GENERALIZATION OF NON-ITERATIVE NUMERICAL METHODS FOR DAMAGE-PLASTIC BEHAVIOUR MODELING

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1 VIII Inrnaional Confrnc on Fracur Mchanics of Concr and Concr Srucurs FraMCoS-8 J.G.M. Van Mir, G. Ruiz, C. ndrad, R.C. Yu and X.X. Zhang (Eds) GENERLIZTIN F NN-ITERTIVE NUMERICL METHDS FR DMGE-PLSTIC EHVIUR MDELING R. GRÇ-E-CST*, J. LFITE, D. DIS-D-CST ND L. J. SLUYS *ICIST, and Dparmn of Civil Enginring, ISE, Univrsidad do lgarv, Campus da Pnha, , Faro Porugal rcosa@ualg.p ICIST, and Dparmn of Civil Enginring, Insiuo Suprior Técnico, Tchnical Univrsiy of Lisbon, v. Rovisco Pais, Lisbon, Porugal alfaia@civil.is.ul.p INESC, and Dparmn of Civil Enginring, Univrsiy of Coimbra Rua Luís Ris Sanos, Coimbra, Porugal dias-da-cosa@dc.uc.p Dparmn of Civil Enginring and Goscincs, Dlf Univrsiy of Tchnology, Th Nhrlands l.j.sluys@udlf.nl Ky ords: Non-iraiv Enrgy asd Mhod, Damag-Plasic Unloading, Discr Crac pproach, Non-iraiv analysis bsrac: Modlling fracur in concr or masonry is non o b problmaic rgarding h robusnss of iraiv soluion procdurs and, h us of non-iraiv mhods (or ha minimiz h us of iraions) in quasi-bril marials is no undr srong dvlopmn, du o h ncssiy o obain ffciv rsuls in fini lmn analysis [1, 2] hr srong non-linariis mrg ha ar ohris unildy o modl. In h proposd lcur, o n mhods dsignad as Non-Iraiv Enrgy basd Mhod (NIEM) and uomaic prsnd in [1, 3] ar applid ih xnsion o modlling damag-plasic bhaviour. Th n mhods combin an incrmnal-oal procdur ih h prfrnial us of incrmnal sps, siching o h oal approach only a criical bifurcaion poins. Th dvlopmn of h load-unloading abiliis is allod by hs incrmnal/oal mhods, hich a advanag of ping h marial s srss/srain mmory du o h prfrnial us of an incrmnal procdur. n approach o h unload-load cycls is usd in h scop of a noniraiv procdur, hich ill miiga h numrical difficulis inhrn o cyclic loading. Th formulaion for boh mhods for srucurs ih boh sofning and hardning bhaviour is prsnd and a srucural xampl hr h numrical rsuls ar compard ih xprimnal rsuls. 1 INTRDUCTIN Th us of oal approachs basd on a scan siffnss allos for h corrc modlling of concr and masonry srucurs undr monoonic soliciaion [2]. Hovr, in h oal approach only scan unloading is assumd (pah 4 in Figur 1) and rloading ill rcovr h posiion on h marial nvlop using h sam scan siffnss. This 1

2 is a simplificaion of h ral bhaviour, sinc hr is no valuaion of h rsidual srains or crac opnings. In fac, modlling h bhaviour of srucurs undr rvrsd loading, such as cyclic acions or a load dcras on a givn srucural mmbr, is no corrcly simulad by h scan unloading-rloading (damag) consiuiv la; insad, plasic, plasicdamag (Figur 2) or hysric las should b adopd. o h angn consiuiv modulus hras K is h scan modulus. Exprimnally, sl rinforcmn xhibis unloading-rloading bhaviour hich is modlld by mans of an lasic branch, sinc h rinforcing sl is assumd as an lasic-plasic marial. s for concr, his assumpion consiss on a simplificaion in boh h comprssiv and h nsil srss sas, sinc h unloadingrloading cycl is associad ih an incras in damag. Undr comprssion, his simplificaion is possibl on lo lvls of pos-pa damag and for slo unloadingrloading cycls, sinc damag incras also dpnds on hs facors. nohr simplificaion is o subsiu h hysric cycls by linar lasic-damag branchs li h xampls in Figur 3. Figur 1: Srss-displacmn pahs In Figur 2, 1D rprsnaions of h hr yps of unloading-rloading las adopd ih h n non-iraiv mhods ar prsnd for sofning. In his figur, d rfrs o an isoropic damag paramr, ha asily ranslas h currn siffnss cofficin of any ingraion poin i as a funcion of h iniial lasic siffnss D : j di 1 (1) D i j1 j ih, 0d i 1 nd is usd o dfin h marial srss/srain hisory. Th plasic-damag paramr d h (Figur 2 c)), is a numrical paramr ha as ino accoun h rsidual srain/displacmn jump (d h d). Th unloading-rloading cycls ar approximad ih a linar branch, iniially dfind by h lasic siffnss, ih h possibiliy of volving according o h plasicdamag d h funcion. In h folloing, D rfrs (1-d) rsidual rsidual (1-d h ) a) Damag b) plasic c) plasicdamag Figur 2: Unloading bhaviour ll abov rfrrd modls, simplifid or no, canno b implmnd in a pur oal approach. Nvrhlss, h inroducion of n mhods combining an incrmnal procdur ih a oal approach opnd h possibiliy of modlling h unloading bhaviour aing ino accoun rsidual dformaions. a) nsil s b) comprssiv s Figur 3: Exprimnal cyclic loading rsuls and damag branch simplificaion (adapd from [4]) 2

3 2 NN-ITERTIVE PRCEDURES Th n non-iraiv procdurs proposd in [1] us h combind incrmnal and oal approachs. This combind procdur allos for h propr racing of h marial loading hisory, convrsly o purly oal mhods. Furhrmor, informaion obaind from h prvious incrmnal soluion is usd o prdic h consiuiv la adopd in h oal approach. In his ay: i) hr is no nd o adjus h marial paramrs o nsur msh objciviy; and ii) h consisncy condiion is saisfid. 2.1 Soluion conrol In h incrmnal approach, hnvr a bifurcaion poin is rachd, h pah choic is basd upon h signal of h paricular load incrmn lading o h largs incrmnal nrgy dissipaion. I is assumd ha, for a load incrmn applid on a srucur, h corrsponding voluion on h srss-srain la of h marial poin should follo h pah hich ould rlas h largs amoun of nrgy. Using an incrmnal approach, h scond ordr nrgy rlas in a fini lmn, G, is givn by: T T G ε σd d, (2) d hr is h bul and d sands for all disconinuiis, and ar h srains and h srsss in h bul, rspcivly, and and ar h jump displacmns and h racions, rspcivly, a h disconinuiis. Sinc mulilinar consiuiv rlaions ar adopd, a criical load facor ( cri ) is firs valuad in a rial sp, in ordr o rach h nars marial poin conncing o linar branchs. frards, h ru sp is nforcd such ha: ΔP ΔP (3) ru, j cri rial, j 2.2 Criical bifurcaion poins Whn bifurcaion poins ar rachd on h marial la, o possibiliis occur: incras of damag or unloading. In Figur 4, four pahs on h uniaxial racion-displacmn d curv ar displayd: pah 1 corrsponds o incras of damag, pahs 2 and 3 ar unaccpabl sinc hy viola h marial la, and pah 4 corrsponds o scan unloading Figur 4: 1D rprsnaion of possibl racion-jump pahs Whnvr a marial poin undrgos unloading, mmory is p unil i rloads bac o h nvlop. Whn h currn sa is unloading his poin is nvr criical xcp for prvning ovrlapping of crac facs bu, hn rloading occurs, h load facor is simad, similarly o ha is don for all poins ha rmain on h nvlop. Whn svral nonlinariis aris during h analysis du o h xisnc of many bifurcaion poins, pah sarching chniqus mus b applid. If no soluion is obaind, i.., if no singl admissibl pah is obaind, a ransiion o h oal approach is mad. This aspc is common o boh mhods hrin prsnd. 2.3 uomaic mhod In h auomaic mhod, hnvr a criical bifurcaion poin is rachd, i bcoms impossibl o incrmnally drmin h ffciv pah, a oal mhod is adopd in hich h scan marial siffnss is usd. This scan siffnss is hn rducd by a prdfind facor as don in h Squnially Linar pproach (SL) dvlopd by Ros[2] (Figur 5): 3

4 Figur 5: 1D xampl of h uomaic mhod K j j j (4) hr sp j is h currn (valid) sp and racion and displacmn jumps ar dfind in Figur 5. In h folloing sp j+1 h scan siffnss is rducd, using h sandard SL, such ha: f0 rd j NSL (5) hr rd is h n srss on h nvlop, f 0 is h nsil srngh and N SL is h prdfind numbr of SL rducion sps. Th nx sp is prformd similar o h SL, in hich usually only on of h poins ill bcom criical and rachs h original nvlop. ll ohr poins ill rmain on h currn scan pah. In h folloing sp h incrmnal approach is rcovrd using h angn siffnss marix D. No ha h scan siffnss is alays adopd in h oal approach, hich has a dirc corrspondnc o h adopd nvlop. In lasoplasic marials i is also possibl o nforc h corrc unloading pah using h sam oal approach; in his cas, h scan siffnss is only adopd o rach n quilibrium posiions on ihr: i) h loading surfac or ii) h unloading surfac. Mor dails can b found in [1]. 2.4 Non-Iraiv Enrgy basd Mhod (NIEM) In h auomaic mhod h spis dcras of h scan siffnss mus b dfind a priori, ihou a clar physical maning. In ordr o avoid his siuaion a n mhod, dsignad NIEM, as inroducd, hich allos for siching bn h incrmnal and h oal approach ihou imposing a prdfind numbr of rducions of h scan siffnss. In h rial sp (Figur 6), all non-criical poins ar rad in h usual ay, such ha a criical load facor λ cri is compud. Th goal of his sp is solly o sima h damag lvl ha ould b rachd if his as a valid sp; in his ay, h scan siffnss upda for h nx sp ill mrg from his prdicion. Nvrhlss, sinc on sp j som ingraion poins ould follo invalid pahs, his sp is null. Th scan siffnss K j+1 is simad according o: j 1 K j 1 (6) j rial, j Figur 6: 1D xampl of h NIEM Thus, in sp j no chang occurs; only h valuaion of h n scan siffnss is prformd, hich ill b adopd in sp j+1 using a oal approach. In h folloing sp h incrmnal approach is rcovrd, similar o h auomaic mhod. Mor dails can b found in [1]. 3 FRMULTIN FR SFTENING EHVIUR In h folloing, mod-i fracur is assumd, alhough adapaion for mod-ii fracur is sraighforard. 4

5 Th modlling of non-scan unloading prsns no problm in an incrmnal approach as long as no criical bifurcaion poins ar aaind. Hovr, hn a criical bifurcaion poin is found, a ransiion o h oal approach is mad. This is a consqunc of h formulaion prsnd in [1], hr i as noicd ha an invrsion on h jump incrmn ovr h sofning branch lads o an inadmissibl pah: a null sp is adopd and a ransiion o a scan siffnss marix occurs, similarly in boh n mhods. In Figur 7, his inadmissibl pah occurs in sp u, in hich an unloading branch is dfind bn poins u and, ih siffnss D u. Th iniial siffnss is assumd lasic. Hovr, in h scop of a discr crac approach, h iniial branch usually corrsponds o a pnaly funcion. In his cas, is limid o an accpabl maximum valu (an as 10 3 N/mm 3 in all h xampls prsnd). Nx, h unloading siffnss D u is dfind according o: D (1 d ) D (7) u h hr d h is h plasic-damag scalar funcion (d h d). K u f 0 N u K u+1 D u K u+1 D u u u 1 a) loading siuaion b) unloading siuaion K u rsuling incras in damag is in agrmn ih h xprimnal vidnc. In h folloing oal sp u+1, h prviously simad scan siffnss valu is usd and hr siuaions may occur: i) h ingraion poin is criical and rloads bac o h nvlop a poin. In his siuaion, in h folloing sps h usual incrmnal algorihm is follod ovr h original nvlop; ii) h ingraion poin is no criical and is posiiond byond h unloading branch (poin in Figur 7 a)), maning ha h ffciv bhaviour of his ingraion poin is loading and so, i ill also rload bac o h nvlop iii) a poin in fuur sps; h ingraion poin is no criical and lis bfor h unloading branch (poin in Figur 7 b)); in his cas, i is assumd ha an ffciv unloading siuaion has occurrd and a n nvlop is gnrad. Th n nvlop in iii) is dfind by h inrscion of h unloading branch ih h original nvlop, originaing a cu in h abscissa, liminaing h xising original surfac bfor poin (Figur 8). In his ay, h compuaional algorihm prviously gnrad for h o n mhods can b asily xndd o h prsn formulaion by suprssing h branchs bfor poin and adding h n branch. suprssd ara of h diagram Figur 7: Sofning curv: dfiniion of h unload branch and drminaion of load siuaion Similarly o h usual NIEM and uomaic formulaions, a scan siffnss is dfind for h folloing sp. In Figur 7, h scan is dfind by racion upda, dividing h nsil valu by N u. This rducion paramr N u ill corrspond o N SL hn using h uomaic mhod or a larg valu (say 50 o 100) hn using h NIEM. In his ay, a non-convrgn cyclic bhaviour is avoidd. Furhrmor, h u n branch Figur 8: Sofning curv: xampl of h insrion of an unload branch in a mulilinar sofning nvlop In h sps folloing sp u+1, h n nvlop is assumd for his ingraion poin and svral siuaions may dvlop unil poin is again rachd (in rloading cas), ihr 5

6 du o h bhaviour of his paricular ingraion poin alon or by influnc of h rmaining ons. Th firs siuaion occurs hn h currn sa is shor of (poin in Figur 8 b)) h inrscion bn h currn scan and h unloading branch (poin ). In his cas h scan siffnss is follod in boh unloading and loading, in h inrval bn h origin and h limi poin. If unloading o h origin occurs, crac closur is assumd; convrsly, if poin is rachd, a chang in siffnss from u+1 o D u is adopd and h n nvlop is assumd. frards, hn h currn srss sa is on h unloading branch (poin in Figur 9a)), h incrmnal approach is follod similarly o h original formulaion of h mhod. Loading and unloading cass ar allod on h n firs branch of h ransformd nvlop and, in loading cass, h pah ill b follod incrmnally unil h nx vrx on h linarisd curv is found (Figur 9 a)) or, if an unloading siuaion occurs, h load facor is compud oards poin u. In furhr oal sps, unil poin is rachd, h load facor is compud such ha h n nvlop is aimd a h n lasic branch (poin C in Figur 9 b)). Th locaion of poin C is simad using h original NIEM algorihm. u D=Du a) unloading/rloadi ng on h unloading branch K j+1 K j u C rial,j (*) D=D u b) rloading f 0 (**) N SL siuaion in a oal sp: Figur 9: Sofning curv: incrmnal/oal algorihm ih adapd n nvlop In h uomaic mhod h simaion of h posiion of poin C is prformd by using h usual spis scan siffnss upda. Similarly o h firs oal sp afr his siffnss upda, h scan siffnss K j+1 ill b follod unil inrscion ih h unloading branch occurs. 4 DMGE PRMETERS If non-scan unloading is assumd, apar from h scalar damag valu d prviously, an addiional paramr (d u ) is adopd. Thus, hroughou h hol procss h valu of d is fixd, ping h mmory of h nvlop srss sa, hras d u is a mporary variabl usd for h dfiniion of h scan siffnss, unil full rloading is achivd. fr rloading bac o h nvlop, d ill hn b normally updad ih damag incras and d u ill b clard unil anohr unloading cycl occurs. This ay, in Figur 10, h fixd damag valu of h unloading poin is: d 1 D (8) nd, h damag of h subsqun oal sp ill b: du, 1 D (9) ih, u bing h jump incras du o h nforcd scan upda. D D u u u u K = (1-d) D K = K =(1-d ) D Figur 10: Sofning curv: valuaion of h damag paramrs s prviously sad, if hr is no an ffciv unloading siuaion, h algorihm is abandond and h original nvlop ill again b follod; for insanc, hn poin is rachd, h marial damag is mad qual o h plasic-damag paramr a poin. u, 6

7 d d u (10), 5 FRMULTIN FR HRDENING EHVIUR n Scion 2, h formulaion usd for unloading/rloading as prsnd in h scop of h sofning bhaviour. No, h xnsion o hardning bhaviour is almos sraighforard. Firs, in hardning hr is no nd o prvn an ovrsimad iniial siffnss valu as don in h discr crac approach. Thus, h iniial modulus is h Young s modulus ( = D ). Morovr, h scan modulus is no updad by mans of spis srss upda, bu, by spis srain incras or siffnss rducion, hich ar mor suiabl procdurs as xplaind in [1]. Th rfrnc poins prviously usd in h xampls ar no adapd o hardning curvs in Figurs 11 o 14. K u+1 u D u K u K u+1 a) loading siuaion b) unloading u D u K u siuaion In Figur 13 h valu is simad according o h addiional rial sp in h NIEM, such as implmnd ih sofning; hn using h uomaic mhod i mus b valuad by mans of spis srain incras or siffnss rducion, obaining h associad paramrs from h linar branch. u D=D u a) load/unloading ovr h branch K j+1 K j C (*) D=D u b) unloading siuaion Figur 13: Hardning curv: incrmnal/oal algorihm ih adapd n nvlop Undr hardning bhaviour, h variabl plasic-damag paramr d u is simad similarly o h prsnd bfor, by using h original nvlop for valuaion (Figur 14). In his cas, d u is compud from h srain variaion rsuling from h inrscion bn h scan valu of h currn rload siuaion and h original nvlop u. Figur 11: Hardning curv: dfiniion of h unload branch and drminaion of load siuaion Th procdur adopd for h dfiniion of h n nvlop is similar o h prvious on, suprssing h ara prcding poin. D u K u= (1-d) D D u suprssd ara of h diagram u n branch Figur 12: Hardning curv: xampl of h insrion of an unloading branch in a mulilinar sofning nvlop K = K =(1-d ) D u u, Figur 14: Hardning curv: Evaluaion of h damag paramrs In Figur 15 a flochar of h 1D algorihm is prsnd, hr h prviously rfrnc poins and indxs ar usd. Th schm is prsnd using a srss-srain 7

8 rfrnial bu i is asily convrd o a racion-jump procdur. 6 STRUCTURL EXMPLE In his Scion, a srucural xampl is prsnd in ordr o valida h proposd mhods and discuss h diffrn opions inroducd for compur analysis. ilinar fini lmns ar usd for h simulaion of h bul, hras cracing is simulad using srong mbddd disconinuiis [5, 6]. Thus, cracs ar as disconinuiis mbddd insid h fini parn lmn insrd hn h nsil srngh f 0 is aaind. sp = u+2 D j=du+2 sp u+1 Unload Ys D=Di Compu o D u+2=ku+1 unld. branch No sp = u+3 Criical? Unload d K u+1=(1-d u,)d K u+1 Ys D u+3=du Ys Rurn o nvlop K j in accordanc ih h usd mhod Criical? sp = j No u+1 No? Compu o nvlop compu o unld. branch No Load Compu D u+2=ku+1 o nvlop Rurn o nvlop d Ys ll in. ps. ih admissibl pahs? No Ys D j =Du D j =Du ox yps: Incrmnal sp Toal sp Compu o u compu o nvlop Figur 15: Flochar ih non-scan 1D unload algorihm Th displacmn jumps ar obaind a addiional dgrs of frdom, locad a h mbddd crac. In his modl, hs dgrs of frdom ar global, giving ris o coninuous displacmn jumps and racions across lmn boundaris. Th us of mbddd disconinuiis allos for h simulaion of cracing on quasi-bril marials, ih h advanag of avoiding rmshing. Th us of his idsprad chniqu has bn applid o concr in nsion [6], undr srong comprssion and o spliing failur. This yp of crac simulaion can also b adopd in ohr marials, li masonry. s for h comprssiv bhaviour of concr, is modlling can bcom rahr complx, sinc nonlinariis ar prsn pracically from h bginning of h loading and h marial undrgos sofning afr pa load. Crushing is simulad using a simplifid laso-plasic modl hich limis h comprssiv srsss. I as found in prvious analyss [7] ha h considraion of a nonlinar pr-pa rlaionship dos no sm paricularly imporan on ss hr concr crushing is no h dominan failur mod. Nvrhlss, h pos-pa sofning rspons may induc lss baring capaciy han h assumd prfc plasic bhaviour. Th main rasons hy h comprssiv sofning bhaviour is no adopd hr ar: i) h lac of xprimnal vidnc, hich suppors h dfiniion of a fracur nrgy undr comprssion as a marial paramr; ii) msh siz inobjciviy, hich is a consqunc of modlling sofning ihin a coninuum, unlss rgularisaion approachs ar usd, hich li ousid h scop of h prsn analysis. In 2005, a sudy as prsnd ih rinforcd concr bams, applying h (SL) o modls hr cracing as simulad using srong mbddd disconinuiis. This sudy as summarisd in [7]. Ths modls r basd on a sing campaign, and had h goal of simulaing h bhaviour of svral xprimnal ss prformd in rinforcd concr bams, loadd unil a prviously dfind damag lvl. fr his iniial damag, h xising cracs r rpaird by poxy glu injcion, ih posrior bonding of a sl pla consising of addiional xrnal 8

9 Graça--Cosa, lfaia, Dias-da-Cosa and Sluys rinforcmn. Nx, h bams r subjcd o addiional loading unil failur. This s campaign, includd h sing of a rfrnc bam, hich as loadd unil failur ih hr unloading cycls. Th aim of h prsn xampl is o illusra h us of h unloading-rloading algorihm prsnd on h prvious scions. Th modl is a four poin bnding rinforcd concr bam (Figur 16) subjcd o circular bnding a h cnral span (no shar) and i is composd of concr ih f c = N/mm 2, f 0 = 2.4 N/mm 2, G FI = N/mm. Sl rinforcmn is composd of o 10 mm bars on h boom fac and o 6 mm bars on h op fac. Sirrups r usd on h xprimnal ss, bu hy r no modlld in h fini lmn msh. 2 Ø 6 2 Ø P P Figur 16: Unloading-rloading cycls in a rinforcd concr bam: srucural schm, load and boundary condiions (200 mm idh, dimnsions in mm) Th fini lmn modl is a srucurd msh (Figur 17), composd of 360 bilinar lmns, ih a mod-i bilinar sofning la ( 1 = mm and f 1 = 1.84 N/mm 2, ul = 0.13 mm), Cracing is modlld by a discr crac approach, using srong mbddd disconinuiis [5]. Th racion-displacmn jump rlaionship is characrisd by a mod-i bilinar sofning la; h shar siffnss is rducd proporionally o zro as mod-i sofning volvs unil a srss-fr crac is obaind. Unloading-rloading cycls in sofning ar modlld by mans of a linar branch ih a siffnss qual o 10 3 N/mm, and h lasic siffnss in hardning. 900 Figur 17: Unloading-rloading cycls in a rinforcd concr bam: fini lmn msh muli-linar bhaviour la for concr undr comprssion is adopd, hich consiss of h linarisaion of h MC90 [8] concr bhaviour funcion: Eci c c E f, for E ci c 1 2 Ec1 c1 c1 c1 c1 c c i c1 2 (11) hr: Ec1 is h iniial angn modulus (32000 N/mm 2 ); E f / ; c ci c is h comprssion srain; c is h srain valu associad ih f c. Convrsly o h MC90, no sofning bhaviour is dfind afr c1. s prviously sad, his simplificaion is inroducd in ordr o avoid h corrsponding msh siz dpndncy. Thus, h adopd nvlop approximas h MC90 funcion unil h srain valu c1 is aaind, afr hich, a pur plasic bhaviour is adopd (Figur 18). Unloading-rloading follos h lasic branch, paralll o h iniial angn modulus. Figur 18: MC90 and adopd comprssion modls for concr undr comprssion Sl bhaviour is modlld, by mans of 72 Linar lmns, ih an laso-plasic la (f y = N/mm 2, E = N/mm 2 ). Th unloading-rloading siffnss is h lasic siffnss. ond-slip is modlld using 72 9

10 inrfac lmns undr mod-ii fracur, using h MC90 local bond slip modl. Exprimnally, h bam as loadd unil failur ih 3 cycls; h firs on unil 10.0 N, procdd ih unloading and rloadd unil h scond loading sag of 15.0 N. fr his scond sag, h bam as unloadd and rloadd unil 19.6 N, afr hich h load as p fixd for a crain priod in hich crp dvlopd. Sinc crp is no simulad hr, h plaau on h chars of Figurs 19 and 20 is arificially inroducd by adding h corrsponding displacmn. Th load-displacmn curv, using h uomaic mhod, follos accuraly h xprimnal curv, ih h o firs unloading branchs slighly lss siff han h xprimnal unload-rload branchs. Using h NIEM h curvs ar smoohr, ih similar unloading-rloading bhaviour. Th rsuls of boh mhods do no simula h damag incras du o h unloadingrloading loops sinc hy r no modlld. Hovr, h good agrmn bn h numrical and h xprimnal unloading branchs and h lac of numrical difficulis opns h possibiliy of modlling mor complx cycls. of h analysd ingraion poins prsnd blo. Figur 20: Unloading-rloading cycls in a rinforcd concr bam: load-displacmn curv obaind using h NIEM mhod. Figur 21: Unloading-rloading cycls in a rinforcd concr bam: crac suprposiion in h dformd fini lmn msh using h NIEM a h iniiaion of h firs unloading sag = 3.06 mm (displacmns amplifid 5 ims, crac idh amplifid 25 ims) Figur 19: Unloading-rloading cycls in a rinforcd concr bam: load-displacmn curv obaind using h uomaic mhod. Th dformd shaps and crac parn ar prsnd in Figurs 21 o 23. Crac localisaion is somha diffrn bn h mhods. failur, a diagonal crac is formd du o lac of sirrups, bu h dformaion limi of 12 mm prvnd shar failur. n hs figurs h circls rprsn h locaion Figur 22: Unloading-rloading cycls in a rinforcd concr bam: crac suprposiion in h dformd fini lmn msh using h NIEM a failur sag = mm (displacmns amplifid 5 ims, crac idh amplifid 25 ims) In ordr o visualiz h voluion of h concr bhaviour, h obaind srss-srain diagrams ar plod in Figur 24 for h bul in h mos comprssd fini lmn, hras in Figurs 25 and 26 h racionjump rlaions a h ip of h mos opnd crac ar prsnd. For bul comprssion h plod nvlop is obaind ih h NIEM, 10

11 hich is similar o h on obaind ih h uomaic mhod. f 0 oal sps incrmnal sps unloading sags ul Figur 23: Unloading-rloading cycls in a rinforcd concr bam: crac suprposiion in h dformd fini lmn msh using h uomaic mhod a failur sag = mm (displacmns amplifid 5 ims, crac idh amplifid 25 ims) unloading sags Figur 25: Unloading-rloading cycls in a rinforcd concr bam: marial nvlop of h mos opnd crac ip lmn using h uomaic mhod. incrmnal sps f 0 oal sps unloading sags ul Figur 26: Unloading-rloading cycls in a rinforcd concr bam: marial nvlop of h mos opnd crac ip lmn using h NIEM. Figur 24: Unloading-rloading cycls in a rinforcd concr bam: marial nvlop of h mos comprssd concr lmn using h NIEM. Similarly, on h nsil sofning diagrams, h unloading sags ar clarly idnifiabl, and du o h lo valu of h limi racion jump, only on or o sags occur on ach fracur ingraion poins. In h cass prsnd in Figurs 25 and 26, h las unloading sag occurrd afr h analysd ingraion rachd full sofning, hus, h unload sps follod h horizonal (zro siffnss) branch. Th scond unloading sag ld o ngaiv racion valus, bu no crac closur is obaind. In h sl lmns Figur 27, du o h us of an lasoplasic bhaviour la, h o firs unloading sags occur hil h lasic branch is bing follod. In his ay, i is only possibl o visualis on unloading branch mrging from h plasic horizonal sag. gain, h nvlop is naurally rrivd afr rloading. unloading sag Figur 27: Unloading-rloading cycls in a rinforcd concr bam: marial nvlop of h mos yildd sl lmn using h NIEM mhod. 11

12 I can b concludd ha h proposd algorihm is valid and saisfacorily dpics h ral unloading-rloading siuaions, boh in hardning and in sofning bhaviour. Morovr, h rurn o h sa prvious o unloading is guarand by mans of h damag variabls dfind in Scion 3. Sinc, in oal sps, h marial poins may lays on h scan branchs mporarily, bfor aaining h unloading/rloading branchs, a sofr unloading bhaviour can b obaind. n procdur is no bing implmnd o ovrcom his issu. 7 CNCLUSINS In modls in hich no criical bifurcaion poins aris, h folloing conclusions can b dran: i) h us of an incrmnal approach ih linarisd curvs has provn suiabl in prsnd xampl; and ii) h nrgy soluion conrol ffcivly allos for h corrc loading pah choic. oh n mhods ld o good rsul. Th NIEM shos a good agrmn ih h marial las hn compard o h uomaic mhod ih an accpabl incras in compuing im. Non of h prsnd mhods is msh-dpndn in h scop of discr crac approach[1]. Th possibiliy of boh mhods soring h marial srss/srain or racion/jump hisory, by mans of a damag paramr, allos a prfc corrlaion bn oal (scan) and incrmnal (angn) approachs, during a compl analysis, hus opning h possibiliy o h modlling of ohr yp of unloading pahs (ohr han scan), such as lasic and damag-plasic unload or vn hysric cycls, sinc h coordinas of h rurn poin on h surfac ar asily obaind by mans of h currn damag valu. Th prsnd las, ffcivly modl cyclic bhaviour on boh hardning and sofning, alloing h us of a damag prdicion bn unloadingrloading siuaions. Finally, h o mhods r capabl of good prdicions of h xprimnal rsuls, convrsly o h classic iraiv mhods hich ofn fail o convrg. REFERENCES [1]. Graça--Cosa, R., lfaia, J., Diasda-Cosa, D., and Sluys, L.J., Non-iraiv pproach for h Modlling of Quasi-bril Marials. Inrnaional Journal of Fracur (accpd), (CFRC2011). [2]. Ros, J.G., lli,., and Invrnizzi, S., Robus modling of RC srucurs ih an "vn-by-vn" sragy. Enginring Fracur Mchanics, (3-4): p [3]. Graça--Cosa, R., lfaia, J., No, P., Dias-da-Cosa, D., and Sluys, L.J., Gnralisaion of non-iraiv mhods for h modlling of srucurs undr non-proporional loading. Inrnaional Journal of Fracur (submid), [4]. midi,. and Lofi, V., Numrical analysis of cyclically loadd concr undr larg nsil srains by h plasic-damag modl. Inrnaional Journal of Scinc and Tchnology (Scinia Iranica), (3): p [5]. lfaia, J., Simon,., and Sluys, L.J., Non-homognous displacmn jumps in srong mbddd disconinuiis. Inrnaional Journal of Solids and srucurs, : p [6]. Dias-da-Cosa, D., lfaia, J., Sluys, L.J., and Júlio, E., Toards a gnralizaion of a discr srong disconinuiy approach. Compur Mhods in pplid Mchanics and Enginring, (47-48): p [7]. Graça--Cosa, R. and lfaia, J., Th numrical analysis of rinforcd concr bams using mbddd disconinuiis. SDHM - Srucural Durabiliy & Halh Monioring : p [8]. CE, CE-FIP Modl Cod 1990, d. C.E.-I.d. on. 1991: Thomas Tlford. cnoldgmns This or is suppord by FEDER funds hrough h praional Programm for Compiivnss Facors CMPETE and by Porugus funds hrough FCT Porugus Foundaion for Scinc and Tchnology undr Projc No. FCMP FEDER (FCT rf. PTDC/ECM/119214/2010). 12