A static and free vibration analysis method for non-prismatic composite beams with a non-uniform flexible shear connection

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Delft University of Technology

A static and free vibration analysis method for prismatic composite beams with a

non-uniform flexible shear connection

Nijgh, Martin; Veljkovic, Milan

DOI

10.1016/j.ijmecsci.2019.06.018

Publication date

2019

Document Version

Final published version

Published in

International Journal of Mechanical Sciences

Citation (APA)

Nijgh, M., & Veljkovic, M. (2019). A static and free vibration analysis method for non-prismatic composite

beams with a non-uniform flexible shear connection. International Journal of Mechanical Sciences, 159,

398-405. https://doi.org/10.1016/j.ijmecsci.2019.06.018

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International Journal of Mechanical Sciences 159 (2019) 398–405

ContentslistsavailableatScienceDirect

International

Journal

of

Mechanical

Sciences

journalhomepage:www.elsevier.com/locate/ijmecsci

A

static

and

free

vibration

analysis

method

for

non-prismatic

composite

beams

with

a

non-uniform

ﬂexible

shear

connection

Martin

Paul

Nijgh

∗

_,

_Milan

_Veljkovic

Faculty of Civil Engineering and Geosciences , Delft University of Technology, Stevinweg 1, 2628 CN, Delft, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Keywords: Composite beam Analytical modelling Eigenfrequency Tapered beam Sustainability

a

b

s

t

r

a

c

t

Steel-concretecompositebeamsarewidelyusedinpracticebecauseoftheireconomiccross-sectiondesign.As sustainabilitybecomesmoreandmoreimportantintheconstructionindustry,thedesignofcompositebeamsmust beadaptedtomeettherequirementsofthecirculareconomy.Thiscallsfordemountabilityandreusabilityofthe structuralcomponents,aswellasoptimizeduseofmaterials,forexamplebyusingnon-prismaticbeams. Linear-elasticdesignandthe(optimized)useofdemountableshearconnectorsarekeyinthedesignofreusablecomposite structures.Inthispaper,analyticalpredictionmodelsfortheelasticbehaviourandthefirsteigenfrequencyof non-prismaticcompositebeamswithnon-uniformshearconnectorarrangementsarederived.Theapproachis basedon6th_and₂nd_order_{differential}_equations_used_to_define_matrix_equations_for_a_finite_number_of_linearized

compositebeamsegments.Theanalyticalmodelsarevalidatedusingexperimentalandnumericalresultsobtained withasimplysupportedtaperedcompositebeam.Theanalyticalmodelsaresuitableforcomprehensivestructural analysisofnon-prismaticcompositebeamswithnon-uniformshearconnection.

1. Introduction

Compositeactionbetweenasteelbeamandaconcretedeckis tradi-tionallyachievedusingweldedheadedstuds.Themechanicalbehaviour of welded headed studsis well established in literature (e.g. [1–4]) andthereforeincludedindesigncodes.Althoughtheuseofthewelded headedstudiswidespread,themaindrawbackofthistypeofshear con-nectoristhatitdoesnotallowfornon-destructiveseparationofthesteel beamandconcretedeck[5,6].Oncethebuildinghasbecomeobsolete, demolitionistheonlyoptiontotakethebuildingapart.

Demountableshearconnectorsareincreasinglygaininginterestin theresearch fieldof compositestructures, astheydoallow for non-destructiveseparationofthesteelbeamandconcretedeckandthereby scorecomparativelybetterinsustainabilityassessment. Demountabil-ityoftheshearconnectionoffersthepossibilitytoreusethestructure, eitherbychangingthefloorplantoallowfordifferentfunctionaluse and/orbyre-erectingtheentirestructure atanotherlocation.By de-signingastructuretobesuitablefordemountabilityorreusability,its servicelifetimeisnolongercontrolledbyitsfunctionallifetimeona specificconstructionsitebutbyitstechnicallifetime[7].

Themainbarrierstodesigningdemountableandreusablecomposite structureshavebeenidentiﬁedbyTingly&Davison[8]as:

• Perceivedriskinspecifyingreusedmaterials,

• Additionalcostsrelatedtothemeasuresrelatedtodemountability,

∗_{Corresponding}_author.

E-mailaddress:M.P.Nijgh@tudelft.nl (M.P.Nijgh).

• Compositeconstruction,

• Lackofareusedmaterialmarket,

• Longerdeconstructiontime.

Thepotentialbarriersmustbemitigatedtoallowthe implementa-tionofdemountableandreusablestructuresin practice.The feasibil-ity ofconstructionandexecutionof demountableandreusable com-posite beamswasrecentlydemonstratedbyNijghetal.[9]byusing largeprefabricatedconcretedecks,ataperedsteelbeamand demount-ableshearconnectors,incombinationwithoversizedholesand resin-injectedbolts.Inaddition,investigationsareon-goingwhich address the(de)constructiontimeandadditionalcostsrelatedtodemountable andreusablecompositestructures.

Steel-concrete compositebeamsaregenerally prismatic,i.e.their cross-sectiondoesnotvaryalongthebeamlength.However,tapered compositebeamsoﬀerbothstructuralandfunctionaladvantages com-paredwithprismaticcompositebeams.Recently,Nijghetal.[9] con-ductedexperimentstodeterminetheelasticmechanicalbehaviourof taperedcompositebeamswithvariousarrangementsofdemountable shearconnectors.Itwasfoundthattheelasticbehaviourofsimply sup-portedcompositebeamscouldbeoptimisedbyconcentratingtheshear connectorsnearthesupports.Thisﬁndingisinlinewiththetheoretical predictionsbyRoberts[10]andLinetal.[11].

Thedesignof compositebeamsis governedeitherby serviceabil-itycriteriaorbyitsresistanceintheultimatelimitstate.Inboth de-signcases,acompositebeamcanbedesignedtobedemountableand

https://doi.org/10.1016/j.ijmecsci.2019.06.018

Received31January2019;Receivedinrevisedform29May2019;Accepted8June2019 Availableonline9June2019

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M.P. Nijgh and M. Veljkovic International Journal of Mechanical Sciences 159 (2019) 398–405

reusableaslongastheelasticlimitsarenotexceeded.Inaddition,the perceptionofhumancomfortmustbeconsideredbydesigningfora suf-ﬁcientlyhighﬁrsteigenfrequency.

InthecomparativestudyofRanzietal.[12],fourdiﬀerentmodelling methodsforcompositebeamsareoutlined:

1 Exactanalyticalmethods 2 Finitediﬀerencemethod 3 Finiteelementmethod 4 Directstiﬀnessmethod

Exactanalyticalmethodsarebasedonsolvingdifferentialequations obtained byconsideringthe straindiagramandinternal equilibrium of compositebeams. Theelasticmechanical behaviour of composite beamswith flexible(non-rigid) shear connectors was firstdescribed analyticallybyNewmarketal.[13].TheNewmarkmodelconsistsof twoEuler-Bernoullibeams(onerepresentingthesteelbeam,andthe otherrepresentingtheconcretedeck)whicharecoupledatthe inter-faceusingauniformlydistributedshearconnection.Girhammar&Pan

[14]andGirhammar[15]studiedtheelasticbehaviourofcomposite beamsusingtheNewmarkapproach,whereasXu&Wu[16]andSchabl etal.[17]alsoimplementedsheardeformationintheirmodelsbyusing Timoshenkobeamtheory.Yam&Chapman[18]extendedtheoriginal Newmarkmodeltoaccountfornonlinearmaterialandshear connec-torbehaviour.Theexactanalyticalmethodsarenotdirectlysuitable foraccountingfornon-uniformshearconnectorarrangements.An at-tempttomodelnon-uniformshearconnectorarrangementsusing ana-lyticalmethodswasmadebyLawsonetal.[19]byassumingtheslip distributiontobecosinusoidal.However,theshapefunctionoftheslip distributionalongthebeamlengthmightnotbereadilypredeﬁnedfor non-prismaticcompositebeamswith(highly)non-uniformshear con-nectorarrangements.

Finitediﬀerencemethodsapproximatethebehaviourofcomposite beamsnumericallybyassumingderivativesintheformofalgebraic ex-pressions.Thismodellingmethodhasbeendeveloped extensivelyby Adekola[20],Roberts[10],andRoberts&Al-Amery[21].

Finiteelementmethodsprovidenumericalsolutionsandarerobust andreliableincasesuitableshapefunctionsarechosen[12]to approx-imatethedisplacements.Theﬁniteelementformulationsarebasedon Euler–Bernoullibeamtheory(e.g.[22,23]),Timoshenkobeamtheory (e.g.[24,25])orhigherorderbeamtheories(e.g.[26,27]).

Thedirectstiﬀnessmethodisbasedonaninitiallyundeformed el-ementthatissubjectedtoaunitrotationortranslationinoneofthe degreesoffreedom(DOF),whilstrestrainingallotherDOFs.Thedirect stiﬀnessmethodispresentedintheworkofRanzietal.[28],andlater extendedbyRanzi&Bradford[29]toaccountfortime-dependent ef-fects.

Inthiswork,analyticalpredictionmodelsfortheelasticbehaviour andtheﬁrst eigenfrequencyof non-prismatic compositebeamswith non-uniform shear connector arrangement are derived. The starting pointforthepredictionmodelsistodiscretisethecompositebeaminto segments,whichindividuallyfulﬁlthebasicassumptionsofthe analyt-icalNewmarkmodel.Theresultsoftheproposedanalyticalmodelsfor non-prismaticcompositebeamswithnon-uniformshearconnector

ar-rangementsarecomparedwiththeresultsofactualexperimentsand/or theresultsofﬁniteelementanalysis.

2. Theoreticalbackground

Thestartingpointfortheanalyticalmodelsfortheelasticmechanical behaviourandeigenfrequencyofnon-prismaticcompositebeamsisthe partialdiﬀerentialequation,Eq.(1)[16],validforprismaticcomposite beamswithuniformlydistributedﬂexibleshearconnectorssubjectto bendingdeformation.ForthederivationofEq.(1),thereaderisreferred totheworkofXu&Wu[16].Otherresearchers(e.g.Girhammeretal.

[30])havealsoderivedEq.(1),althoughwithdiﬀerentnotations.

𝜕6_𝑤 𝜕𝑥6 −𝛼 2𝜕4𝑤 𝜕𝑥4 +𝛽 2_𝛾 1 𝜕 4_𝑤 𝜕𝑥2_𝜕_𝑡2−𝛼 2_𝛾 1𝜕 2_𝑤 𝜕t2 =− 𝛼2 𝐸𝐼∞ 𝑞+ 1 𝐸𝐼0 𝜕2_𝑞 𝜕𝑥2. (1) 𝛼2₌ 𝐾⋅ 𝑟 𝐸𝐼0 ( 1− 𝐸𝐼0 𝐸𝐼∞ ) ; 𝛽2₌ 𝐸𝐼∞ 𝐸𝐼0 ;𝛾1= 𝑚 𝐸𝐼∞ . (2) 𝐾=𝑘sc 𝑠 ; 𝑚=𝜌s𝐴s+𝜌deck𝐴deck (3)

InEq.(1),wisthedeﬂectionfunctionand𝛼2_,_𝛽2_and_𝛾

1are

geometri-calandshearconnectionparametersdeﬁnedinEq.(2).Thedistributed load(forceperunitlength)actingonthebeamisdenotedbyq.EI∞

and𝐸𝐼0denotethebendingstiﬀnessincaseofrigidandnoshear

con-nection,respectively.Thedistancebetweentheelasticneutralaxesof theconnectedmembersundertheassumptionofnoshearinteraction isrepresentedbyr.Thesmearedshearconnectionstiffnessisdenoted byK,andisdefinedastheshearconnectorstiffnesskscdividedbythe

(uniform)connectorspacings.Themassperunitlengthisdenotedby

m.TheconventionofinternalandexternalactionsisdefinedinFig.1. Thesheardeformationoriginatingfromthetransversalloadisnot includedintheanalysis,becausedeflectionduetobendingisdominant forcompositebeamswithtypicalspanoverdepthratios.Therotational inertiaisalsodisregardedbecauseitsinfluenceonthelower eigenfre-quenciesisnegligible[16].

Eq.(1)isonlyvalidforprismaticbeamswithuniformlydistributed shearconnectors.AdiscretisationofthebeamintoJsegmentsis per-formedalongthelengthofthecompositebeamtoaccountonthe non-uniform shear connector arrangements andvarying geometryof the compositebeam.SuchadiscretisationprocesswasfirstadoptedbyTaleb &Suppiger[31]tomodelthefreevibrationsofnon-compositebeams, butsuchanapproachhasnotyetbeenappliedtocompositebeamswith aflexibleshearconnection.Thediscretisationprocesscreatesastepped beamwithdifferentgeometricalandmechanicalpropertiesineach seg-ment,asillustratedinFig.2.Thegeometricalandmechanicalproperties ofasegmentaredeterminedbasedonthemagnitudesoftheinfluencing variablesinthesegment’scentre.Ineachsegment,theshearconnection isassumedcontinuous(smeared)overthesegmentlength.Itisassumed thatallmaterialsbehaveelasticallyandthatthecurvatureofthe con-stituentmembersisequalineachcross-section.Therefore,eachbeam segmentfulfilsthebasicassumptionsoftheNewmarkmodel.

Fig.1.Conventionofinternalactionsinthediﬀerentialelementofacompositebeamwithaﬂexibleshearconnection.Theresultantofthenormalforceiszero undertheassumptionthatnoexternalaxialloadisapplied.

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Fig.2. Discretizationofataperedcompositebeam(dashedline)intoJprismatic compositebeamsegmentsofequallength,subjecttoauniformlydistributed load.

2.1. Staticanalysis

Forcompositebeamssubjecttostaticuniformlydistributedloads,

Eq.(1)reducesto 𝑑6_𝑤 𝑑𝑥6 −𝛼 2𝑑4𝑤 𝑑𝑥4 =− 𝛼2 𝐸𝐼∞𝑞 + 1 𝐸𝐼0 𝑑2_𝑞 𝑑𝑥2. (4)

DiscretisingthebeamintoJsegmentsofequallength(seeFig.2), andassumingthattheappliedloadperunitlengthqisconstantineach beamsegment,Eq.(4)furtherreducesto

𝑑6_𝑤 𝑗 𝑑𝑥6 −𝛼 2 𝑗 𝑑4_𝑤 𝑗 𝑑𝑥4 =− 𝛼2 𝑗 𝐸𝐼∞,𝑗 𝑞𝑗, (5)

with1≤j≤J.Thesolutiontothissixthorderlineardiﬀerentialequation isgivenby 𝑤𝑗(𝑥)=𝐶1,𝑗 e𝛼𝑗𝑥 𝛼4 𝑗 +𝐶2,𝑖 e−𝛼𝑗𝑥 𝛼4 𝑗 +𝐶3,𝑗𝑥3+𝐶4,𝑗𝑥2+𝐶5,𝑗𝑥+𝐶6,𝑗+ 1 24 𝑞𝑗𝑥4 𝐸𝐼∞,𝑗. (6) ExpressionsforthebendingmomentM,shearforceV,shearﬂowVs,

normalforceN₁andinterlayerslipΔuaregivenbyEqs.(7)–(11), re-spectively[14,16,30]. 𝑀𝑗=𝐸𝐼∞,𝑗 𝛼2 𝑗 [ −𝑑 4_𝑤 𝑗 𝑑𝑥4 +𝛼 2 𝑗 𝑑2_𝑤 𝑗 𝑑𝑥2 + 𝑞𝑗 𝐸𝐼0,𝑗 ] (7) 𝑉𝑗=−𝑑_𝑑𝑥𝑀𝑗 = 𝐸𝐼_𝛼∞2,𝑗 𝑗 [ 𝑑5_𝑤 𝑗 𝑑𝑥5 −𝛼 2 𝑗 𝑑3_𝑤 𝑗 𝑑𝑥3 − 1 𝐸𝐼0,𝑗 𝑑𝑞𝑗 𝑑𝑥 ] (8) 𝑉s,𝑗=_𝑟1 𝑗 [ 𝑉𝑗+𝐸𝐼0,𝑗 𝑑3_𝑤 𝑗 𝑑𝑥3 ] (9) 𝑁1,𝑗= 𝐸𝐼∞,𝑗 𝛼2 𝑗𝑟𝑗 [ −𝑑 4_𝑤 𝑗 𝑑𝑥4 +𝛼 2 𝑗 ( 1− 𝐸𝐼0,𝑗 𝐸𝐼∞,𝑗 )_𝑑2_𝑤 𝑗 𝑑𝑥2 + 𝑞𝑗 𝐸𝐼0,𝑗 ] (10) Δ𝑢_𝑗= 𝑑𝑁1,𝑗 𝑑𝑥 1 𝐾𝑗 (11)

The6Jintegrationconstants(C1,1,C2,1...C5,J,C6,J),resultingfrom

the J segments in which Eq. (6) is deﬁned, can be solved by im-posing boundary conditions at x₀ and x_J, and interface conditions at x1...xJ−1. For a beamsimply supported at x0=0 andxJ=L, the

sixboundaryconditionsarew1(0)=0,wJ(L)=0,M1(0)=0,MJ(L)=0,

w′′′′1 (0)=q1/EI0,1andw′′′′J (L)=qJ/EI0,J.Forasymmetricalsimply

supportedcompositebeam, theboundaryconditionscan alsobe ex-pressedatxJ=L/2as𝑤′_𝐽(𝐿∕2)=0,VJ(L/2)=0,andΔuJ(L/2)=0.Other

typesofsupportingconditionscanbeincludedbymodifyingthe bound-ary conditions appropriately. The equilibrium of shear force, bend-ingmomentandnormalforce,aswellasthecontinuityofdeﬂection, slopeandslipisenforcedattheinterface ofneighbouringsegments. Theseinterfaceconditionsareexpressedasw_j(x_j)=w_j+1(xj),𝑤′𝑗(𝑥𝑗)=

𝑤′

𝑗+1(𝑥𝑗), Mj(xj)=Mj+1(xj), Δuj(xj)=Δuj+1(xj), Vj(xj)=Vj+1(xj), and

N_1,_j(x_j)=N_1,_j+1(xj).Anyconcentratedforcescanbeappliedbyimposing

theseintheinterfaceconditionsrelatedtotheverticalforceequilibrium.

2.2. Freevibrationanalysis

Then-theigenfrequencyofaprismaticbeamwithaspanL,a uni-formlydistributedmassmandconstantbendingstiﬀnessEIisgivenby

𝑓𝑛=𝐾₂_𝜋𝑛

√

𝐸𝐼

𝑚𝐿4, (12)

inwhichKnisaconstantdependingontheboundaryconditions.The

mostimportantobservationfromEq.(12)isthat𝑓𝑛∼

√

𝐸𝐼.Assuming thatthebendingstiﬀnessofthebeamAwiththen-theigenfrequency

fn,AisEIA,andthatthemassperunitlengthandthespanofthebeam

AandBareequal,then-theigenfrequencyofthebeamBequals

𝑓𝑛,B=𝑓𝑛,A

√

𝐸𝐼B

𝐸𝐼A.

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Fortaperedcompositebeams,thebendingstiffnessisnotconstant along thebeamaxis,andtherefore thedeflectionat mid-spanunder theself-weightcanbeassumedtobeameasureforthebeamstiffness instead.Theprecedingleadstothehypothesisthatthenaturalfrequency ofataperedcompositebeamcanbedeterminedusingtheexpression

𝑓𝑛=𝑓𝑛,∞

√

𝑤m,∞

𝑤m

, (14)

inwhichfn,∞isthen-thnaturalfrequencyofthe(non-prismatic)

com-posite beamundertheassumptionofrigidshearconnection.wmand

w_m,∞denotethedeﬂectionatmidspanbecauseoftheself-weight

im-postedalongthebeamaxisincaseofﬂexibleandrigidshearconnection, respectively.Themagnitudesofwmandwm,∞foragivenbeamdesign

canbecomputedusingtheanalyticalmethodpresentedinSection2.1. Thenaturalfrequencyfn,∞ofthecompositebeamwithrigidshear

connectioncanbedeterminedusingEq.(15)[31],whichisbasedon Euler-Bernoullibeamtheory.

𝐸𝐼∞𝜕 4_𝑤₍_𝑥,_𝑡₎

𝜕𝑥4 =𝑚

𝜕2_𝑤₍_𝑥,_𝑡₎

𝜕𝑡2 (15)

Eq.(15) canbe simpliﬁedbyusing theprincipleof separationof variablesintheform𝑤(𝑥,𝑡)=𝑤̃(𝑥)exp(𝑖𝜔_𝑛𝑡).Insertingthisexpression intoEq.(15)gives

𝑑4_𝑤_̃ 𝑑𝑥4 −𝜁 4_𝑤_̃₌₀_, ₍₁₆₎ inwhich 𝜁4₌ 𝑚𝜔 2 𝑛,∞ 𝐸𝐼∞ . (17) InEq.(17),𝜔_𝑛,∞isatrialsolutionfortheangulareigenfrequencyof

thefull-interactioncompositebeam.Theangulareigenfrequency𝜔_𝑛,_∞ andtheeigenfrequencyf_n_,∞arerelatedtoeachotherby

𝑓𝑛,∞=

𝜔𝑛,∞

2𝜋 . (18)

ThegeneralsolutionofEq.(16)canbeexpressedintheform

̃

𝑤(𝑥)=𝐶1sin(𝜁𝑥)+𝐶2cos(𝜁𝑥)+𝐶3sinh(𝜁𝑥)+𝐶4cosh(𝜁𝑥). (19)

Thegeneralsolutionineachofthebeamsegmentsiswrittenas

̃

𝑤𝑗(𝑥)=𝐶1,𝑗sin(𝜁𝑗𝑥)+𝐶2,𝑗cos(𝜁𝑗𝑥)+𝐶3,𝑗sinh(𝜁𝑗𝑥)+𝐶4,𝑗cosh(𝜁𝑗𝑥), (20)

with 𝜁𝑗4= 𝑚𝑗𝜔2𝑛,∞ 𝐸𝐼∞,𝑗 . (21) 400

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Fig.3. (a)OverviewofthecompositebeampresentedintheworkofNijghetal.

[9] .(b)Cross-sectionaldimensionsofthetaperedsteelbeam.

Thecorrespondinginternalactionsaregivenby

𝑀𝑗=𝐸𝐼∞,𝑗

𝑑2_𝑤_̃ 𝑗

𝑑𝑥2 , (22)

𝑉𝑗=𝑑_𝑑𝑥𝑀𝑗. (23)

The4•Jintegrationconstants,resulting fromtheJsegmentsin whichEq.(20)isdeﬁned,canbesolvedbyimposingboundary condi-tionsatx0andxJ,andinterfaceconditionsatx1...xJ−1.Forasimply

supportedbeamsupportedatx0=0andxJ=L,theboundaryconditions

atthesupportsarew₁(0)=0,w_J(L)=0,M₁(0)=0andM_J(L)=0.The in-terfaceconditionsareexpressedas𝑤̃𝑗(𝑥𝑗)=𝑤̃𝑗+1(𝑥𝑗),𝑤̃′𝑗(𝑥𝑗)=𝑤̃′𝑗+1(𝑥𝑗),

Mj(xj)=Mj+1(xj),andVj(xj)=Vj+1(xj).Itshouldbenotedthatthefull

beammustbemodelledtoﬁndalleigenfrequenciesand–modes:if

sym-metryconditionsareused,onlytheodd-numberedeigenfrequenciesand -modes(n=1,3,5,…)canbefound.

Byinsertingtheboundaryconditionsintothegeneralsolutions,a systemofJhomogeneousequationsisobtained.Thesystemof homoge-nousequationscanbewrittenas

[A]{c}={0}, (24)

in which {c}=[C1,1C2,1...C5,JC6,J] and [A] is thecoeﬃcient matrix.

Non-trivialsolutionsofEq.(24)canonlybefoundifthedeterminantof thecoeﬃcientmatrixiszero,henceif

det[A]=0. (25)

Incasedet[A]=0,theangulareigenfrequency𝜔_𝑛wasassumed cor-rectly in Eq.(21). Incase det[A]≠ 0, anothertrialsolution mustbe adoptedtofindtheangulareigenfrequency.Wuetal.[32]proposed tofindtheangulareigenfrequencybysteppingthroughasequenceof smallincrementsof𝜔_𝑛andcomputingthesignforthedeterminantof [A].Ifthesignofthedeterminantof[A]changes,anapproximationfor theangulareigenfrequencyisobtained,which canbefurtherrefined usingthebisectionmethod.

Afterdetermining𝜔_𝑛suchthatdet[A]=0,theeigenfrequencyofthe compositebeamwithrigidshearconnectioncanbedeterminedbased onEq.(18).Theeigenfrequencyforacompositebeamwithaﬂexible shearconnectioncanthenbedeterminedusingtheproposedexpression inEq.(14).Theparameterswmandwm,∞inEq.(14)canbedetermined

usingtheanalyticalmethodpresentedinSection2.1.

Fig.4. Cross-section(side-view)ofcompositebeamstudiedbyNijghetal.[9] .Twoprefabricatedsolidconcretedecksaresupportedbytwotaperedsteelbeams withaspanof14.4m.Loadsareappliedat4.05mfromthesupports.Thec.t.c.distancebetweenthesteelbeamsis2.6m.

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M.P. Nijgh and M. Veljkovic International Journal of Mechanical Sciences 159 (2019) 398–405 Fig.5.Shearconnectorarrangements consid-ered in the work of Nijgh et al. [9] . Each colouredboxindicatesapairoffasteners(one persteelbeam).Resin-injectedboltsprovide shearconnection;normalboltsareplacedonly topreventverticalseparationofthedeckand beam. “U” denotesuniform connector spac-ing,“C” denotesconcentratedconnector spac-ingnearthesupports.Thebeamissymmetric intheplaneatx=L/2.(Forinterpretationof thereferencestocolourinthisﬁgurelegend, thereaderisreferredtothewebversionofthis article.)

3. Comparisonwithexperimental/numericalresultsand discussion

3.1. Staticanalysis

TheanalyticalmethodpresentedinSection2.1isvalidatedagainst actualbeamtests performedbyNijghetal.[9]on tapered prefabri-catedcompositebeamswithvariousshearconnectorarrangements.The simply-supportedcompositebeam(seeFig.3a)consistsoftwo prefabri-catedsolidconcretedecksof7.2mby2.6m,connectedtotwo symmetri-callytaperedsteelbeamsusingdemountableshearconnectors.The com-positebeamspans14.4mandissubjectedtobendingbyapplyingpoint loadsat4.05mfromthesupports.Aschematicdrawingofthe speci-menisshowninFig.4.Theheightofthesymmetricallytaperedsteel beamsvarieslinearly betweenthesupports,hs(x=0;x=L)=590mm,

andmidspan,h_s(x=L/2)=725mm.Thecross-sectionaldimensionsof thetaperedsteelbeamarepresentedinFig.3b.Theconcretedeckhasa constantthicknessof120mmalongitslengthanditisassumedthat

E_deck=33GPa[33]. Theshear connector stiﬀness k_sc was previously determinedas55kN/mm[9].Theshearconnectorarrangements pre-sentedinFig.5wereconsideredintheexperimentalprogramme.

Asensitivitystudyiscarriedouttodeterminetheminimumnumber ofbeamsegmentsperhalf-span(J)thatarenecessarytobemodelled, suchthatthedeﬂectionatmidspanbasedonJsegmentsconvergesto thevalueobtainedforJ→ ∞.Thisanalysisisperformedunderthe as-sumptionofauniformlydistributedloadandauniformlydistributed shearconnectionwithK=367kN/mm2_(equivalent_to_shear_connector

arrangementU-24)forthecompositebeampreviouslyintroduced.The resultsofthissensitivitystudyarepresentedinFig.6,indicatingthata smallnumberofsegmentsissufficientforconvergenceofthedeflection atmidspan.ForJ≥3theerrorregardingmidspandeflectioncompared withJ→ ∞ issmallerthan1%.Astheexperimentalbeam[9]offersthe possibilitytoinstall24pairsofshearconnectorsineachhalf-span,the theoreticalbeamisconvenientlysubdividedintoJ=24segmentsper half-span.

Theresultsobtainedusingtheproposedanalyticalmethodarelisted inTable 1, togetherwiththeexperimentalandfinite-elementresults obtainedbyNijghetal.[9].Theresultsareexpressedintermsofthe effectivebendingstiffnessandtheeffectiveshearstiffnessofthe

com-positebeam,respectivelydeﬁnedas

𝑘b,ef f=

Δ𝐹

Δ𝑤(𝑥=𝐿∕2);𝑘s,ef f= Δ𝐹

Δ𝑢(𝑥=0). (26) InEq.(26),ΔFistheforceincrement,ΔW(x=L/2)isthedeﬂection incrementatmidspanandΔu(x=0)istheslipincrementatthesupports. Theseparameterswereevaluatedatlinear-elasticloadlevels.

Fig.7,Fig.8andTable1clearlyshowthattheproposedanalytical modelandthefiniteelementresults[9]areingoodagreement regard-ingtheeffectivebendingstiffnessandtheeffectiveshearstiffnessforall theconsideredshearconnectorarrangements.Thenumericaland ana-lyticalpredictionscloselymatchtheexperimentalresultsregardingthe effectivebendingstiffness,withaveragedeviationsof only0.4%and 2.4%,respectively.Largedeviationexistsregardingtheeffectiveshear stiffness,aswasalreadyobservedin[9].Onaverage,theactualend-slip is47%smallerthanpredictedusingtheproposedanalyticalmodel.The sourceofthisdeviationisnotconsideredinthispaper.Theproposed analyticalmodel,however,showsgoodagreementwiththefinite ele-mentmodel[9]withanaveragedeviationofonly6%.Thisindicates thatthelargedeviationbetweenanalyticalmodelandexperimental re-sultsislikelyrelatedtotheexperimentsandnottothepresentanalytical modelnortothefiniteelementanalysis[9].

Fig.6. Relativedeﬂectionofthetaperedcompositebeamsubjecttoauniformly distributedload,asafunctionofthenumberofsegmentsJperhalf-span. 402

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M.P. Nijgh and M. Veljkovic International Journal of Mechanical Sciences 159 (2019) 398–405 Table1

Resultsobtainedusingproposedanalyticalmodel(presentstudy)andtheexperimentallyandnumerically obtainedresultsbyNijghetal.[9] regardingeffectivebendingstiffnessandeffectiveshearstiffnessfor theconsideredshearconnectorarrangements.

kb,eﬀ(kN/mm) ks,eﬀ(kN/mm)

Arrangement Analytical model Experiment FE model Analytical model Experiment FE model

U-24 7.25 6.89 7.13 285 514 301 C-12 7.04 6.69 6.96 269 487 294 C-6 6.31 6.18 6.28 190 389 188 U-12 6.60 6.35 6.53 177 330 190 U-6 5.90 5.82 5.87 120 199 128 U-0 3.96 4.10 4.07 46 98 51

Fig.7. Theeffectivebendingstiffnessparameterobtainedusingtheproposed analyticalmethod,comparedwiththeexperimentallyandnumericallyobtained results[9] ,forthedifferentshearconnectorarrangements.

Fig.8. Theeffectiveshearstiffnessparameterobtainedusingtheproposed an-alyticalmethod,comparedwiththeexperimentallyandnumericallyobtained results[9] ,forthedifferentshearconnectorarrangements.

Presentstudyconﬁrmsthatthedeﬂectionandshearconnectorslip canbeminimizedbyconcentratingtheshearconnectorsnearthe sup-portsofasimplysupportedbeam.Byoptimizingtheshearconnector ar-rangementthetotalnumberofshearconnectorscanbereduced,which hasapositiveimpactonthespeedof(de)constructionandthematerial andlabourcosts.Bothparametersarekeytothesuccessful implemen-tationofdemountableandreusablestructureswithintheconstruction industry[8].

Theslipdistributionalongthetaperedcompositebeamsubjectedto auniformlydistributedloadq(F=0)isshowninFig.9forthediﬀerent shearconnectorarrangements.Thisloadcasecorrespondstoapractical beamapplication.Fig.9indicatesthattheassumptionofLawsonetal.

[19]ofacosinusoidalshapefunctionfortheinterlayerslipisnot gener-icallyvalid:particularlyin thecaseofshearconnectorsconcentrated nearthesupports(arrangementsC6andC12)amorereﬁnedanalysis usingthepresentmethodmustbecarriedouttodeterminetheactual slipdistributionandthecorrespondinginternalactions.

Fig.9. Shapeofslipdistributionalongthelengthofthetaperedcompositebeam subjectedtoauniformlydistributedloadq(F=0),comparedtothecosinusoidal distributionassumedbyLawsonetal.[19] .

3.2. Freevibrationanalysis

Theﬁrsteigenfrequencyofthetaperedcompositebeampresentedin

Section3.1isdeterminedfortheshearconnectorarrangementslisted in Fig.5andforvariousmagnitudesof theshear connectorstiﬀness

k_sc (25,55and100kN/mm).Inthisanalysis,thecompositebeamis regardedaspartofalargerstructureandthatthereforeonetapered compositebeameﬀectivelyconsistsofonetaperedsteelbeamandtwo prefabricatedconcretedecks.

Theresultsofasensitivitystudytodeterminetheminimumnumber ofbeamsegmentsperhalf-spanJtoensureanaccuratelypredictionof thefirsteigenfrequencyarepresentedinFig.6.Theanalysishasbeen conductedwiththesameassumptionsasfor thesensitivitystudy re-gardingthedeflection.AlsointhiscaseitisfoundthatforJ≥3the convergenceerrorintermsoffirsteigenfrequencyissmallerthan1%. AllcalculationsarecarriedoutbysubdividingthebeamintoJ=24 seg-mentsperhalf-spantomatchthesegmentationusedinSection3.1.

Theanalyticalmethodtodeterminetheeigenfrequenciesofa non-prismaticcompositebeamwithﬂexibleshearconnectorsisvalidatedby

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M.P. Nijgh and M. Veljkovic International Journal of Mechanical Sciences 159 (2019) 398–405 Table2

FirsteigenfrequenciesofthetaperedcompositebeampresentedinSection 3.1 forvariousshear con-nectorarrangementsandshearconnectorstiﬀness,obtainedusingtheproposedanalyticalmodeland usingﬁniteelementanalysis.

Arrangement

fn,∞,FEA fn,∞,analytical ksc 𝑤 m,∞∕ 𝑤 _m fn,analytical fn,FEA fn,analytical / f n,FEA

(Hz) (Hz) (kN/mm) (-) (Hz) (Hz) (-) U-24 5.39 5.51 25 0.82 4.99 5.00 0.998 55 0.9 5.23 5.17 1.012 100 0.94 5.34 5.25 1.017 C-12 25 0.79 4.88 4.92 0.992 55 0.87 5.14 5.10 1.007 100 0.92 5.27 5.20 1.013 C-6 25 0.69 4.59 4.71 0.975 55 0.78 4.88 4.89 0.997 100 0.84 5.03 5.00 1.007 U-12 25 0.73 4.71 4.81 0.978 55 0.84 5.04 5.02 1.002 100 0.9 5.22 5.15 1.013 U-6 25 0.64 4.41 4.61 0.957 55 0.75 4.78 4.85 0.987 100 0.83 5.02 5.00 1.004 U-0 0 0.44 3.65 3.67 0.995 Average 0.997

Fig.10. Eigenmodesofthetaperedcompositebeam.

ﬁniteelementanalysis.Thesimplysupportedcompositebeamis mod-elledinABAQUS/Standardusingfour-nodeshellelements(S4)forthe taperedsteelbeamandconcretedeck.Theshearconnectorsare mod-elledusingmesh-independent,point-basedfastenerswithaspring stiﬀ-nessequaltoksc.

Theresultsobtainedusingtheanalyticalandnumericalmodelsare presentedinTable2.Onaverage,thefirsteigenfrequencyobtainedby theproposedanalyticalmodelis0.3%lowerthanpredictedbythe fi-niteelementmodel.Thefirsteigenfrequencyisunderestimatedincase ofaweakshearconnectionandoverestimatedincaseofastrongshear connection,withamaximumdeviationof4.3%fortheU-6casewith

ksc=25kN/mm.Theproposedanalyticalmodelisthereforeconsidered

suitablefordeterminationoftheﬁrsteigenfrequencyofatapered com-positebeamwithnon-uniformshearconnectorarrangements.

Fig.10showsthefirstfivenaturalvibrationmodesofthetapered compositebeam.Good agreementbetween theanalyticalmodeland finiteelementmodelisobservedintermsofmodalshape.Thehigher

Table3

Natural frequencies for the ta-pered composite beam with a rigidshearconnection.

n fn,∞,analytical √ 𝑓1,∞,analytical 𝑓𝑛,∞,analytical (-) (Hz) (-) 1 5.51 1.00 2 21.2 1.96 3 47.9 2.94 4 84.7 3.92 5 132.5 4.90

ordernaturalfrequenciesofthetaperedcompositebeamarelistedin

Table3.Itisobservedthatthehigher-ordernaturalfrequenciesarenot equaltof1n2,asisthecaseforprismaticbeams,butareslightlysmaller

becauseofthenon-uniformmassandbendingstiﬀnessdistributions.

5. Conclusions

Themainoutcomesofthetheoreticalandnumericalassessmentof thedeﬂectionandﬁrsteigenfrequenciesof(reusable)taperedcomposite beamsareasfollows:

• Theproposedanalyticalmethodsprovideaneasytouseformulation toassessthestructuralresponseofacompositebeamintheelastic stage.

• Theproposedanalyticalmethodrequiresdiscretisationintoa lim-itednumberofsegmentsalongthebeamlengthtoobtainaccurate results.Discretisingthebeaminto3segmentsperhalf-spanleadsto convergenceofthedeﬂectionandtheﬁrsteigenfrequencyforthe compositebeamstudiedinthiswork.

• Theproposedanalyticalmethodaccuratelypredictsthedeflectionof taperedcompositebeams.Onaverage,thedeviationoftheproposed analyticalmethodregardingmidspandeflectionis2.4%compared withtheexperimental resultsand0.4%compared withthefinite elementresultsofNijghetal.[9].

• Predictionsregardingendslipobtainedusingtheproposed analyti-calmethodareinlinewithﬁniteelementanalysis,withanaverage deviationof6%.Theanalyticalandnumericalmodeldonot repro-ducetheendslipobtainedintheexperimentalworkofNijghetal.

[9].Discussionofreasonsforsuchscatteringisleftoutofthescope ofthiswork.

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• Theshapeoftheslipdistributionalongthelengthofanon-prismatic compositebeamisnotnecessarilycosinusoidal,particularlyfor non-uniformshearconnector arrangements.Thislimitsthevalidityof theLawsonmodel[19].Therefore,itisrecommendedtousepresent methodtodeterminetheactualslipdistributionandthe correspond-inginternalactions.

• Verygood agreementis found between theeigenfrequencies ob-tained using ﬁnite element analysis and the proposed analytical model.Onaverage,theproposedanalyticalmodelunderestimates theﬁrsteigenfrequencyby0.3%.

Acknowledgment

ThisresearchwascarriedoutunderprojectnumberT16045inthe frameworkoftheResearchProgramoftheMaterialsinnovationinstitute M2i(www.m2i.nl)supportedbytheDutchgovernment.

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