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

Abstract

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.

Full text

45
Designoftimbermemberssubjectedto
axialcompressionorcombinedaxial
compressionandbendingbasedon2nd
ordertheory

AndreaFrangi,ETHZurich,InstituteofStructuralEngineeringIBK,Zurich,Switzerland
RenéSteiger,Empa,MaterialsScienceandTechnology,StructuralEngineering
ResearchLaboratory,Dübendorf,Switzerland
MatthiasTheiler,dspIngenieure&PlanerAG,Greifensee,Switzerland


Keywords:Timberstructures,columns,compressionparalleltothegrain,stability,
globalbuckling,P‐deltaeffect,2ndorderstructuralanalysis


1 Introduction
Axialcompressionorcombinedaxialcompressionandbendingareencounteredin
manytypesoftimbermemberssuchascolumns,framestructuresorcompression
membersoftrussgirders.Thebehaviourofthesestructuralmembersisprimarily
characterisedbythenon‐linearincreaseofthedeformationduetotheincreasing
eccentricityoftheaxialload(P‐deltaeffect).Inadditiontothisgeometricnon‐linear
behaviour,thenon‐linearmaterialbehaviouroftimbermemberssubjectedto
compressionparalleltothegrainhastobeaccountedfor.TheinfluenceoftheP‐
deltaeffectontheload‐bearingcapacityoftimbermemberssubjectedtoaxial
compressionwasinvestigatedfirstbyTetmajer(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(1975)confirmedtheresultsobtained
byTetmajer.Theexperimentalinvestigationshowedthegreatinfluenceof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(1987and1991)performedMonte
Carlosimulations.Thebucklingcurvesgivenindifferentdesigncodes(SIA265:2012,
EN1995‐1‐1:2004andDIN1052:2008)werederivedfromtheseinvestigations.
INTER / 48 - 02 - 02

46
Fortimbermemberssubjectedtocombinedaxialcompressionandbending,
Buchanan(1984and1985)developedanumericalmodelcapableofinvestigatingthe
influenceofthenon‐linearmaterialbehaviouronthemoment–axialforce
interaction.InadditionBuchananinvestigatedtheinfluenceofthesizeofthe
member.CurrentdesigncodessuchasEurocode5(EN1995‐1‐1:2004),theSwiss
nationalcodeforthedesignoftimberstructures(SIA265:2012)orthewithdrawn
Germancode(DIN1052:2008)provide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
isamethodwhich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,the2ndorderanalysisisoftenunderstoodasatheorybasedonlinear
elasticmaterialbehaviourandtheeffectscausedbythenon‐linearityofthematerial
areneglected.Eventhedesigncodes(SIA265:2012,EN1995‐1‐1:2004andDIN
1052:2008)onlyproviderulesforthis2ndorderlinearelasticanalysisofthestructure.
Inthispaper,acleardistinctionbetweenthe2ndorderlinearelasticanalysisandthe
generalised2ndorderanalysisismade.
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(Kesseletal.2005and2006;
Möller2007;Köhleretal.2008).Thediscussioninparticularshowedthatthereare
inconsistenciesconcerningtheconsiderationoftheeffectofmoisturecontent(MC)
anddurationofload(DOL)aswellasinconsistenciesconcerningtheimplementation
ofthe2ndorderlinearelasticanalysisinthedesigncodes.Whilerecentresearchon
theload‐bearingbehaviouroftimbermemberssubjectedtoaxialcompressionor
combinedaxialcompressionandbendingwasmainlyfocusedonMCandDOL(Kessel
etal.2005and2006;Möller2007,BeckerandRautenstrauch2001,Hartnacketal.
2002)thispaperdealswiththeinfluenceofthenon‐linearmaterialbehaviourand
withtheimplementationof2ndorderlinearelasticanalysisinthedesigncodessuch
thatthereareonlyminordifferencesbetween2ndorderlinearelasticanalysisandthe
EffectiveLengthMethod.Theresultspresentedhereareonlyvalidforshort‐term
responseunderloadatconstantinteriorclimate.Infact,MCandDOLandin
particularthecreepbehaviourandtheclimatetakeamajorimpactontheload‐
bearingbehaviouroftimbercolumnsandshouldalsobeconsideredforthedesignof
timbermemberssubjectedtocompressionorcombinedcompressionandbending
(Hartnack2004,HartnackandRautenstrauch2005,Becker2002).
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47
2 Designoftimbermembersincompression
paralleltothegrain
Ingeneral,theEffectiveLengthMethodisusedforsimpledesignsituations(e.g.
verificationofstabilityofsinglemembers)whilethe2ndorderlinearelasticanalysisof
thestructureprovidessomeadvantagesformorecomplexdesignsituations(e.g.
impactofstiffnessofmembersandconnectionsontheforcedistributionintrussand
framestructures,designofbracing).
2.1 EffectiveLengthMethod(ELM)
ThesimplifiedcalculationmodelisbasedontheEffectiveLengthMethod(ELM).The
bucklingproblemofastructuralsystemisreducedtothatofanequivalentsimply
supported(pinned)column(Blass,1995).Forthedesign,theinternalforcesand
momentsarecalculatedbasedonasimple1storderanalysisandthenon‐linearP‐
deltaeffectistakenintoaccountbymeansofabucklingfactorkc.Thisfactor
describestheratiobetweentheaxialstressatbucklingfailureofamembersubjected
toaxialcompressionanditscompressivestrengthparalleltothegrain.kcdependson
theeffectivelengthofthestructuralsystemwhichcanbeexpressedbythe
slendernessratio

.
Thebucklingfactorkcasgivenindifferentdesigncodes(EN1995‐1‐1:2004,SIA
265:2012andDIN1052:2008)isbasedonextensiveinvestigationsperformedbyBlaß
(1987).Inordertodeterminethecharacteristicvalue(i.e.5thpercentile)oftheload‐
bearingcapacityoftimbercolumnsBlassperformedMonteCarlosimulations.The
numericalmodelandtheparameterstudyconsideredtheP‐deltaeffect,the
variabilityofthestrengthandthestiffnesspropertieswithinthetimbermembers,
thegeometricimperfectionofthetimbermembersandthenon‐linearmaterial
behaviouroftimberwhensubjectedtocompressionparalleltothegrainand
bending.
Fortheultimatelimitstateanalysis,thedesigncodes(EN1995‐1‐1:2004,SIA
265:2012andDIN1052:2008)recommendusingalinearinteractionmodelfor
combinedaxialcompressionandbending.Inthisinteractionmodel,thebuckling
factorkcisusedtoreducethecompressivestrengthparalleltothegrainofthe
timbermemberinordertoaccountforbuckling.
2.2 2ndorderlinearelasticanalysis
AsanalternativetothecalculationmodelbasedontheELM,timbermembers
subjectedtoaxialcompressionorcombinedaxialcompressionandbendingcanbe
designedbyperforminga2ndorderlinearelasticanalysis.The2ndorderlinearelastic
analysisisamethodwhichtakesintoaccountthegeometricnon‐linearitybystudying
theequilibriumofthedeformedstructuralmember.Aninitialdeformationis
introducedintothecalculationinordertoaccountforthegeometricimperfectionof
thememberase.g.deviationfromaperfectlystraightshape.
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48
Forasimplysupported,axiallyloadedcolumnthe2ndorderlinearelasticanalysiscan
easilybeperformed,assumingsinusoidaldistributedinitialdeformations.Theinitial
deformationincombinationwiththeaxialloadleadstoaninitialbendingmoment
MI.TheP‐deltaeffectcausesamagnifiedmomentMII.MIIcanbecalculatedby
multiplyingtheinitialbendingmomentMIwithamagnificationfactor

(Bazantand
Cedolin1991):
III MM 

(1)
Euler
N
N


1
1

and2
2
cr
Euler
EI
N



(2)
With
MII: magnifiedbendingmoment(2ndorderlinearelastictheory,deformedstructure)
MI: initialbendingmoment(1stordertheory,undeformedstructure)

: magnificationfactor
N: normalforceactingonthecolumn
NEuler:Eulerbucklingload
E: modulusofelasticity(MOE)
I: 2ndmomentofinertia

cr: effectivelength
Timbermemberssubjectedtocombinedaxialcompressionandbendingtendto
developnon‐lineardeformationsofthecompressionzonebeforefailureoccurs.This
non‐linearityleadstoacurvedshapeofthemoment–axialforceinteractiondiagram
dependingontheratiobetweenthetensilestrengthft,m,0andthecompressive
strengthfc,m,0paralleltothegrain(Buchanan1985,SteigerandFontana2005).For
theultimatelimitstateanalysisoftimbercolumnsthedesigncodes(SIA265:2012,
EN1995‐1‐1:2004andDIN1052:2008)considerthisnon‐linearinteractionbehaviour
bysquaringthecompressionpartintheinteractionmodel(Eq.(3)).However,the
non‐linearmaterialbehaviouralsoinfluencesthedeformationsofthestructural
system,andasaconsequence,alsothemagnifiedmomentMIIisinfluencedbythe
non‐linearmaterialbehaviour.However,theseeffectsareneglectedwhen
performinga2ndorderlinearelasticanalysis.
0.1
,
,,
2
,0,
,0, 








dm
dIIm
dc
dc
ff

(3)
With

c,0,d:designvalueoftheactingcompressivestressparalleltothegrain
fc,0,d:designcompressivestrengthparalleltothegrain

m,II,d:designvalueoftheactingbendingstressfroma2ndorderstructuralanalysis
fm,d:designbendingstrength

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3 Strain‐basedmodel
Inordertopredicttheglobalbucklingbehaviouroftimbercolumns,anumerical
strain‐basedmodelhasbeenimplemented.Strain‐basedmodelsarewidelyusedin
thedesignofstructuralmembersmadefromotherconstructionmaterialsthan
timber.E.g.forreinforcedconcretecolumns,astrain‐basedmodelissuggestedin
CEB/FIPManual(1978).Duetothefailuremechanismintimberbeinginfluencedby
thedistinctnonlinearstress‐strainrelationshipandleadingtoamorecomplex
calculationprocedure,uptonow,onlyafewapplicationsofthesemodelstotimber
structuresarereportedinliterature(Buchanan1984,Hörsting2008).
Figure1showsthecalculationprocedureofthestrain‐basedmodel.Onthelefthand
side,thecalculationoftheinternalforceNiandbendingmomentMiisillustrated.
Thecalculationstartswithselectingvaluesforthestrain

0atthemasscentreofthe
cross‐sectionandforthecurvature

y.Thesetwoparametersdefinethestrain
distributionwithinthewholecross‐section,whenassumingthatplanesections
remainplane.Basedonthestraindistribution,thestressdistributioniscalculated
usingtherelationshipgivenbythestress‐straincurve.Anyshapeofstress‐strain
curvecanbeappliedinthecalculation.Finally,theinternalforceNiandmomentMi
areestimatedbyintegratingthestressesoverthewholecross‐section.Theright
handsideofFigure1showsthecalculationoftheexternalforceNeandbending
momentMe.TheexternalbendingmomentMedependsontheexternalforceNeas
wellasonthedeformationofthecolumnduetotheinitialimperfectionsandtheP‐
deltaeffect.Sincethecurvatureisequaltothe2ndderivationofthedeflectioncurve,
themaximaldeflectioneIIofthecolumnduetotheP‐deltaeffectcanbecalculatedas
afunctionofthecurvature

y.BoththeinternalmomentMiandtheexternal
momentMedependonthecurvature

y.Hence,equilibriumbetweeninternaland
externalforcesandmomentscanbeobtainediteratively.

Figure1.Calculationprocedureinthestrain‐basedmodel(Theileretal.2013)

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50
Thestrain‐basedmodelhasbeenusedforstudyingtheinfluenceofvarious
parameters.E.g.itcanbeshownthattheplasticbehaviouroftimberwhensubjected
tocompressionparalleltothegrainconsiderablyinfluencesthebucklingbehaviourof
columns(Theileretal.2013).Therefore,theapplicationofanadequatematerial
model(stress‐strainrelation)isessentialwhenmodellingthebehaviouroftimber
memberssubjectedtocompression.Inthepresentstudy,themodelproposedby
Glos(1978)hasbeenused,sinceitappearstobemoresuitablethanothermaterial
modelsbecauseitisbasedonextensiveexperimentalinvestigationsonsolidtimber
boards.Inaddition,Glos(1981)developedthemodelfortimbermemberssubjected
tocompressionandbendingwhileothermodelsaremainlyfocusedontimber
memberssubjectedtopurebending.Glos’smodelaccountsforthereductionof
stiffnessbeforereachingtheultimatecompressionstrengthaswellasforthe
subsequentsoftening.Figure2qualitativelyshowsthestress‐strainrelationship
proposedbyGlos.Thedescriptionofthefullcurveinamathematicalformasksforsix
parameters(Figure2right).

Figure2.Qualitativerepresentationofthestress‐strainrelationshipinthematerialmodelproposed
byGlos(1978).
4 Experiments
Anexperimentalcampaignongluedlaminatedtimbermemberssubjectedto
eccentriccompressionhasbeenperformedatETHZurich(TheilerandFrangi2015).
Theaimoftheexperimentalinvestigationswastocreateadatabase,whichcouldbe
usedtovalidatetheoreticalcalculationmodelsandtoassesstheaccuratenessofthe
designapproachesgivenincodesforthedesignoftimberstructures.Thespecimens
wereproducedusinglamellasmadeofNorwayspruce(piceaabies)grownin
Switzerland.Atotalof336lamellaswereavailable.Inthefirststep,non‐destructive
testsonthelamellaswereperformed.Thesetestsaimedatcollectingdatainorderto
characterisetherawmaterial.Inthesecondstep,thelamellaswerestrengthgraded.
Theaimofthegradingprocesswastoselecttwoclassesoflamellasforthe
productionofthetestspecimens.Thelamellaswereselectedsothattheywere
suitabletoproducegluedlaminatedtimberofstrengthclassesGL24handGL32h.
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Withinthegradingprocess,visualgradingcriteriaaswellasmachinegradingcriteria
(dynamicMOE)wereused.Specimensforfivetestsserieswereproduced,three
seriesofgluedlaminatedtimberGL24handtwoseriesofgluedlaminatedtimber
GL32h(Table1).Eachofthetestseriesconsistedoftenspecimens.Thelengthofthe
timbermemberswasvariedbetweenthedifferenttestseries:L=1’400mm,L=
2’300mmandL=3’200mm.Thecross‐sectionwas140mmx160mm.
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
oftests
Strength
class
Cross‐section
[mm]
LengthL
(Slenderness)
Meanvalueofaxialstress
atfailure(COV)
110GL24h140x1601'400mm(30.3)25.6N/mm2(0.07)
210GL24h140x1603'200mm(69.3)15.3N/mm2(0.11)
310GL24h140x1602'300mm(49.8)20.3N/mm2(0.12)
410GL32h140x1601'400mm(30.3)31.1N/mm2(0.09)
510GL32h140x1603'200mm(69.3)18.1N/mm2(0.10)

Duringtheglulamproduction,thesetupofthetestspecimenswasrecorded.Hence,
thepositionandtheorientationofeverylamellawithineachtestspecimenwere
documented.Finally,thegluedlaminatedtimbermembersweresubjectedto
bucklingtests.Thetestspecimenswereloadedwithaneccentric(15mm)
compressionforceuptofailure.Duringthetests,theappliedloadsaswellas
horizontalandverticaldeformationswererecorded.Forasubsampleof20test
specimens,additionallocaldeformationmeasurementswereperformedusingan
opticalmeasurementsystem.

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
ThegraphsinFigure3showtheresultsofalltestsperformedincomparisonwiththe
assumedlognormaldistributionestimatedfromthetests.Agoodagreement
betweentestresultsandestimatedlognormaldistributioncanbeseen.The
coefficientofvariation(COV)wasintherangeof10%foralltestseries(Table1).All
detailsofthetestsarepresentedinatestreport(TheilerandFrangi2015).The
resultsofthetestshavebeenusedforthevalidationofthestrain‐basedmodel.
5 Numericalsimulations
Inordertoaccountforthevariationinmaterialproperties,MonteCarlosimulations
wereperformed.Columnsofdifferentslendernessanddifferentstrengthgrades
weremodelledwiththestrain‐basedmodelbyassigningthemrandomlyselected
materialproperties.Sixdifferentparametersareneededtodescribethefullstress‐
strainrelationship(Figure2).Thedistributionsofthepropertiesintermsof
probabilitydensityfunctionPDFaswellasthecorrelationbetweenthedifferent
propertieshavetobetakenintoaccount.Thestudywasperformedfortwodifferent
gradesofsolidtimber(C24andC30)andgluedlaminatedtimber(GL24handGL32h).
CharacteristicvaluesgiveninEN338:2009andEN14080:2013wereconsidered.
However,thecharacteristicvaluesarenotsufficientforthestochasticmodelling.
Furtherinformationonthevariabilityofthemechanicalpropertiesisrequiredthat
forexamplecanbefoundintheJCSSProbabilisticModelCode(2007).Inaddition,
Glos(1978)investigatedvariabilityandcorrelationofthemodelparameters.Using
theseinvestigationsandthecharacteristicvaluesasabasis,foreachmaterial
propertyameanvalue,astandarddeviationandaprobabilisticdensityfunctionwas
estimated(Table2).
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)
Materialproperty C24C30GL24hGL32h
Et,0,Ec,0[N/mm2]
Meanvalue11’00012’00011’50014’200
Standarddeviation2’2002’4001’5001’850
PDFLognormalLognormal
ft,m,0[N/mm2]
Meanvalue38.848.634.045.5
Standarddeviation9.712.25.16.8
PDFLognormalLognormal
fc,m,0[N/mm2]
Meanvalue29.932.630.440.2
Standarddeviation5.35.63.95.2
PDFLognormalLognormal
εc,0[‐]
Meanvalue3.40*10‐33.40*10‐33.27*10‐33.51*10‐3
Standarddeviation6.80*10‐46.80*10‐44.25*10‐44.57*10‐4
PDFLognormalLognormal
fc,m,u,0[N/mm2]
Meanvalue25.427.725.633.9
Standarddeviation3.84.22.63.4
PDFLognormalLognormal
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Startingfromthestochasticallymodelledmaterialproperties,2ndordersimulations
werecarriedoutwiththestrain‐basedmodel.Foreachtimbergradeandslenderness
ratio10’000simulationswereperformedallowinganaccurateestimationofthe
meanvalueandthe5%fractilevalueoftheload‐carryingcapacity.Altogether,about
twomillionsimulationswereperformedinthisresearchproject.

Figure4.Comparisonbetweenexperimentaldataandnumericalsimulationsforgluedlaminated
timberofstrengthclassesGL24andGL32h(Theiler2014).
Figure4showstheresultsofthenumericalsimulationsperformedfortheanalysisof
thetestresultsforgluedlaminatedtimberofstrengthclassesGL24handGL32h.It
canbeseenthatthevariationinthenumericalpredictionislargerforstockycolumns
thanforslenderones.Thiscanbeexplainedbythevariationoftheinputparameters.
Theload‐bearingcapacityofstockycolumnsisgovernedbythecompressionstrength
paralleltothegrainwhiletheload‐bearingcapacityofslendercolumnsisgoverned
bythemodulusofelasticityMOE.Therefore,thevariationinthenumericalprediction
isadirectconsequenceofthevariationoftheseparameters.Forcolumnsof
intermediateslendernessvariousmaterialpropertiesaswellastheinitialdeflection
influencetheload‐bearingcapacity.
6 AssessmentandimprovementoftheEurocode5
designapproach
TheMonteCarlosimulationsperformedallowcheckingtheaccuratenessofthe
designapproachesgiveninEurocode5.InFigure5theresultsofthenumerical
simulationsusingasinputparametersthevaluesgiveninTable2arecomparedto
analyticalcalculationsbymeansofELMand2ndorderlinearelasticanalysisforglued
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54
laminatedtimberGL24handGL32h.Resultsofthenumericalsimulationsforsolid
timberC24andC30canbefoundinTheiler(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.
Itcanbeseenthatthe2ndorderlinearelasticanalysismayleadtoanoverestimation
oftheload‐bearingcapacityespeciallyforcolumnsofintermediateandhigh
slenderness(

>50).Forslendercolumns(

>100),thecharacteristicvalues
obtainedfrom2ndorderlinearelasticanalysisareintherangeofthe25thpercentile
ratherthanintherangeofthe5thpercentile.Thisindicatesthatthedesignrulesfor
the2ndorderlinearelasticanalysisgiveninEurocode5donotensureanaccurate
designoftimbermemberssubjectedtocompressionandthereforeshouldbe
modified.However,thisconclusionisonlyvalidforsimplysupportedcolumns.
InEurocode5andotherdesigncodesthe2ndorderlinearelasticanalysisisbasedon
theassumptionoflinearelasticmaterialbehaviour,whichmeansthatthereduction
instiffnesscausedbythenon‐linearmaterialbehaviourisneglectedandtheload‐
bearingcapacityisoverestimated.Inordertoreachabetteragreementacorrection
factorhastobeintroducedinthedesignequation.Possibilitiestomodifythedesign
approacharetoenlargetheinitialdeformationsortoreducethedesignstiffness.
Inparticular,thereductionofthedesignstiffnessseemstobeapracticablesolution,
sinceitdescribesthephysicalphenomenaaccurately.In1889Engesserintroduced
thedesignconceptofreductioninstiffnessforsteelcolumnsandsuggestedtousea
tangentmodulusinsteadoftheMOE.Engesser’stheorywasfirstquestionedbyother
scientistsbutShanleyshowedin1947,thatEngesser’smethodwasavaluable
possibilitytoaccountfornon‐lineardeformationsinthecompressionzone.

INTER / 48 - 02 - 02

55
Basedontheresultsoftheresearchproject(Theiler2014)averygoodagreement
between2ndorderlinearelasticanalysisandnumericalsimulationsisobtainedwhen
usingabucklingmodulusdefinedasfollows:
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

forsolidtimberand 0.4
T

forgluedlaminatedtimber.


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.
AsimilarapproachwasalreadyproposedbyRošandBrunner(1931)andusedinthe
previousstandardSIA164:1981(Dubas1981).Theinvestigationshaveshownthat
thestrength‐dependentreductionofthestiffness(Eq.(5)and(6))leadstoavery
goodagreementbetweenthe2ndorderlinearelasticanalysisandthenumerical
simulations(Figure6).Ontheotherhand,thestrength‐dependentreductionofthe
stiffnessforhighloadlevelsleadstomorelaboriousdesignprocedure.Thebuckling
modulushastobedeterminedbymeansofiterationandcanbedifferentfor
differentdesignsituation.Additionalcalculationshaveshownthatforpracticaldesign
theestimationofthebucklingmodulusTk,daccordingtoEquation5isareasonable
solutionevenforhighloadlevels(

c,0,d
f
c,0d
0.5
)makingthedesigneasierforthe
engineersasnoiterationisneeded.
Thisstudyconcentratesonthebehaviourofsimplysupportedtimbercolumns.For
structuralsystemssuchasframestructuresthebehaviourisdifferentduetothe
distributionoftheaxialloadinthesinglemembers.Sincetheaxialloadinfluencesthe
INTER / 48 - 02 - 02

56
reductionofthestiffnessduetotheplasticdeformations,thebucklingbehaviour
dependsonthedistributionoftheaxialloadand,asaconsequence,theresults
obtainedwiththeELMortheadjusted2ndorderlinearelasticanalysis(Eq.(5)and
(6))wouldbetoosafeasshowninareliabilityassessmentperformedinaprevious
analysis(Köhleretal.2008).
7 Conclusions
Whendesigningtimbermemberssubjectedtosimultaneouslyactingaxial
compressionandbendingmoment,theincreaseofthebendingmomentduetothe
eccentricityoftheaxialforceandduetothenon‐linearmaterialbehaviouroftimber
subjectedtocompressivestresshastobetakenintoaccount.Thecurrentdesign
codesprovidetwodifferentapproachesforthedesignofrespectivemembers
(simplifiedanalysisbasedontheEffectiveLengthMethodand2ndorderlinearelastic
analysisofthestructure).However,thetwodesignapproachesarenotconsistent
andcanleadtodifferentresults.Basedontheinvestigationsperformed,the
followingconclusionscanbedrawn:
 Theload‐bearingcapacityofstockycolumns(

<20)isgovernedbythe
compressionstrengthparalleltothegrain.Forslendercolumns(

>100)the
modulusofelasticity(MOE)isthedominantmaterialproperty.Forcolumnsof
intermediateslendernessratio(50<

<100),thecompressionstrengthparallelto
thegrain,theMOEandthenon‐linearmaterialbehaviourimpacttheload‐bearing
capacity.
 Whenperforminga2ndorderlinearelasticanalysis,thenon‐linearmaterial
behaviouroftimbercannotbetakenintoaccount.Consequently,anadjustmentof
theresultsobtainedwiththismethodisrequired.Thiscanbedonebyreducingthe
designstiffnessofthestructuralmember.Theuseofabucklingmodulusfor
columnsappearstobeanappropriatesolution.
 Whendesigningsinglecolumnsorbeam‐columnsusinga2ndorderlinearelastic
analysis,thecalculationofthedesignvalueofthebucklingmodulusTk,dshouldbe
basedon5thpercentilevaluesofthemodulusofelasticityE0,05.Theinvestigations
haveshownthatthestrength‐dependentreductionofthestiffness(Eq.(5)and(6))
leadstoaverygoodagreementbetweenthe2ndorderlinearelasticanalysisand
thenumericalsimulations.However,theestimationofthebucklingmodulusTk,d
accordingtoEquation5isareasonablesolutionevenforhighloadlevels
(

c,0,d
f
c,0d
0.5
)makingthedesigneasierfortheengineersasnoiterationis
needed.
 Whendesigningstructuralsystemsusinga2ndorderlinearelasticanalysis,the
calculationofthedesignvalueofthebucklingmodulusTk,dshouldbebasedon
meanvaluesofthemodulusofelasticityE0,mean,astheuseof5thpercentilevalues
wouldleadtotoosaferesults.
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57
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