Sliding frame‐solid interaction using BEM/FEM coupling · Wilson Sergio Venturini b Humberto...

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AbstractIn thiswork it is presented a coupling between theBoundary Ele‐mentMethod BEM andtheFiniteElementMethod FEM fortwo‐dimensionalelastostaticanalysisofframe‐solidinteraction.TheBEMisused tomodel thematrixwhile thereinforcement ismodeledbytheFEM.Regardingthecouplingformulationathirddegreepolyno‐mial is adopted to describe the displacement and rotations of thereinforcement, while a linear polynomial is used to describe thecontact force among the domain matrix and the reinforcement.Perfect bounding contact forces are improved bymeans of redun‐dant equations and Least squaresmethod. Slip‐boundingwith twoandthreepathswrittenasfunctionofrelativedisplacementareusedto calculate the transmitted contact forces. Examples are used todemonstratethattheproposedslip‐boundingprocedureregularizesthecontactforcebehavior.KeywordsBoundaryElementMethod,FiniteElementMethod,BEM/FEMcou‐pling,adherencemodels.

Slidingframe‐solidinteractionusingBEM/FEMcoupling

1 INTRODUCTION

Thecombinationof the finiteelementmethod FEM andboundaryelementmethod BEM tosolvestructuralanalysisisattractivebecauseitallowsforanoptimalexploitationoftherespectiveadvantageofthemethods Zienkiewiczetal.,1977 .ThemainstrengthoftheBEMforboundary‐valueproblemsgoverned by linear, homogenous, and elliptic differential equationswith constant coefficients is thereductionof thedimensionalityof theproblembyoneunit for linearconstitutiverelations Brebbia,1978;Brebbia,1980 .Particularly,BEMisusefultomodelspecialsituationssuchasverylargeorun‐boundeddomains,geometricalsingularities e.g.cracks ortoobtainveryaccurateresultsinregionsofcomplicatedshape Aliabadi,1997;Bonnet,1999;Frangietal.,2002 .Thus,couplingtheBEMandtheFEMallowsexploitingtheircomplementaryadvantages.Bytheotherhand,theFEMisappropriatetosolvealotofproblems,includinge.g.thosewithheterogeneousornon‐linearconstitutiveproperties,orfinitedeformations. The idea of combining these twomethods goes back to Zienkiewicz et al. 1977 .OnebranchofBEM/FEM coupling is the iterative coupling in which the individual sub‐domains are treated inde‐pendentlybyeithermethod.Theprocedurestartswithaninitialguessoftheinterfaceunknownsthatwillbeimprovedbysolvingeachsub‐domainandreturnedtointerface.Thisprocedurerepeatsuntilanerror tolerance is achieved.Although this iterative coupling is very attractive to softwaredesign, itsconvergence commonly depends on relaxation parameters which are rather empirical Estorff andHagen,2005 .Forthisreason,adirectcouplingapproachisadoptedinthiswork. StandardBEMformulationstodealwithsolidsstiffenedbybarsorfibersarederivedbycombiningtheBEMandFEMalgebraicequations.Thedomain continuumormatrix isanalyzedbyBEM,while

FabioCarlosdaRochaaWilsonSergioVenturinibHumbertoBrevesCodaca,b,cUniversityofSãoPaulo,SchoolofEngineeringofSãoCarlos,DepartmentofStructuralEngineering,Av.TrabalhadorSãoCarlense,400,13566‐590SãoCarlos‐SP,Brazil;aFederalUniversityofSergipe,Depart‐mentofCivilEngineering,Av.MarechalRondon,s/n,49100‐000SãoCristovão‐SE,Brazil;afabcivil@sc.usp.brbventurin@sc.usp.brchbcoda@sc.usp.br

F.C.Rocha,W.S.VenturiniandH.B.Coda/Slidingframe‐solidinteractionusingBEM/FEMcoupling1377

LatinAmericanJournalofSolidsandStructures11 2014 1376‐1399

finite elements areused to represent inclusions bars for instance .The coupling is alwaysdonebyenforcingdisplacementcompatibilityandtractionequilibriumatinterfacenodes.Practicalapplicationsfornon‐slippingorslippingcouplingusingBEM/FEMcouplingarepresented,forinstance,inBeerandWatson 1996 , Coda and Venturini 1995 , Coda 2001 , Leite et al. 2002 , Botta and Venturini2005,2003 ,Leonel 2009 ,Rocha 2010 . Ascontribution, thepresentwork isable tocapturebendingeffects in theanalysisof frame‐solidinteraction,which includesbending stiffness in reinforcements immersed in 2D continuum.For thispurpose,inthisstudyitisadoptedaRissner‐Mindlintypeframefiniteelementwiththirdorderofap‐proximation forbothdisplacementandrotationsresulting in fournodesand12degreesof freedom.However,thecontactforceismodeledbylinearapproximationresultinginonly4independentvalueswhich leads toanot square forcematrix.Theseapproximations displacementsand tractions wereusedbecauseitisthelowestorderthatsatisfiesthedifferentialequationgoverningtheproblem.Inthisapproach, theboundaryelement force linesarebuild inacompatibleway, that is thereare4sourcepointsgeneratingathirdorderapproximationfordisplacements,butthecontactforceapproximationislinear,alsoresultinginanotsquareforcematrix. The leastSquares technique isused toeliminate thedependentequationsdue to theabovemen‐tioneddifferenceinapproximationorderfordisplacementandtractions.Moreoversomeauthors,BottaandVenturini 2005,2003 andLeonel 2009 ,claimthatthisprocedurereducescontactforceoscilla‐tions. Thispaper isorganizedas follows. It ispresented insection2 theFEMformulation tomodel theframestructure,which isshownthekinematicsadopted. Insection3 isshowntheBEMformulationadaptedtodomainmodeling.Insection4itispresentedtheproposedcouplingformulationbetweenBEMandFEMconsideringbothperfectbondinganddebondingcases.Thissectionisdividedinsubsec‐tion4.1inwhichtherearepresentedthebasicequationstoperfectbondingandanexampletoverifythis formulation. In sub‐section4.3 it is presented the coupling formulation considering the slip be‐tweenreinforcementanddomain.Debondingmodels,basicequationsandthenon‐linearformulationtosolveslippingarepresentedinsub‐sections4.3.1,4.3.2and4.3.3,respectively.Thesub‐section4.4presentstwoexamples.Thefirstsimulatesapullouttestandthesecondsolveasoil‐structureinterac‐tioncase.Finally,insection5thefinalremarksandconclusionsaregiven.2 THEFRAMEELMENTMODELING–FEMFORMULATION

Asmentioned before, the FEM is used tomodel frame elements and structures. Here, elementswhichhavethreedegreesoffreedompernodeandcubicapproximationfordisplacementandrotationareemployed.Thisway,theelementshavefournodesandthesenodespresenttwotranslations verti‐calandhorizontal andonerotation.Moreover,thedistributedappliedforceswillfollowlinearapprox‐imation.2.1Kinematics

Foranypointonthestructure,thehorizontalandverticalcomponentsofdisplacementsaregivenby,respectively:

( ) ( ) ( )( ) ( )

U x y U x x y

V x y V x

where x and y arethereferencesysteminthecenterofthelayer,asshowninfigure1and pU and

pV arethedisplacementsofpoint P .

1378F.C.Rocha,W.S.VenturiniandH.B.Coda/Slidingframe‐solidinteractionusingBEM/FEMcoupling

Figure1:Kinematicsofpoint“P”.

Fromequation 1 ,onecanapplythedifferentialoperatortoobtainthelinearstraincomponents:

( )( ) ( ) ( )

( )( )

( )( ) ( )

( )( )

¶ ¶ ¶= = +

¶ ¶ ¶¶

æ ö æ ö¶ ¶ ¶÷ ÷ç ç÷ ÷ç ç= + = +÷ ÷ç ç÷ ÷÷ç ç÷¶ ¶ ¶è øè ø

, ,1 1, .

U x y U x xx y y

x x xV x y

x yyU x y V x y V x

x y xy x x

Applyingtheconstitutivelawfortheisotropicmaterials,thestresscomponentsatthepoint“p”areobtained:

( ) ( )( ) ( )

( ) ( )

( ) ( ) ( )( )

æ ö¶ ¶ ÷ç ÷ç= = + ÷ç ÷÷ç ¶ ¶è ø= =

æ ö¶ ÷ç ÷ç= = + ÷ç ÷÷ç ¶è ø

, , ,2

U x xx y E x y E y

x y E x y

V xGx y G x y x

whereEandGarethelongitudinalandshearelasticmoduli,respectively. TowritetheequilibriumequationitisusedthePrincipleofMinimumTotalPotentialEnergy.Usingequations 2 and 3 onewritestheTotalPotentialEnergyequationas,

( )( ) ( ) ( )

= -æ é ù öæ ö æ ö ÷ç ¶ ¶ ¶ê ú ÷÷ ÷ç çç ÷÷ ÷ç ç= + + +ç ê ú ÷÷ ÷ç çç ÷÷ ÷÷ ÷ê ç ç ú¶ ¶ ¶ç ÷è ø è ø ÷çè øê úë û

2 21 0 0 0

V UEU G y dA d

U t U t V dA

x x q xq x x

x x x 4

where eU and pU aretheinternalandpotentialenergyofexternalforces, xt and yt arethecompo‐

nentsof thedistributed loading contact tractions appliedtothestructure, L and A arethe lengthandcrosssectionalareaoftheframeelement,respectively.Forapproximateunknowns 0U , 0V e 0q

cubicindependentapproacheswereused,asshown:

( ) ( ) ( )

( ) ( )= = =

å å å

0 0 0 0 0 01 1 1

i i ii i i

j jx j x y j y

U u V v

t t t t

j x j x q j x q

y x y x

with 0

iu , 0iv e 0

iq beingthenodalvalues unknown .Since ( )ij x and ( )jy x areshapefunctions:

( ) ( ) ( ) ( ) ( )æ öæ ö æ ö÷ ÷ ÷ç ç ç= - + - - = + + - -÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è øè ø è ø1 2

9 1 1 27 11 , 1 1

16 3 3 16 3j x x x x j x x x x

( ) ( ) ( ) ( ) ( )æ ö æ öæ ö÷ ÷ ÷ç ç ç= - + + - = + + - +÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷è ø è øè ø3 4

27 1 9 1 11 1 , 1

16 3 16 3 3j x x x x j x x x x

( ) ( )- += = - £ £1 2

1 1, 1 1

2 2and

x xy x y x x 6

Minimizingtheenergyfunctional,equation 4 ,onefindsthealgebraicequilibriumsystemgivenby:

= +3 ,3 3 ,1 3 ,2 3 ,12 ,1extr extr

E E E E

NF NF NF NF NF NFNF

K U G f F7

where: EK is stiffnessmatrix, EG is equivalence forcematrix, EU unknownvectorwithdisplace‐ments and rotations, Ef is the vector containing thenodal valuesof thedistributed load, F is theconcentratedforce.LabelsNF and extrNF arenodenumbersforthedisplacements androtations ,

fourperelement,andnodenumbersforforces twoperelement ,respectively3 THEDOMAINMODELING–BEMFORMULATION

LetusconsiderthedomainW anditsboundary G .ForanelasticbodydefinedbythedomainW ,theequilibriumequation,writtenintermsofdisplacements,isgivenby:

( ) + ⋅ + =-

1 2 Gnb

whereurepresentsthedisplacementvector,G istheshearmodulusandn isthePoisson’sratio. ForadomainW withboundary G ,theintegralrepresentationofdisplacementsisderivedbyap‐plyingreciprocitytheorem orGreen’ssecondidentity .

G G⋅ = - ⋅ G + ⋅ Gò ò* *d dc u P u U p 9

Wherethesymbols“*”isusedtoindicatefundamentalsolution equation10 and p representbound‐arytractionvalues.

( )( )

( )( ) ( ) ( )( ){ }

ì ü-ï ïï ïé ù= - - - Ä +í ýë ûï ï- ï ïî þé ù= - ⋅ - + Ä + - Ä - Äë û-

7 81ˆ ˆ3 4 ln

1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ1 2 2 1 2

n np n

U I r r I

P n r I r r n r r n

where ,ˆ i ir r r= =r with r beingthedistancebetweenthesourceandfieldpoints, ( )1 2i ir rr= , ir are

componentsofthevector r , n̂istheboundarynormalunitvectorand I isthesecondorderidentitytensor orKroneckerdelta . For numerical solution, u and p are approximate by polynomial functions over boundary ele‐ments,inthisworklinearpolynomialsareusedforboundaryelements internalforcelinesarefurtherdiscussed ,andtheintegralequation 9 isconvertedintoanequivalentalgebraicsystemasfollows:

=U PH G 11wherematrix H isobtainedfromthelefttermsinequation 9 andmatrixG fromthetermsontherightside.U isavectorwhichcontainsthenodalvaluesofdisplacementsforallboundarynodesandP isthenodaltractionvector. Aftersubstitutionoftheprescribedboundaryconditions,thealgebraicequationsmaybewrittenas

= =X YA B F 12Thevector X containsalltheunknownboundarydisplacementsandtractions, A isacoefficientma‐trixwhichisusuallynon‐symmetricanddenselypopulated,andB isamatrixwhichcontainsthecoef‐ficientscorrespondingtotheprescribedboundaryconditionsY . Onecandifferentiateequation 9 toderivetheintegralrepresentationofstrainsandthenapplytheHooke’slawtoobtainthestressintegralequation,writtenforinternalpoints,asfollows,

G G= - ⋅ G + ⋅ Gò ò* *d dS u D ps 13

Where *S and *D arewellknownthird‐ordertensorforthestressequationobtainedbyapplyingtheHooke’slawonthefundamentalsolutionatthesourcepoint BrebbiaandDomingues,1992 .4 BEM/FEMCOUPLINGFORMULATION

Inthissection,additionaltermsinsertedintheclassicalformulationsofBEMandFEMaswellasthecouplingbetweenBEMandFEMareshown.4.1Basicequations–perfectbonding

Inthissubsectiontheperfectboundingsituationisdescribed. It impliesadirectcompatibilitybe‐tweendisplacementsandcontactforceequilibrium orcontinuity .Thus:

= -R Df f 14

=D RU QU 15

Where RU and DU arevectors containingnodaldisplacements for frameelementanddomain, re‐

spectively; Rf and Df arenodaldistributedforcevectorappliedontheframefiniteelementsandon

theforcelineboundaryelement inthe2Ddomain ,respectively.Oncevector RU containsthreecom‐

ponents twotranslationsandonerotation andvector DU contains twocomponents two transla‐tions ,thecorrelationbetweenthesetwovectorsisdonebyQ matrix.

Thecomponentsofthevectors , ,R D RU U f and Df areshowninfigure2.

Figure2:Forceanddisplacementapproximationatframeinterface.

For3Dsolidstheunknownforces, Df ,wouldbedistributedover internalsurfaces.For2Dsolidsthisforcesappearsdistributedalonginternallines,which,roughly,workasinternal“boundaries”dedi‐catedonlyfortractions.Thus,theintegralequation 9 ismodifiedtoincludetheadditionalterm:

G G G⋅ = - ⋅ G + ⋅ G + ⋅ Gò ò ò* * *

DRd d dc u P u U p U f 16

where Df is the internal forceactingalong the interface, GR ,between the twomaterialsandrepre‐

sents the fibereffectapplied in thedomain.Similarly, the integralequation 13 including thisaddi‐tionaleffectiswrittenas:

G G G= - ⋅ G + ⋅ G + ⋅ Gò ò ò* * *

DRd d ds S u D p D f 17

Selectingapropernumberofcollocationpoints sourcepoints attheboundaryandattheframeelement,twosetsofalgebraicequationsare,respectively,writtenfromequation 16 ,asfollows:

= + Dbb b bb b brU P fH G G 18

= - + +D D

rb b rb b rrU U P fH G G 19

whereindexb denotesboundaryandindex r denotesframe,respectively.Additionally,thefirstindexdenotessourcepointsandthesecondindexdenotesfieldpoints,which,areonboundaryorloadline. The superimposed frame element reinforcement and load line representations can be seen infigure3 inwhich n elements finiteand line areemployed.Thedisplacementnodes foreach finiteelement are represented by squares and crosses represent the collocation points of equation 19 wheredisplacementsarecalculated.ItisobservedthatforfiberendsBEMequationsarenotwritteninthesamepositionofthefiniteelementnodestoavoidsingularities;howeverdisplacementsareextrap‐olated to thenodalpositions inorder tomake themcompatible.Moreover, thismodelingallows thereinforcement frame endstoreachthebodyboundarywithoutinterferinginbasicBEMequations.

Figure3:CompatibilitybetweenBEMandFEMnodes.

Therefore,thedisplacementcompatibilityisrewrittenas:

D RU TU

T TQ20

Where T̂ is thematrix that relates thenodalpositionsanddirectionof the DU vectorwith the RU vector.Furthermore,the

T matrixhasdimension 3NF columnsby int2N rows,forwhich intN isthe

amountofBEMinterfacepoints. Accordingtofigure3,theamountoffiniteelementnodesisequaltotheamountofboundaryinter‐facenodes,i.e. intNF N= ,thereforethedeterminationofthecouplingparametersandtheboundary

valuescanbesummarizedbythefollowingsystemofequations:

ìïïï = +ïïïïïïï + = +íïïïï= +

intintint int

int int int int int

2 ,12 ,2 2 ,1 2 ,2 2 ,22 ,1

2 ,1 2 ,12 ,22 ,2 2 ,1 2 ,2 2 ,1

3 ,3 3 ,1 3 ,3 3 ,13 ,1

extrextr

extr extr

Dbb b bb b br

NN N N N N N NND D

rb b rb b rrN NFN NFN N N N N N

R R R R

N N N N NF NNF

U U P f

K U G f F

ïïïïïï

The 1N valuesof bU and 2N valuesof bP areknownon 1G and 2G ( )1 2G=G +G respectively,

hence there are only 2N unknowns in the system of equations 21 . As usual, to introduce theseboundaryconditionsinto 21 onehastorearrangethesystembymovingcolumnsof bbH , rbH with

bbG , rbG fromonesidetotheother,respectively.Onceallunknownsarepassedtotheleft‐handside

andapplyingconditionsofequations 14 and 20 onecanwritethenewsystemofequationsas:

ìïïï = +

+ = +í

int int intint intint

int int int int int

2 ,1 2 ,12 ,2 2 ,2 2 ,22 ,1

2 ,3 3 ,12 ,1 2 ,12 ,2 2 ,22 ,2 2 ,1

3 ,3 3 ,1 3 ,2 3 ,12 ,1

extrextr

extr extr

Dbb bb b br

N NN N N N N NNR D

rb rb b rrN N NN NFN N N NFN N NR R R D

N N N N NF NNF

X T U F f

K U G f F

ïïïïïïïïïïïïïïïïïî

Orinmatrixform,

{ } { }

+ ++ + + + +

ì üé ù é ù ì üï ï- ï ïï ï ï ïê ú ê úï ï ï ïï ïê ú ï ï ï ïê ú= +í ý í ýê ú ê úï ï ï ïê ú ê úï ï ï ï- ï ï ï ïê ú ê úï ï ï ïë û î þë û ï ïî þ

int iint int int

2 5 ,2 2 52 5 ,2 3 2 2 3 2 ,1

extr extr

bb br bbR R R

bD Nrbrb rr

N N N N NN N N N NF N N NF

K G U F F

Asdescribedabove,atthefiberelementlevel,thenumberofdisplacementvaluesislargerthanthenumberofbonding orcontact forcenodalvalues.Thisoccursbecauseinequation 19 thenumberofalgebraic relations ismuch larger than thenumberof forcevalues in Df . To reduce thenumberofequationstothesameasthenumberofunknownsonecanapplytheLeastSquareMethod LSM .InthisworktheLSMisappliedoverequation 21 or 19 ,asfollows:

intint int int int intint int

2 ,1 2 ,12 ,2 2 ,2 2 ,2 2 ,2 2 ,22 ,2 2 ,1 2 ,2 2 ,1 extrextr extr extr extr extr

D Drr rb b rr rr rb b rr rr

N NFNF N NF N NF N NF N N NFN N N N N N

U U P fG H G G G G G 24

int int intint int int int int intint

2 ,3 3 ,12 ,1 2 ,12 ,2 2 ,2 2 ,2 2 ,2 2 ,2 2 ,22 ,2 2 ,1 extrextr extr extr extr extr

R Drr rb rr rr rb b rr rr

N N NN NFNF N NF N NF N N N NF N N NFN N N

X T U F fG A G G B G G 25

Wherethematrix

rrG definedforeachfiberand

rrG isthetransposematrixof rrG ,i.e. =

rr rrG G .

Therefore,thesystemofequations 23 turnsintoasquaresystemas:

+ ++ + + + + +

ì üé ù é ùï ï- ï ïê ú ê úï ïï ïê ú ê úï ï =í ýê ú ê úï ïê ú ê úï ïï ï-ê ú ê úï ï ë ûë û ï ïî þ

intint int int2 3 2 ,2 3 2 ,2 3 2 2 3 2 ,1

extrextr extr extr

bb br bbR R R

Drr rb rr rr rr rr rb

N N NFN N NF N N NF N N NF

G A G G G G B

{ } { }

ì üï ïï ïï ïï ï+ í ýï ïï ïï ïï ïî þint

2 3 2 ,12

N N NFN

F F 26

4.2Numericalexample–perfectbonding

In this section,onenumerical example isanalyzed to check theperformanceandaccuracyof theproposedBEM/FEMcouplingfortwo‐dimensionalreinforcedsolids. The reinforced simple supported beam subjected to homogeneous transversal surface load

710q N= - ,showninfigure4,isanalyzed.Thisplanestructureisfivemeterlength ( )L ,onemeter

height ( )H andthepositionofthereinforcementis25centimeter 0( )h fromthelowerpartofthema‐

trixandisfourmeterslong 0L .Thefollowingpropertiesfordomain ( )D andreinforcement ( )R are

considered: Elasticity modulus 2.8DE = 10 210 N m and 2.8RE = 11 210 N m , Poisson’s ration

0.2Dn = and 0.0n = , inertiamoment 1.78891RI = 7 410 m- andcross‐sectionalarea 1.29RS =2 210 m- .

Figure4:Geometryofthestructureunderanalysis.

Thebeamisdiscretizedby120 linearboundaryelementswiththesame length.Threediscretiza‐tionsareemployedtomodelthereinforcement,with25,50and100finiteelements.Thesamenumberof force line BEM elements is employed tomodel the interface. The results are comparedwith thecommercial softwareANSYSemploying100BEAM3elements tomodel the reinforcement and2800PLANE422Dsolidelementstomodelthedomain.Figures5,6and7compareresults axial,transversedisplacementsandrotations achievedusingtheBEM/FEMcouplingdiscretizationsandANSYS.

Figure5:Axialdisplacementgraphicsoninterface,BEM/FEM.

Figure6:Transversedisplacementgraphicsoninterface,BEM/FEM.

Figure7:Graphicshowstherotationoninterface,BEM/FEM.

TheaboveresultsmakeevidentthateventhepoorestBEM/FEMcouplingdiscretizationhasgoodaccuracywhencomparedwiththereferenceresult. Regardingtractionsattheinterface,theBEM/FEMresultscanbeseeninFigures8and9.

Figure8:Axialcontactforcegraphicsoninterface,BEM/FEM.

Figure9:Transversecontactforcegraphicsoninterface,BEM/FEM.

According toFigures8 and9, forperfect bonded elastic reinforcement, it’s observed that for thethreeBEM/FEMdiscretizationsthere isaperturbationinthetractionvaluesattheendsof thefiber.Thebehaviorofaxialforces,asshowninFigure8,hadbeenobservedbyBottaandVenturini 2005 ,however,thebehaviorshowninFigure9hasnotbeenreportedbefore.Moreover,theuseofLSMpar‐tiallysmoothesthebehaviorofcontactforcewhencomparedwithresultsthatdonotapplythisstrate‐gy for the coupling, Leite et al. 2003 .At thispoint an important resultof this study shouldbe ad‐vanced; when debondig is allowed always occurs in a small vicinity of infinity contact stress theaboveperturbationdisappears. Whenusingnonlinearconstitutiverelationsformatrix,thereareevidencesofsmoothsolutionsforthiskindofproblemswhichwerereportedbyCoda 2001 .Moreover,onemaynote,inFigure9,thattheextensionofthecontactforceperturbationreducesasdiscretizationincreases. 4.3Couplingformulation–debonding

Inthissection,thepreviouscouplingisextendedtoaccomplishdebonding.Theadoptedmodelstorepresent debonding aswell as the nonlinear formulation to simulate the slip betweendomain andframeelementsarepresented.Thefeasibilityofthisformulationisshownthroughnumericalexamples.4.3.1Debondingmodels

Fibersembeddedinthedomainplayanimportantroletoimprovesolidstiffnessandloadingcapaci‐tyifenoughinternalforcesalongtheinterfacecanbesustained.Slidingalongtheinterfacemaybeal‐lowedwhenacertainamountofstrengthispreserved.Theidealsituationforwhichperfectbondingisassumed,asshowninprevioussection,isimpossibleinpractice;atleastinthevicinityoffiberends,asthe interface forcesapproachthe infinity Radtkeetal,2011 .Therefore,acertainamountofslidingoccursaccordingtothebondingcarryingcapacity. Tomodeltheslipthatmayoccurinthefiber‐domaininterface,adebondingcriterionshouldbecon‐sidered.Inthiswork,twomodelswereimplementedtogetherwiththeproposedBEM‐FEMcoupling. ThecurvesshowninFigure10and11representthedebondingcriterionthatrelatesthebondingforce f withtherelativedisplacementatinterface slip s .Thefollowingparametersdefinethetwomodels:model1dependsonlyonthemaximumbondingforce maxf andmodel2dependsonthemax‐

imumforce maxf ,residualbondingforce resf andslipcharacteristicvalues 1s and 2s .

FromtheFigure10and11,thefollowingrelationshipsarewrittenfortheadoptedmodels,respec‐tively: Model1

max 0f f for s= > 27 Model2

max 1[0, ]f f for s= 28

max 1 max 21 2

1 2 1 2[ , ]res resf f f s f s

f s for s ss s s s

- -= +

- - 29

2resf f for s s= > 30

Figure10:DebodingModel1

Figure11:DebodingModel2

4.3.2Basicequations

Theslipconsideration introducesanewvariable inequations 20 .Thisnewvariable representstherelativedisplacement, s ,betweenthedomainandframeelements.Thecompatibilityequationsarenowexpressedby:

( )= -D Rf f s 31

D RU TU TS 32Wherevector S contains thenodal relativedisplacementvalues,T relates thenodalpositions DU with S .Furthermore,T matrixhasdimension 2 extrNF columnsby int2N rows.Theothertermsin

equations 31 and 32 havealreadybeenexplained,butnowtractionsoninterfacedependonrela‐

tivedisplacements s .Inthiswork, DU and RU havebeenapproximatedbycubicpolynomialand S bylinearpolynomial. From the introduction of relative displacements, the equilibrium equation 19 of the boundaryelementmethodmayberewrittenas:

+ = - + +R D

rb b rb b rrTU TS U P fH G G 33

Therefore,theBEMcouplingequationcanberewrittenas,:

+ + = +

int int int int

intint int

int int in

2 ,2 2 ,1 2 ,2 2 ,22 ,1 2 ,1

2 ,3 3 ,1 2 ,2 2 ,1 2 ,22 ,2 2 ,1 2 ,2 2 ,1 2 ,1

3 ,3 3

extr extr

extr extrextr extr

Rbb b bb b br

N N N N N N NFN NFR R

rb b rb b rrN N N N NF NF N NFN N N N N N NF

U P f s

U T U T S P f s

ìïïïïïïïïïïíïïïïïï = +ïïïïî

t int int,1 3 ,2 3 ,12 ,1

extrextr

N NF NNF

G f s F

Applyingboundaryconditions,theequation 34 results:

+ + = +

int int int int

int intint

int int int

2 ,12 ,2 2 ,2 2 ,22 ,1 2 ,1

2 ,3 3 ,1 2 ,2 2 ,12 ,1 2 ,2 2 ,22 ,2 2 ,1 2 ,1

3 ,3 3 ,

extr extr

extr extrextr extr

Rbb bb b br

NN N N N N NFN NFR R

rb rb b rrN N N N NF NFN N N N NFN N N NF

X F f s

X T U T S F f s

ìïïïïïïïïïïíïïïïïï = +ïïïïî

int int1 3 ,2 3 ,12 ,1

extrextr

N NF NNF

G f s F

If ( )Rf s isknown,thematrixformof 35 is:

( ){ }

++ + + + + +

é ù é ù é ù é ùê ú ê ú ê ú ê úê ú ê ú ê ú ê ú= +ê ú ê ú ê ú ê úê ú ê ú ê ú ê úê ú ê ú ê ú ê úë ûë û ë û ë û

intint int int int

2 5 ,22 5 ,2 3 2 2 3 2 2 5 ,2

extr extr extr

bb br bbR R R R

NNFrr rbrb

N N NN N N N NF N N NF N N NF

K U G f s F

é ùê úê ú+ ê úê úê úë û

2 5 ,1

Bytheotherhand,ifS isknown,onewrites:

+ ++ + + + +

é ùé ù é ù é ù- ê úê ú ê ú ê úê úê ú ê ú ê ú- = +ê úê ú ê ú ê úê úê ú ê ú ê ú- ê úê ú ê ú ê úë û ë ûë û ë û

int iint int int

2 5 ,2 2 52 5 ,2 3 2 2 3 2 ,1

extrextr extr

bb br bbR R R

R NF rbrb rr

N N NF N NN N N N NF N N NF

K G U S

TT f s

é ùê úê ú+ ê úê úê úë û

nt int

,2 2 5 ,1

F F 37

As can be seen, in equations 36 and 37 , there are more equations than unknowns, since

int extrN N³ . Thus, to reduce thenumber of equations to be equal to thenumber of unknowns the

LeastSquareMethod LSM isappliedoverinternalpointequations 35 .Thisway:

int int int intint int intint

int int int int

2 ,3 3 ,1 2 ,2 2 ,12 ,12 ,2 2 ,2 2 ,22 ,2

2 ,2 2 ,2 2 ,2 2 ,22 ,1 2 ,1

extr extrextr extr extr

extr extr extr extr

Rrr rb rr rr

N N N N NF NFNNF N NF N NF NN NR

rr rb b rr rr

NF N N N NF N N NFN NF

X T U T S

G A G G

G B G G 38

rrG isthetransposematrixof rrG .

Therefore,equations 36 and 37 becomesquare,asfollows:

( ){ }

+ + + + + + + +

é ù é ù é ùê ú ê ú ê úê ú ê ú ê ú=ê ú ê ú ê úê ú ê ú ê úê ú ê ú ê úë û ë û ë û

int int int int

2 3 2 ,2 3 2 2 3 2 2 3 2 ,2

extr extr extr extr extr

bb brR R R R

NFrr rb rr rr rr rr

N N NF N N NF N N NF N N NF NF

K U G f s

G A G G G G

+ ++ +

é ù é ùê ú ê úê ú ê ú+ +ê ú ê úê ú ê úê ú ê úë ûë û

intint

2 3 22 3 2 ,2

extrextr

Nrr rb

N N NFN N NF N

( )+ ++ + + + + +

é ùé ù é ù- ê úê ú ê úê úê ú ê ú- =ê úê ú ê úê úê ú ê ú- ê úê ú ê úë ûë û ë û

intint int int2 3 2 ,22 3 2 ,2 3 2 2 3 2 ,1

extr exextr extr extr

bb brR R R

Rrr rb rr rr rr rr

N N NF NFN N NF N N NF N N NF

T Tf s

G A G G G G

+ ++ +

é ù é ùê ú ê úê ú ê ú+ +ê ú ê úê ú ê úê ú ê úë ûë û

intint

2 3 22 3 2 ,2

Nrr rb

N N NN N NF N

4.3.3Non‐linearformulation

Asonecansee,equations 39 and 40 arenonlinearregardingslip s .Tosolvethem,onehastotake into account the non‐linear relationship described by the debondingmodel presented in item4.3.1,inwhichtherelationbetweenthedebondingforce f andtheslip s isestablished.Theequilibri‐umequation 39 isthenrewrittenintermsofthevariableincrements,asfollows:

( ){ } { }é ù é ù é ù é ù é ùDê ú ê ú ê ú ê ú ê úê ú ê ú ê ú ê ú ê úD = D D + D + Dê ú ê ú ê ú ê ú ê úê ú ê ú ê ú ê ú ê úDê ú ê ú ê ú ê ú ê úë ûë ûë û ë û ë û

bb i br bbR R R R i

i i i b i

irr rb rr rr rr rr rr rb

K U G f s F F

G A G G G G G B

IsolatingD iX andD R

iU infirstandsecondequation,respectively,of 41 ,onehas:

( ){ } { }- -é ù é ù é ù é ù é ùD = D D + Dë û ë û ë û ë û ë û1 1R i

i bb br i i bb bb bX f s FA G A B 42

{ } ( ){ } { }- -é ù é ù é ùD = D D + Dê ú ê ú ê úë û ë û ë û

1 1R R R R Ri i i iU K G f s K F 43

whichcanbereplacedinequation 41 ,resulting:

( ) ( ){ } { } { } { }é ù é ù é ù é ùD = D D + D + D + D =ë û ë û ë û ë û1 2 3 4 0R ii i i b i iY s M f s M F M F M S 44

é ù é ùé ù é ù é ùé ù é ù é ù é ù= + -ê ú ê úë û ë û ë û ë û ë ûë û ë û ë û ë ûé ù é ùé ù é ù é ù é ù é ù= -ë û ë û ë û ë û ë ûë û ë û

é ùé ù é ùé ù = ê úë û ë ûë û ë ûé ùé ù é ù=ë û ë ûë û

R Rrr rb bb br rr

rr rb bb bb rr rb

M T K G I

G A A G G

G A A B G B

Where I istheidentitymatrix.

Equation 44 represents a non‐linear system of equations given in terms of the slip increment{ }D iS .ItcanbesolvedbyapplyingtheiterativeNewton‐Raphsonscheme.Then,fromtheiterationn

thenexttry, 1n + ,forthetimeincrement itD isgivenby:

+D = D + D1n n ni i iS S Sd 46

Linearizingequation 44 andusingthefirsttermoftheTaylor’sexpansion,results:

( )( )

0nin n

Y sY s s

¶ DD + D =

¶D 47

Thederivativethatappearsinequation 47 isdirectlyobtainedfromequation 44 usingthedebond‐ingmodelrelationshipsgivenbyequations 27 ‐ 30 .Then,onehas:

( ) ( ){ }¶ D ¶ D Dé ù é ù é ù= + =ë û ë û ë û¶D ¶D

n R nCTOi i i

n ni i

Y s f sM M W

Thematrix é ùë ûCTO

W ,inequation 48 ,istheconsistenttangentoperatoroftheproposedalgorithm.

Thederivativesontherighthandsideofequation 48 dependontheupdatedslipvalue,computedappropriately according to the adoptedmodeldefined inequations 27 ‐ 30 .Thesederivatives arelocallydefinedby:Tomodel1

( )¶ D= >

¶D0 0

R ni i

f sfor s

Tomodel2

( )¶ Dé ù= ë û¶D

10 0,R ni i

f sfor s

( )¶ D - é ù= ë û-¶Dmax

1 21 2

,R ni i res

f s f ffor s s

s ss51

( )¶ D= >

R ni i

f sfor s s

Reachingtheconvergenceinequation 44 forthetimeincrement itD aftern iterations,onehastocomputetheslipvariable s tostartthenextincrement,asfollows:

+ = + D1n

i i is s s 53

After finding +D = -1i i is s s , othervariables aredirectlyobtained.The internal displacements, and

boundary tractions and displacements are computed from equation 41 . The debonding forces are

computedfromtheconstitutiverelation ( )D DRi if s .

4.4Numericalexample–debonding

Inthisitem,twonumericalexamplesareanalyzedtoexaminetheperformanceandaccuracyoftheproposedBEM/FEMcombinationusingtwomodelstoconsiderthedebondingbetweenastraightbarandatwo‐dimensionalsolid.4.4.1Example1

Inthisexamplethecapabilityoftheformulationtomodelthebondingshearcontactforcedistribu‐tionalongthebar‐matrixinterfaceduringaclassicalpullingtestisanalyzed.Infigure12,abarispar‐tiallyembedded intoa2Ddomainandasmallpart to thebar isnot immersedtoallowapplyingthepulling force. The adopted geometric dimensions are 1.0H m= , 0 4.0L m= and 5.0L m= , see

figure12.Nulldisplacementsareprescribedalongtheleftverticalsideofthetwo‐dimensionaldomain.Whereasattheoppositesidetheloadisappliedbyprescribingthedisplacement, 4.0U = 710 m- ,atthebarextremity;the2Ddomainrightendisfreetomove.AsU isapplied,itsconjugateforce P iscalculated. Theadopteddomainelasticpropertiesare:Young’smodulus, 2.8DE = 10 210 N m andPoisson’s

ratio 0.0n = . The bar properties are: Young’s modulus, 2.8RE = 11 210 N m , inertia moment,

1.79RI = 7 410 m- andcrosssectionalarea 1.29RA = 2 210 m- .Forthisexampleitisconsideredthe

debonding model 2, with the following parameters: 1 1.0s = 910 m- , 2 1.0s = 810 m- , max 1.40f =2 210 N m and 1.30resf = 2 210 N m .

Figure12:Pullouttest.

Aboundarymeshwith120 linearelements isadoptedtoapproximate thematrix,while100uni‐formcubicfiniteelementswereadoptedtomodelthesinglebar.Finermesheshavebeentestedtocon‐firmthatthediscretizationadoptedwasenoughfinetogiveaccurateresults.

IntheFigure13itispresentedthetractioncurves ( )N m alongtheinterface atbar toninedif‐

ferent imposed displacements, which shows the evolution of the bar pullout. As one can see, thedebondedregionkeepstheconstantvalue, resf ,atallincrements.Whenthepullingoutiscompleted

thefinalloadisexactlytheexpected.

Figure13:Shearcontactforcealongtheinterface,BEM/FEM.

Figure 14 shows thedisplacement curves, inmeter, at the bar‐domain interface. It is possible toverifyadecreaseontheslopeofdisplacementsforthepointslocatedneartheloadapplication,astheloadingprogresses.

Figure14:Evolutionofthedomaindisplacementsalongtheinterface,BEM/FEM.

Figure15shows therelativedisplacementsbetweenbaranddomain.According to thedefinitiongiveninequation 32 ,thebardisplacementsareobtainedbysubtractingtheresultsoffigure14fromfigure15. Theevolutionofdecouplingisillustratedinfigure16whereinthedomaindisplacementsareune‐qualtothebardisplacements.Atthefirstdisplacementincrementitisverifiedthatalmostallnodesareperfectlycoupled,excepttheendsnodes.Asthefreeenddisplacementisincreased,othernodesbegintodecoupleuntiltheninthincrementsituationinwhichthebardisplacementscontinueincreasinganddomaindisplacementdecreasing.

Figure15:Relativedisplacement s

Figure16:Decoupleevolutionbetweendomainandbar.

One important aspect shownby this example,more evident in figure13, is that the limitationofadherenceforcebythedebondingprocessregularizestheshearcontactforceandothervariables.

4.4.2Example2

ThestructureanalyzedinthisexampleisshowninFigure17,itisimportanttonotethatthisexam‐plecannotbesolvedwithoutconsideringbendingstiffnessandtransversecontactforcesasdidinthiswork.Thisstructureisa2Ddeepfoundation,i.e.,apileembeddedinaninfinitesoil.Inordertosimu‐late thepresenceofnearby structuresanda rigid supporting rockmass, somesoildisplacement re‐

strictionsareimposed.Thepileisconsideredinclinedatanangleof10 ( )a .Asoilregion,with5me‐

ters length ( )L and20metersdepth ( )H , is considered.Thepilehas4meters in length ( )0L anda

prescribedloadof 0.11xP kN= , 1.2yP kN= and 0.003 .M N m= - atitstop.Loadsincreasefrom

zerotothereferencevaluesin50equalincrements. Ameshof300linearelementsisusedtomodelthesoiland100finiteelements lineelements are

usedtomodelthepile.Thesoilelasticpropertiesare: 2.8DE = 10 210 N m and 0.2Dn = .Thepile

properties are: 2.8RE = 11 210 N m , 0.0Rn = , 1.79RI = 2 210 m- and 1.29RA = 2 210 m- . Both

adherence models are considered. The model 2 parameters are: 71 10s m-= , 2 8.0s = 610 m- ,

max 3.0f = 2 210 N m and 2.7resf = 2 210 N m .Formodel1itisadopted: max 2.0f = 2 210 N m .

Figure17:Inclinedpileembeddedininfinitedomain.

Figure17showstheanalyzedstructureandthereferencenodes forwhichresultsarepresented.Figure18showscurvesofthepile/soilinterfaceforcesatreferencenodes.Thisfigurealsoshowsthedevelopmentofthoseinterfaceforcesastheloadingstepsareincreasedforbothslip’smodels.

Figure18:Shearcontactforcealongthepile.

Figures19‐21showthedisplacementintheaxialandtransversedirections,aswellastherotationalongthepileforeachreferencenode.

Figure19:Axialdisplacementattheinterfacepile/soil.

Figure20:Transversedisplacementattheinterfacepile/soil.

Figure21:Rotation, q ,ofthepile.

As can be seen in figures 19‐21, the pile behavior for model 1 presents severe changes. Thesechangesoccurwhenthemaximumloadcapacityisreached increment35 ;thenslipoccurswithoutfurthergainofresistance.Itisimportanttomentionthatovertheloadcapacitythesolutionisunstableandthesystemlossobjectivity. Formodel2astheinterfaceforcesdonotreachtheresidualpartofthemodelthemaximumloadcapacity isnotachievedandnoabruptchangeoccurs in thepilebehavior.However, if the total loadcapacityisreached allcontactpointsreachtheresidualpartofthemodel thecollapseoccurs. Figure22 a and b show thedisplacements in the y direction, inmeters, for the soil internal

pointsconsideringthemodels1and2,respectively.Figures23 a and b showthestressvalues, ys ,

forthesoilinternalpointsformodels1and2.

Figure22:Soildisplacementintheydirection

Figure23:Soilstress, ys ,fordifferentadherencemodels.

5 CONCLUSIONS

InthisworkaBEM/FEMcouplingamongframebarsand2Dcontinuumissuccessfullydevelopedandimplemented.ThedomainismodeledbyBEMandreinforcementismodeledbyFEM.Thecombi‐nationof the twomethods ismadebywritingdisplacement compatibility and interfaceequilibrium.Themostimportantfeatureoftheformulationistheconsiderationofslidingatframe/continuuminter‐face.Thisprocedureisabletomodel,forexample,theprogressivefailureofpile‐soilinteractionuntilreachingthecollapse load,or theprogressive failureof fiberreinforcedbodiesconsideringthe influ‐enceofshearandnormalcontactforces. Regardingthesolutionbehavioranimportantconclusionshouldbestated.Asboundaryelementsareabletomodelhighstressconcentrations,thevaluesofcontactforcesatthebeginningorendingofanyperfectboundingregionpresentastrongperturbation.Theuseofredundantalgebraicequationsandtheleastsquaresmethodaretestedhereandresultinasmallimprovementofthisphenomenon.Theincreasingofdiscretizationreducestheextensionofperturbationbutincreasesthenearsingularstresses. Thecompletesolutionfor thisproblemresultswhenusingamorerealisticmodel thatallowsthenaturalstressrelaxationatsingularities.Thedevelopednonlinearbehaviorofcontactforces,consider‐ingtheslidingordecouplingbetweenreinforcementandcontinuum,completelyregularizesthecontactforcebehavior,leadingtoreliablesolutionsforloworhighloadsituations.Furtherdevelopmentsaretheconsiderationofnon‐linearbehaviorforbothcontinuumandframemedia.AcknowledgementsAuthorswouldliketoacknowledgeCNPq NationalCounselofTechnologicalandScientificDevelopment andFAPESP SãoPauloResearchFoundation forthefinancialsupport.References

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