Elasto-plastic multi-fixed smeared crack model for concrete

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1 Elasto-plasti multi-ixed smeared ak model or oete José Sea Cruz, Joauim Barros ad Álvaro Azevedo Report 04-DEC/E-05 Date: Marh o 006 N. o pages: 70 Keywords: Coete, umerial model, smeared ak, plastiity Versio:.0 Uiversidade do Miho Esola de Egeharia Departameto de Egeharia Civil

2 Elasto-plasti multi-ixed smeared ak model or oete INDEX Idex... Notatio...4 Itrodutio...6 Crak oepts...8. Smeared ak oept Crak strais ad ak stresses Coete ostitutive law Costitutive law o the ak Costitutive law o the aked oete Crak rature parameters.... Multi-ixed smeared ak oept Crak iitiatio Crak evolutio history Algorithmi aspets Stress update Crak status Sigularities Model appraisal Plastiity Basi assumptios Itegratio o the elasto-plasti ostitutive euatios Evaluatio o the taget operator Elasto-plasti oete model Yield surae Hardeig behavior Retur mappig algorithm Cosistet taget ostitutive matrix Model appraisal Uiaxial ompressive tests Biaxial ompressive test Elasto-plasti multi-ixed smeared ak model Yield surae...49 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

3 Elasto-plasti multi-ixed smeared ak model or oete 4. Itegratio o the ostitutive euatios Costitutive euatios rom the multi ixed smeared ak model Costitutive euatios rom the elasto plasti model Retur mappig algorithm Method proposed by de Borst ad Nauta Cosistet taget ostitutive matrix Model appraisal Numerial tests Beam ailig by shear Colusios...6 Reerees...63 APPENDIX I: hardeig/soteig law or oete...67 APPENDIX II: Cosistet taget operator...69 José Sea Cruz, Joauim Barros ad Álvaro Azevedo 3

4 Elasto-plasti multi-ixed smeared ak model or oete NOAION I D II D D e D e D ep D E G G Mode I stiess modulus Mode II stiess modulus Crak ostitutive matrix Elasti ostitutive matrix Elasto-aked ostitutive matrix Elasto-plasti ostitutive matrix Youg's modulus o oete Shear modulus o oete Mode I rature eergy o oete rasormatio matrix o a ak ( κ, ) = 0 Yield surae t h h m p Compressive stregth o oete esile stregth o oete Crak bad-width, Hardeig modulus Salar parameter that ampliies the plasti strai vetor Number o itial ak status hages Combiatio Number o distit smeared ak orietatios at eah itegratio poit Hydrostati pressure Iteratio ε Iemetal strai vetor ε l l α th β Iemetal ak strai vetor (i CrCS) Iemetal ak stress vetor (i CrCS) hreshold agle Shear retetio utio γ t Crak shear strai 4 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

5 Elasto-plasti multi-ixed smeared ak model or oete ε Strai vetor ε Crak strai vetor ε l Crak strai vetor (i CrCS) ε Crak ormal strai θ κ Agle betwee the x global axis ad the ak ormal axis Hardeig parameter Crak ormal stress τ t Crak shear stress Stress vetor l Yield stress Crak stress vetor (i CrCS) ν Poisso's ratio o oete José Sea Cruz, Joauim Barros ad Álvaro Azevedo 5

6 Elasto-plasti multi-ixed smeared ak model or oete INRODUCION he iite elemet method is the basis o a powerul omputatioal tool, whih a be used to simulate the respose o strutures, strutural ompoets ad materials, whe submitted to a speiied load. his tool has bee extesively used to assess the behavior o oete strutures. I order to simulate the strutural respose o oete strutures uder the iite elemet ramework, a mathematial idealizatio o the material behavior is reuired. his mathematial approah is ommoly amed ostitutive or material model, ad provides the relatio betwee the stress ad strai tesors i a material poit o the body. I order to predit with high auray the behavior o oete strutures, appropriate ostitutive models must be used. hese ostitutive models must be apable o simulatig the most relevat oliear pheomea o the iterveig materials. he oliear rature mehais theory has bee used to simulate the uasi-brittle ailure o oete (ACI 99, ACI 997). he disete ad the smeared ak oepts are the most used to model the oete rature uder the ramework o the iite elemet method. For oete strutures with a reioremet ratio that assures ak stabilizatio, the smeared ak approah is more appropriate tha the disete approah, sie several aks a be ormed i the struture. he disete approah is espeially suitable to simulate oete strutures where the ailure is govered by the ourree o a small umber o aks with a path that a be predited. he disete approah is ot treated i the preset work. Nevertheless, a omprehesive desiptio o the disete approah a be oud elsewhere, e.g., Ngo ad Sordelis (967), Hillerborg et al. (976), Rots (988) ad Bitteourt et al. (99). I smeared ak models, the rature proess is iitiated whe the maximum priipal stress i a material poit exeeds its tesile stregth. he ak propagatio is maily otrolled by the shape o the tesile-soteig diagram ad the material rature eergy. I order to assure mesh objetivity, the eergy dissipated i the ak propagatio proess is assoiated with a harateristi legth o the iite elemet (Bazat ad Oh 983). I the origial smeared ak or sigle-ixed smeared ak oept, the orietatio o the ak, i.e., the diretio whih is ormal to the ak plae is oiidet with the maximum priipal stress orietatio at ak iitiatio, ad remais ixed 6 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

7 Elasto-plasti multi-ixed smeared ak model or oete throughout the loadig proess. However, due to aggregate iterlok ad dowel atio o the reioremet (Che 98), the priipal stresses a hage their orietatio ad, oe more, exeed the tesile stregth. I this ase, the sigle-ixed smeared ak approah predits a umerial respose that is stier tha the experimetal observatios. o avoid this ioveiee, rotatig sigle smeared ak or multi-ixed smeared ak models have bee developed. I the ormer, the loal ak oordiate system is otiuously rotatig with the modiiatio o the diretio o the priipal axes. I the multi-ixed smeared ak models, several ixed smeared aks are allowed to orm, aordig to a ak iitiatio iterio. Plastiity theory has bee extesively used to model the oete behavior, partiularly uder ompressive states o stress (ASCE 98, Che ad Ha 988). Plastiity theory is based o a miomehaial or a pheomeologial approah. I the miomehaial approah, also amed udametal approah, the ostitutive relatios are established or the miostrutural behavior. I otrast, the pheomeologial approah, also kow as the mathematial theory o plastiity, establishes the ostitutive model diretly based o observed eatures rom experimetal tests. Plastiity theory is a atural ostitutive desiptio or metals (Hill 950), but it a also be used or emetitious materials. I the 980s several tools were developed or mathematial plastiity, e.g., impliit Euler bakward algorithms ad osistet taget operators (e.g., Ortiz ad Popov 985, Simo ad aylor 985), whih made this theory eve more attrative to model the oete behavior. Hybrid models derived rom rature mehais ad plastiity theories have bee proposed by several researhers. I these models, rature mehais theory is used to simulate the tesile post-akig behavior o oete, whereas plastiity theory is used to simulate its ompressive behavior. Elasto-plasti multi-ixed smeared ak models seem to be suitable or the simulatio o oete strutures, but due to their oeptual omplexities ad severe omputatioal diiulties, oly a ew researhers were suessul i the implemetatio o these models (de Borst ad Nauta 985, Crisield ad Wills 989, Barros 995). he preset report details the developed elasto-plasti multi-ixed smeared ak model. he desiptio o the model is divided i three parts: the irst part deals with the José Sea Cruz, Joauim Barros ad Álvaro Azevedo 7

8 Elasto-plasti multi-ixed smeared ak model or oete smeared ak model; the seod desibes the elasto-plasti model; ad, ially, the third part presets the elasto-plasti multi-ixed smeared ak model. he developed umerial model is validated with results available i the literature. CRACK CONCEPS I this setio, irstly, the sigle-ixed smeared ak oept is desibed, ollowed by the geeralizatio to the multi-ixed smeared ak oept. he most relevat algorithmi aspets are detailed. Fially, the developed umerial model is validated usig results available i the literature.. Smeared ak oept Ater ak iitiatio, the basi assumptio o smeared ak models, is the deompositio o the iemetal strai vetor, ε, ito a iemetal ak strai vetor, ε, ad a iemetal strai vetor o the oete betwee aks, o ε : o ε = ε + ε () he deompositio expressed by () has bee adopted by several researhers (Litto 974, Bazat ad Gambarova 980, de Borst ad Nauta 985, Rots et al. 985, Rots 988)... Crak strais ad ak stresses Figure shows the morphology o a ak or the ase o plae stress. wo relative displaemets deie the relative movemet o the ak lips: ak opeig displaemet, w, ad ak slidig displaemet, s. Axes ad t deie the loal oordiate system o the ak (CrCS), beig ad t the ak ormal ad tagetial diretios, respetively. CrCS, I the smeared ak approah w is replaed with a ak ormal strai deied i ε, ad s is replaed with a ak shear strai i CrCS, γ t. he same approah a be applied to the iemetal ormal ad shear ak strais ( ad ε γ t ). he iemetal ak strai vetor i CrCS, ε l, is deied by 8 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

9 θ Elasto-plasti multi-ixed smeared ak model or oete ε = ε γ () l t x s t τ t Crak τ t w x Figure Crak stresses, relative displaemets ad loal oordiate system o the ak. he iemetal ak strai vetor i the global oordiate system (GCS), ollowig three ompoets, ε, has the (3) ε = ε ε γ he trasormatio o the iemetal ak strai vetor rom CrCS to GCS reads ε os θ siθosθ ε ε = si θ siθosθ γ γ siθosθ os θ si θ t (4) or ε = ε (5) l José Sea Cruz, Joauim Barros ad Álvaro Azevedo 9

10 Elasto-plasti multi-ixed smeared ak model or oete beig the ak strai trasormatio matrix ad θ the agle betwee x ad (see Figure ). he iemetal loal ak stress vetor, l, is deied by = τ (6) l t where ad τ t are the iemetal ak ormal ad shear stresses, respetively. he relatioship betwee l ad the iemetal stress vetor (i GCS),, a be deied as os θ si θ siθosθ = τ siθosθ siθosθ os θ si θ t τ (7) or = (8) l.. Coete ostitutive law Assumig liear elasti behavior or the oete betwee aks (udamaged oete), the ostitutive relatioship betwee o ε ad is give by, o o = D ε (9) where o D is the ostitutive matrix aordig to Hooke's law, D E ν ν 0 o = ν 0 ( ν ) 0 0 (0) beig E ad ν the Youg's modulus ad Poisso's ratio o plai oete, respetively. 0 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

11 Elasto-plasti multi-ixed smeared ak model or oete..3 Costitutive law o the ak I a similar way, a relatioship betwee l ad ε l is established to simulate the ak opeig ad the shear slidig usig a ak ostitutive matrix, D, = D ε () l l where D is a matrix iludig mode I ad mode II ak rature parameters...4 Costitutive law o the aked oete Combiig the euatios preseted i the previous setios, a ostitutive law o the aked oete is obtaied. Hee, iorporatig euatios () ad (5) ito (9) yields, ( l ) o = D ε ε () Pre-multiplyig both members o euatio () by leads to o o = D ε D ε (3) l Substitutig (8) ito the let side o (3) yields o o + D ε = D ε (4) l l Iludig () ito the let side o (4), the ollowig euatio deiig the iemetal ak strai vetor i CrCS is obtaied ( ) o o ε = D + D D ε (5) l he ilusio o (5) i () leads to the ostitutive law o the aked oete, whih reads José Sea Cruz, Joauim Barros ad Álvaro Azevedo

12 Elasto-plasti multi-ixed smeared ak model or oete ( ) o o o o = D D D + D D ε (6) or o = D ε (7) where o D is the ollowig ostitutive matrix or the aked oete ( ) o o o o o D = D D D + D D (8)..5 Crak rature parameters I the preset model, the ak ostitutive matrix, D, is assumed to be diagoal D DI 0 = 0 DII (9) I this matrix the ak behavior. D I ad D II are the mode I ad mode II stiess modulus assoiated with he ak-dilatay eet ad the shear-ormal stress ouplig is ot osidered i the preset approah. he shear-ormal stress ouplig, however, may be simulated idiretly, allowig o-orthogoal aks to orm ad relatig strai (Rots 988). his strategy is adopted i the preset model. D II with the ak ormal he ak iitiatio i the preset model is govered by the Rakie yield surae (see Figure ), i.e., whe the maximum priipal stress, I, exeeds the uiaxial tesile stregth, t, a ak is ormed. his assumptio is justiied by the experimetal results obtaied by Kuper et al. (969) whe the tesile akig is ot aompaied by sigiiat lateral ompressio. José Sea Cruz, Joauim Barros ad Álvaro Azevedo

13 Elasto-plasti multi-ixed smeared ak model or oete Aordig to Rots (988), the most suitable approah to simulate the ak propagatio uder the iite elemet ramework is by takig ito aout the oete rature parameters, amely, the shape o the tesile-soteig diagram ad the rature eergy. II I Figure Rakie yield surae i the D priipal stress spae. wo distit tesile-soteig diagrams are available i the developed omputatioal ode: tri-liear ad expoetial diagrams (see Figure 3). he tri-liear diagram show i Figure 3(a) is deied by the ollowig expressios ( ε ) t + DI,ε i 0 < ε ξε ult, α + D ( ε ξε ) i ξε < ε ξε = α + D ( ε ξε ) i ξε < ε ε 0 i ε > εult, t I, ult, ult, ult, t I,3 ult, ult, ult, (0) with, D Ii, h = () t ki G where José Sea Cruz, Joauim Barros ad Álvaro Azevedo 3

14 Elasto-plasti multi-ixed smeared ak model or oete k k k 3 = = = ( α )( ξ + αξ αξ + α ) ξ ( α α)( ξ+ αξ αξ + α) ( ξ ξ) α( ξ+ αξ αξ + α) ( ξ ) () he ultimate ak ormal strai, ε, ult, is give by, G ε ult, = k4 (3) t h where k 4 = ξ + αξ αξ + α (4) t t D I, D I α t D I, α t ξ ξ ε,ult ε,ult D I,3 ε,ult ε ε,ult ε (a) Figure 3 esile-soteig diagrams: tri-liear (a) ad expoetial (b). (b) he expoetial soteig diagram proposed by Corelisse et al. (986) (see Figure 3(b)) is deied by 4 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

15 Elasto-plasti multi-ixed smeared ak model or oete ( ε ) t = + C exp C + C exp( C) i 0 < < = 0 i ε ε 3 ε 3 ε ε ( ) ε ε ult, εult, εult, εult, ult, (5) where C = 3.0 ad C = he ultimate ak ormal strai, ε, ult, is obtaied rom, G ε ult, = (6) k h t where k = + + C C C C C C C C C C 3 C exp ( ) ( ) (7) he mode I ak stiess modulus is alulated with the ollowig expressio ε C ε I t 3 exp ε ult, ε ult, ε ult, D = C C ε ε ε + C exp C C C exp ( C) ε ult, ε + ult, ε ult, εult, (8) he oete rature eergy, G, is the eergy reuired to propagate a tesile ak o uit area. Geerally, G is assumed to be a material parameter ad aordig to the CEB-FIB model ode (993) it a be estimated rom the oete ompressive stregth,, ad maximum aggregate size. I the smeared ak approah, the rature zoe is distributed i a ertai width o the iite elemet, whih is desigated ak bad-width, h, as idiated i Figure 4. I this model a ostat strai distributio i the width h is assumed. o assure mesh José Sea Cruz, Joauim Barros ad Álvaro Azevedo 5

16 Elasto-plasti multi-ixed smeared ak model or oete objetivity, the oete rature eergy ad the ak bad-width must be mesh depedet. Several researhers have proposed dieret ways to estimate h (Bazat ad Oh 983, Rots 985, Leibegood et al. 986, Oñate et al. 987, Dahlblom ad Ottose 990, Oliver et al. 990, Cerveka et al. 990, Rots 99, Feestra 993). I the preset umerial model, the ak bad-width a be estimated i three dieret ways: eual to the suare root o the area o the iite elemet, eual to the suare root o the area o the itegratio poit or eual to a ostat value. o avoid sap-bak istability, the ak bad-width is subjeted to the ollowig ostrait (de Borst 99), GE h (9) b t where b max{ k} = or tri-liear soteig ad b k C ( C 3 ) exp( C ) expoetial soteig. i ( ) = + + or t t Disete approah Smeared approah w h G (a) w G /h (b) ε Figure 4 wo distit approahes to model the tesile-soteig diagram: dis ete (a) ad smeared (b) ak models. Applyig the strai deompositio oept to the ak rature mode II, yields o γ = γ + γ (30) 6 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

17 Elasto-plasti multi-ixed smeared ak model or oete or, = + (3) β G D G II resultig D II β = G β (3) D II is the mode II ak rature stiess modulus. he parameter β is alled the shear retetio ator ad its value depeds o the ak ormal strai ad o the ultimate ak ormal strai (Rots 988, Póvoas 99, Barros 995), ε β = ε ult, p (33) I this euatio p is a iteger parameter that, urretly, a assume the values o, or 3 (Barros 995). Whe ε = 0 (losed ak) a ull iterlok is assumed. For a ully ope ak ( ε ε, ult ) the shear retetio ator is eual to zero, resultig i a ull ak shear stiess that orrespods to a egligible aggregate iterlok.. Multi-ixed smeared ak oept I the previous setios the oept o the ixed smeared ak model was desibed. I this model oly oe ixed smeared ak was allowed to orm at eah itegratio poit. o be apable o simulatig the ormatio o more tha oe ixed smeared ak, as well as, to be ot restrited to the partiular ase o two orthogoal aks (Azevedo 985, Póvoas 99), the ormulatio was exteded, resultig i the multi-ixed smeared ak model. José Sea Cruz, Joauim Barros ad Álvaro Azevedo 7

18 Elasto-plasti multi-ixed smeared ak model or oete o deal with the evetual ormatio o smeared aks at eah itegratio poit, the geeralized ak trasormatio matrix, adopt the ollowig ormat, ad the ak ostitutive matrix, D, ( θ ) ( θ ) ( θ ) = (34) D D D 0 = 0 0 D (35) θ ad I these matries, ( ) i i D i orrespod to the ak trasormatio matrix ad to the ak ostitutive matrix o the i-th ak, respetively. Matrix D is diagoal sie the sub-matries D i have ull o-diagoal terms (see Setio..5)... Crak iitiatio Crakig ours whe the maximum priipal stress exeeds the oete uiaxial tesile stregth, t. Ater ak iitiatio, ad assumig that the shear retetio ator is o-ull, i.e., the ak shear stresses a be traserred betwee the ak lips, the values ad the orietatio o the priipal stresses a hage durig the loadig proess. For this reaso the maximum priipal stress i the oete betwee aks a also exeed t. I the preset work a ew ak is iitiated whe the ollowig two oditios are satisied simultaeously: the maximum priipal stress, I, exeeds the uiaxial tesile stregth, t ; the agle betwee the diretio o the existig aks ad the diretio o I, θ I, exeeds the value o a predeied threshold agle, α th. ypially, the threshold agle varies betwee 30 ad 60 degrees (de Borst ad Nauta 985). Whe the seod oditio is ot veriied (whih meas that the ew ak is ot iitiated) the tesile stregth is updated i order to avoid iosisteies i the ak 8 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

19 Elasto-plasti multi-ixed smeared ak model or oete iitiatio proess. With this strategy the updated tesile stregth a sigiiatly exeed the origial oete tesile stregth (Rots 988)... Crak evolutio history I a previously aked itegratio poit, the ouplig betwee o-orthogoal aks is simulated with rature parameters assoiated to the ew aks. he rature eergy available or the ext ak, ext G, is alulated with (Barros 995) p ext α =, a +, a ( ) G G G G (36) π where p is a iteger parameter ad a assume the values o, or 3, α is the agle (i radias) betwee the ext ad the previous ak ad G, a is the available rature eergy i the previous ak. Its value is alulated subtratig the rature eergy osumed by prev the previous ak, G,, rom the oete rature eergy (see Figure 5), G = G G (37) prev, a,, t, () α () t ε,, g g,,, ε Figure 5 Frature eergy available or the ext ak. José Sea Cruz, Joauim Barros ad Álvaro Azevedo 9

20 Elasto-plasti multi-ixed smeared ak model or oete.3 Algorithmi aspets I a multi-ixed smeared ak model the osideratio o all the ak status hages that a our durig the loadig proess o a oete elemet, reuires the implemetatio o several omputatioal proedures. Otherwise the model beomes ureliable ad ieiiet or pratial use (de Borst ad Nauta 985, Rots 988, Crisield ad Wills 989, Barros 995, Hostetter ad Mag 995). he implemetatio o these algorithms i the FEMIX omputer ode (Azevedo et al. 003) is desibed below..3. Stress update Whe the strai ield i a aked itegratio poit is submitted to a iemetal strai, ε m, the stress state o the itegratio poit is also modiied ad must be updated ( m ). he iemetal relatioship (8) a be writte i terms o total stresses, = (38) l, m m m his euatio is euivalet to ( ) + = + (39) l, m l, m m m m Iludig () i (39) yields ( ( )) o ε ε, + =, + l m l m m m D m m l, m (40) Euatio (40) a be writte as ( ) o o + + = 0 l, m l, m l, m m m l, m m m m m ε D ε D ε (4) where l, m depeds o ε l, m. he ompoets o the iemetal strai ak vetor, ε l m,, are the ukow variables o the oliear euatios (4). his vetor otais the two loal strai ompoets o the ative aks (o-losed aks). o solve this 0 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

21 Elasto-plasti multi-ixed smeared ak model or oete euatio two dieret methods were implemeted: the Newto-Raphso ad the ixedpoit iteratio methods (Chapra ad Caale 998). he methods exhibitig uadrati overgee, suh as the Newto-Raphso method, are usually very eiiet, but i some ases the solutio aot be obtaied. I theses ases the Newto-Raphso method is replaed with the ixed-poit iteratio method whih exhibits liear overgee. I the ollowig algorithms the irst member o euatio (4) is reerred as a utio o i.e., ε l, ( ) ( ) o o l, m l, m l, m l, m m m l, m m m m m ε = + ε + D ε D ε (4) with this assumptio, euatio (4) beomes ( ε l m ) =., 0 Figure 6 shows the lowhart o the Newto-Raphso method adapted to the solutio o (4). he alulatio o the iitial solutio (step () i Figure 6) is perormed with euatio (4), osiderig, ( ε l l, ) eual to D m m m ε l, m, where D m is the tagetial ak ostitutive matrix o the previous overged stress state. I step 3 oler 6 = 0, where is the oete ompressive stregth. he symbol meas the iiite orm o the vetor, i.e., the maximum absolute value oud i vetor. he irst derivatives o i order to the iemetal ak strai vetor a be deied as ( ε l, ) ε l, m = D + ˆ + m Dm md m m o (43) where Dˆ m ˆ D, 0 0 m 0 ˆ D, m 0 = 0 0 ˆ D, m (44) José Sea Cruz, Joauim Barros ad Álvaro Azevedo

22 Elasto-plasti multi-ixed smeared ak model or oete ad 0 0 ˆ D im, = τtm, 0 (45) ε m, i () Zero the iteratio outer: 0 () Calulate the iitial solutio: ( ) 0 ε l, m (3) (( ε l, m ) ) < oler? Yes (4) No Update the outer: + (5) Calulate the variatio o the ukows as the solutio o the ollowig system o liear euatios: m ε l, m ( εl, m) + δ = 0 (6) Update the urret solutio: ( εl, m) ( εl, m) + δ( εl, m) (7) END Figure 6 Flowhart o the Newto-Raphso method. José Sea Cruz, Joauim Barros ad Álvaro Azevedo

23 Elasto-plasti multi-ixed smeared ak model or oete Whe euatio (33) is adopted to deie (3) the o-ull term o (45) is p ε, ε m + m, τ ε Gp tm, ult, εm, εm, + εm, εm, + εm, ε, ult εult, εult, = γtm, p p ε, ε m + m, ε G p ult, γ p ε, ε, ε, ε m + m m + m, ε, ε ult, ε ult ult, tm, (46) Whe the overgee is ot obtaied usig the Newto-Raphso method, the ixed-poit iteratio method, show i Figure 7 is tried. José Sea Cruz, Joauim Barros ad Álvaro Azevedo 3

24 Elasto-plasti multi-ixed smeared ak model or oete () Zero the iteratio outer: 0 () Calulate the iitial solutio:( ε ) 0 l, m (3) (( ε l, m ) ) < oler? Yes (4) No Update the outer: + (5) Calulate the iemetal ak strai vetor ( ε l ) as the solutio o the, m ollowig system o liear euatios: ( ) ( ε l, ) l, m l m, l, m + ε + o m m m D o mm md ε m = 0 (6) END Figure 7 Fixed-poit iteratio method..3. Crak status Depedig o the ollowed ε path, a ak a assume oe o six ak statuses as show Figure 8. he irst () was amed iitiatio ad orrespods to the ak iitiatio. he opeig status ours whe the ak is i the soteig brah (). I the preset model a seat brah is assumed to simulate the uloadig (3) ad the reloadig (5) phases. he losig status desigates the uloadig phase while the reopeig ak status is attributed to the ak i the reloadig phase. his assumptio does ot orrespod to the most realisti approah, sie yli tests reveal the ourree o a hystereti behavior 4 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

25 Elasto-plasti multi-ixed smeared ak model or oete (Hordijk 99). Sie the preset model was developed to simulate the behavior o oete strutures uder mootoi loadig, this simple approah is suiietly aurate. I a ak loses, i.e., ε = 0, the ak status reeives the desigatio o losed (4). he ully ope (6) ak status ours whe i the ak the mode I rature eergy is ully exhausted. - INIIAION - OPENING 3 - CLOSING 4 - CLOSED 5 - REOPENING 6 - FULLY OPEN ε Figure 8 Crak status. he stress update proedure desibed i the previous setio is oly applied to the ative aks, i.e., whe ε > 0. Whe a ak iitiates ( I > t ad θi α ), whe a ak loses ( ε < 0 ) or whe a losed ak reopes ( > 0 ), the iemetal strai vetor ε must be suessively deomposed i order to aurately simulate the ak status evolutio (see Figure 9). hese three ak status hages were amed itial ak status hages. his deompositio is eessary sie the otet o D ad matries depeds o the umber o ative aks. For istae, whe a ew ak is ormed the size o these matries must be exteded i order to aommodate ew terms (see euatios (34) ad (35)). José Sea Cruz, Joauim Barros ad Álvaro Azevedo 5

26 Elasto-plasti multi-ixed smeared ak model or oete () Zero the outer o itial ak status hages: m 0 () Calulate the umber o ative aks (3) Extrat the iormatio related to the ative aks rom the historial data (4) Calulate the ew stress vetor usig the urret iemetal strai vetor: ε m (5) New ak iitiates or a losed ak reopes or a ope ak loses? No Yes (6) Update the outer o itial ak status hages: m m+ (7) Calulate the trasitio poit orrespodig to: a ew ak iitiatio (i appliable): k ewm, a losed ak reopeig (i appliable): a ope ak losure (i appliable): k reopem, k losem, (8) Calulate the trasitio poit orrespodig to the irst k mi k ; k ; k ak status hage: m { ewm, reopem, losem, } (9) Calulate the ew stress vetor usig the urret iemetal strai vetor: ε k m m () Update the historial data (0) Update the historial data (3) END () Update the iemetal strai vetor: ε ( k ) ε m m m Figure 9 Algorithm used or the deompositio o the iemetal strai vetor. 6 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

27 Elasto-plasti multi-ixed smeared ak model or oete he alulatio o the umber o ative aks (step () i Figure 9) is based o the otet o the database otaiig the historial data. his database stores, or eah itegratio poit ad or eah iteratio o the iemetal-iterative proedure, all the itial parameters suh as the stress ad strai vetors, the umber o aks, the ak stress ad strai vetors, the ak statuses, the ak orietatio ad data assoiated with the ak evolutio history. he stress update proedure, desibed i the Setio.3., is perormed i step (4) o Figure 9. Whe oe itial ak status hage ours, the urret iemetal strai vetor, ε, must be deomposed. o alulate the trasitio poit orrespodig to ak iitiatio, k, ak reopeig, k reopem,, or to a ope ak losure, k, losem ewm, to a losed, two algorithms were implemeted: the Newto-Raphso method (Figure 0) ad the bisetio method (Figure ). he last oe is used whe the irst ails. able otais the deiitio o the utio ( k ), the iitial solutio ad the parameter oler or some ak status hages. hese utios ad parameters are used i the algorithms show i Figure 0 ad Figure. able Deiitio o the utio used i the algorithms show i Figure 0 ad Figure, the iitial solutio ad the respetive overgee iterio parameter. Critial ak status hages New ak iitiatio Closed ak reopeig Ope ak losure ( k ) t I ( k ) ( k ) ε ( k ) 0 k p t I εm, p I I εm, εm, p ( ) ε 0.5 oler Besides the ak iitiatio oditios desibed i Setio.. (tesile stregth ad threshold agle), a additioal hek is reuired. Whe a ew ak is iitiatig, k ewm, is alulated (see Figure 9). At this phase, the ew ak is oly osidered as potetial José Sea Cruz, Joauim Barros ad Álvaro Azevedo 7

28 Elasto-plasti multi-ixed smeared ak model or oete ak. At the ed o the irst part o the iemetal strai vetor, k, ε ormal stress, ε ewm m, the ak, is eual to the urret tesile stregth, t, ad its ormal ak strai,, has a ull value (poit i Figure 8). For the remaiig part o the iemetal strai vetor, ( k ewm) ε,, the potetial ak is already osidered i euatio (4). o m beome a deiitive ak, ε o the potetial ak must be positive durig the evaluatio o euatio (4). I this oditio is ot ulilled, the ak iitiatio proedure is aborted ad the tesile stregth is replaed with the value o the urret maximum priipal stress. Ater the determiatio o the trasitio poit orrespodig to the irst itial ak status hage (step 7), the stress vetor is alulated, ad the historial data o the aks ad the iemetal strai vetor are updated. he deompositio o the iemetal strai vetor eds whe o more itial ak status hages our (see Figure 9). I this setio, m is the outer o itial ak status hages, reuirig a deompositio o the vetor lariied. ε. I able the meaig o previous iteratio m is able Meaig o m value. m value Algorithmi strategy (PD or PI) Meaig o m m = 0 Path depedet Path idepedet Previous Newto-Raphso iteratio Previous overged ombiatio m > 0 Path depedet or path idepedet Previous iteratio i the algorithm o Figure 9 8 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

29 Elasto-plasti multi-ixed smeared ak model or oete () Zero the iteratio outer: 0 () Calulate the iitial solutio: 0 k (3) ( k ) < oler? No Yes (4) Update the iteratio outer: + (5) Calulate the variatio o k: k ( ) (6) Update the urret solutio: k k + k (7) END Figure 0 Calulatio o the trasitio poit by the Newto-Raphso method. José Sea Cruz, Joauim Barros ad Álvaro Azevedo 9

30 Elasto-plasti multi-ixed smeared ak model or oete () Zero the iteratio outer: 0 () 0 0 Deie the iitial rage: ka, kb [ 0.0,.0] (3) Calulate midpoit o the rage: ka + kb k (4) ( k )? < oler No Yes (5) Update the iteratio outer: + Yes (6) ( A ) ( ) k k > 0? No ka k (7) (8) k k B B k k B A k k A (9) ( A ) ( k ) ( B) ( kb ) k k (0) ( B ) ( k ) ( A) ( ka ) k k () END Figure Calulatio o the trasitio poit by the bisetio method. 30 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

31 Elasto-plasti multi-ixed smeared ak model or oete.3.3 Sigularities Whe two ully ope orthogoal aks our at a itegratio poit, it a be show that, i the system o oliear euatios (4), the shear euatios related to these aks are liear depedet. his situatio a be illustrated with the ollowig example. Cosiderig two orthogoal aks, beig oe horizotal ( θ = 90º ) ad the other vertial ( θ = 0º ), ad osiderig that both are ully ope, the variatio o the ak stress vetor, l, is ull. Assumig that i the previous state l, m = mm, euatio (4) leads to o o ε ε = 0 m m, m m m D D (47) l resultig i, ε = ε,, + γ t γt = γ, ε = ε,, γt + γt =+ γ (48) where, ε,, γ t,, ε ad, γ t are the ormal ad shear ak strai variatios o the ak ad, respetively. he system o euatios (48) aot be solved sie the seod ad ourth euatios are liearly depedet. A physial iterpretatio o this situatio is preseted i Figure. he ak ormal strai variatios a be obtaied diretly rom the global strai variatios. José Sea Cruz, Joauim Barros ad Álvaro Azevedo 3

32 Elasto-plasti multi-ixed smeared ak model or oete ε () (), ε / ε () ε γ () ε, / ε,/ ε ε, / γ (a) (b) Figure Crak strai variatio: (a) ormal strai ad (b) shear ompoet. he solutio o (4) or the ase o ully ope orthogoal aks reuires the itrodutio o the ollowig additioal oditio i,, j γ + γ = t t 0 (49) where, γ i t ad, γ j t are the ak shear strais variatios o a pair orthogoal aks. o alulate the stiess matrix o a elemet, K, the ostitutive matrix, D, is reuired. he alulatio o D o a aked oete itegratio poit reuires the iversio o the matrix that results rom the evaluatio o the ollowig expressio (see setio..4, euatio (8)) o D = D + D (50) Whe a itegratio poit has two ully ope orthogoal aks, D is ull resultig i a sigular D matrix. o overome this problem the ollowig residual value is assiged to D, D 6 II = 0 G (5) 3 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

33 Elasto-plasti multi-ixed smeared ak model or oete.4 Model appraisal he perormae o the developed multi-ixed smeared ak model is assessed usig results published by other researhers. Sie the oete plasti deormatio is ot osidered i the ormulatio desibed, the example seleted to validate the model exhibit a liear behavior i ompressio. hree-poit bedig tests are ommoly used to evaluate the oete tesile stregth ad the rature eergy (RILEM 985). he tests arried out by Kormelig ad Reihardt (983) are simulated usig the implemeted umerial model. he adopted mesh (see Figure 3) was omposed o 4-ode Lagragia plae stress elemets with Gauss-Legedre itegratio sheme. I order to obtai a well-deied ak patter at mid-spa, ahead the oth, Gauss-Legedre itegratio rule was used i the elemets oss the eter lie. hikess = 00 mm F/ F/ Itegratio rule Figure 3 Nothed beam: geometry, mesh, loadig oiguratio ad support oditios. Note: all dimesios are i millimeters. he oete properties used i the preset simulatio are listed i able 3. hree dieret types o tesile-soteig diagrams were used: liear, tri-liear ad expoetial. he beam weight was iluded i the simulatio. Figure 4 presets the respose obtaied usig the three dieret types o tesilesoteig diagrams desibed above. he experimetal results are also displayed. It a be observed that all umerial simulatios have the same pre-peak respose, up to 050 kn. he maximum umerial peak load was obtaied with the liear soteig diagram. he José Sea Cruz, Joauim Barros ad Álvaro Azevedo 33

34 Elasto-plasti multi-ixed smeared ak model or oete tri-liear ad the expoetial tesile-soteig diagrams lead to a idetial respose i the post-peak phase, i good agreemet with the experimetal results. Figure 5 shows the ak patter at the ial stage, or the ase o tri-liear diagram. A well-deied ak above the oth a be observed. Spurious aks with losig status were ormed i the eighborhood o the rature surae. able 3 Coete properties used i the simulatio o the three poit bedig test. Desity 6 3 ρ =.4 0 N/mm Poisso's ratio ν = 0.0 Iitial Youg's modulus Compressio stregth esile stregth E = N/mm = 48.0N/mm =.4 N/mm t ri-liear soteig parameters ξ = 0.4 ; α = 0.6 ; ξ = 0.8 ; α = 0. Frature eergy Parameter deiig the mode I rature eergy available to the ew ak G = p = 0.3N/mm Shear retetio ator Expoetial ( p = ) Crak bad-width hreshold agle α th = 30º Suare root o the area o the elemet Liear ri-liear Expoetial Load [kn] Kormelig et al. (983) Deletio [mm] Figure 4 Iluee o the type o tesile-soteig diagram o the load-deletio respose. 34 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

35 Elasto-plasti multi-ixed smeared ak model or oete INIIAING CLOSING OPENING CLOSED FULLY OPEN REOPENING Figure 5 Numerial ak patter at the ial stage usig the tri-liear diagram. 3 PLASICIY he plastiity theory has bee used by may researhers i the simulatio o the behavior o strutures built with materials exhibitig irreversible deormatios, suh as oete (Che 98), soils (Che ad Mizuo 990) or masory (Loureço 996). A extesive study o this subjet a be oud i the literature (Lemaitre ab Cabohe 985, Lublier 990, Crisield 997, Simo ad Hughes 998). I the simulatio o ompressed oete, a model based o the plastiity theory is adopted. his model is desibed i the ollowig setios. Results available i the literature are used to assess the perormae o the model. 3. Basi assumptios he basi assumptio o the plastiity theory, i the otext o small strais, is the deompositio o the iemetal strai, ε, i a elasti reversible part, e ε, ad a irreversible or plasti part, p ε : e p ε = ε + ε (5), he elasti ostitutive matrix, e D, is used to obtai the iemetal stress vetor, José Sea Cruz, Joauim Barros ad Álvaro Azevedo 35

36 Elasto-plasti multi-ixed smeared ak model or oete ( ) = D e ε e = D e ε ε p (53) Plastiity based models deped o the oepts o yield surae, low rule ad hardeig (or soteig) law. he yield surae, deied i stress spae, limits the elasti behavior domai. I geeral, this surae is a utio o the stress state i a poit,, ad o some iteral variables, a ad κ, that deie the evolutio o the yield surae. he geeral euatio o the yield surae is ( a ),, κ = 0 (54) he bak-stress vetor, a, loates the origi o the yield surae ad κ is the salar hardeig parameter, whih deies the amout o hardeig or soteig. Depedig o the evolutio o the yield surae durig the loadig proess, three basi hardeig types a be deied (see Figure 6): isotropi hardeig (Odvist 933), kiemati hardeig (Prager 955) ad mixed hardeig (Hodge 957). he iteral variables ivolved i these hardeig rules are idiated i able 4. II II II (3) () (3) () () (3) () () () I I I (a) (b) () Figure 6 Basi hardeig rules: (a) isotropi hardeig, (b) kiemati hardeig ad () mixed hardeig. 36 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

37 Elasto-plasti multi-ixed smeared ak model or oete able 4 Basi hardeig rules. Hardeig rule Variables ivolved No hardeig (ideal plastiity) ( ) Isotropi hardeig (Figure 6(a)) ( κ, ) Kiemati hardeig (Figure 6(b)) (, a) Mixed hardeig (Figure 6()) (, a, κ ) I the geometri represetatio show i Figure 6 a deies the loatio o the origi o the yield surae whereas κ otrols the size ad shape o the yield surae. Good results a be obtaied with the isotropi hardeig whe loadig is mootoi. However, more omplex hardeig rules are reuired whe the material is submitted to yli loadig. Sie the aim o the preset model is to simulate the behavior o oete strutures uder mootoi loadig, the bak-stress vetor will ot be osidered as a yield surae parameter. With these assumptios the yield oditio adopted or the preset model is the ollowig ( κ, ) = 0 (55) he evolutio o the plasti strai is give by the ollowig low rule g λ p ε = (56) where λ is a o-egative salar desigated by plasti multiplier ad g is the plasti potetial utio i stress spae. Whe g ad oiide, the low rule is amed assoiated. Otherwise, a o-assoiated low rule is obtaied. he yield utio ad the plasti multiplier are ostraied by the ollowig oditios 0, λ 0 ad λ = 0 (57) he variatio o the hardeig parameter, κ, oiides with the euivalet plasti strai variatio eps ε (strai hardeig) or with the plasti work variatio p W (work José Sea Cruz, Joauim Barros ad Álvaro Azevedo 37

38 Elasto-plasti multi-ixed smeared ak model or oete eps hardeig). Whe the irst hypothesis holds ( κ = ε ), the hardeig parameter is deied by ( ) eps p p κ = ε = ε ε (58) he assumptio o = 3assures that the plasti strai i the loadig diretio o eps p a uiaxial test is eual to the euivalet plasti strai variatio, i.e., ε = ε ad ε = ε = ε (Owe ad Hito 980). p p p 3 he euivalet plasti strai variatio a also be deied as a utio o the plasti work per uit volume, W p, resultig p eps W p κ = ε = = ε (59) where is the uiaxial yield stress whih depeds o the hardeig parameter, ad is urretly amed hardeig law. Whe the variatio o the hardeig parameter is deied p with the work hardeig hypotheses ( κ = W ), the ollowig relatio holds p p κ = W = ε (60) 3. Itegratio o the elasto-plasti ostitutive euatios he itegratio o the elasto-plasti ostitutive euatios over a iite step i a osistet maer is oe o the mai halleges i omputatioal plastiity. At the previous step p, the stress state ad the iteral variables are kow (, κ, ε, ε ), ad the mai task is the alulatio o the urret values o these variables whe a strai variatio ours, ε. his problem a be solved with a impliit Euler bakward itegratio algorithm. he stability ad auray o this algorithm has bee demostrated by several researhers (Ortiz ad Popov 985, de Borst ad Feestra 990, Shellekes ad de Borst 990). he algorithm has two phases: a elasti preditor phase ad a plasti 38 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

39 Elasto-plasti multi-ixed smeared ak model or oete orretor phase. I the ormer ull plasti low is assumed whih leads to a disete set o euatios e = + D ε κ = κ = (, κ ) = 0 (6) Whe the elasti trial stress,, lies out the yield surae, plasti low must be osidered ad the plasti orretor phase o the algorithm is used to id a admissible stress state. Otherwise, the load step is osidered liear elasti. he algorithm used to id a admissible stress state is amed retur-mappig algorithm ad osists i the solutio o the ollowig system o oliear euatios, e e g D ( ) λ + = 0 κ κ κ = 0 (, κ ) = 0 (6) he irst euatio o the system o oliear euatios is obtaied rom the euatio ( ) = + D ε ε = D ε (63) e p e e p where ε is replaed with the right-had side o euatio (56). he Newto-Raphso p method is used to solve the system o oliear euatios (6), where the ukows., κ ad λ are 3.3 Evaluatio o the taget operator I the preset work, the Newto-Raphso method is used to alulate the solutio o the system o oliear euatios resultig rom the oliear iite elemet aalysis. he oliear problem is overted ito a seuee o liear iteratios util overgee is José Sea Cruz, Joauim Barros ad Álvaro Azevedo 39

40 Elasto-plasti multi-ixed smeared ak model or oete reahed. he liearized orm o the euatios depeds o a taget stiess matrix, whih plays a uial role i the perormae ad robustess o the Newto-Raphso method. I the otext o the mathematial plastiity, ad aordig to Simo ad aylor (985), the taget stiess matrix must be obtaied by osistet liearizatio o the stress update resultig rom the retur-mappig algorithm at the ed o the iteratio i. K, he elasto-plasti osistet taget ostitutive matrix a be determied rom the total dieretials d, p dε ad d (Hostetter ad Mag 995) or rom part o the Jaobia matrix used i the Newto-Raphso method o the retur-mappig algorithm (Loureço 996). 3.4 Elasto-plasti oete model Several elasto-plasti models have bee proposed to simulate the oete behavior. hese models dier rom eah other, maily, i the shape o the yield surae ad i the hardeig ad low rules. he model desibed i this setio is suitable to simulate the oete ompressive behavior uder mootoi loadig, admittig that the tesile stresses do ot exeed the oete tesile stregth Yield surae he yield surae proposed by Owe ad Figueiras (983) was adopted i the preset model. Its mai harateristi is the osideratio o paraboli meridias. his yield surae is deied with the ollowig euatio ( κ) ( ) ( κ), = P + = 0 (64) where P is the projetio matrix, give by a b 0 P = b a (65) 40 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

41 Elasto-plasti multi-ixed smeared ak model or oete ad is the projetio vetor deied by [ ] = d = d 0 (66) he parameters a, b, ad d a be obtaied with A a = + B, A B b =, = 3B, A d = (67) where the salars A ad B assume the values that result rom the ittig proess betwee the preset model ad the experimetal results obtaied by Kuper et al. (969). I these irumstaes, A ad B assumes the values o (Owe ad Figueiras 983) A = ad B =.355 (68) Figure 7 represets the iitial ad the limit yield suraes. his iitial yield surae is the limitig surae or elasti behavior. Experimetal results obtaied by Kuper et al. (969) are also iluded. 0. II/ α I / -0. Iitial yield surae α Kuper et al. (969) Limit yield surae Figure 7 Yield surae or oete. José Sea Cruz, Joauim Barros ad Álvaro Azevedo 4

42 Elasto-plasti multi-ixed smeared ak model or oete 3.4. Hardeig behavior Figure 8 represets the relatioship betwee the yield stress,, ad the hardeig parameter, κ, used to simulate the hardeig ad soteig phases o the oete behavior. hree poits deie the trasitios betwee the brahes o the urve. he loatio o these poits is obtaied rom uiaxial ompressio tests: 0 = α0, p = ad lim = 0.5. he euivalet plasti strai orrespodig to the peak ompressive stregth, κ p, with the ollowig euatio, κ = ε E (69) p where ε is the total strai at the peak ompressive stregth. he α 0 parameter deies the begiig o the plasti behavior. I most ases α 0 a assume the value 0.3. For the hardeig brah, ( ) adopted, whereas or the soteig phase, ( κ ) ad 3 ( ) κ, the relatioship used by Loureço (996) was κ, the post-peak relatioship proposed by CEB-FIB (993) or the uiaxial ompressive behavior was used. he expressios o the hardeig (ad soteig) behavior laws are iluded i APPENDIX I. _ p (κ) _ (κ) _ lim 0 3 (κ) κ p κ lim κ Figure 8 Hardeig/soteig law or oete. 4 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

43 Elasto-plasti multi-ixed smeared ak model or oete he plasti strai variatio is desibed by the ollowig expressio whih is assumed to be valid whe a assoiated low rule is osidered g = = p ε λ λh (70) he salar utio h is iluded i this euatio i order to ampliy the otributio o λ to p ε. he utio h depeds o the hydrostati pressure, p, ad reads (Abaus 00) ( ) h h = = + 0 p (7) A value o or 0 was obtaied based o the oditio that uder biaxial ompressio, with eual ompressive stress i both diretios, the plasti strai at ailure is, aordig to Kuper et al. (969), approximately.8 times the plasti strai at ailure uder uiaxial ompressio Retur-mappig algorithm Assumig the strai- hardeig hypothesis, κ = λ (Cahim 999, Abaus 00), the system o oliear euatios (6) a be redued to the ollowig pair o euatios, ( κ ) e ( ) e, D κh,, = + = 0 =, = 0 (7) Figure 9 shows the retur-mappig algorithm urretly implemeted i the omputer ode. he orm deied i step (4) is give by José Sea Cruz, Joauim Barros ad Álvaro Azevedo 43

44 Elasto-plasti multi-ixed smeared ak model or oete r e e D ( ) + κ h, = (, κ ) (73) where the supersipt orrespods to the iteratio outer. he Jaobia matrix used i step (6) is deied by the ollowig our bloks J κ = κ e h D + κ + h h κ = (74) where P = + ; h ( P ) = p ; 0 P P P = 3 ( P ) ( P ) d = = h κ dκ (75) 44 José Sea Cruz, Joauim Barros ad Álvaro Azevedo

45 Elasto-plasti multi-ixed smeared ak model or oete () Zero the iteratio outer: 0 () Calulate the iitial solutio: e = + D ε κ = 0 D ( ) (3) r Calulate the residue: e e + κ h, = (, κ ) (4) r 6 0 < 6 0 re? Yes No (5) Update the outer: + (6) Calulate the variatio o the stress vetor ad the hardeig parameter: J δ = r δκ (7) Update the urret solutio: + δ κ κ + δκ (8) END Figure 9 Retur-mappig algorithm o the elasto-plasti model. José Sea Cruz, Joauim Barros ad Álvaro Azevedo 45

46 Elasto-plasti multi-ixed smeared ak model or oete Cosistet taget ostitutive matrix he osistet taget ostitutive matrix adopted i the preset umerial model is dedued i APPENDIX II, resultig D ep H H = H h+ H (76) where e H = D + h λ (77) 3.5 Model appraisal he perormae ad the auray o the developed elasto-plasti model are assessed usig results available rom the literature. All the seleted examples are govered by the ompressive behavior Uiaxial ompressive tests he uiaxial ompressive tests 3B-4 to 3B-6, arried out by Va Mier (984), were seleted or a ompariso with the proposed model. Oe sigle 4-ode Lagragia plae stress iite elemet with Gauss-Legedre itegratio sheme was used to simulate the experimetal results. he dimesios o the iite elemet oiides with those o speime ( mm 3 ). able 5 shows the adopted oete properties. he umerial ad the experimetal results are ompared i Figure José Sea Cruz, Joauim Barros ad Álvaro Azevedo

47 Elasto-plasti multi-ixed smeared ak model or oete able 5 Coete properties used i the simulatio o the uiaxial ompressive test. Poisso's ratio ν = 0.0 Iitial Youg's modulus Compressio stregth Strai at peak ompressio stress E = N/mm = 43.4N/mm ε = Parameter deiig the iitial yield surae α 0 = Numerial model Experimetal: Va Mier (984) Stress [N/mm ] Strai [mm/m] Figure 0 Stress-strai relatioships: experimetal ad umerial results. Up to peak stress, the model mathes with high auray the experimetal results. I the soteig phase, ad or strais higher tha 4.5, the model estimates a residual stregth that is lower tha those experimetally obtaied. his idiates that the soteig brah o the uiaxial ompressive behavior proposed by CEB-FIB (993), maily the seod soteig brah, ( ) 3 κ, may ot be suitable to reprodue this type o test Biaxial ompressive test o evaluate the importae o the h parameter i the low rule, the biaxial ompressive tests arried out by Kuper et al. (969) were seleted. Oe sigle 4-ode Lagragia plae stress elemet with Gauss-Legedre itegratio sheme was used i the umerial José Sea Cruz, Joauim Barros ad Álvaro Azevedo 47

48 Elasto-plasti multi-ixed smeared ak model or oete model. able 6 shows the properties adopted or the oete ad or the yield surae. I Figure the umerial simulatio with 0 = 0 ( h =.0) ad 0 = are ompared with the experimetal results. able 6 Coete properties used i the simulatio o the biaxial ompressive test. Poisso's ratio ν = 0.0 Iitial Youg's modulus Compressio stregth E = N/mm = 3.06 N/mm Strai at peak ompressio stress ε =. 0 Parameter deiig the iitial yield surae α 0 = Priipal strai [mm/m] Numerial model with 0 = Numerial model with 0 = Experimetal: Kuper et al. (969) -0.5 I / Figure Iluee o the parameter i umerial respose. 0 he results show i the Figure idiate that the respose obtaied with 0 = 0 is stier i the hardeig phase ad too brittle ater the peak stress. A good agreemet with the experimetal results was obtaied with 0 = José Sea Cruz, Joauim Barros ad Álvaro Azevedo

49 Elasto-plasti multi-ixed smeared ak model or oete 4 ELASO-PLASIC MULI-FIXED SMEARED CRACK MODEL I the preset setio a elasto-plasti multi-ixed smeared ak model is proposed. his model orrespods to the ouplig o the multi-ixed smeared ak model desibed i Setio ad the elasto-plasti model preseted i Setio 3. I the ollowig setios the implemeted model is desibed. 4. Yield surae wo types o yield surae were ombied i the proposed umerial model: the Rakie iterio (desibed i Setio ) or oete i tesio, ad the Owe ad Figueiras (983) yield surae (desibed i Setio 3) or oete i ompressio. Figure represets the iitial ad the limit yield suraes. Experimetal results rom Kuper et al. (969) are also iluded. 0. II/ α I/ -0. Iitial yield surae α Kuper et al. (969) Limit yield surae Figure Yield suraes adopted i the elasto-plasti multi-ixed smeared ak model. José Sea Cruz, Joauim Barros ad Álvaro Azevedo 49

50 Elasto-plasti multi-ixed smeared ak model or oete 4. Itegratio o the ostitutive euatios he iemetal strai vetor is deomposed i a iemetal ak strai vetor, ε, ad a iemetal strai vetor o the oete betwee aks, o ε. his vetor is deomposed i a elasti reversible part, resultig e ε, ad a irreversible or plasti part, p ε, o e p ε = ε + ε = ε + ε + ε (78) he ostitutive euatios o the preset model ollow the multi-ixed smeared ak model ad the elasto-plasti model ad are dedued i the ollowig setios. 4.. Costitutive euatios rom the multi-ixed smeared ak model he iemetal stress vetor a be omputed rom the iemetal elasti strai vetor, = D e ε e (79) m m Iorporatig (79) ito (39) leads to ( ) + = + D ε (80) e e l, m l, m m m m Substitutig (78) ito (80) yields ( ) e p e + = + l, m l, m m m m m m m m l, m D ε ε D ε (8) ad iludig (70) i (8) results i ( ) e l, m l, m l, m md m l, m m m + ε + ε D h e m εm λm m, = 0 m (8) 50 José Sea Cruz, Joauim Barros ad Álvaro Azevedo