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Design of timber members subjected to axial compression or combined axial compression and bending based on 2nd order theory

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The structural behaviour of timber members subjected to axial compression or combined axial compression and bending is characterised by a non-linear increase of the deformations due to geometric and physical non-linearity. The geometric non-linear behaviour results from an increasing eccentricity of the axial load (P-delta effect). The physical non-linearity is caused by the material behaviour of timber when subjected to compression parallel to the grain. The influence of the P-delta effect on the load-bearing capacity of timber members subjected to axial compression was investigated first by Tetmajer in 1896. Tetmajer’s studies set up the basis for the design of timber members subjected to axial compression for a long time. Tests performed by Larsen and Pedersen confirmed the results obtained by Tetmajer. The experimental investigation showed the great influence of the varying material properties on the load-bearing capacity. In order to account for these variations and hence to estimate the resistance of glued laminated timber members subjected to compression more accurately, Blass performed Monte Carlo simulations. The buckling curves given in different design codes were derived from these investigations. For timber members subjected to combined axial compression and bending, Buchanan developed a numerical model capable of investigating the influence of the non-linear material behaviour on the moment – axial force interaction. Eurocode 5 provides two different approaches for the design of centrically and eccentrically loaded timber columns: – a simplified calculation model based on the Effective Length Method (ELM), – 2nd order analysis of the structure. In ELM, the buckling problem of a structural system is reduced to that of an equivalent simply supported (pinned) column. The 2nd order analysis of the structure takes into account the non-linearity by studying the equilibrium of the deformed structural system. In general, non-linearity caused by the increasing eccentricity of the external load as well as non-linearity caused by the non-linear material behaviour of timber subjected to compression should be considered. However, current design codes only provide rules for a 2nd order linear elastic analysis of the structure and the effects caused by the non-linearity of the material are neglected. The two approaches (ELM and 2nd order linear elastic analysis of the structure) given in the codes are not consistent and can lead to different results. This situation led to controversial discussions in the scientific community. The paper examines the behaviour of structural timber members subjected to axial compression or combined axial compression and bending. Based on experimental and numerical investigations, the accuracy of the existing approach in Eurocode 5 for the design of timber members subjected to axial compression or combined axial compression and bending is assessed and modifications are suggested. By means of extensive experimental investigations, a data base was created for the validation of calculation models and for the assessment of design concepts. In order to assess the behaviour of timber members subjected to axial compression or combined axial compression and bending, strain-based calculation models were developed. The models base on equilibrium conditions for the deformed shape of the member and their efficiency is demonstrated by benchmarking them to the results of the experimental investigations. Using the developed models, Monte Carlo simulations were performed with the aim of investigating the influence of the varying material properties. The mechanical models were combined with a probabilistic model which accounts for the variation in material properties. It turned out, that the variation of the material properties influences the load-bearing behaviour of timber members subjected to compression or combined compression and bending significantly. The investigations indicate that the existing approach of Eurocode 5 based on 2nd order analysis can lead to an overestimation of the load-bearing capacity. Hence, a modified design approach was developed which agrees with the results of the Monte Carlo simulations very well and thus ensures a safe and economical design of timber members subjected to compression or combined compression and bending. The modified design approach is based on a “buckling” modulus Tk,d as a function of the axial compression. The “buckling” modulus Tk,d is used for the calculation of the Euler buckling load Fc,crit,d that is necessary for the determination of the 2nd order magnification factor.
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45
Designoftimbermemberssubjectedto
axialcompressionorcombinedaxial
compressionandbendingbasedon2nd
ordertheory
AndreaFrangi,ETHZurich,InstituteofStructuralEngineeringIBK,Zurich,Switzerland
RenéSteiger,Empa,MaterialsScienceandTechnology,StructuralEngineering
ResearchLaboratory,Dübendorf,Switzerland
MatthiasTheiler,dspIngenieure&PlanerAG,Greifensee,Switzerland
Keywords:Timberstructures,columns,compressionparalleltothegrain,stability,
globalbuckling,Pdeltaeffect,2ndorderstructuralanalysis
1 Introduction
Axialcompressionorcombinedaxialcompressionandbendingareencounteredin
manytypesoftimbermemberssuchascolumns,framestructuresorcompression
membersoftrussgirders.Thebehaviourofthesestructuralmembersisprimarily
characterisedbythenonlinearincreaseofthedeformationduetotheincreasing
eccentricityoftheaxialload(Pdeltaeffect).Inadditiontothisgeometricnonlinear
behaviour,thenonlinearmaterialbehaviouroftimbermemberssubjectedto
compressionparalleltothegrainhastobeaccountedfor.TheinfluenceoftheP
deltaeffectontheloadbearingcapacityoftimbermemberssubjectedtoaxial
compressionwasinvestigatedfirstbyTetmajer(1896).Tetmajer’sstudiessetupthe
basisforthedesignoftimbermemberssubjectedtoaxialcompressionforalong
time.TestsperformedbyLarsenandPedersen(1975)confirmedtheresultsobtained
byTetmajer.Theexperimentalinvestigationshowedthegreatinfluenceofvarying
materialpropertiesontheloadbearingcapacity.Inordertoaccountforthese
variationsandhence,toestimatetheresistanceofgluedlaminatedtimbermembers
subjectedtocompressionmoreaccurately,Blaß(1987and1991)performedMonte
Carlosimulations.Thebucklingcurvesgivenindifferentdesigncodes(SIA265:2012,
EN199511:2004andDIN1052:2008)werederivedfromtheseinvestigations.
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46
Fortimbermemberssubjectedtocombinedaxialcompressionandbending,
Buchanan(1984and1985)developedanumericalmodelcapableofinvestigatingthe
influenceofthenonlinearmaterialbehaviouronthemomentaxialforce
interaction.InadditionBuchananinvestigatedtheinfluenceofthesizeofthe
member.CurrentdesigncodessuchasEurocode5(EN199511:2004),theSwiss
nationalcodeforthedesignoftimberstructures(SIA265:2012)orthewithdrawn
Germancode(DIN1052:2008)providetwodifferentapproachesforthedesignof
centricallyandeccentricallyloadedtimbercolumns:
asimplifiedcalculationmodelbasedontheEffectiveLengthMethod(ELM),
2ndorderanalysisofthestructure.
InELM,thebucklingproblemofastructuralsystemisreducedtothatofan
equivalentsimplysupported(pinned)column.The2ndorderanalysisofthestructure
isamethodwhichtakesintoaccountthenonlinearitybystudyingtheequilibriumof
thedeformedstructuralsystem.Ingeneral,nonlinearitycausedbytheincreasing
eccentricityoftheexternalloadaswellasnonlinearitycausedbythenonlinear
materialbehaviouroftimbersubjectedtocompressionshouldbeconsidered.
However,the2ndorderanalysisisoftenunderstoodasatheorybasedonlinear
elasticmaterialbehaviourandtheeffectscausedbythenonlinearityofthematerial
areneglected.Eventhedesigncodes(SIA265:2012,EN199511:2004andDIN
1052:2008)onlyproviderulesforthis2ndorderlinearelasticanalysisofthestructure.
Inthispaper,acleardistinctionbetweenthe2ndorderlinearelasticanalysisandthe
generalised2ndorderanalysisismade.
Thetwoapproaches(ELMand2ndorderlinearelasticanalysisofthestructure)given
inthecodesarenotconsistentandcanleadtodifferentresults.Thissituationledto
controversialdiscussionsinthescientificcommunity(Kesseletal.2005and2006;
Möller2007;Köhleretal.2008).Thediscussioninparticularshowedthatthereare
inconsistenciesconcerningtheconsiderationoftheeffectofmoisturecontent(MC)
anddurationofload(DOL)aswellasinconsistenciesconcerningtheimplementation
ofthe2ndorderlinearelasticanalysisinthedesigncodes.Whilerecentresearchon
theloadbearingbehaviouroftimbermemberssubjectedtoaxialcompressionor
combinedaxialcompressionandbendingwasmainlyfocusedonMCandDOL(Kessel
etal.2005and2006;Möller2007,BeckerandRautenstrauch2001,Hartnacketal.
2002)thispaperdealswiththeinfluenceofthenonlinearmaterialbehaviourand
withtheimplementationof2ndorderlinearelasticanalysisinthedesigncodessuch
thatthereareonlyminordifferencesbetween2ndorderlinearelasticanalysisandthe
EffectiveLengthMethod.Theresultspresentedhereareonlyvalidforshortterm
responseunderloadatconstantinteriorclimate.Infact,MCandDOLandin
particularthecreepbehaviourandtheclimatetakeamajorimpactontheload
bearingbehaviouroftimbercolumnsandshouldalsobeconsideredforthedesignof
timbermemberssubjectedtocompressionorcombinedcompressionandbending
(Hartnack2004,HartnackandRautenstrauch2005,Becker2002).
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47
2 Designoftimbermembersincompression
paralleltothegrain
Ingeneral,theEffectiveLengthMethodisusedforsimpledesignsituations(e.g.
verificationofstabilityofsinglemembers)whilethe2ndorderlinearelasticanalysisof
thestructureprovidessomeadvantagesformorecomplexdesignsituations(e.g.
impactofstiffnessofmembersandconnectionsontheforcedistributionintrussand
framestructures,designofbracing).
2.1 EffectiveLengthMethod(ELM)
ThesimplifiedcalculationmodelisbasedontheEffectiveLengthMethod(ELM).The
bucklingproblemofastructuralsystemisreducedtothatofanequivalentsimply
supported(pinned)column(Blass,1995).Forthedesign,theinternalforcesand
momentsarecalculatedbasedonasimple1storderanalysisandthenonlinearP
deltaeffectistakenintoaccountbymeansofabucklingfactorkc.Thisfactor
describestheratiobetweentheaxialstressatbucklingfailureofamembersubjected
toaxialcompressionanditscompressivestrengthparalleltothegrain.kcdependson
theeffectivelengthofthestructuralsystemwhichcanbeexpressedbythe
slendernessratio
.
Thebucklingfactorkcasgivenindifferentdesigncodes(EN199511:2004,SIA
265:2012andDIN1052:2008)isbasedonextensiveinvestigationsperformedbyBlaß
(1987).Inordertodeterminethecharacteristicvalue(i.e.5thpercentile)oftheload
bearingcapacityoftimbercolumnsBlassperformedMonteCarlosimulations.The
numericalmodelandtheparameterstudyconsideredthePdeltaeffect,the
variabilityofthestrengthandthestiffnesspropertieswithinthetimbermembers,
thegeometricimperfectionofthetimbermembersandthenonlinearmaterial
behaviouroftimberwhensubjectedtocompressionparalleltothegrainand
bending.
Fortheultimatelimitstateanalysis,thedesigncodes(EN199511:2004,SIA
265:2012andDIN1052:2008)recommendusingalinearinteractionmodelfor
combinedaxialcompressionandbending.Inthisinteractionmodel,thebuckling
factorkcisusedtoreducethecompressivestrengthparalleltothegrainofthe
timbermemberinordertoaccountforbuckling.
2.2 2ndorderlinearelasticanalysis
AsanalternativetothecalculationmodelbasedontheELM,timbermembers
subjectedtoaxialcompressionorcombinedaxialcompressionandbendingcanbe
designedbyperforminga2ndorderlinearelasticanalysis.The2ndorderlinearelastic
analysisisamethodwhichtakesintoaccountthegeometricnonlinearitybystudying
theequilibriumofthedeformedstructuralmember.Aninitialdeformationis
introducedintothecalculationinordertoaccountforthegeometricimperfectionof
thememberase.g.deviationfromaperfectlystraightshape.
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48
Forasimplysupported,axiallyloadedcolumnthe2ndorderlinearelasticanalysiscan
easilybeperformed,assumingsinusoidaldistributedinitialdeformations.Theinitial
deformationincombinationwiththeaxialloadleadstoaninitialbendingmoment
MI.ThePdeltaeffectcausesamagnifiedmomentMII.MIIcanbecalculatedby
multiplyingtheinitialbendingmomentMIwithamagnificationfactor
(Bazantand
Cedolin1991):
III MM
(1)
Euler
N
N
1
1
and2
2
cr
Euler
EI
N
(2)
With
MII: magnifiedbendingmoment(2ndorderlinearelastictheory,deformedstructure)
MI: initialbendingmoment(1stordertheory,undeformedstructure)
: magnificationfactor
N: normalforceactingonthecolumn
NEuler:Eulerbucklingload
E: modulusofelasticity(MOE)
I: 2ndmomentofinertia
cr: effectivelength
Timbermemberssubjectedtocombinedaxialcompressionandbendingtendto
developnonlineardeformationsofthecompressionzonebeforefailureoccurs.This
nonlinearityleadstoacurvedshapeofthemomentaxialforceinteractiondiagram
dependingontheratiobetweenthetensilestrengthft,m,0andthecompressive
strengthfc,m,0paralleltothegrain(Buchanan1985,SteigerandFontana2005).For
theultimatelimitstateanalysisoftimbercolumnsthedesigncodes(SIA265:2012,
EN199511:2004andDIN1052:2008)considerthisnonlinearinteractionbehaviour
bysquaringthecompressionpartintheinteractionmodel(Eq.(3)).However,the
nonlinearmaterialbehaviouralsoinfluencesthedeformationsofthestructural
system,andasaconsequence,alsothemagnifiedmomentMIIisinfluencedbythe
nonlinearmaterialbehaviour.However,theseeffectsareneglectedwhen
performinga2ndorderlinearelasticanalysis.
0.1
,
,,
2
,0,
,0,
dm
dIIm
dc
dc
ff
(3)
With
c,0,d:designvalueoftheactingcompressivestressparalleltothegrain
fc,0,d:designcompressivestrengthparalleltothegrain
m,II,d:designvalueoftheactingbendingstressfroma2ndorderstructuralanalysis
fm,d:designbendingstrength
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49
3 Strainbasedmodel
Inordertopredicttheglobalbucklingbehaviouroftimbercolumns,anumerical
strainbasedmodelhasbeenimplemented.Strainbasedmodelsarewidelyusedin
thedesignofstructuralmembersmadefromotherconstructionmaterialsthan
timber.E.g.forreinforcedconcretecolumns,astrainbasedmodelissuggestedin
CEB/FIPManual(1978).Duetothefailuremechanismintimberbeinginfluencedby
thedistinctnonlinearstressstrainrelationshipandleadingtoamorecomplex
calculationprocedure,uptonow,onlyafewapplicationsofthesemodelstotimber
structuresarereportedinliterature(Buchanan1984,Hörsting2008).
Figure1showsthecalculationprocedureofthestrainbasedmodel.Onthelefthand
side,thecalculationoftheinternalforceNiandbendingmomentMiisillustrated.
Thecalculationstartswithselectingvaluesforthestrain
0atthemasscentreofthe
crosssectionandforthecurvature
y.Thesetwoparametersdefinethestrain
distributionwithinthewholecrosssection,whenassumingthatplanesections
remainplane.Basedonthestraindistribution,thestressdistributioniscalculated
usingtherelationshipgivenbythestressstraincurve.Anyshapeofstressstrain
curvecanbeappliedinthecalculation.Finally,theinternalforceNiandmomentMi
areestimatedbyintegratingthestressesoverthewholecrosssection.Theright
handsideofFigure1showsthecalculationoftheexternalforceNeandbending
momentMe.TheexternalbendingmomentMedependsontheexternalforceNeas
wellasonthedeformationofthecolumnduetotheinitialimperfectionsandtheP
deltaeffect.Sincethecurvatureisequaltothe2ndderivationofthedeflectioncurve,
themaximaldeflectioneIIofthecolumnduetothePdeltaeffectcanbecalculatedas
afunctionofthecurvature
y.BoththeinternalmomentMiandtheexternal
momentMedependonthecurvature
y.Hence,equilibriumbetweeninternaland
externalforcesandmomentscanbeobtainediteratively.
Figure1.Calculationprocedureinthestrain‐basedmodel(Theileretal.2013)

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50
Thestrainbasedmodelhasbeenusedforstudyingtheinfluenceofvarious
parameters.E.g.itcanbeshownthattheplasticbehaviouroftimberwhensubjected
tocompressionparalleltothegrainconsiderablyinfluencesthebucklingbehaviourof
columns(Theileretal.2013).Therefore,theapplicationofanadequatematerial
model(stressstrainrelation)isessentialwhenmodellingthebehaviouroftimber
memberssubjectedtocompression.Inthepresentstudy,themodelproposedby
Glos(1978)hasbeenused,sinceitappearstobemoresuitablethanothermaterial
modelsbecauseitisbasedonextensiveexperimentalinvestigationsonsolidtimber
boards.Inaddition,Glos(1981)developedthemodelfortimbermemberssubjected
tocompressionandbendingwhileothermodelsaremainlyfocusedontimber
memberssubjectedtopurebending.Glos’smodelaccountsforthereductionof
stiffnessbeforereachingtheultimatecompressionstrengthaswellasforthe
subsequentsoftening.Figure2qualitativelyshowsthestressstrainrelationship
proposedbyGlos.Thedescriptionofthefullcurveinamathematicalformasksforsix
parameters(Figure2right).
Figure2.Qualitativerepresentationofthestress‐strainrelationshipinthematerialmodelproposed
byGlos(1978).
4 Experiments
Anexperimentalcampaignongluedlaminatedtimbermemberssubjectedto
eccentriccompressionhasbeenperformedatETHZurich(TheilerandFrangi2015).
Theaimoftheexperimentalinvestigationswastocreateadatabase,whichcouldbe
usedtovalidatetheoreticalcalculationmodelsandtoassesstheaccuratenessofthe
designapproachesgivenincodesforthedesignoftimberstructures.Thespecimens
wereproducedusinglamellasmadeofNorwayspruce(piceaabies)grownin
Switzerland.Atotalof336lamellaswereavailable.Inthefirststep,nondestructive
testsonthelamellaswereperformed.Thesetestsaimedatcollectingdatainorderto
characterisetherawmaterial.Inthesecondstep,thelamellaswerestrengthgraded.
Theaimofthegradingprocesswastoselecttwoclassesoflamellasforthe
productionofthetestspecimens.Thelamellaswereselectedsothattheywere
suitabletoproducegluedlaminatedtimberofstrengthclassesGL24handGL32h.
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Withinthegradingprocess,visualgradingcriteriaaswellasmachinegradingcriteria
(dynamicMOE)wereused.Specimensforfivetestsserieswereproduced,three
seriesofgluedlaminatedtimberGL24handtwoseriesofgluedlaminatedtimber
GL32h(Table1).Eachofthetestseriesconsistedoftenspecimens.Thelengthofthe
timbermemberswasvariedbetweenthedifferenttestseries:L=1’400mm,L=
2’300mmandL=3’200mm.Thecrosssectionwas140mmx160mm.
Table1.Overviewoftestseriesongluedlaminatedtimbermemberssubjectedtoeccentric
compressionperformedatETHZurich(TheilerandFrangi2015)
Test
series
Number
oftests
Strength
class
Crosssection
[mm]
LengthL
(Slenderness)
Meanvalueofaxialstress
atfailure(COV)
110GL24h140x1601'400mm(30.3)25.6N/mm2(0.07)
210GL24h140x1603'200mm(69.3)15.3N/mm2(0.11)
310GL24h140x1602'300mm(49.8)20.3N/mm2(0.12)
410GL32h140x1601'400mm(30.3)31.1N/mm2(0.09)
510GL32h140x1603'200mm(69.3)18.1N/mm2(0.10)
Duringtheglulamproduction,thesetupofthetestspecimenswasrecorded.Hence,
thepositionandtheorientationofeverylamellawithineachtestspecimenwere
documented.Finally,thegluedlaminatedtimbermembersweresubjectedto
bucklingtests.Thetestspecimenswereloadedwithaneccentric(15mm)
compressionforceuptofailure.Duringthetests,theappliedloadsaswellas
horizontalandverticaldeformationswererecorded.Forasubsampleof20test
specimens,additionallocaldeformationmeasurementswereperformedusingan
opticalmeasurementsystem.
Figure3.Measuredload‐bearingcapacityforalltestsperformedincomparisonwiththeassumed
lognormaldistributionestimatedfromthetestresults(Theiler2014).
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52
ThegraphsinFigure3showtheresultsofalltestsperformedincomparisonwiththe
assumedlognormaldistributionestimatedfromthetests.Agoodagreement
betweentestresultsandestimatedlognormaldistributioncanbeseen.The
coefficientofvariation(COV)wasintherangeof10%foralltestseries(Table1).All
detailsofthetestsarepresentedinatestreport(TheilerandFrangi2015).The
resultsofthetestshavebeenusedforthevalidationofthestrainbasedmodel.
5 Numericalsimulations
Inordertoaccountforthevariationinmaterialproperties,MonteCarlosimulations
wereperformed.Columnsofdifferentslendernessanddifferentstrengthgrades
weremodelledwiththestrainbasedmodelbyassigningthemrandomlyselected
materialproperties.Sixdifferentparametersareneededtodescribethefullstress
strainrelationship(Figure2).Thedistributionsofthepropertiesintermsof
probabilitydensityfunctionPDFaswellasthecorrelationbetweenthedifferent
propertieshavetobetakenintoaccount.Thestudywasperformedfortwodifferent
gradesofsolidtimber(C24andC30)andgluedlaminatedtimber(GL24handGL32h).
CharacteristicvaluesgiveninEN338:2009andEN14080:2013wereconsidered.
However,thecharacteristicvaluesarenotsufficientforthestochasticmodelling.
Furtherinformationonthevariabilityofthemechanicalpropertiesisrequiredthat
forexamplecanbefoundintheJCSSProbabilisticModelCode(2007).Inaddition,
Glos(1978)investigatedvariabilityandcorrelationofthemodelparameters.Using
theseinvestigationsandthecharacteristicvaluesasabasis,foreachmaterial
propertyameanvalue,astandarddeviationandaprobabilisticdensityfunctionwas
estimated(Table2).
Table 2. Mean value, standard deviation and probability distribution function PDF used for the
numericalsimulations(seefigure2forthedefinitionofthematerialproperties)(Theiler2014)
Materialproperty C24C30GL24hGL32h
Et,0,Ec,0[N/mm2]
Meanvalue11’00012’00011’50014’200
Standarddeviation2’2002’4001’5001’850
PDFLognormalLognormal
ft,m,0[N/mm2]
Meanvalue38.848.634.045.5
Standarddeviation9.712.25.16.8
PDFLognormalLognormal
fc,m,0[N/mm2]
Meanvalue29.932.630.440.2
Standarddeviation5.35.63.95.2
PDFLognormalLognormal
εc,0[]
Meanvalue3.40*1033.40*1033.27*1033.51*103
Standarddeviation6.80*1046.80*1044.25*1044.57*104
PDFLognormalLognormal
fc,m,u,0[N/mm2]
Meanvalue25.427.725.633.9
Standarddeviation3.84.22.63.4
PDFLognormalLognormal
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Startingfromthestochasticallymodelledmaterialproperties,2ndordersimulations
werecarriedoutwiththestrainbasedmodel.Foreachtimbergradeandslenderness
ratio10’000simulationswereperformedallowinganaccurateestimationofthe
meanvalueandthe5%fractilevalueoftheloadcarryingcapacity.Altogether,about
twomillionsimulationswereperformedinthisresearchproject.
Figure4.Comparisonbetweenexperimentaldataandnumericalsimulationsforgluedlaminated
timberofstrengthclassesGL24andGL32h(Theiler2014).
Figure4showstheresultsofthenumericalsimulationsperformedfortheanalysisof
thetestresultsforgluedlaminatedtimberofstrengthclassesGL24handGL32h.It
canbeseenthatthevariationinthenumericalpredictionislargerforstockycolumns
thanforslenderones.Thiscanbeexplainedbythevariationoftheinputparameters.
Theloadbearingcapacityofstockycolumnsisgovernedbythecompressionstrength
paralleltothegrainwhiletheloadbearingcapacityofslendercolumnsisgoverned
bythemodulusofelasticityMOE.Therefore,thevariationinthenumericalprediction
isadirectconsequenceofthevariationoftheseparameters.Forcolumnsof
intermediateslendernessvariousmaterialpropertiesaswellastheinitialdeflection
influencetheloadbearingcapacity.
6 AssessmentandimprovementoftheEurocode5
designapproach
TheMonteCarlosimulationsperformedallowcheckingtheaccuratenessofthe
designapproachesgiveninEurocode5.InFigure5theresultsofthenumerical
simulationsusingasinputparametersthevaluesgiveninTable2arecomparedto
analyticalcalculationsbymeansofELMand2ndorderlinearelasticanalysisforglued
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54
laminatedtimberGL24handGL32h.Resultsofthenumericalsimulationsforsolid
timberC24andC30canbefoundinTheiler(2014).
Figure5.ComparisonbetweenELM,2ndorderlinearelasticanalysisandnumericalsimulationsfor
gluedlaminatedtimberGL24handGL32h.DesignequationstakenfromEurocode5.
Itcanbeseenthatthe2ndorderlinearelasticanalysismayleadtoanoverestimation
oftheloadbearingcapacityespeciallyforcolumnsofintermediateandhigh
slenderness(
>50).Forslendercolumns(
>100),thecharacteristicvalues
obtainedfrom2ndorderlinearelasticanalysisareintherangeofthe25thpercentile
ratherthanintherangeofthe5thpercentile.Thisindicatesthatthedesignrulesfor
the2ndorderlinearelasticanalysisgiveninEurocode5donotensureanaccurate
designoftimbermemberssubjectedtocompressionandthereforeshouldbe
modified.However,thisconclusionisonlyvalidforsimplysupportedcolumns.
InEurocode5andotherdesigncodesthe2ndorderlinearelasticanalysisisbasedon
theassumptionoflinearelasticmaterialbehaviour,whichmeansthatthereduction
instiffnesscausedbythenonlinearmaterialbehaviourisneglectedandtheload
bearingcapacityisoverestimated.Inordertoreachabetteragreementacorrection
factorhastobeintroducedinthedesignequation.Possibilitiestomodifythedesign
approacharetoenlargetheinitialdeformationsortoreducethedesignstiffness.
Inparticular,thereductionofthedesignstiffnessseemstobeapracticablesolution,
sinceitdescribesthephysicalphenomenaaccurately.In1889Engesserintroduced
thedesignconceptofreductioninstiffnessforsteelcolumnsandsuggestedtousea
tangentmodulusinsteadoftheMOE.Engesser’stheorywasfirstquestionedbyother
scientistsbutShanleyshowedin1947,thatEngesser’smethodwasavaluable
possibilitytoaccountfornonlineardeformationsinthecompressionzone.

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Basedontheresultsoftheresearchproject(Theiler2014)averygoodagreement
between2ndorderlinearelasticanalysisandnumericalsimulationsisobtainedwhen
usingabucklingmodulusdefinedasfollows:
M
dk
E
T
05,0
,for 5.0
0,,0,
dcdc f
(5)
T
dc
dc
M
dk f
E
T
121
,0,
,0,05,0
,for 5.0
0,,0,
dcdc f
(6)
With 0.3
T
forsolidtimberand 0.4
T
forgluedlaminatedtimber.
Figure6.Comparisonbetween2ndorderlinearelasticanalysisbasedonabucklingmoduslusTk,dand
numericalsimulationsforgluedlaminatedtimberofstrengthclassesGL24handGL32h.
AsimilarapproachwasalreadyproposedbyRošandBrunner(1931)andusedinthe
previousstandardSIA164:1981(Dubas1981).Theinvestigationshaveshownthat
thestrengthdependentreductionofthestiffness(Eq.(5)and(6))leadstoavery
goodagreementbetweenthe2ndorderlinearelasticanalysisandthenumerical
simulations(Figure6).Ontheotherhand,thestrengthdependentreductionofthe
stiffnessforhighloadlevelsleadstomorelaboriousdesignprocedure.Thebuckling
modulushastobedeterminedbymeansofiterationandcanbedifferentfor
differentdesignsituation.Additionalcalculationshaveshownthatforpracticaldesign
theestimationofthebucklingmodulusTk,daccordingtoEquation5isareasonable
solutionevenforhighloadlevels(
c,0,d
f
c,0d
0.5
)makingthedesigneasierforthe
engineersasnoiterationisneeded.
Thisstudyconcentratesonthebehaviourofsimplysupportedtimbercolumns.For
structuralsystemssuchasframestructuresthebehaviourisdifferentduetothe
distributionoftheaxialloadinthesinglemembers.Sincetheaxialloadinfluencesthe
INTER / 48 - 02 - 02
56
reductionofthestiffnessduetotheplasticdeformations,thebucklingbehaviour
dependsonthedistributionoftheaxialloadand,asaconsequence,theresults
obtainedwiththeELMortheadjusted2ndorderlinearelasticanalysis(Eq.(5)and
(6))wouldbetoosafeasshowninareliabilityassessmentperformedinaprevious
analysis(Köhleretal.2008).
7 Conclusions
Whendesigningtimbermemberssubjectedtosimultaneouslyactingaxial
compressionandbendingmoment,theincreaseofthebendingmomentduetothe
eccentricityoftheaxialforceandduetothenonlinearmaterialbehaviouroftimber
subjectedtocompressivestresshastobetakenintoaccount.Thecurrentdesign
codesprovidetwodifferentapproachesforthedesignofrespectivemembers
(simplifiedanalysisbasedontheEffectiveLengthMethodand2ndorderlinearelastic
analysisofthestructure).However,thetwodesignapproachesarenotconsistent
andcanleadtodifferentresults.Basedontheinvestigationsperformed,the
followingconclusionscanbedrawn:
Theloadbearingcapacityofstockycolumns(
<20)isgovernedbythe
compressionstrengthparalleltothegrain.Forslendercolumns(
>100)the
modulusofelasticity(MOE)isthedominantmaterialproperty.Forcolumnsof
intermediateslendernessratio(50<

<100),thecompressionstrengthparallelto
thegrain,theMOEandthenonlinearmaterialbehaviourimpacttheloadbearing
capacity.
Whenperforminga2ndorderlinearelasticanalysis,thenonlinearmaterial
behaviouroftimbercannotbetakenintoaccount.Consequently,anadjustmentof
theresultsobtainedwiththismethodisrequired.Thiscanbedonebyreducingthe
designstiffnessofthestructuralmember.Theuseofabucklingmodulusfor
columnsappearstobeanappropriatesolution.
Whendesigningsinglecolumnsorbeam‐columnsusinga2ndorderlinearelastic
analysis,thecalculationofthedesignvalueofthebucklingmodulusTk,dshouldbe
basedon5thpercentilevaluesofthemodulusofelasticityE0,05.Theinvestigations
haveshownthatthestrengthdependentreductionofthestiffness(Eq.(5)and(6))
leadstoaverygoodagreementbetweenthe2ndorderlinearelasticanalysisand
thenumericalsimulations.However,theestimationofthebucklingmodulusTk,d
accordingtoEquation5isareasonablesolutionevenforhighloadlevels
(
c,0,d
f
c,0d
0.5
)makingthedesigneasierfortheengineersasnoiterationis
needed.
Whendesigningstructuralsystemsusinga2ndorderlinearelasticanalysis,the
calculationofthedesignvalueofthebucklingmodulusTk,dshouldbebasedon
meanvaluesofthemodulusofelasticityE0,mean,astheuseof5thpercentilevalues
wouldleadtotoosaferesults.
INTER / 48 - 02 - 02
57
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... For columns made of spruce glulam GL 24h and GL 32h FRANGI ET AL. (2015) and for beech glulam GL 40h, GL 48h and GL 55h EHRHART ET AL. (2019) experimentally and numerically determined lower load-bearing capacities (up to 18 %) than those obtained by ELM with βc = 0.1 according to EN 1995EN -1-1(2004. This is partly due to the larger assumed imperfections (ey = L/380 to L/570) and partly because of a different ratio of E0,05 : fc,0,k : fm,k for beech glulam. ...
... Additionally to the experimental investigations, numerical simulations of the buckling resistance of beech glulam columns were conducted. Investigations by Blaß [3,4], Theiler et al. [5,14] and Frangi et al. [15] have shown that appropriate numerical models can complement experimental tests and contribute to a better understanding of the buckling behaviour of columns. ...
Article
Full-text available
This article presents experimental and numerical investigations on the buckling behaviour of glulam columns made of European beech (Fagus sylvatica L.) timber and a design proposal. First, the compressive strength parallel to the grain (f_c,0) and the modulus of elasticity parallel to the grain (E_c,0) were experimentally determined in tests on stocky columns (slenderness ratio lambda = 12.5) of strength classes GL 40h, GL 48h, and GL 55h. Subsequent experimental buckling tests on slender columns with buckling lengths of 2.40 m (lambda = 41.5) and 3.60 m (lambda = 62.3) allowed investigating the buckling behaviour and quantifying the influence of the buckling length on the buckling resistance. Applicability of the effective-length method, which is the design method for columns in Eurocode 5 (EN 1995-1-1: Design of timber structures - Part 1-1: General - Common rules and rules for buildings. European Committee for Standardization, Brussels, 2010), was evaluated and a proposal for input parameters valid for columns made of European beech glulam is made. Numerical simulations revealed buckling resistances very close to the experimental results, confirming the proposed critical relative slenderness ratio lambda _rel,0 = 0.25 and the fitted straightness factor beta _c = 0.25. In addition, the numerical simulations allowed for an extension of the scope of the investigations.
... Additionally to the experimental investigations, numerical simulations of the buckling resistance of beech GLT columns were conducted. Investigations by Blaß (1987aBlaß ( , 1987b, Theiler et al. (2012Theiler et al. ( , 2014 and Frangi et al. (2015) have shown that appropriate numerical models can complement experimental tests and contribute to a better understanding of the buckling behaviour. Mentioned authors used a strain-based ...
Conference Paper
Full-text available
The paper reports and discusses full-scale test results of beech GLT timber columns of various slenderness ratios and strength classes and checks the validity of the formulae currently implemented in Eurocode 5. Firstly, the compression strength and the modulus of elasticity parallel to the grain of short beech GLT columns was determined according to EN 408. The raw material used for producing the columns of strength classes GL 40h, GL 48h, and GL 55h was strength graded as described. The moisture content of all specimens tested amounted to 8 ± 2%. Subsequently, 16 beech GLT columns of strength classes GL 40h and GL 48h with a cross-section of 200 × 200 mm2 and buckling lengths of ℓk = 2.40 m (typical column length in residential buildings, λ = 41.5) and ℓk = 3.60 m (typical column length in industrial buildings, λ = 62.3) were tested. The columns were subjected to eccentricities of 5 to 8 mm (≈ ℓk / 500). The vertical displacements of the columns were measured over a length of 600 mm on each side of the specimens. Additionally, the horizontal displacement (curvature) in the buckling plane was measured at five evenly distributed points along the column’s height. The tests were conducted using pinned supports and applying a deformation-controlled loading protocol.
Thesis
Das Langzeittragverhalten von Stützen aus Holz wird entscheidend von der Größe der Dauerlast und dem umgebenden Klima beeinflusst. Im Gegensatz zu Biegeträgern haben diese Effekte auch Einfluss auf den Grenzzustand der Tragfähigkeit und sind unmittelbar Gegenstand von Sicherheitsaspekten. Der Eurocode 5 beachtet dies überhaupt nicht, während in DIN 1052 (2004:08) Hinweise zur Bemessung gegeben sind. Da der Baustoff Holz infolge seines Wuchscharakters stark streuende Materialparameter aufweist, ist es kaum möglich, Experimente in ausreichendem Umfang durchzuführen. Gegen eine solche experimentelle Untersuchung sprechen auch die zu erwartenden hohen Kosten sowie die langandauernden Versuche unter Klimabeanspruchung. Aus diesem Grund wird auf virtuelle Versuche mit Hilfe des Computerprogramms ISOBEAM zurückgegriffen. Dazu sind allerdings abgesicherte Materialmodelle wichtig, die hier an Experimenten mit Klein- und Kleinstproben sowie mit Versuchkörpern baupraktisch relevanter Abmessungen angepasst wurden. Mit diesem verifizierten Modell war es möglich, gezielt Parameterstudien durchzuführen. Der Einfluss der Einbauholzfeuchte auf das hygrothermische Langzeittragverhalten wurde genauso untersucht wie der der Nutzungsklasse und der Querschnittsabmessungen. Die Ergebnisse der virtuellen Versuche dienten zum einen der Überprüfung der Vorgehensweise nach DIN 1052 (2004:08) und zum anderen zur Anpassung neuer Vorschläge zur Bemessung. Es wurde Wert darauf gelegt, dass sich die Vorschläge neuer Bemessungskonzepte an die bestehenden normativen Bemessungsregeln anlehnen. Zum einen wurde das Bemessungsverfahren nach Theorie II. Ordnung um eine weitere Ausmitte zur Berücksichtigung des Kriechens ergänzt, zum anderen wurde beim Ersatzstabverfahren der Beiwert entsprechend modifiziert. Alternativ ist es möglich, den Modifikationsbeiwert neu an die virtuellen Versuchsergebnisse anzupassen. Die Einbauholzfeuchte wurde ebenfalls über einen zusätzlichen Modifikationsfaktor berücksichtigt.
Article
The behaviour of timber members subjected to axial compression or combined axial compression and bending is characterised by the non-linear increase of the deformation due to the increasing eccentricity of the axial load and due to the non-linear material behaviour. The paper presents a strain-based model taking into account these effects.Design approaches given in timber structures design codes are compared and differences in the results obtained with the different approaches are identified. Furthermore, a strain-based model to analyse the load-bearing capacity of centrically and eccentrically loaded timber columns is described and its power is assessed. It is shown that in particular the non-linear behaviour of timber when subjected to compression parallel to the grain considerably influences the load-bearing capacity.The model is validated on the basis of experimental investigations on solid Norway spruce beams loaded in combined axial compression and bending. A good agreement was found between the estimated values using the strain-based model and the experimentally derived values.A comparison of the model with the design approaches given in the codes shows that the load-bearing capacity can be overestimated under certain conditions. Finally, it is illustrated how the design approaches can be modified in order to reach a more consistent design.