Meso- and Macroscopic Models for Fiber-Reinforced Concrete

Transcription

1 Meo- an Maroopi Moel or Fiber-Reinore Conree Sonia M. Vreh & Guillermo e CONICT, Univeriie o Tuuman an Bueno Aire, Argenina. Günher Mehke Iniue or Sruural Mehani, Ruhr Univeriy Bohum, Bohum, Germany. Anonio Caggiano & nzo Marinelli Darmen o Civil ngineering, Univeriy o Salerno, Fiiano (SA), Ialy. ABSTRACT: Fiber reinore onree i analyze an moele a wo ieren level o obervaion. On he one han, a maroopi ormulaion bae on he non-linear miroplane heory i preene. Following approahe reenly propoe in Pieruzzak & Winniki (23) an Manzoli e al. (28), he mixure heory i ue o eribe he ouple aion beween onree an he iber reinoremen. The paraboli Druker- Prager maximum rengh rierion i oniere a he miroplane level. Po-peak behavior i ormulae in erm o he raure energy releae uner moe I an/or II ailure moe. On he oher han, a meoopi moel o iber reinore onree (FRC) i alo preene whih i bae on hree oniuen: aggregae, morar an aggregae-morar inerae. Aggregae are oniere o be elai while rak are rreene in a iree orma by mean o inerae elemen. The preene o eel iber i oniere wihin he ramework o he mixure heory. Conequenly, morar-morar inerae aoun or boh iber-morar eboning an owel ee aoring o he iber volume onen. Aer eribing he oniuive moel he paper oue on numerial analyi o FRC ailure behavior inluing re-analyze o he experimenal e o Haanzaeh (199). The apabiliie an horoming o boh approahe or FRC ailure analye are evaluae. 1 INTRODUCTION Funamenal eiienie o emen-bae maerial like onree an morar uh a low enile rengh an brilene an be miigae by aing eel iber ino he marix. Fiber play a major role in he po-raking behavior o iber reinore morar ompoie (FRMC) by briging he rak an proviing reiane o he rak opening proe. Aually, FRMC may ahieve quai-uile repone exhibiing rain-harening repone wih muliple rak an relaively large energy aborpion prior o raure loalizaion. In hi ae he ompoie ake he name o high perormane eel iber reinore morar ompoie (HPFRMC). Regaring ruural behavior o onree member, he aiion o iber lea alo o igniian improvemen in he uiliy in pre- an po-peak regime a well a in he enile peak re. Thi i a onequene o he inreae o iipaion aribue o he aion o he iber briging opening mirorak an he reuion o volumeri expanion in he low oninemen regime. Oher relevan avanage aribue o FRMC i he reue waer permeabiliy. Dieren approahe on ieren ale have been propoe or he moeling o FRMC. They an be broaly aegorize a ollow: Miro-ale moel (more orrely enoe a meo-ale moel): moel whih eribe he ineraion among he phae o he ompoie maerial, i.e. iber, marix, ine an oare aggregae, an ineraial zone beween hem on he ale o he iniviual iber. Maro-ale moel: he iber an he marix inie he FRMC a hi ale o obervaion are iniinguihable. In hi onex FRMC i oniere a a homogeneou maerial. Among oher, we reer o he maroale moel or FRMC by Hu e al. (23) who propoe a ingle mooh biaxial ailure urae or eel iber reinore onree (SFRC), he propoal by Seow & Swaiwuhipong (25) bae on a ive parameer ailure rierion or FRC wih raigh an hooke iber an ha o Minelli & Vehio (26) bae on a moiiaion o he ompreion iel heory. Oher relevan work

2 are hoe by Guema (23), Pyl (23), e. Sruural-ale moel: hee moel apure he eene o he maerial behavior a he ruural level, or example, roeional momen veru urvaure behavior or panel hear ore veru laeral iplaemen. Semi-analyial moel, or he lexure behavior o iber-reinore onree (FRC) maerial, bae on he equilibrium o ore in he riial rake eion, have been propoe by Zhang & Sang (1997). Sang & Oleen (1998) preen a eign approah or iber reinore onree ruure. Lee & Barr (23) haraerize he omplee loa/re-eormaion urve, uner variou loaing oniion, uing one oninuou our-exponenial union. Muli-ale moel: he perormane o hee moel are bae on oupling ingreien o ieren ale moel: miro, meo, maro an ruural moel (e.g. Kabele (22)). The preen work eal wih ailure analyi o FRC maerial a he maro an meoopi level o obervaion. Boh approahe ue he ompoie heory a a bai or he imulaion o he ineraion beween marix an eel iber. The main aim o hi reearh i he evaluaion o unamenal properie a he meoopi level onrolling he mehanial repone behavior o FRMC uring monooni loaing beyon he elai range. To hi en, maroopi oniuive ormulaion whih inorporae inormaion on he ibre-morar-ineraion a he meo-level i ue. For he 2D meoopi analyi in hi work a new mehoology i ollowe bae on oniering FRMC a a hree phae maerial: aggregae, morar an he inerae beween he oher wo oniuen. The non-linear behavior o eel iber reinore morar i apure by mean o zero-hikne join elemen. To hi en he original inerae moel by Carol e al. (1997) i reormulae o inlue he ineraion beween eel iber an morar bae on he ompoie heory by Manzoli e al. (28). The maroopi approah i bae on he miroplane heory ombine wih he low heory o plaiiy an he paraboli Druker-Prager maximum rengh rierion. In hi ae, he ompoie heory aoun or he ineraion beween eel iber an onree (inluing he ee o aggregae a he maroopi level). The numerial analyi preene in hi work inlue preiion o boh approahe o FRC ailure behavior in ire hear an uniaxial raion. Alo he re pah o experimenal e are oniere. The reul emonrae he poenial o he meoopi approah in hi work o evaluae relevan ape o FRC uh a he inluene o he aggregae maximum ize, o he raio beween hi ize an he iber lengh, an o he ailure mehanim o hi omplex maerial. On he oher han, he miroplane bae heory or maroopi evaluaion o FRC ailure behavior allow he onieraion o arbirary an muliple ireion or eel iber an, evenually, o non-homogeneou maerial. The nex o hi reearh onier he inluion o rain graien in he oniuive ormulaion o he miroplane o apure he ize ee o FRC an, pariularly, he inluene o he raio beween maximum aggregae ize an he iber lengh. 2 CONSTITUTIV MSO-MCHANICAL MODL N Fraure analyi o FRMC on he meo-level ollow he approah propoe by Vonk (1992) an hen alo ue by Lopez e al. (28a,b). While oninuum morar elemen are aume a linear elai, he non-linear iipaive behavior o plain an iber reinore morar i ully loalize in he inerae elemen. Aoring o he bai hypohei o he mixure heory, he ompoie i oniere a a oninuum in whih eah ininieimal volume i ieally oupie imulaneouly by all oniuen. In he plane o he inerae, all oniuen are ubjee o he ame rain iel an he orreponing ompoie ree are given by he weighe (in erm o he volume raion) um o he oniuen ree. Being u he inerae relaive iplaemen veor, axial iplaemen o he iber (in n ireion) i given by u = u n, while in he ranveral ireion, u T = u nt, being he uni veor perpeniular o n, ee Figure (1). Conequenly, he axial an angular iber rain are given by ε = u / l an γ T T repeively, being iber. = u / l N l he lengh o he Figure 1: Shemai oniguraion o join roe by one iber N

3 Similarly o he low heory o plaiiy, he inerae oniuive moel i ormulae in rae orm. Aoring o he ompoie heory, he rae o he re veor & = [ σ &, τ] & in he inerae plane i alulae a he um o eah oniuen weighe by he volumeri raion, k & m m k & + k σ& N + ( ε& ) n k & τ (& γ )( n ) = (1) The uperrip m an reer o morar marix an iber, repeively, while mean he ranpoiion operaion or enor. The inremenal re-iplaemen relaion o he propoe ompoie join moel an be expree in ompa orm a & = u& (2) where he oniuive angen marix T T i m = k C + k n n + k (3) Wih G C / l G l ( nt ) nt = / u, = σ / ε N an = τ γ a angen operaor eine in he T ollowing eion. General ape o he oniuive ormulaion, whih moel he ioninuiy behavior a he inerae in preene o eel iber, are he ollowing: Fraure energy-bae join oniuive law: he oniuive moel i ormulae in erm o normal an hear ree on he inerae plane, an orreponing relaive iplaemen. The ailure rierion ( F = ) o he inerae moel i eribe by he hrearameer hyperbola by Carol e al. (1997). Main eaure o he elaoplai inerae moel ormulaion are ummarize in he Seion 2.1. Fiber bon-lip ee are oniere a a ombinaion o he uniaxial elao-plai moel, or iber, an a uniaxial eboning iipaive moel or he inerae morariber, reuling in a global oniuive moel or he oniuen lipping iber. Dowel aion o reinoremen hor iber roing rak in morar i moele in a meare manner. Beam on elai ounaion heory i uilize o erive he owel ore-iplaemen law, whih i expree in erm o owel re an rain in orer o be ompaible wih he previouly iniae oniuive law. Boh he bon-lip axial moel o he iber an he owel aion o hor iber reinoremen roing rak in morar/onree are imilarly oniere in boh meo- an maroopi approahe. Thee moel are eaile in Seion Fraure-bae inerae oniuive moel In hi eion, he inerae moel, originally propoe by Gen e al. (1988), i ummarize. The elao-plai ormulaion o he inerae moel in rae orm i eine by u & & + & u el r = u u (4) el 1 & = C & r ( u& u& ) (5) & = C (6) where u & = [ u &, v& ] i he rae veor o relaive iplaemen, eompoe ino he elai an rak el r opening omponen u& an u&, repeively. C eine a ully unouple normal/angenial elai ine a he inerae kn C = (7) k The yiel-loaing oniion o he inerae oniuive moel i eine a F ( σ an φ) 2 + ( + χ anφ) 2 = τ 2 (8) wihτ an σ a he inerae re omponen. The enile rengh χ (verex o he hyperbola), he hear rengh (oheion rengh) an he inernal riion angle φ are moel parameer. In q. (8), wo limi iuaion an be iinguihe: (a) raking uner pure enion, wih zero hear re (Moe I), when he yiel urae i reahe along he horizonal axi, an (b) raking uner hear an very high ompreion, when he yiel urae i reahe in i aympoi region, where he hyperbola approahe a Mohr-Coulomb rierion. The la one i alle aympoi Moe II (or Moe IIa). The evoluion o raure proe i riven by he raking parameer χ an, whih en on he energy releae uring inerae egraaion W r. Deail are given in Carol e al. (1997). 3 MACROSCOPIC MODL BASD ON MICROPLAN THORY For he meoopi analyi o FRC ailure behavior a raure energy-bae elaoplai miroplane moel wa evelope. Inea o he exiing pherial miroplane moel, ee a. o. Beghini e al. (27), Carol e al. (21), Kuhl e al. (21), he propoe oniuive heory onier a 2D re an rain iel uing ik miroplane aoring o he propoal by Park & Kim (23). A a reul, a reue number o miroplane i require. Aoring o hi approah,

4 he iber-reinore onree i iealize a a ik o uni raiu an onan hikne b, whih agree wih ha o he analyze maerial pah. The ollowing aumpion are oniere: Maroopi ree are uniorm in he ik an are equilibrae by he urae raion on he miroplane. Miroopi rain in normal (ε ) an angenial ( γ r ) ireion o eah miroplane wih normal ireion n, are obaine rom maroopi rain ε (kinemai onrain) ε = n n ε (9) i j ij [ n i δ jr + n j ir 2nin j nr ] ε ij 1 γ r = δ (1) 2 Normal an angenial miroopi ree = [ σ, τ r ] are obaine rom he miroopi ree energy poenial σ = mi [ ρψ ] [ ψ ], ρ mi ε τ r = γ r (11) wih ρ enoing he maerial eniy. The maroopi ree-energy poenial per uni ma o maerial in iohermal oniion, ( ε,κ) ma ψ, wih κ a e o hermoynamially onien inernal variable, reul ma 1 mi ψ = ψ ( ε, κ) V (12) bπ V wih V a he ik volume. 3.1 Miroplane oniuive law The miroplane oniuive law i bae on he mixure heory by Trueell & Toupin (196) an, imilarly o he inerae moel ormulaion ue in he analyi a he meoopi level o obervaion, on he hypohei o FRC o be rreene by a ompoie moel (Oliver e al. (28)). Then, he rae o re veor & a eah miroplane i obaine by q. (1) where uperrip m reer now o he onree marix variable. The oniuive moel o he onree marix i bae on he paraboli Druker-Prager rengh rierion. The yiel oniion in harening/oening regime i eine by he uniie equaion F (,k) = β J + α p ( K( )) (13) 2 κ in erm o he preure p, he eon invarian o he eviaori re enor J 2, he iipaive plai re K in erm on he rain-like inernal variable κ, an he riion an oheion parameer α an β, repeively, ha are eine a union o he uniaxial ompreion an enile rengh,, repeively, aoring o ( ) 3 3 α =, = an β (14) The evoluion o he inernal variable i eine in erm o he plai parameer rae λ & a F & κ = = & λ K (15) A non-aoiae low i aope in orer o avoi he exeive inelai ilaany. The plai poenial i bae on a volumeri moiiaion o he yiel oniion in he ompreive regime. Then, he graien enor m o o he plai poenial an be obaine by a moiiaion o he graien enor n σ o he yiel urae a nσ i p > m = p (16) 1 nσ i < p < pil pil where p il i a moel parameer rreening he value a whih he ilaany vanihe, ee Figure (2). In he po-peak regime he evoluion o he iipaive re, ue o miro-raure proe a he miroplane level, i eine hrough he homogenizaion proe o he raure energy releae uring rak ormaion wih he plai iipaion o an equivalen oninuum, imilarly o he raure energy-bae plaiiy moel by Willam e al. (1985) an e & Willam (1994), a I = h G K& 1 exp 5 ε& IIa r (17) ur G wih he equivalen raure rain & ε & r = m I κ (18) where h rreen he haraerii lengh aoiae wih he aive raure proe an, more peiially, he iane or araion beween mirorak. Moreover, u r rreen he maximum rak opening iplaemen in moe I ype o ailure. G an G are he raure energie in moe I I IIa an II o ailure, repeively. The MAuley brake in q. (18) iniae ha only enile prinipal plai rain onribue o he raure rain uring raure proe. In he peial ae o uniaxial enion ae, he evoluion o he iipaive re an be obaine wih he impliie expreion

5 Figure 2: Maroopi oniuive moel. = h K& 1 exp 5 ε& r (19) ur 4 FIBR-MORTAR/CONCRT INTRACTION In hi eion he moel or he ineraion beween eel iber an morar (meoopi moel), a well a eel iber an onree (maroopi moel), oniering he bon-lip an owel ee, are preene. 4.1 Bon-lip axial moel o he iber The uniaxial behavior o he eel iber i approimae by mean o a imple 1D elao-plai moel. The ollowing e o equaion i oniere & ε & + & el p = ε ε (2) & σ el & = (21) ε p (& ε & ε ) & σ = (22) where he rae o he axial iber rain ε& i eompoe ino a elai par an a plai omponen, ε& el p an ε&, repeively. rreen an equivalen uniaxial elai moulu inluing he uniaxial repone o he eel an he bon-lip ee o he hor reinoremen. σ& i he rae o bon-lip axial re o he eel iber. The yiel rierion, in enion a well a in ompreion, i rreene by he ollowing expreion ( Q ) F = σ σ + (23) y, where σ y, i he elai limi. The evoluion, in he po-elai regime o he 1D urae i riven by he re-like inernal variable Q, given in inremenal orm a Figure 3: Uniaxial bon-lip moel or he iber. & = & λ H (24) Q p // wih & ε & λ F / σ = & λ ign[ σ ] = rreening he plai low law, & λ i he non-negaive plai muliplier an H i he harening/oening moulu. The inremenal re-rain relaionhip i & σ = & ε (25) where he elao-plai angen moulu ake he wo ollowing iin ollowing value, ee Figure (3) = = 1 /H + 1 lai repone (26) lao - plai regime I i aume ha he iber rain ε i eompoe in wo aiive par, one ue o he inrini iber uniaxial eormaion ε an anoher one aoiae wih he inerae eboning ε ε = ε + ε (27) Auming a erial moel oniue by he iber an he iber-morar join he orreponing oal eormabiliy 1 / i given by 1 / 1 / 1 / = + (28) where an are he eel Young moulu an an equivalen elai moulu o marix-iber inerae, repeively. Two limi iuaion an be reognize: : he ine o he omplee ruure beome null an he ee o he iber vanihe. : rreen he ae o a pere aherene beween marix-iber.

6 In hi onex, an o omplee he bon-lip axial oniuive moel preene by he q. (2) o (26), he ollowing maerial parameer are eine σ = min[ σ, ] (29) y, y, σ y, H H I σ y, < σ = H oherwie y, y, (3) in whih σ y, an σ y, are he maerial yiel re an he equivalen inerae elai limi, repeively; while he uper-inie an reer o eel an eboning, repeively. The parameer, σ an H require or he bon/lip moel haraerizaion an be alibrae rom a imple pull-ou e (Oliver e al. 28). 4.2 Dowel aion o reinoremen hor iber roing rak in morar marix. The owel ee o iber roing rak in morar, i aken aoun in he join moel by mean o a 1D hear re-rain elao-plai oniuive moel, imilar o he previouly menione one or he axial re-rain. In hi ae, he ollowing equaion are uilize & γ & γ + & γ el pl = (31) & τ el & = (32) γ G pl (& γ & ) & τ = (33) G γ being γ& he rae o he hear iber rain, whih i eompoe ino a elai an a plai par, el γ& an pl γ&, repeively. G rreen he hear moulu, while τ& i he rae o owel hear re o he ineraion beween iber-marix. The moel i omplee wih: yieling rierion, imilar o q. (23); harening/oening law, imilar o q. (24). The inremenal hear re-rain relaion, an be wrien a & τ = & γ (34) G where, in a imilar manner a in he bon moel, he angen hear moulu G ake he wo iin ollowing value G G = G = G 1 G /H ow + 1 elai repone plai regime (35) Figure 4. Dowel ee bae on Winkler beam heory. ow H i he harening/oening moulu o he uniaxial owel moel, ommonly aume a H ow =. Dowel ee an be analyze reaing eah reinoremen iber a beam on elai ounaion (Winkler heory) o eal wih he ineraion beween he iber an he urrouning morar (He an Kwan, 21, Rumanu an Mehke, 21). The iber an be reae a a emi-ininie beam on he elai ounaion, loae wih a onenrae loa a one exreme rreening he owel reulanv. The analyial oluion o he beam on elai ounaion in Figure (4) reul in he ollowing ore-iplaemen relaionhip V - 3 V = I λ (36) 4 in whih I = π / 64 i he momen o ineria o he iber ( iameer o he iber) an λ parameer rreening he relaive ine beween he iber an he ounaion, eine a k λ 4 (37) = 4 I Thereby i k he ounaion moulu o he urrouning morar ha govern he owel ine. xperimenal aa, available or RC peimen (Dei Poli e al. 1992), how ha he ame oeiien ake 3 value ranging rom 75 o 45 N/mm. Oher e (Sorouhian e al. 1987) how ha he oeiien k inreae a he rengh o urrouning morar inreae an when he volume raion o he reinoremen inreae. An equivalen hear elai moulu an be alulae a 3 L 3 V = I λ = G A G = I λ (38) L A 2 where A = π / 4 i he ro area eion o he bar. A he limi age, loal ruhing o he urrouning morar an/or yieling o he owel bar our. Bae on experimenal reul or RC peimen, he

7 ollowing equaion ha been propoe by Dulaka (1972) or he owel ore a he limi age V u = k σ (39) ow 2 y, In (39) k ow i a non-imenional oeiien ( k ow = 1. 27, or RC-ruure), while i he ompreive rengh o he onree or he urrouning morar. Finally, he equivalen hear yiel re, reul a V τ y, u τ y, = (4) A 5 NUMRICAL ANALYSIS In hi eion numerial analye are perorme wih boh he meoopi an maroopi moel an oniering ailure behavior o FRMC an FRC, repeively. 5.1 Meoopi evaluaion o FRMC ailure Behavior To evaluae he preiive apabiliie o he propoe non-linear iipaive inerae moel or FRMC he elemen pahe hown in Figure (5) are oniere. Thereby, one inerae elemen i plae beween 2D plane re ioparameri our noe elemen ubjee o he iniae bounary oniion in erm o impee iplaemen. A iniae in he ame Figure (5) ix ieren ae are oniere wih one, wo, hree, ive, even an nine eel iber roing he inerae elemen (all o hem having he ame iameer ). Firly, a uniaxial enile e i perorme by impoing homogeneou verial iplaemen o all our noe o he upper quarilaeral elemen. The reul in erm o verial nominal re v. verial iplaemen are hown in Figure (6) when he e i perorme wih ieren amoun o iber roing he inerae ( = onan =.8mm), an in Figure (7) where he peii ae o 5 iber are oniere bu wih ix ieren value o. The reul in Figure (6) an (7) iniae ha he propoe moel i able o rroue he eniiviy o FRMC regaring peak loa an po-peak uiliy o boh he amoun o iber an he iameer o he iber (when equal number o iber are oniere). Moreover, he eniiviy o he po-peak uiliy i igniianly more imporan han ha o Figure 5: Inerae oniguraion wih: (a) one, (b) wo, () hree, () ive, (e) even an () nine eel iber ( = :8mm). he peak loa wih repe o iber amoun an iameer. Po-peak repone in boh igure how inreaing reloaing ee wih he inremen in he amoun an iameer o he iber. Thi i ue o he inreaing ompoie (inhomogeneiy) ee ha reul by enlarging he amoun or iameer o he iber. Figure (8) an (9) how he moel perormane when ompare again he experimenal reul by Haanzaeh (199) (inlue wih oe line). Thee e are haraerize by impoing ombine normal an hear relaive iplaemen o a eveloping rak in a primai onree peimen 2 o.7.7 m quare ro eion wih a.15m e noh. During ir par o he numerial e, a pure enion re ae i impoe unil he peak rengh i reahe. From ha poin, normal an hear relaive iplaemen are applie imulaneouly in a ixe proporion haraerize by a onan value o he relaion an θ = u/v, wih u an v he normal an angenial relaive inerae iplaemen, repeively. Thee e were reanalyze wih θ = 3 an θ = 6, boh or =.8mm. Moel parameer value ue in hee numerial analyi are: k N = 2 MPa/m, k T = 2 MPa/m, an φ =.9, χ = 2.8MPa, I = 7.MPa, G =.1N/mm, σ il = 3 MPa, α χ = an α =, or he inerae, while ν =. 2 an m = 25 MPa, were ue or he elai moulu an he Poion raio, repeively, or he morar. Reul in Figure (8a) how, a expee, ha he angenial an normal iplaemen onrol in he eon par o he Haanzaeh e or θ = 3 i reponible or a ronger oening o he normal enile re han in ae o he pure enion e.

8 Figure 6: Normal re v. verial iplaemen perorme wih ieren amoun o iber wih = onan =.8mm. Figure 8: Single rak roe by iber: numerial e wih θ = 3, = onan =.8mm an ieren number o iber: (a) normal re v. relaive iplaemen an (b) hear re v. relaive iplaemen. Figure 7: Normal re v. verial iplaemen perorme wih ix ieren value o ( n = onan = 5 ). The iber onen ae mainly he la porion o he urve in he reloaing zone. Wih oher wor, he rong po-peak ereae (onnee wih evere raking) in he ir porion o he oening regime o hi e praially uppree he iber onribuion o he uiliy (o ompare he reul wih ha orreponing o plain morar). In onra, reul in erm o hear re v. angenial relaive iplaemen in Figure (8b) how relevan iber onribuion in he pre- an po-peak regime a well a in he maximum rengh. When omparing hee urve wih reul in Figure (9a) an (9b) we oberve, a expee, le evere oening boh in normal an hear re omponen. Moreover, only in ae o plain morar or low onen o iber he oening branh lea o zero in he ompreive regime o he normal re. Wih hree iber (or more hen hree) he oening regime ully remain in he enile regime iniaing a igniian inreae o he uiliy a ompare o plain morar ae. In onluion, he propoe inerae moel or meoopi analye o FRMC ailure behavior eem o provie realii preiion o peak ree, uiliy an po-peak behavior o hi maerial when ieren iber ireion an iber onen are oniere roing a ingle rak. 5.2 Maroopi valuaion o Failure Behavior o FRC To evaluae he apabiliie o he propoe maroopi moel bae on miroplane heory o imulae ailure behavior o FRC, preliminary numerial uie oniering a uniireional an an ioropi iribuion o iber, are perorme. A ingle elemen problem in plane rain oniion an ubjee o a homogeneou re/rain ae i oniere. The onree marix i haraerize by he ollowing maerial parameer = 19MPa, ν =. 2, = 22MPa, = 3.MPa, h = 18mm, u r =.127 mm, G /G = 1, p il / = 1. 2 Fiber maerial parameer are hoe o Subeion 5.1. Firly, a uniaxial enile e i perorme. Reul or iber oriene in he loaing ireion an wih ioropi iribuion o iber are illurae in Figure (1) an (11). Dieren volume onen V IIa I. o he iber are oniere inluing he exreme ae o plain onree ( V = ).The elai ine inreae in ae o bia iber a ompare o he plain onree ae an wih he inremen o V.

9 Figure 1: Uniaxial enile e or miroplane moel. Fiber oriene in loaing ireion. Figure 9: Single rak roe by iber: numerial e wih θ = 6, = onan =.8mm an ieren number o iber: (a) normal re v. relaive iplaemen an (b) hear re v. relaive iplaemen. In boh ae o iber orienaion a ligh inreae o he peak enile rengh an a re-iening ee in po-peak regime are oberve. Figure (12) an (13) how he reul obaine in he uniaxial ompreion e when boh uniaxially oriene iber an an ioropi iribuion o iber ireion are oniere. In ae o bia iber, a expee, he ine in he pre-peak regime an he overall iipae energy uring po-peak regime inreae wih V. 6 CONCLUSIONS Meoopi an maroopi moel or iber reinore emen ompoie maerial were preene. The meoopi moel ake ino aoun a hree phae meoruure ompoe by elai aggregae, morar an morar-aggregae inerae. Thi moel alo inlue morar-morar inerae o imulae he iipaive repone behavior o hi oniuen. The maroopi moel i ormulae wihin he heoreial ramework o miroplane heory. Boh he inerae moel an he miroplane moel or meo- an maroopi analye, repeively, are bae on low rule o plaiiy, mixure heory by Trueell & Toupin (196) an ompoie Figure 11: Uniaxial enile e or miroplane moel. Ioropi iribuion o iber. moel by Oliver e al. (28). The ineraion beween eel iber an morar/onree aoiae wih eboning an owel ee are oniere in boh moel. Preliminary numerial uie preene in hi paper emonrae (parially on a onual level) heir apabiliie o rroue he mo relevan ape o ailure behavior o eel iber reinore onree uner enile, hear an ompreive ree. 7 ACKNOWLDGMNTS The ir wo auhor aknowlege inanial uppor or hi work by FONCYT (Argenine ageny or reearh & ehnology) hrough Gran PICT1232/6, an by CONICT (Argenine ounil or iene & ehnology) hrough Gran PIP621/5. 8 RFRNCS Beghini, A., Bazan, Z.P., Zhou, Y. Gouiran, O. an Caner,

10 Figure 12: Uniaxial ompreion e or miroplane plaiiy. Fiber oriene in loaing ireion. Figure 13: Uniaxial ompreion e or miroplane plaiiy. Ioropi iribuion o iber. F.C. 27. Miroplane Moel M5 or Muliaxial Behavior an Fraure o Fiber-Reinore Conree. Journal o ngineering Mehani. 133: Carol, I., Pra, P.C. an Lopez, C.M Normal/hear raking moel: Appliaion o iree rak analyi. Journal o ngineering Mehani. 123(8): Carol, I., Jiraek, M. an Bazan, Z. 21. A hermoynamially onien approah o miroplane heory. Par I. Free energy an onien miroplane ree. Inernaional Journal o Soli an Sruure. 38: Dei Poli, S., Di Prio, M. an Gambarova, P.G Shear repone, eormaion, an ubgrae ine o a owel bar embee in onree. ACI Sru. J. 89(6): Dulaka, H Dowel aion o reinoremen roing rak in onree. ACI J. 69(12): e, G. an Willam, K A raure energy-bae oniuive heory or inelai behavior o plain onree. J. ngineering Mehani, ASC. 12, Gen, A., Carol, I. an Alono, An inerae elemen ormulaion or he analyi o oil-reinoremen ineraion. Compu. Geoehni. 7: Guema, T. B. 23 in Beirag zur realianahen Moellierung un Analye von ahlaerverarken Sahlbeon un Sahlbeonlahenragwerken. PhD Thei. Univ. Kael. Haanzaeh, M Deerminaion o raure zone properie in mixe moe I an II. ngineering Fraure Mehani. 35 (4/5): He, X. an A. Kwan (21). Moeling owel aion o reinoremen bar or inie elemen analyi o onree ruure. Compuer an Sruure 79, Hu, X. D., Daz, R. an Dux, P. 23. Biaxial ailure moel or iber reinore onree. Journal o maerial in ivil engineering. 15(6): Kabele P. 22. quivalen oninuum moel o muliple raking. ngineering Mehani. 9(1/2): Kuhl,., Seinmann, P. an Carol, I. 21 A hermoynamially onien approah o miroplane heory. Par II. Diipaion an inelai oniuive moeling. Inernaional Journal o Soli an Sruure. 38: Lee, M. K. an Barr, B. I. G. 23. A ourexponenial moel o eribe he behavior o ibre reinore onree. Maerial an Sruure. 37 (7): Lopez, C. M., Carol, I. an Aguao, A. 28a. Meo-ruural uy o onree raure uing inerae elemen. I: numerial moel an enile behavior. Maerial an Sruure. 41: Lopez, C. M., Carol, I. an Aguao, A. 28b. Meo-ruural uy o onree raure uing inerae elemen. II: ompreion, biaxial an Brazilian e. Maerial an Sruure. 41: Manzoli, O.L., Oliver, J., Huepe, A.. an Diaz, G. 28. A mixure heory bae meho or hree-imenional moeling o reinore onree member wih embee rak inie elemen. Compuer an Conree. 5(4): Minelli, F. an Vehio, F. J. 26. Compreion Fiel moeling o iber-reinore onree member uner hear loaing. ACI Sruural Journal. 16(2): Oliver, J., Linero, D.L., Huepe, A.. an Manzoli, O.L. 28. Two-imenional moeling o maerial ailure in reinore onree by mean o a oninuum rong ioninuiy approah. Compu. Meho Appl. Meh. ngrg. 197: Park, H. an Kim, H. 23. Miroplane moel or reinoreonree planar member in enion-ompreion. Journal o Sruural ngineering. 129: Pyl, T. 23 Tragverhalen von Sahlaerbeon. PhD. Thei. igenihen Tehnihen Hohhule Zrih. Swizerlan. Pieruzzak, S. an Winniki, A. 23. Coniuive Moel or Conree wih mbee Se o Reinoremen. Journal o ngineering Mehani. 129(7): Rumanu,. an Mehke, G. 21. Homogenizaion-bae moel or reinore onree. Compuaional Moeling o Conree Sruure (URO-C 21), in prin. Seow, P.. C. an Swaiwuhipong, S. 25. Failure Surae or Conree uner Muliaxial Loa - a Uniie Approah. Journal o Maerial in Civil ngineering. 17(2): Sorouhian, P., Obaeki, K., Roja, M.C Bearing rengh an ine o onree uner reinoring bar. ACI Maer. J. 84(3): Sang, H. an Oleen, J. F On he inerpreaion o bening e on FRC maerial. Fraure Mehani o Conree Sruure. 1: Trueell, C. an Toupin, R The laial iel heorie. Hanbuh er Phyik, Springer Verlag, III/I, Belin. Vonk, R Soening o onree loae in ompreion. Ph.D. hei, Tehnihe Univeriei inhoven, Pobu 513, 56 MB inhoven, he Neherlan. Willam, K., Hurbul, B. an Sure, S xperimenal an oniuive ape o onree ailure. In US-Japan Seminar on F..Anal. o R.C.Sru. ASC-Speial Publi Zhang, J. an Sang, H Appliaion o Sre Crak Wih Relaionhip in Preiing he Flexural Behavior o Fibre - Reinore Conree. Cemen an Conree Reearh. 28 (3):