A beam theory fracture mechanics approach for strength analysis of beams with a hole

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1 A beam theory fracture mechanics approach for strength analysis of beams with a hole Danielsson, Henrik; Gustafsson, Per-Johan Published in: International Network on Timber Engineering Research - Proceedings Meeting 48 Published: 5-8- Document Version Publisher's PDF, also known as Version of record Link to publication Citation for published version (APA): Danielsson, H., & Gustafsson, P-J. (5). A beam theory fracture mechanics approach for strength analysis of beams with a hole. In International Network on Timber Engineering Research - Proceedings Meeting 48 [INTER / 48-9-] Karlsruhe, Germany. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. L UNDUNI VERS I TY PO Box7 L und +4646

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3 Abeamtheoryfracturemechanics approachforstrengthanalysisof beamswithahole HenrikDanielsson,DivisionofStructuralMechanics,LundUniversity,Sweden PerJohanGustafsson,DivisionofStructuralMechanics,LundUniversity,Sweden Keywords:Glulam,Hole,FractureMechanics,StrengthAnalysis. Introduction INTER / Aholeinamemberconstitutesasuddenchangeinthecrosssectionandconcen tratedperpendiculartograintensilestressandshearstressappearinthevicinityof thehole,seefigure..thisstresssituationmayforrelativelylowexternalloads leadtocrackpropagationinthefiberdirection.lookingatthedesignapproachesfor beamswithaholeintimberengineeringdesigncodesoverthelastdecades,itcanbe seenthattheissuehasbeentreatedinmanydifferentways.thetheoreticalback groundsonwhichthedesignapproachesarebasedshowfundamentaldifferences andtherearealsomajordiscrepanciesbetweenthestrengthpredictionsaccordingto thedifferentcodesaswellasbetweentestsandpredictionsaccordingtocodes (Aicher&Höfflin4,Höfflin5,Danielsson&Gustafsson8,Danielsson& Gustafsson). Thereisatthemomentnofullyaccepteddesignmethodbasedonacompletelyra tionalmechanicalbackground.thereareforexamplenodesignequationsforbeams withaholeinthecontemporaryversionofeurocode5(4).however,design equationsarefoundinthegermanandaustriannationalannexestoec5.thisde signapproachoriginatesfromtheworkpresentedbykolbandepple(985),although simplificationsandempiricalmodificationshavebeenaddedovertime

4 Forapplicationswithconcentratedperpendiculartograintensilestressandshear stress,conventionalmaximumstressfailurecriteriaareseldomofanyusebutin steadstrengthanalysisbymeansoffracturemechanicsismorerelevant.theec5 designequationsforendnotchedbeamsanddoweltypeconnectionsloadedatan angletograinareexamplesoffracturemechanicsbaseddesignequations. Concerningthecaseofendnotchedbeams,anequationfortheenergyreleaserate duringcrackextensionandthecorrespondingbeamstrengthwasderivedbycompli anceanalysisusingbeamtheoryalmost3yearsago(gustafsson988).initsec5 implementation,thegustafssondesigncriterionisexpressedasacomparisonofa nominalshearstressandthereducedshearstrength,althoughthedecisivematerial propertiesfromthefracturemechanicsapproacharefractureenergyandstiffness. Inspiredbytherelativesimplicityofthedesignequationforendnotchedbeams,a similarbutgeneralizedmethodwasoutlinedbygustafsson(5).thisgeneralized methodisinthepresentpaperindetaildevelopedandpresentedforthecaseof beamswithahole.theapplicationofabeamtheoryfracturemechanicsapproach forbeamswithaholeismorecomplexthanforendnotchedbeamsdueto: thesignificantinfluenceofshearmakesitnecessarytoconsidertherespective contributionsofmodesiandiitothetotalenergyreleaserate, thenormalforceactingonthecrosssectionpartsaboveandbelowtheholemust beconsideredand thecrosssectionforcesandmomentsactingonthepartsaboveandbelowthe holearestaticallyindeterminate. Inthispaper,abeamtheoryfracturemechanicsapproachforstrengthanalysisof beamswithaholeispresented.thisapproachisbasedonassumptionsofanortho tropicmaterialbehavior,abeammodelaccordingtotimoshenkobeamtheoryanda mixedmodefracturecriterionbasedonlinearelasticfracturemechanics(lefm). Thestrengthanalysisapproachisevaluatedbymeansofacomparisonbetweenthe oreticallypredictedbeamstrengthsandstrengthsfoundfromexperimentaltests.for thispurpose,testresultsforbeamswithcircularholes(höfflin5,aicher&höfflin 6)areusedaswellastestresultsofbeamswithsquareholes(Danielsson8). Thetestsofbeamswithsquareholesarefurtheralsopresentedanddiscussedin (Danielsson&Gustafsson8)and(Danielsson&Gustafsson). Furtherevaluationiscarriedoutbymeansofcomparisontootherstrengthanalysis methods,includingbothcodetypemethodsandmoregeneralmethodsbasedonlin earandnonlinearfracturemechanicsapproachescarriedoutusingdand3dfinite element(fe)analysis. 336

5 Theoryandmodelformulation Thebasicconceptofthepresentapproachforanalysisofbeamswithaholerelates torectangular/squareholesandbyassumptionalsocircularholesareanalyzed.cross sectionforcesandmomentsinthebeampartsaboveandbelowtheholearedeter minedbyequilibriumconsideringkinematicassumptionsaccordingtotimoshenko beamtheory.aninfinitesimallyshortpartofthebeamattheendoftheholeisthen considered.theforcesandmomentactingacrossahorizontalsectionoftheinfinites imallyshortpartofthebeam,dividingitintoonepartaboveandonepartbelowan assumedcrack,aredeterminedbyequilibrium.theenergyreleaseratesformodesi andiiarethenobtainedbyusingthemethodofworkofcrackclosurewithconsidera tiontothedeformationsoftheinfinitesimalpartsaboveandbelowtheassumed crack.thebeamstrengthwithrespecttocrackingisthenfoundbyusingamixed modefracturecriterion. Inordertofacilitateaconvenientformulation,whichisconsistentforcircularand rectangular/squareholeswithorwithoutroundedcorners,anumberofassumptions andsimplificationsareintroduced.thesearepartlybasedonengineeringapproxima tionsandpartlybasedonexperiencefromexperimentaltests.. Basicassumptionsanddefinitionofgeometry Definitionsandnotationfortheparametersusedtodescribethebeamandthehole geometryarefoundinfigure..acircularholewithdiametermayformallybe regardedasarectangular/squareholewith= =.Theparameterdefinesthe holeplacementwithrespecttobeamheightandisgivenbythecoordinateofthe centreofthehole.,andrefertovaluesattheedgeoftheholeonitsrighthand side,althoughthefigurebelowmayseemtoindicateanotherlocation. Theformulationofthepresentapproachisgeneralinthesensethatitallowsforany combinationofcrosssectionalforces,and.dependingonloadingconditions andthesignsof,and,differentareasintheholevicinityareexposedtoper pendiculartograintensilestressandmayexperiencecracking,seefigure..the mostcommoncaseforpracticalengineeringpurposesishoweverbelievedtobe transversalloadinggivingacombinationofshearforceandbendingmomentonly. a INTER / h/ h/ y x r s h u h d h l M V N h b 337

6 Thelocationofthepointofcrackinitiationisassumedtobeknownaprioriandcrack propagationisassumedtooccurintheparalleltograindirection,whichisassumed tobealignedwiththebeamlengthdirection(thedirection).thechosenpredefined cracklocationisbasedonresultsfromtheexperimentaltestsmentionedaboveof beamsloadedinbending,havingeitheracircularorasquarehole.thetestswith squareholeshavingroundedcorners(. )showedthatcrackingcommonly startedatthetoprightcornerofthehole.crackinitiationcommonlyoccurredwithin theroundedpartoftheholecorner.thetestswithcircularholesshowedcrackinitia tionattheholeperipheryatanangleofapproximately45 withrespecttothebeam centralaxis,forholesplacedcentricallywithrespecttothebeamheight.basedon thesefindingsfromexperimentaltests,thelocationofthecrackisinthepresentap proachassumedtobeatanangleof45 fromthebeamlengthdirectionaccordingto Figure.a),yieldingadistancefromtheupperhorizontalholeedgetothecrack planeof.3. Thegeometryofthebeampartsaboveandbelowtheholeareinthecalculations simplifiedinthesensethattheseareassumedtobeprismatic.asillustratedinfigure.b)e),thereareanumberoffeasibleinterpretationsoftheholegeometrywhich resultinthissimplification.theinterpretationusedforallanalysespresentedhereis accordingtofigure.b).interpretationofholegeometryaccordingtofigures. c)d)orsomeothersimplificationmayhoweverpossiblyyieldequalorbetteragree mentbetweenmodelstrengthpredictionsandexperimentallyfoundbeamstrength. a) b) r 45 o.3r assumed crack locaon r 45 o c) d) e) 338

7 . Elementcrosssectionalforcesandmoments Thecrosssectionalforces, and (=,)inthebeampartsabove()andbe low()theholearedeterminedbyequilibriumconsiderationsofthestaticallyinde terminatesystemshowninfigure.3.kinematicassumptionsaccordingtotimo shenkobeamtheoryareusedforbeamelementsand.theverticalcrosssections onthetwosidesoftheholeareassumedtoremainplaneatloading.withthesimpli ficationgiveninfigure.b),thebeampartsaboveandbelowtheholeareassumed tohaveconstantheightsaccordingto () () Toaccountfortheelasticclampingofthebeamelementsinanapproximateway, theseareinthecalculationsgivenalengthslightlylongerthantheactuallengthof thehole,=+.5,basedonnumericalresultspresentedbypetersson(974). L e h M e N h V Accordingtothekinematicassumptions,theinfinitesimallythincrosssectiontothe rightoftheholeremainsplanepriortocrackinitiationandpropagation.atcrack propagationhowever,itissplitintotwocrosssectionsaccordingtofigure.4.the crosssectionsectionalforces, and forcingthecrosssectiontoremain planeintheuncrackedstate maybedeterminedfromequilibriumgiving (3) (4) (5) where, and aretheresultingforcesandthemomentfromthenormal andshearstressesactingontherightsideofthecrosssectionpartabovethecrack planeaccordingto (6) (7) (8) 339

8 h N M V crack locaon h N M V V M N N s M s V s central axis.3 Relativedisplacementsatcrackpropagation Atcrackinitiationandpropagationinthebeamlengthdirection,theoriginalbeam crosssectionissplitintotwoseparatecrosssections.thetworespectivecrosssec tionsarestillassumedtoremainplanebutsincetheyareseparated,theymaybede formedindifferentways.therelativedisplacementsandandtherelativerota tionofthetwocrosssections,aboveandbelowthecrackplane,areillustratedin Figure.5.Therelativedisplacement,relatedto,correspondstomodeIcrack deformation.therelativedisplacementandtherelativerotationareinfluenced byboth and withcorrespondingtomodeiicrackdeformationandassumed tocontributetomodeicrackdeformation. dx aer crack propagaon before crack propagaon -v -θ h crack locaon -u h-h Therelationsbetweentherelativedisplacementsandandtherelativerotation andthecrosssectionsectionalforcesandmoment, and maybeexpressedas (9) wherethedisplacementandforcevectorsaregivenby () () andthecompliancematrixisgivenby 34

9 () Thecomponentsofthecompliancematrixaredeterminedbasedonthecompliance ofthecrosssectionsofinfinitesimallengthaboveandbelowthecrackplane.consid eringkinematicassumptionsaccordingtotimoshenkobeamtheorygives (3) (4) (5) (6) where istheparalleltograinstiffnessand istheshearstiffness..4 Crackclosurework,energyreleaserateandfracturecriterion AccordingtoLEFM,seee.g.textbook(Hellan985),crackpropagationisgovernedby crackpropagationcriteriawhichmaybeformulatedbasedon,forexample,thecon ceptofstrainenergyreleaserateorstressintensityfactors.thestrainenergyrelease ratecan,accordingtolefm,alsobecalculatedintermsof,i.e. theworkrequiredtocompletelycloseapropagatedcrack.thecrackclosurework mayforthepresentapplicationbeexpressedas (7) where and refertothecrackclosureworkrelatedtomodeiandiirespec tively.indices,andrefertotherelativedisplacementsandtherotationillus tratedinfigure.5above.accordingtolefm,thestressintensityfactors and areproportionaltotheappliedload.theprincipleofsuperpositionmaybeusedfor multipleloadcasesgivingcontributionsinthesamemodeofdeformation.forthe presentapplicationthismeansthatthemodeistressintensityfactormaybeex pressedas (8) TherelationshipbetweenthemodeIandmodeIIstressintensityfactors andthe correspondingenergyreleaserates (=I,II)isgivenby (9) where (=I,II)isameasureofthestiffnessofthematerialwithrespecttothecor respondingmodeofdeformation(gustafsson). 34

10 Theenergyreleaserateatcrackextensionoveranarea=isequaltothecor respondingcrackclosureworkgivingforthepresentcase () and () () (3) (4) Inordertodeterminethebeamstrengthwithrespecttocracking,acrackpropaga tioncriterionneedstobechosen.inthepresentapplications,withingeneralmixed modeofloading,thefollowinginteractioncriterionhasbeenused (5) where and aretheenergyreleaseratesinmodeiandiigivenaboveandwhere and arethecorrespondingcriticalenergyreleaserates(orfractureenergies). ResultspresentedinSections4and5,relatingtoanalysisofbeamswithaholeusing thepresentapproach,arebasedon==.5andmaterialpropertyparametersac cordingtotable.thechoiceofcrackpropagationcriterionisnotobvious,andother criteriacouldpossiblybemoresuitable.incaseofnegativevaluesof and/or, thenegativecontributionisignoredandfractureinpuremodeioriiisconsidered. Parameter Notation Value Paralleltograinstiffness MPa Shearstiffness 6MPa Fractureenergy,modeI 3Nm/m Fractureenergy,modeII 9Nm/m 3 Comparisontoendnotchbeamequation Thepresentanalysisapproachforbeamswithaholehasmanyfeaturesincommon withthelefmbaseddesignapproachfornotchedbeams,originallypresentedby Gustafsson(988)andwithmodificationsincludedinEC5.Thepresentapproachfor strengthanalysisofbeamswithaholemayalsobeusedforanalysisofnotched beams,sincethisisessentiallyonlyaspecialcaseintermsofthegeneralgeometry illustratedinfigure.. Asmentionedintheintroduction,therearehoweversomedifferencesinthegeneral formulationbetweenthetwoapproaches.fortheendnotchedbeamapproach,no 34

11 distinctionbetweenmodesiandiiismadeintermsofthecrackpropagationcrite rionandthematerialischaracterizedbythemodeifractureenergyonly.another differencerelatestotheeffectofelasticclampingofthereducedbeampartsatthe sectionwherethenotchorholecornerislocated,whichisconsideredpartlydiffer entinthetwoapproaches.theexpressionforthenominalbeamstrengthwithre specttocrackingatarightanglednotchderivedbygustafsson(988)reads (6) where isthecrackshearforce, isthenetcrosssectionareaatthereduced crosssectionandwhereandaredefinedinfigure3.. Illustrationsofthepredictedbeamstrengthsasinfluencedbythenormalizedbeam heightandthenormalizednotchlengthaccordingtothetwoapproachesare showninfigure3..thecomparisonisbasedonabeamofheight=5mm,with = =3Nm/m andvaluesoftheexponentinequation5as==.for thischoice,i.e.for = and==,isthemixedmodecrackpropagationcrite rionthesameforbothcalculations.moreorlessslightdifferencesinpredicted strengthsaretobeexpectedduetothedifferencesinthemodelformulationswith respecttotheinfluenceoftheelasticclampingofthereducedcrosssection..5.5 αh.5h Normalized beam height α h 4 Comparisontoexperimentaltests Normalized notch length β Examplesoftheoreticallypredictedstrengthsforbeamswithaholearepresentedin Figures4.and4.,wherealsoresultsofexperimentaltestsonbeamswithcircular holes(höfflin5,aicherandhöfflin6)andwithsquareholes(danielsson8) areshown.theoreticalbeamstrengthsrefertothecrackpropagationloadpredicted accordingthetheorypresentedinsection,usingequation5with==.5and withmaterialdataaccordingtotable.thenetcrosssectionareaisdefinedas =( )andthestrengthfromexperimentaltestscorrespondstotheloadatthe instantofcrackpropagationacrosstheentirebeamwidth h βh h 343

12 Beamandholegeometriesarefortheseillustrationschosenbasedonavailabilityof testresults,yieldingpartlydifferentgeometriesforthecaseofcircularandsquare holesrespectively.resultspresentedinfigure4.relatetotheinfluenceofbeam height,holesizeandbendingmomenttoshearforceratioforholescentricallyplaced withrespecttobeamheight(=).resultspresentedinfigure4.relatetotheinflu enceofholecornerradiusandholeplacementwithrespecttobeamheight Test results a = hd = r =.3h M/(Vh) = Beam height h [m] Test results h = 45 mm M/(Vh) =.65 ± Normalized hole height hd/h [ ] Test results a = hd = h/3 r =.hd M/(Vh) = Beam height h [m] Test results a = hd = h/3 r =.hd h = 63 mm M(Vh) = Normalized hole height hd/h [ ] V Test results a = hd =.3h h = 45 mm Moment to shear force ratio M/(Vh) [ ] 4 6 Moment to shear force ratio M/(Vh) [ ] Test results a = hd = h/3 r =.hd h = 63 mm 344

13 5 Comparisonofstrengthanalysisapproaches Anoverviewoftheratiobetweenexperimentallyfoundbeamcapacityandtheoreti callypredictedcapacityaccordingtosomedifferentapproachesforstrengthanalysis isgiveninfigure5..theexperimentaltestsusedforthiscomparisonconsistof beamswitheithercircularholes(höfflin5,aicher&höfflin6)orsquareholes (Danielsson8)alsousedforcomparisoninSection4. Thestrengthanalysismethodsincludedare:. PFM AProbabilisticFractureMechanicsmethodbasedonDFEanalysisand considerationoffractureductilityaccordingtoageneralizedlefmapproachin combinationwithconsiderationofmaterialvariabilityaccordingtoweibulltheory (Danielsson&Gustafsson).. Weibull ClassicalWeibulltheorybasedonDFEanalysis(Danielsson&Gus tafsson). 3. NLFM Anonlinearfracturemechanicsapproach(acohesivezonemodel)based on3dfeanalysis(danielsson&gustafsson4). 4. Presentapproach seeprevioussections. 5. GlulamHandbook endnotchedbeamanalogyapproachforbeamswithahole foundintheoldversionoftheswedishglulamhandbook(carling)andalso includedinadraftversionofec5(). 6. DINEN995/NA SemiempiricalapproachfoundinGermanandAustrian NationalAnnexestoEC5. 7. Aicheretal DesignapproachbasedonWeibulltheorypresentedbyAicher,Höf flinandreinhardt(7).originallysuggestedtobeusedforcircularholesonly buthereusedalsoforsquareholesassuming= = Test results a = hd =.3h h = 45 mm.5 M/(Vh) =.65 s = Normalized hole corner radius r/hd [ ] Shear strength Vc/Anet [Pa] Test results, h = 8 mm Test results, h = 63 mm a = hd = h/3.5 r =.hd M/(Vh) = Hole placement w.r.t beam height s/h [ ] 345

14 Theoretical capacity / Experimental capacity [ ] ( ) ( ) ( ) FE based methods ( ) ( ) code type methods ( ). PFM. Weibull 3. NLFM 4. Present approach 5. Glulam Handbook 6. DIN EN 995 /NA 7. Aicher et al Forthegeneralapproaches(items4above),experimentallyfoundmeanvaluesare comparedtopredictedstrengthbasedonassumedmeanvaluesofmaterialproperty parameters.forthenlfmapproach(item3above),onlyresultsforthebeamswith squareholesarepresentedsinceanalysisofbeamswithcircularholeswerenotin cludedintheworkpresentedin(danielssonandgustafsson4). Thestrengthaccordingtothethreecodetypeapproaches(items57above)are basedoncharacteristicmaterialstrengthvalues =3.5MPaand =.5MPa, validforstrengthclassgl3haccordingtossen48:3.thepredicted strengthsareforthesemethodscomparedtocharacteristicvaluesoftheexperimen tallyfoundstrengths;see(danielsson&gustafsson)forfurtherinformation. Alltestresultsincludedinthecomparisonrelatetoholeswithlargerheight than allowedtobeusedwithoutreinforcementaccordingtothegermanandaustrianna tionalannexestoec5. 6 Discussion Thepresentedapproachisgeneralinthesensethatitallowsforstrengthanalysisin cludingnotonlyloadingintermsofshearforceandbendingmoment,butalsoaxial force.dependingonrelativeloadlevelsandthesignoftheaxialforce,theassumed locationofthecrack(seefigure.)mayhoweverneedtobeadjusted. 346

15 Beamswithholesareinpracticaldesignsituationsoftenreinforced.Thereasonfor thisislikelyrelatedtotwodifferentfactors:(i)theactualstrengthreductiondueto theholeand(ii)theuncertaintyandlackofknowledgerelatedtostrengthanalysis anddesignofbeamswithahole.thecrosssectionsectionalforces, and (see Equations35)couldpossiblybeofinterestinrelationtodesignofinternalorexter nalbeamreinforcementintermsoffullythreadedscrews,gluedinrodsorgluedon panels. AnewversionoftheSwedishGlulamHandbookisplannedtobereleasedduring 5,withinwhichtheendnotchbeamanalogyapproachforbeamswithaholewill beremovedinfavourofthedesignapproachedfoundinthegermanandaustrian NationalAnnexestoEC5. 7 Conclusions Fromtheworkpresentedinthispaper,thefollowingconclusionsaredrawn: ThepresentapproachisconsistentwiththeLEFMbaseddesignapproachforend notchedbeamsfoundinec5. Thepresentapproachappearstobeabletocapturethestrongbeamsizeinflu encefoundfromexperimentaltestsofbeamswithaholefairlywell. Thepresentapproachappearsfairlyaccurateincapturingtheabsolutevaluesof thebeamstrengthforbothcircularandsquareholesofvarioussizesandlocations. Althoughbasedonbeamtheoryanalysis,thepresentapproachisnotsuitablefordi rectincorporationintotimberengineeringdesigncodesofpracticeinitscurrent form,becauseofrathercomplexequations.forcertainapplicationsandloadconfigu rations,moreuserfriendlydesignexpressionsmaypossiblybederivedandcould thenserveasabaseforimproveddesignrecommendations.furtherworkregarding verificationandcalibrationishoweverneededbeforethiscanberealized. 8 References AicherS,HöfflinL(4):Newdesignmodelforroundholesinglulambeams.In: Proceedingsof8 th WorldConferenceonTimberEngineering,vol,Lahti,Finland. AicherS,HöfflinL(6):TragfähigkeitundBemessungvonBrettschichtholzträgern mitrundendurchbrüchen SicherheitsrelevanteModifikationenderBemes sungsverfahrennacheurocode5unddin5.materialprüfungsanstalt(otto GrafInstitute),UniversityofStuttgart,Germany. AicherS,HöfflinL,ReinhardtHW(7):RundeDurchbrücheinBiegeträgernaus Brettschichtholz.Teil.TragfähigkeitundBemessung.Bautechnik84(): CarlingO():Limträhandbok(Glulamhandbook).SvensktLimträAB,Print&Me diacenterisundsvallab,sweden. 347

16 DanielssonH(8):Strengthtestsofglulambeamswithquadraticholes Testre port.reporttvsm753,structuralmechanics,lunduniversity,sweden DanielssonH,GustafssonPJ(8):Strengthofglulambeamswithholes Testof quadraticholesandliteraturetestresultcompilation.in:proceedingsofcibw8 Meeting4,StAndrews,Canada,Paperno.CIBW8/44. DanielssonH,GustafssonPJ():Aprobabilisticfracturemechanicsapproachand strengthanalysisofglulambeamswithholes.eurjwoodprod69:4749. DanielssonH,GustafssonPJ(4):Fractureanalysisofgluedlaminatedtimber beamswithaholeusinga3dcohesivezonemodel.engngfractmech4 5:895. DINEN995/NA:38(3):GermanNationalAnnextoEC5 EN48:3(3):Timberstructures Gluedlaminatedtimberandgluedtimber Requirements.CEN. Eurocode5(4):DesignoftimberstructuresPart:General Commonrules andrulesforbuildings.cen.(en995). Eurocode5():DesignoftimberstructuresPart:General Commonrules andrulesforbuildings.finaldraft9.(pren995). GustafssonPJ(988):Astudyofstrengthofnotchedbeams.In:ProceedingsofCIB W8Meeting,Parksville,Canada,Paperno.CIBW8/. GustafssonPJ():Meanstressapproachandinitialcrackapproach.In:AicherS, GustafssonPJ(ed)HallerP,PeterssonH:Fracturemechanicsmodelsforstrength analysisoftimberbeamswithaholeoranotch areportofrilemtc33.report TVSM734,StructuralMechanics,LundUniversity,Sweden. GustafssonPJ(5):Mixedmodeenergyreleaseratebyabeamtheoryappliedto timberbeamswithahole.in:abstractbookof th InternationalConferenceon Fracture,Turin,Italy. HellanK(985):Introductiontofracturemechanics.InternationalEdition.McGraw Hill. HöfflinL(5):RundeDurchbrücheinBrettschichtholzträger Experimentelleund theoretischeuntersuchungen.phdthesis,materialprüfungsanstalt(ottografin stitute),universityofstuttgart,germany. KolbH,EppleA(985):VerstärkungvondurchbrochenenBrettschichtholzbindern. VorschungsvorhabenI.4348,ForschungsundMaterialprüfungsanstaltBaden Württemberg,Germany. PeterssonH(974):Analysisofloadbearingwallsinmultistoreybuildings:stresses anddisplacementscalculatedbyacontinuummethod.phdthesis,chalmersuni versityoftechnology,sweden. ÖNORMB995:4(4):AustrianNationalAnnextoEC5. 348

17 Discussion The paper was presented by H Danielsson H Blass commented that the stress distribution shown in slide 8 close to the hole is very different from reality. He questioned why the results are so good. H Danielsson agreed but stated that the beam theory is exact for normal stresses but might be different for shear stresses. PJ Gustafsson added that the solution should be exact in terms of energy release rate. As the crack propagated, the normal forces due to the moment should be exact but shear would not be true. BJ Yeh asked how close the hole can be to the support and if there were two holes, how close can they be. H Danielsson responded the distances should be such that the support would not influence the hole. This would also apply to the multiple holes cases. K Malo asked if there would be a limitation to the hole size. H Danielsson said that there would be no limit and the analysis would also work for notched beams. I Smith commented that good results need accurate analysis, real structure, and accurate material properties; when results do not agree, maybe some of these are not working. P Dietsch and H Danielsson discussed the cases of round and square holes and limitations to the model. R Jockwer received clarifications that at this moment there is no recommendation for the size of the hole without reinforcement. 349