeepbeamandifthedepthofopeningis morethanthatorliesintheshearspanitishighlyrecommendedtostrengthentheopening internallybyadditionalreinforcementarounditsperimeter.

©2019TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

*Correspondingauthor.

E-mailaddress:[email protected](I.G. Shaaban).

https://doi.org/10.1016/j.cscm.2019.e00328

2214-5095/©2019TheAuthor(s).PublishedbyElsevierLtd.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/ 4.0/).

ContentslistsavailableatScienceDirect

Case

Studies

in

Construction

Materials

andf2

ft Ultimateuniaxialtensilestrength fc Ultimateuniaxialcompressivestrength

fcc Concretepeakstress

fcb Ultimatebiaxialcompressivestrength

f1 Ultimatecompressivestrengthforastateofbiaxial compressionsuperimposed onhydrostatic stress state

f2 Ultimatecompressivestrengthforastateof uni-axial compression superimposed on hydrostatic stressstate

fcu Cubecompressivestrengthofconcrete fc0 Cylindercompressivestrength

a/d Shearspan-to-depthratio

fsand

e

s The average stress and strain of steel bars, respectively

fyand

e

y Theyieldstressandstrainofsteelbars, respective-ly

Es Theyoung'smodulusofsteelreinforcement

fcr Thecrackingstrengthofconcrete

e

c Concretestrain

e

c1 Concretestrainatultimatestress

e

cc1 Concretestrainatpeakstress

e

n Straininsteelbars

FEM Finiteelementmethod

h Thebeamheight

d Thebeamdepth

b Thebeamwidth

a Theshearspan

a1 TheheightofnodeN1

a2 TheheightofnodeN2

b1 Thelengthofbearingplate1

b2 Thelengthofbearingplate2

Ld TheinternalleverarmbetweenthetieforceT,and compressionstrutS2

S TheforceinstrutS

N1 ThenodeN1 N2 ThenodeN2

W11 The width of strut S at node N1, measured

perpendiculartostrutcenterline

W12 The width of strut S at node N2, measured

perpendiculartostrutcenterline

W1av TheaveragewidthofstrutW11andW12;

L Thebeameffectivelength

α TheinclinationofstrutS

C Theconcretecover

1.Introduction

Deepbeamsareusedinspecialstructuressuchastransferfloorsofhigh-risebuildings,offshorestructuresandcomplex foundationsystems.Theshearcapacityisthegoverningdesignfactorfordeepbeams.Insimpledeepbeams,thezoneof highshearcoincideswiththedistrictoflowmoment.Differentvaluesofthespantodepthratio(Le/d)andtheshearspan

todepthratio(a/d)areproposedbydifferentdesigncodestodefinedeepbeams[35,40].Openingsinthewebareaare oftenprovidedforcriticalservicesandaccessibility.Theventilatingslotsaretheidealexamplefortheopeningindeep beams.Suchopeningsmayaffectthecapacityorthestressesdistribution,particularlywhenopeningsexistinthecritical regions.Theshearcapacityofbeamswithopeningsisbasedonnumerousfactors,suchas:location,dimensionsofthe opening, andpropertiesof used materials (e.g.concrete andsteel)[1,2].Simplified designmethodsfor deep beams withoutspecialconsiderationtotheinfluenceofwebopeningsaredescribedinmostoftheavailableinternationaldesign codes[35,40].

Extensiveanalytical,numerical,andexperimentalinvestigationshavebeencarriedoutforstudyingdeepbeamswith webopenings[3–11].MansurandAlwist[12]experimentallytested12reinforcedfiberconcretedeepbeams(1300650 mm)withsmallsizeopenings(175125mm).Theirresultsindicatedthattheamountofwebreinforcement(fibersor continuoussteelreinforcement),andthelocationofopeningaretheprincipalparametersthataffectthestrengthofdeep beams. They predicted the strength using equations for non-fiber concrete deep beams with reasonable accuracy. ShanmugamandSwaddiwudhipong[13]developedanempiricalformulatopredicttheultimatestrengthofexperimentally testedfiberreinforcedconcretedeepbeamscontainingopenings.Theirresultsshowedthattheultimatestrengthprimarily depends upontheextenttowhichtheopeningintercepts thenatural loadpath.Ashour[14]developeda modelfora mechanismofshearfailureofanalyzedspecimens.Themodelpresentedthata/dratiohasahigherinfluenceontheshear capacitythanthatoftheLe/dratio.

Sahooetal.[15]investigatedtheperformanceoftwoRCandtwosteel_fiber-reinforcedconcrete(SFRC)deepbeams withlargeopeningsundermonotonicallyincreasedconcentratedloads.Theboundaryregionsnearthesupportsoftwo studiedspecimens werestrengthenedwith steelcagesformedbysteelreinforcementbars. Theyfoundthatthe RC specimen with strengthened boundariesexhibited a ductile mode of failure and hadsignificantly higher ultimate strengththanthatpredictedbyStrutandTieModels(STMs).TheyfoundalsothattheSFRCspecimenswith1.5%volume fraction of fibers reached much higher strength than the design load and exhibited significant postpeak residual strengthandaductilemodeoffailure.Dohetal.[16]carriedoutaparametricstudyforhighstrengthconcretedeep beamswithvariouswebopeningsconfigurationsusingnonlinear-layeredfiniteelementmethod(LFEM).Theirresults confirmed thatthe current designmethodsare inadequatein predicting the maximumshear strength whenweb openingsarepresent.

Abduljalil [17]carried outanexperimental worktostudyshearresistanceof reinforcedconcretedeepbeamswith openingstrengthenedbyCFRPstrips.HefoundthatexternallyCFRPstripssignificantlyincreasedtheultimateshearcapacity andtheylimitedtheshearcrackwidthofthedeepbeamswithopenings.Adametal.[18]carriedoutexperimentalandfinite elementstudyforself-compactedconcretesoliddeepbeamswith_fibers.Theyfoundthatbothverticalandhorizontalweb reinforcementare efficient in shearcapacity enhancementof studiedspecimens. Theyalsofoundthat ultimateshear capacitywasincreasedbyabout47%withincreasingthelongitudinalsteelratiofrom1.0%to2.2%.Hussain[19]developeda finiteelementmodelusingANSYSsoftwarerelease12.0programtostudytheultimateloadandcrackpropagationfor reinforcedNSCspecimensprovidedwithopenings.Hisresultspresentedacceptableagreementwithexperimentalresultsof ultimatebeamcapacity,correspondingmidspandeflection,anddetectedinclinedcracks.

TheuseofLWCstructures,especiallydeepbeams,isincreasingwidely.Thesebeamsareefficientsincetheirultimate strengthcanbecomparabletoNWCcounterpartsatanapproximateweightofonlysixtypercentofthatforNWCdeep beams.Huangetal.[20]studiedexperimentalshearbehavioroffulldimensionLWCsoliddeepbeamspecimens.Their resultsconcludedthatthefailuremodesofLWCbeamsaresimilartothoseofNWCbeams,includingshear-compression failureandshear-tensionfailure.Sathiyamoorthy[21]foundthatshearstrengthofLWCbeamsincreasedwiththedecrease ofa/dratio.HereportedthatLWCbeamsshowedhighernumberofcracksandwidercrackwidthatfailurecomparedtotheir NWCcounterparts.Healsomentionedthattheinternationaldesignbuildingcodes[35,40]areconservativeinpredicting shearstrengthofshear/non-shearreinforcedLWCbeams.Backin1973,KongandSharp[22]studiedtheshearstrengthof

’

i Thelongitudinalsteeldiameter

n Thenumberofsteellayers

s Thespacingbetweensteellayers

As Theareaofthereinforcement

V Theshearforceatsupport

fs_{ce The effective concrete compressive strength for}

strutS

fN1_{ce EffectiveconcretecompressivestrengthatnodeN1}

LWCdeepbeamswithsmallsizeoftheiropeningsandtheydevelopedasemi-empiricalmethodfortheanalysisofdeep beamswithsmallopenings.ItisworthmentioningthatLWCdeepbeamsstudiedinliteratureandmentionedaboveare thosecontaininglightweightaggregates.

Itcan be seen from the review above that further research is needed for accurate prediction of strength and behaviourofLWCdeepbeamscastbypartiallyreplacingcoarseaggregatesbypolystyrenefoamballswithopeningsof large dimensions. The authors of the current investigation started a project in 2015, funded by two Egyptian universities, namely: Ain Shams and October6, to study the LWC deep beams containing polystyrene balls with openings[23,24].Thecurrentstudyinvestigatestheeffectofopeningsize,locationandnumberofopeningsonthe shearbehaviorofbothLWCandNWCdeepbeamssubjectedtoconcentratedloads.Otherstudiedvariablesincludethe transversereinforcementratioanda/dratio.Experimentalworkiscarriedoutandtheoreticalworkincludedprediction of experimentalresults using therational method, Strut-and-Tie Model (STM), andthe more accurateone, three-dimensionalFiniteElementModeling(FEM).STMapproachfollowsFosterandLanGilbert[25]whileFEMisdeveloped usingANSYSpackage(ANSYS13.0)topredicttheresultsandtoevaluateitssensitivitytothestudiedparameters.In addition,aparametricstudyiscarriedouttofurtherstudytheeffectofincreasingthelongitudinalmainreinforcement ratio,theinternalstiffeningoftheopeningperimeterusingadditionalreinforcementontheshearbehaviorofLWCdeep beamswithopenings.Ultimately,recommendationsareintroducedfortheanalysisanddesignofLWCdeepbeamswith openings.

2.Experimentalprogram

2.1.Preparationoftestspecimens

TheprogramincludesthirteenLWCandNWCdeepbeams.Beamsweretestedunderasinglemidspanconcentrated load.Deformedsteel,grade40/60wereusedforlongitudinaltopandbottomreinforcement.Thebottombarsconsisted offourdeformedbarsof16mmdiameterintwolayerswhilethetopreinforcementconsistedoftwobarsof10mm diameter.Thespecimensweredesignedtoensurethatshearfailureoccurs.Adequateanchoragewasprovidedtothe longitudinalbars. Mildsteel,grade 24/35,of 6-mmand8-mmdiameter wasusedas horizontalandvertical shear reinforcement.Additionalprecautionsweretakenatthesupportsbyplacingbearingplates(10010015mm)under theloadpositioninordertopreventlocalfailurebybearing.Specimensweredividedintothreegroupsasindicatedin

Table1.Allthetestedspecimenshadthesamerectangularcross-sectionof80mmwidthand400mmtotalheightas showninFigs.1–2.Mixdesign,ofbothofNWCandLWC,wascarriedouttoachievesimilartargetcubestrengthafter28 days.Lightweightconcrete(LWC)weremadebypartiallyreplacingcoarseaggregatebypolystyrenefoamballs.This resultedinareductionofoverallweightoftheLWCtestbeamsbyapproximately30%comparedtotheircounterpartsof NWCones. Table2 displays themixdesignfor LWCand NWC.Mechanical propertiesof LWC andNWCmixes are recordedinTable3.

BNWC8 1.63 160 6 0.442 1B22

BNWC9 2.08 160 6 0.442 1B22

2.2.Descriptionoftestspecimens

Table1andFig.2showthattheopeningsarrangementswereeithersinglerowofopenings,havingaheightequalsto20% ofthebeamtotalheight;ordoublerowsofopeningshavingaheightequalsto40%ofthebeamtotalheight.Threedifferent locationsoftheopeningsweretested;location1,2and3.Forlocation1,theopeningliesbetweenthefirststirrupatthe supportandthesecondone.Forlocation2,theopeningliesbetweenthesecondstirrupfromthesupportandthethirdone. Regardinglocation3,theopeningliesbetweenthethirdstirrupfromthesupportandthefourthone.Thelocationsofweb openingswereselectedtotestthreedifferentloadflowstothesupport.Spacing(Sv)betweentheverticalwebreinforcement

was100mmand200mm.Threedifferentvaluesofa/dratio;0.97,1.63and2.08,wereconsidered.Thebeamnotation,as indicatedinTable1,includedfourparts.Thefirstpartreferstothenumberofopeningsintheshearspan(1or2)andthe secondpartindicatedthesizeofopening(A=widthxheight=8080mmandB=14080mm).Thethirdpartreferredto thewebreinforcementarrangement(1forSv=100mmand2forSv=200mm)andthefourthpartreferredtothepositionof

theopenings(location1,2and3).

2.3.Loadingandtestprocedure

Specimens wereloadedin increments uptofailure.Specimenswereinstrumented tomeasure theirdeformational behavior.Therecordeddataincludemeasurementsofstraininconcrete,mainsteel,transversereinforcement(stirrups)and longitudinalbars strain;deflection and crackpropagation.The straingaugewas shownin Fig.1.Thedeflectionswere recordedusingthreeLVDT,Fig.3.LVDTwerearrangedtomeasurethedeflectiondistribution.Thecracksweretracedateach increment.

3.Experimentalresultsanddiscussion

Crack pattern and failure modes, deflections, and steel strains for horizontal, vertical stirrups and main bars reinforcementwerenotedforeachofthethirteenspecimensandtherelationshipsareplottedinFigs.4–11.Theultimate loadsanddeflectionsarerecorded,Table4.

3.1.Crackpatternandfailuremodes

Fig.4displaysthecrackpatternswhichwereintermsofflexuralandshearcrackingforallthetestspecimens.Forall specimens,theflexuralcrackwasinitiatedatthecentralofthebeamspan.Flexuralcracksweredistributedastheload increased.ForsolidbeamBLWC1,thetensilecracksinitiatedonthetensionsideofthebeamspan.Thecrackspropagated

upwardwiththeincreaseofloading.Diagonalcrackssuddenlydevelopedattheshearspan.Thecracksweredetected parallel to the compressionstrut. The cracks werespread towards the loadingregion and supports. A typical shear compressionfailureofbeamBLWC1occurredsuddenlybycrushingintheconcretecompressionstrutsresultingalsoinaloud

noise.Finally,suddenshearfailureoccurredinstantlyaftermaindiagonalcracksformedwithinoneortwosideoftheshear spanasshowninFig.4.

Forthestudiedbeamswithsmallopeningsize,BLWC2andBLWC3;thefirstobservedcrackswereflexuralonthetension

sideatthemiddleofthebeamspan.Thesecracksrapidlypropagatedwiththeloadincreasetowardsthelowercornersofthe opening.Withtheincreaseoftheappliedload,sheardiagonalcrackswereinitiatedandextendedfromthesupportplatesto theedgesoftheopenings.Athigherloadingstages,thewidthofthemaindiagonalcrackincreasedasshowninFig.4.For

thesebeams,failureoccurredabovethewebopeningwithinclinedcracksstartingformloadedplatetotheuppercornersof theopening.Alsaeq[26]reportedsimilarobservationsfordeepbeamswithopenings.

Forthespecimenswithlargeopenings,BLWC5andBLWC6,inclinedcracksappearedfirstfromloadingpointtillcornerofthe

upperopeningparalleltothecompressionstrut,andthenfurtherdiagonalcrackswereinitiatedatopeningcorners.Then, crackspropagatedtowardsloadingzoneandsupports.Morediagonalcracksappearedparalleltothestrut,passingthrough theopeningcornersandpropagatedtowardstheloadingregionandthesupportingplates.Threea/dratiowerechosen,the firstratiowas0.97,beamBLWC5,thesecondratiowas1.63,beamBLWC8,andthethirdratiowas2.08,beamsBLWC9.The

developmentofflexuralcrackswasfasterinsampleswithalargevalueofa/dratio;beamsBLWC9andBLWC5(seeFig.4).The

recordedultimatestrengthofspecimenBLWC5testedata/dequals0.97wasmorethanthatofSpecimensBLWC8andBLWC9

by11%and18%,respectively.ReferringtoFigs.4and5,itisclearthatthedimensionsandpositionoftheopeningshave majoreffectsonthecrackpattern,ultimatestrengthandfailureofstudiedspecimens.ThisagreeswiththefindingsofJasim etal.[27]whoreportedinhisexperimentalandtheoreticalstudyofdeepbeamswitha/dratioequals1.1thatthelargeweb openingshaveagreateffectontheshearstrengthofdeepbeams.Itcanbeseenfromtheaboveobservationsthatwhenthe openingsinterrupttheloadpathbetweentheloadingandreactionpoints,thecrackpathchangestoamorecomplexone.Itis worthmentioningthatfailureofalltestedspecimenswasshearfailureasshowninFig.4).

3.2.Crackingloadsandultimateloads

Fig.5displaysthecrackingloadsandultimateloadsofthestudiedspecimens.ItcanbeseenthatthecrackingloadofLWC specimens,BLWC6,BLWC7,BLWC8andBLWC9,arenearlyequaltothecrackingloadofNWCspecimens,BNWC6,BNWC7,BNWC8and Table2

Mixproportions.

Concretetype Cement (kg=m3₎

Sand (kg=m3₎

Gravel (kg=m3₎

w/cratio Super-Plasticizer (liter=m3₎

Silicafume (kg=m3₎

PolystyreneFoam (liter=m3₎

Lightweight 420 630 630 0.30 2.8 40 330

Normalweight 350 630 1260 0.5 – – –

Table3

Mechanicalpropertiesofconcrete.

Concretetype Targetfcu

(N/mm2

)

Cubestrength(N/mm2_{) Cylindricalcompressivestrength(N/mm}2₎

28days

7days 28days

LWC 25 19.7 26.3 20.16

NWC 25 17.6 25.6 19.62

BNWC9.Ontheotherhand,ultimateloadsofLWCspecimenswereapproximately90–96%oftheultimateloadsofNWC

counterparts.ThecrackingloadofspecimensBLWC2,BLWC3andBLWC4was70%,65%,and60%ofthecrackingloadsofsolid

specimens.Asfarasthespecimenswithdoublerowsofopeningsofdepth,40%,ofthebeamdepth,theultimateloadsof specimensBLWC6andBLWC7wereapproximately41%and46%oftheultimateloadofthesolidspecimenBLWC1.Generally,

increasingthewidthoftheopeningintheshearzoneledtodropintheultimateloadofthespecimens.Theultimateloadof specimensBLWC6was66%ofthatBLWC4.Itshouldbenotedthatthedimensionsandpositionofthewebopeninghave

significanteffectonthemodeoffailureandultimatestrengthofLWCandNWCdeepbeams.Thepresenceofopeningsinthe shearspanconsiderablyreducedtheultimatestrengthofthespecimens.Similarobservationswerereportedby[2],who

Fig.7. Effectofwideopeningonload-midspandeflectionofspecimens. Fig.5.Crackandultimateloadoftesteddeepbeams.

foundthattheultimateshearstrengthofHSCdeepbeamswithopeningsreducedrapidlywiththepresenceofopeningsin theshearspan.

3.3.Load-midspande_flectionoftestspecimens

Thedeflection atmid-spanwasmeasured andrecordedateachloadincrementduringtestingofeachbeam. Load-de_flectionrelationshipsareshowninFigs.6_–10foralltestbeams.ItcanbeseenfromFig.6thatforspecimenBLWC1,the

beamsbehavedinatrulyelasticmanneratearlystagesofloading.Inaddition,beamswithsmallwebopenings;BLWC2and

BLWC3showedloaddeflectionbehaviorverysimilartothatofthesolidbeam.Fig.7showsthattheultimateloadofspecimens

decreasedwithincreasingthesizeofopenings.Itwasnoticedalsothatspecimenswithsmallopenings,BLW2,BLW3,andBLW4 Fig.9.Effectofnumberofrowsofopeningsontheload-midspandeflectionofspecimens.

Fig.8.Effectofconcretetypesonload-midspandeflectionofspecimens.

(seeFig.6)havestiffnesshigherthanthosewithlargeopenings,BLW5,BLW6,andBLW7,showninFig.7.Forexample,the

ultimateloadofBLWC2equalsto1.06thatofBLWC3withopeningnearthesupport(seeFig.6).Theeffectofconcretetypeon

themidspandisplacementfordeepbeamswithwebopeningsisshowninFig.8.Itcanbeseenfromthefigurethatthe ultimateloadsandstiffnessofLWCstudiedbeams,BLWC6,BLWC7,BLWC8,BLWC9andtheircounterpartsNWCspecimens,BNWC6,

BNWC7,BNWC8,BNWC9,haveasimilarpattern.Fig.9displaystheinfluenceofopeningsonthereductionofstiffnessofthe

studieddeepbeams.Forexample,theultimateloadsofspecimenswithopeningheightrepresent40%ofthedepthofbeams, BLWC4andBLWC6,wereapproximately75%and63%ofthatofthebeamsofopeningheightequals20%ofthebeamdepth.For

Fig.11.Straininsteelreinforcementofdifferentstudiedspecimens.

Table4

Experimentalandfiniteelementanalysisresults.

Group Beam Experimental NLFEA NLFEA/Experimental

Ultimateload(2Vu)KN Deflectionmm Ultimateload(2Vu)KN Deflectionmm Ultimateloadratio Deflectionratio

1 BLWC1 193.81 1.42 185.00 1.36 0.95 0.96

BLWC2 170.64 2.18 170.00 2.03 1.00 0.93

BLWC3 162.00 2.34 155.00 2.20 0.95 0.94

BLWC4 122.00 1.95 117.00 1.89 0.96 0.97

2 BLWC5 130.89 1.98 119.75 1.92 0.92 0.97

BLWC6 82.00 2.13 73.50 1.81 0.89 0.85

BLWC7 83.00 1.5 75.00 1.49 0.90 0.98

BLWC8 125.00 2.43 120.00 2.37 0.96 0.97

BLWC9 110.25 2.99 110.00 2.87 1.00 0.96

3 BNWC6 70.00 1.56 66.80 1.55 0.95 0.99

BNWC7 75.00 1.29 70.00 1.34 0.93 1.04

BNWC8 120.45 2.66 120.00 2.02 1.00 0.75

BNWC9 110.66 2.96 107.26 2.81 0.97 0.95

BLWC6withlargeopening(openingsizenotation,2B22),themidspandeflectionwashigherthanthatofBLWC4withopening

sizenotation,2A12,andBLWC5withopeningsizenotation,1B22.ThisagreeswiththefindingsofIbrahimetal.[28]who

reportedtherelationshipbetweenopeningdimensions,position,thestiffnessandultimatecapacity.Fig.10showsthatthe reductionintheultimateloadofthestudiedspecimenwas16%withincreasinga/dratiofrom0.97to2.08.

3.4.Steelstrains

Straingaugeswereattachedtothetensilesteelreinforcementofspecimenstoexaminethedisparityofstraininbottom reinforcement.Thesegaugeswereattachedatmid-spanofspecimens.Fig.11 displaysatypicalmeasuredstrainofthe specimens.Thestraindeviationintensilereinforcementwasnearlycomparableforalltestbeams.Thedevelopmentofa tie-actionwasdetectedandthestrainincreasedrapidlyinthelocalityofthefirstcrack.Finally,thestrainswereincreasedat almostconstantloadingleveluntilthefailureoccurred.Themaximumstrainin_flexuralreinforcementwaslessthanthatof

Fig.14.Stress-straincurveforconcrete[30].

theyieldstrainvalue.Themeasuredstrainswererangedfrom25%to50%ofyieldstrainofthelongitudinalreinforcement. Therecordedstrainintensionbarsshowedthatthetensionfailurewasprotectedforallofthespecimenstopermitforshear failuremode.Themaximumrecordedstraininlongitudinalbarswas0.0009forSpecimenBLWC8(seeFig.11).Similar

load-tensilesteelreinforcementstrainwasobservedbyIbrahimetal.[28].Fig. 11showsalsothatthestraininverticalstirrupswas recordedatthecriticalshearlocations.Priortotheoccurrenceofthefirstcrack,theinternalshearresistancewasprovidedby thebeamsection.Oncethediagonalcracksoccurred,strainsoftheverticalstirrupswererecorded,representingshear resistancerolebytheverticalstirrups.Themaximumstraininverticalstirrupswasapproximately0.002forSpecimenBLWC9.

Theyieldingofverticalstirrupsoccurredbeforefailureofdeepbeams.Themaximumhorizontalstirrups’strainwas0.00046 forSpecimenBLWC7asshowninFig.11.

4.Finiteelementmodellingofdeepbeams

Theexperimentallytestedthirteenspecimenswerenumericallymodeledusing[29]),packagetopredicttheirresults versustheexperimentalresultsforthemainstudiedparameters,dimensionandpositionofopening,thedesignconcrete compressivestrength,thetransversereinforcementratio,anda/dratio.Detailsofthefiniteelementmodelling,predictions oftheresultsusingthetestedmodelarereportedanddiscussedinthefollowingsections.

4.1.Elementtypes,materialpropertiesandconstitutivemodels

Concreteandsteelarethetwomainmaterialsusedinthenumericalanalysisofthedeepbeaminwhichtheirproperties andconstitutivemodelsarepresented.ThesolidelementusedinthisstudywasSolid65whichisoneoftheelementsin ANSYSprogramtomodelthethree-dimensionalbehaviorofconcrete.Solid65wasassumedtomodeltheconcreteasitis capableofcrackingintensionandcrushingincompression.Thegeometricalcharacteristicsofthe3-DSolid65elementare showninFig.12.Theelementisdefinedbyeightnodesandeachnodehasthreedegreesoffreedom.Theflexuralandshear reinforcementinthetestedbeamswereidealizedusingtheLink8elementasshowninFig.13.Theaxialstressisassumedto beuniformovertheentireelement.Fullbondwasassumedbetweenconcreteandreinforcingsteel.Bothlinearand non-linearbehaviorsoftheconcretewereconsidered.Forthelinearstage,theconcreteisassumedtobeanisotropicmaterialup tocracking.Forthenon-linearsegment,theconcretemayundergoplasticity.Thenumericalsolutionschemeadoptedfor non-linearanalysiswasanincrementalloadprocedurebasedontheiterativesolutionusingNewton-Raphsonmethod.The convergencecriterioncurrentlyusedwasbasedontheiterativenodaldisplacementwhereonlytransitionaldegreesof freedomwereconsidered.

4.1.1.Constitutivemodelingforconcrete

TheconcretematerialmodelassignedforSolid65elementusedthroughoutthisstudyischaracterizedbyitscapabilityto predictthefailureofbrittlematerials.Bothcrackingandcrushingfailuremodesareaccountedfor.Thecriterionforfailureof concreteduetoamulti-axialstressstatecanbeexpressedintheform:

F

fcS0 ð1Þ

Tomodelconcretebehavior,nonlinearstress-straincurveswereusedincompressionandtension[30].Suchmodels accountfor compression&tensionsoftening,tensionstiffening andsheartransfermechanismsincracked concreteas presentedinFig.14.

4.1.2.Constitutivemodelingforsteel

Theaveragestress-straincurvedevelopedearlierbySoroushianandLee[31]forsteelbarsembeddedinconcreteisused inthecurrentresearch(seeFig.15).Thestress-strainrelationshipisexpressedbytwostraightlinesasfollows:

For

e

s

e

n:

fs¼Es

e

s ð2Þ

andfor

e

s

e

n:

fs¼fy ð0:912BÞþ 0:02þ0:25B

e

s

e

y

ð3Þ

Where

e

n¼

e

yð0:932BÞ.

AndtheparameterBisgivenasfcr=fy1:5=r. Therecommendedvalueoffcrisgivenas:

fcr¼0:31

ffiffiffiffiffi

f=c

q

4.2.1.Predictionofcrackpatternsandfailuremode

Fig.18showsacomparisonbetween_finiteelementpredictionandexperimentallyobservedcrackpatternsforselective specimens,whichwerepreviouslyintroducedinFig.5.Inaddition,Table4showsthepredictedvaluesfortheultimateloads andcorrespondingdeflectionforthestudiedbeams.ItcanbeseenfromFig.18 thatthecrackpatternsfordeepbeams predictedby_finiteelementmodelareingoodagreementwiththeexperimentallyobservedones.Atapproximately43%of theultimateloadofcapacityofspecimens,arapidmaininclinedtensioncrackformednearlyinthemiddlepartoftheshear span.Inaddition,inclinedcracksspreadtothebeamsupport.Forbeamswithopenings,thecracksspreadaboveopeningsto thepointload.Thenthecrackextendeddownfromtheopeningstosupports.Finally,failureoccurredinopeningregion. Compressionstresseswereconcentratedalongtheloadpath.Thetensilestresseswereeliminatedinthefiniteelement analysisgeneratingcracksinconcreteandtheywererelocatedtosteelcrossingthisregion.Thestressleveldependsmostly onthedimensionandpositionoftheopening.Inaddition,theconcretestrengthhasagreatereffectwithadecreasinga/d ratio.Thehighercompressivestressesoccurredatnodalzones,whilstareductioninthecompressivestressesoccurredinthe inclinedstrutslinkingthepointsloadandsupports.Thisreductionisduetotheopeningintheloadpath.Fig.19displaysthe deformedshapeandverticaldisplacementbeforefailureforselectedbeams.Theabovecomparisonshowsalsothatthe ultimateloadofLWCisslightlyhigherthanthatofNWCandthismaybeattributedtothefactthatactualconcretecube strengthforLWCspecimenswasslightlyhigherthanthatofNWCcompanionsinthisstudy.

4.2.2.PredictionofLoad-de_flectionrelations

Fig. 20shows comparisons between experimental load-deflection relationships and those predicted numerically. In addition,Table4 showscomparisonsbetweenexperimentalandnumericalresultsforultimateloadsandcorresponding

deflection.Fig.20andTable4revealthatthereisaverygoodagreementbetweenthenumericalandtheexperimentaltest results.Theratioofthepredictedultimateloadstotheexperimentalonesforthetesteddeepbeamsrangedbetween0.89-1.0.

4.2.3.Parametricstudy(Effectofinteriorstrengthening)

Aparametricstudywascarriedoutbyaddingmoreparameters,otherthanthoseusedintheexperimentalstudyinthis research,toexaminetheperformanceofLWCdeepbeamswithopeningshavingdifferentconfigurations.Theextrastudied parametersreportedinTable5werechangingthebottom(tensile)reinforcementratioforsoliddeepbeamsandthosewith openings,addingtopandbottomreinforcementadjacenttotheedgesoftheopening,addingrightandleftreinforcement

adjacenttotheopeningedges,andaddingreinforcementatallsidesoftheopening.Itisworthmentioningthattheopening notationisthesameasthatforexperimentallytestedbeamsinTable1.Fig.21ashowsthenumericalload-de_flectionforsolid deepbeamspecimenswithdifferentbottomreinforcement.Itcanbeseenfromthefigurethattheincreaseoflongitudinal bottomreinforcementledtoareasonableincreaseintheultimatecapacity.Forexample,SpecimenDA3ofhighertensile reinforcementratioof0.54%(4Ø22bottomreinforcement)exhibitedmoreductilebehaviorcomparedtothatofspecimen DA1of0.16%tensilereinforcementratio(4Ø12bottomreinforcement).TheincreaseinultimateloadofspecimensDA3was approximately18%.ThesamechangeinbottomreinforcementwasappliedtoSpecimens,DC1,DC2,andDC3asindicatedin

Table5.Fig.21bshowstheload-deflectionrelationshipsforthosespecimens.Thetensilesteelreinforcementhasaprofound effectonthepost-peakresponseoftheseLWCspecimens.Themaximumincreaseintheultimateloadwas11%asthetensile reinforcementratioincreasedfrom0.16%(4Ø12)to0.54%(4Ø22).

Fig.21cshowsthat,SpecimensDK2andDK3,providedwithadditionalsteelreinforcementaboveandbelowopening, exhibitedmoreductilebehaviorcomparedtothatofSpecimenDC2,whichhadnoadditionalreinforcement.Theincreasein ultimate load of Specimens DK2 and DK3 was approximately 9 % and 15 %, respectively. Adding 2Ø16 additional reinforcementforSpecimenDK3aboveandbelowtheopeningledtoanincreaseintheultimatedeflectionby30%overthat ofSpecimenDC2,whichhasnoadditionalreinforcementaroundedgesoftheopening. Fig.21dshowsloaddeflection relationshipsforspecimenswithadditionalverticalbarsleftandrighttheopening.Theenhancementintheultimateload was13%and23%,forSpecimensDR2andDR3overthatofSpecimenDC2,whichhasnoadditionalreinforcement.Fig.21e showstheload-deflectionrelationshipsforspecimensprovidedwithadditionalreinforcementatallsides.Itcanbeseen fromthefigurethattheloadcarryingcapacityincreaseswithincreasingadditionreinforcementaroundallsideofthe opening.Forexample,theincreaseintheultimateloadandcorrespondingultimatedeflectionforSpecimenDY2were18% and12%,comparedtothose ofSpecimenDC2,whichhasnoadditionalreinforcement.Inaddition,theincreaseinthe ultimateloadandcorrespondingultimatedeflectionofSpecimenDY3were30%and28%overthoseofSpecimenDC2,which hasnoadditionalreinforcement.Theparametricresultsshowedthataddingadditionalreinforcementaroundtheopenings couldbeconsideredasinternalstiffeningofthebeamsaroundopenings.

5.Rationalpredictionofresults(Strut-andTie-modelling)

MostofStrut-and-tiemodels(STM)forsimpledeepbeamswithopeningsinliteraturearebasedonexperimentalresults forcrackpatternsandmodesoffailure[32–36].Inthecurrentresearch,theSTMwasappliedtothestudiedspecimens similartotheanalysiscarriedoutearlierbyFosterandLanGilbert[25];[34];El-Demerdashetal.[36]whohadsingle midspanloading(TypeI)whichissimilartotheappliedloadsinthecurrentresearch.Fig.22showsasimpledeepbeamwith asingletoppointloadatitsmid-spanalongwiththeproposedStrut-and-Tiemodel(Fig.22a).Themodelhastwoconcrete struts,S,onetensiontieT,andthreenodesN1andN2(Figs.22bandc).Theloadtransferreddirectlyfrompointloadto supportthroughtheconcrete,S(Fig.22c).Thestepsforsolvingthebeamsbythenumericalprocedureforoneconcentrated pointLoadsisdetailedasfollows:

(a)Inputdata

Thetermsofbeamsize(h,b,b1,andb2),(a/d),andtheusedconcreteandreinforcementstrength(f0c,andfy)areknown

asinputdata.

(a)TheinternalleverarmLd

a1¼2ðcþ

fstr

Þþ

Xn

i¼1

fi

þs ð5Þ

Thus

Ld¼h0:50ða1þa2Þ ð6Þ

Anda2=0.80a1

(a)InclinationofstrutS

a

¼tan1Ld

a ð7Þ

(a)StrutwidthsW1avandW2:

FromthedetailsofnodeN1,showninFig.22c,W11maybeobtainedfrom:

W11¼a1cos

a

þb1sin

a

ð8Þ

fromequilibriumoftrussforces,

T=Scos(a) (9)

Atnode1

T=0.80fca1b (10)

Wherethevalue(0.80),representstheeffectivenessfactorofthenodalzone(

n

n)[25]atnodeN2:

Scos

a

¼1:0fca2b ð11Þ

Wherethevalue(1.0),representstheeffectivenessfactorofthenodalzone(

n

). a2=0.8a1

Also,fromthegeometryofnodeN2,W12isgivenby:

W12¼a2cos

a

þ

b2

2sin

a

ð12Þ

W1av¼W11þ₂W12 ð13Þ

DR2 1A12 4Ø16 2Ø10 – 2Ø12

DR3 1A12 4Ø16 2Ø10 – 2Ø16

DY2 1A12 4Ø16 2Ø10 2Ø12 2Ø12

(a)Trussforces

Assumingthatthesteelbarsreachtheiryieldstrength(fs=fy),thecompressiveandtensileforcesinthetrussmembers

areasfollows:

T=Asfy (14)

V=Ssinα (15)

(a)Checkofthestresslimits

-Concretestruts:

Theeffectiveconcretecompressivestrengthintheconcretestrutscanbeobtainedfrom:

fce¼vs

f

f

0

c ð16Þ

Wherethevaluesofeffectivenessfactor(

n

s)forstrutsS1andS2arechosen[25],thencheckif

SfsceW1avb ð17Þ

-Nodes:

Theeffectiveconcretecompressivestrengthinthenodalzonescanbeobtainedfrom:

fce¼vn

f

f

0

c ð18Þ

Wherethevaluesofeffectivenessfactor(

n

n)fornodesN1andN2arechosen[25].

CheckthefollowingconditionsatnodeN1:

SfNce1W11b ð19Þ

VfN_ce1b1b ð20Þ

TfNce1a1b ð21Þ

SfNce2W12b ð22Þ

PfNce2b2b ð23Þ

Iftheabove-mentionedchecksaresatis_fied,therequiredultimateshearcapacitycanbeobtained.

Theproposed(STMs)fortesteddeepbeamsareplottedinFig.23wherethedottedlinesindicatethecompression memberswhilecontinuouslinesindicatethetensionmembers.Alltheexperimentallytestedbeamsweresolvedusing STMapproachusingthestepsintheaboveequationsandthemodelsfortestedbeamsaredetailedinFig.23.Results obtainedbytheSTMarerecordedinTable6.ItcanbeseenfromTable6thattheSTMunderestimatestheshearofthe studiedbeamscomparedtoexperimentalresultsfordifferenttestedbeams.Inaddition,theexperimentalresultswerein acceptableagreementwiththoseobtainedusingSTMinmostcases.Moreover,comparingtheresultsinTable4withthose inTable6showsthatthefiniteelementresultsaremoreaccuratethanthoseobtainedbySTM.However,theSTMcanbe usedasarationalapproachfortheanalysisofLWCdeepbeamswithopeningswhichcontainpolystyreneballsascoarse aggregates.

6.Conclusions

Thispapercoversthegapintheliteraturesinceitcontainsmorestudiedvariablessuchaspartialreplacementofcoarse aggregatebypolystyrenefoamballs,thenumberofopenings,webreinforcementratioandpositioningtheopeningsinshear zone.Theauthorscarriedoutexperimentalwork,finiteelementtechnique,andrationalSTMapproachinthisresearchto studytheshearbehaviorofstudiedbeamssubjectedtosinglemidspanconcentratedloads.Thefollowingconclusionscanbe drawnfromthisstudy:

1Theoverallweightofdeepbeamscontainingpolystyrenefoamballsaspartialreplacementofcoarseaggregatewasless thantheircounterpartsofNWCdeepbeamsbyapproximately30%whiletheirshearbehaviorandmodeoffailurewere almostsimilartotheircounterpartsofNWCones.Thisisveryinterestingandpromisingtobuildlighterdeepbeamswith efficientstructuralbehaviour.

2Shearspan-to-depthratio(a/d)hasaconsiderableeffectonthecrackingandtheultimateshearstrengthofdeepbeams withopenings.Thedevelopmentofinitialflexuralcrackswasmorerapidinspecimenswithalargervalueofshear-span-to depthratio.Increasinga/dratiofrom0.97to2.08ledtoreductionofthecrackingandtheultimateshearstrengthsby approximately50%and12%,respectively.

3Thepresenceofopeningsintheshearspanconsiderablyreducedtheultimatestrengthofthespecimens.Sizeoftheweb openingshasamajorimpactonthefailuremodeandultimateshearstrengthofreinforcedLWCdeepbeams.Increasing thedepthoftheopeningfrom20%to40%ofthetotalbeamdepthledtoreductionintheultimateloadby28%and46.4% comparedtothatofsimilarsoliddeepbeams.

4Locatingtheopeninginmidshearspanzoneleadstoahighreductioninshearstrength.Increasingthewidthofthe openingintheshearzoneledtoreductionoftheultimateloadofthespecimensupto66%.

5Applicationoffiniteelementmodellingtotestbeams,yieldedverygoodpredictionofload-carryingcapacities,cracking patternandload-deflectionrelationships.

6Theparametricstudycarriedoutbyfiniteelementshowed thatincreasingtensilesteelreinforcementratio(bottom reinforcement)ledtoanincreaseoftheultimateshearcapacityandultimatedisplacement.Inaddition,internalstiffening ofthebeamsaroundopeningsincreasedtheshearcapacityandthedisplacementductility.Theenhancementofshear capacitywasapproximatelyupto30%.

7Prediction of experimental results using strut-and-tie model was carried out successfully and the agreementwas acceptableinmostcasesbutthefiniteelementresultsweremoreaccuratethanthoseobtainedbySTM.However,theSTM canbeusedasarationalapproachfortheanalysisofLWCdeepbeamswithopeningswhichcontainpolystyreneballsas coarseaggregates.

8Basedontheresultsofthisresearch,itisrecommendedthatthedepthofopeningshouldnotexceed20%ofthedeepbeam depth.Ifthedepthofopeningismorethanthatorliesintheshearspanitishighlyrecommendedforstrengtheningthe openinginternallybyadditionalreinforcementaroundperimeteroftheopening.

DeclarationofCompetingInterest

Thisistodeclarethatalltheauthorshavenoconflictofinterest.

Acknowledgement

October6Universitiesisacknowledgedforfundingthisresearchproject.Theexperimentalworkwascarriedoutinthe reinforcedconcretelaboratoryatAinShamsUniversity.Techniciansandstaffareacknowledgedfortheirvaluableassistance.

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