Finite Difference Energy Method for nonlinear numerical analysis of reinforced concrete slab using simplified isotropic damage model

Transcription

1 Volum 7, Numbr 6 (Dmbr 4) p ISSN Fnt Dffrn Enrgy Mthod for nonlnar numral analyss of rnford onrt slab usng smplfd sotrop damag modl Análs não lnar numéra va MDFE d lajs d onrto armado utlzando a mâna do dano M. V. A. LIMA a mvnosalmda@gmal.om J. M. F. LIMA a lma.jmf@gmal.om P. R. L. LIMA a lma.prl@pq.npq.br Abstrat Ths work prsnts a modl to prdt th flxural bhavor of rnford onrt slabs, ombnng th Mazars damag modl for smulaton of th loss of stffnss of th onrt durng th rakng pross and th Classal Thory of Lamnats, to govrn th bndng of th strutural lmnt. A varatonal formulaton basd on th prnpl of vrtual work was dvlopd for th modl, and thn tratd numrally aordng to th Fnt Dffrn Enrgy Mthod, wth th nd rsult a program dvlopd n Fortran. To valdat th modl thus proposd hav bn smulatd wth th program, som ass of slabs n flxur n th ltratur. Th valuaton of th rsults obtand n ths study dmonstratd th apablty of th modl, n vw of th good prdtablty of th bhavor of slabs n flxur, swpng th path of qulbrum to th ruptur of th strutural lmnt. Bsds th satsfatory prdton of th bhavor obsrvd as postv aspts of th modl to ts rlatv smplty and rdud numbr of xprmntal paramtrs nssary for modlng. Kywords: rnford onrt slabs, damag mhans, fnt dffrn nrgy mthod. Rsumo Nst trabalho é aprsntado um modlo para prvsão do omportamnto à flxão d lajs d onrto armado, ombnando o modlo d dano d Mazars, para smulação da prda d rgdz do onrto durant o prosso d fssuração a Tora Clássa d Lamnados, para rgr a flxão do lmnto strutural. Uma formulação varaonal om bas no prnípo dos trabalhos vrtuas fo dsnvolvda para o modlo, sndo m sguda tratada numramnt sgundo o Método das Dfrnças Fntas Enrgétas (MDFE), tndo omo rsultado fnal um programa dsnvolvdo m Fortran. Para valdar o modlo proposto, foram smulados om o programa dos asos d lajs sob flxão, nontrados na ltratura. A avalação dos rsultados obtdos nas análss dmonstrou a potnaldad do modlo, tndo m vsta a boa apadad d prvsão do omportamnto d lajs sob flxão, varrndo a trajtóra d qulíbro até a ruptura do lmnto strutural. Além da satsfatóra prvsão do omportamnto obsrvou-s, omo asptos postvos do modlo, a sua rlatva smpldad o númro rduzdo d parâmtros xprmntas nssáros à modlagm. Palavras-hav: laj d onrro armado, mâna do dano, método das dfrnças fntas nrgétas. a Unvrsdad Estadual d Fra d Santana, Programa d Pós-graduação m Engnhara Cvl Ambntal, Fra d Santana, Baha, Brasl. Rvd: 5 Sp 3 Aptd: Jul 4 Avalabl Onln: D 4 4 IBRACON

2 M. V. A. LIMA J. M. F. LIMA P. R. L. LIMA. Introduton Th onrt struturs ar usd throughout th world wth many dffrnt applatons, as brdgs, buldngs, roads, t. Howvr, th rduton of natural rsours and th larg onsumpton of nrgy n produton of mnt and stl hav rqurd th us of struturs usng th full potntal of ths matrals. Dspt th dsgn ods prmttng th us of smplfy modls to alulat th rnford onrt struturs, only wth th dvlopmnt of mor sophstatd mthods wll b possbl to obtan an optmzd strutur. Usng an approprat numral mthod, that norporats th nonlnarty of th physal matral, wll b possbl to prdt wth mor auray th strutural bhavor of struturs undr bndng, dntfyng thr stat of rakng and dformaton. Sn 94, th fnt dffrn mthod (FDM) has bn usd n solvng dffrntal quatons for strutural analyss of ontnuous bams or plats loadd n plan or out of plan []. As a omputatonal mthod th FDM, togthr wth fnt lmnt mthod (FEM), orgnatd n th ngnrng ltratur n th md 95s. Many das and thnqus ar ommon to both mthods, and n som ass thy ond. Nvrthlss, wth ts mor systmat us of th varatonal approah, ts gratr gomtr flxblty, and th way t mor asly lnds tslf to rror analyss, th FEM has bom th domnatng approah both among numral analysts and n applatons []. Howvr, rnt studs hav dmonstratd th apablty of th FDM n th analyss of rnford onrt struturs: strutural analyss of slab [3], tnsl mmbrs [4], frams [5] and olumns [6] of rnford onrt has bn dvlopd usng physal or gomtral nonlnarty approah to valuat th rakng, ultmat load, spallng, stlonrt bond-slp, dflton and buklng. Th rsults ndat that: ) th omparsons wth xprmntal rsults onfrm th valdty of th mthod and ) th fnt dffrn analyss produr s both fast-runnng and aurat and most sutabl for dsgn off applaton, ombnng th spd of analyss and th dtal and auray of FE analyss [7]. An voluton of th fnt dffrn mthod s th Fnt Dffrn Enrgy Mthod (FDEM). Fnt dffrn analyss dvds a mmbr nto dsrt sgmnts to fnd an approxmat soluton to th dffrntal quatons of mmbr moton. Th sgmnts n partal dffrn analyss ar dvdd by nods, wth th dsplamnt at ah nod xprssd n trms of th dffrn n dsplamnt of adjant nods. In FDEM ths dffrns ar usd drtly n quatons assoatd to prnpl of vrtual work or to prnpl of potntal nrgy. Ths fat lads to som advantags ovr th tradtonal approah of th fnt dffrn mthod wth lowr ordr drvatvs and hghr auray. Wth th varatonal prnpl, th boundary ondton s takn ar automatally and th matrx of qulbrum quatons s symmtr (n th as of lmnts wthout damag), whh yld onsdrabl savngs n omputatonal ffort. By rlyng on nrgy quatons, th mthod has a smlar strutur wth th fnt lmnt mthod (FEM): th dsrtzaton s onvd n trms of nodal dgrs of frdom and t uss th sam knmat boundary ondtons. On advantag of ths mthod wth rspt to th FEM s a rduton of dgrs of frdom by nod what dmonstrats th grat potntal to mhanal analyss of omplx struturs. In fat, strutural applatons of FDEM hav mrgd n 95s, wth Houbolt [8], Grffn and Varga [9] and Bushnll []. In Brazl th FDEM has bn usd sn 98s to analyz of sotrop bams, plats and shlls [-4]. Rntly th FDEM, n assoaton wth Damag Mhans, was usd to modlng of rnford onrt bams and fbr onrt bams [5] rsultng n good agrmnts wth xprmntal rsults. Ths qualfs th modl for analyss of mor omplx struturs suh as rnford onrt slabs. Th applaton of nonlnar modls for th study of rnford onrt slabs s lss rportd n th thnal ltratur, whn ompard wth studs on bams. Howvr, som mportant ontrbutons to th thortal prdton of th bndng bhavor of slabs an b vrfd as th assoaton btwn lastoplast modl and FEM [6-7]. Th us of Mazars [8] wth Boundary Elmnt Mthod [9] alon or ombnd wth FEM [] has also bn rportd. Fratur Mhan modl and FEM wr usd by Bandra [] to analyz rnford onrt slabs wth ommral softwar DIANA. All rsults ndat that, wth th us of non-lnar modls for modlng th strss-stran bhavor of onrt, t s possbl to prdt satsfatorly th strutural bhavor of rnford onrt slabs undr bndng. Th man objtv of ths work s to vrfy th applaton of th FDEM to nonlnar analyss of rnford onrt slabs usng th Damag Modl of Mazars [8]. Anothr nonlnar modls, basd at xprmntal rsults, has bn usd to dsrb th strss-stran bhavor of onrt [-4] but th applaton of Mazars s Modl showd a good prforman to analyz of rnford onrt bams [5-6] and rqurs fw xprmntal paramtrs. Two rnford onrt slabs valuatd xprmntally by Taylor t al. [7] and MN [8] wr usd to valdat th proposd modl.. Analytal formulaton. Damag modl Th damag modl proposd by Mazars [8] has asy mplmntaton, whn ompard wth anysotrop modl, for xampl. Ths s baus th damag varabl D s assoatd wth th xtnson dformaton of onrt aordng to th followng hypothss: a) Th damag pross of onrt ours n last stat; b) Th onrt s onsdrd sotrop bfor and aftr th damag; ) It s supposd that th damag bgns whn th quvalnt stran rahs a rfrn stran valu ε d, dtrmnd n unaxal tnsl tsts n rlaton to th maxmum strss. In ths analyss, th strss n a gvn man drton (, and 3) an b rprsntd by postv and ngatv: s + s f s > and s + f s s - s f s < and s - f s ³ (a) (b) IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6 94

3 Fnt Dffrn Enrgy Mthod for nonlnar numral analyss of rnford onrt slab usng smplfd sotrop damag modl Thus th tnsl and omprssv strans an b rprsntd usng Hook s Law: + u u s s t - å j E E j (a) whr: å 3 + a t t + v (8a) + u u s s å j E E j (b) a 3 + å + v (8b) whr u s th offnt of Posson of onrt. Th xtnson stat s loally haratrzd by an quvalnt stran that s xprssd by: Whr ε, man stran omponnts, and ε + +, th postv parts, wth, and 3, ar dfnd by (4) and (5), rsptvly: (3) wth + + t t t + + (9a) (9b) t + (4) 3 å v t (9) + + Aordng Mazars [8] th damag pross starts whn th tnsl stran xds th dformaton ε do, rfrrd to th maxmum strss of an unaxal tnson tst (Equaton 6).. As th onrt bhavs dffrntly n tnson and omprsson, th damag varabl D s obtand by ombnng proprly th varabls D t and D, rlatd to tnson and omprsson, rsptvly, as follows: (5) and: αt ; α ; αt + α. D and D ar dfnd as: t D D t ( - A ) do t At - - xp ë ébt ( - A ) ( - ) do A - - xp ë éb do û ù ( - ) do û ù (a) (b) - do (6) whr A t, B t, A and B ar xprmntal paramtrs dfnd rlatd to strss-stran urvs of onrt undr tnson or omprsson, rsptvly. Th onsttutv rlaton of damagd onrt an dfnd thus as: D a D + a D t t (7) s E - ( D ) () whr E rprsnts th last modulus of onrt undamagd. 94 IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6

4 M. V. A. LIMA J. M. F. LIMA P. R. L. LIMA. Flxural modl Th Classal lamnatd plat thory was usd to modl th rnford onrt slab. In omparson, th dsrt approah to modlng onrt and stl rnformnt nlud a larg numbr of funtons and matral paramtrs and nvolv tdous programmng and omputatonal ffort du hghr dgrs of frdom [7]. Howvr, layrd approah s mor smpl sn th matrals ar rprsntd by layrs and th stran and strss of onrt and rnformnt through th thknss of th ross-ston an b montord progrssvly, n ths mannr provdng an aurat rprsntaton of th strutural bhavor. Layrd approah n FEM hav bn wdly usd to analyss of rnford onrt struturs [7, 4, 9-3] wth rlatv suss. Smlar to th Eulr-Brnoull bam thory and th plat thory, th lassal lamnatd plat thory s only vald for thn lamnats wth small dsplamnt n th transvrs drton. It shars th sam lassal plat thory assumptons: [3-33]: ) Th layrs ar prftly bondd togthr; ) Th lamnat s thn ompard to th latral dmnsons and s loadd n ts plan; 3) A straght ln orgnally normal to th undformd md-surfa rmans straght and rotats so as to rman straght and normal to th dformd md-surfa plan; 4) A ln normal to th md-surfa of th plat s nxtnsbl (dos not strth); 5) Lnar gomtr analyss s onsdrd wth smalls dfltons and rotatons whn ompard to unty. Th Krhhoff s thn plat approxmatons an b usd hr to drv rlatonshps btwn n-plan and out-of-plan dsplamnts, urvaturs, twsts, and strans; 6) Th matral of ah layr s lnarly last and has thr plans of matral symmtry (., orthotrop). In fat, th lassal plat thory usually assums that th matral s sotrop, whl a fbr rnford ompost lamnat wth multpl layrs may hav mor omplatd strss-stran rlatons. 7) Loads ar appld n md-surfa plan. Basd assumptons to 5 th dformatons u, v and w an b dsrbd ntrly n trms of th dformaton of th md-surfa plan u, v and w (Fgur ): (a) (b) () w whr w and ar rotatons of a normal th md-plan x y about th x and y axs, rsptvly. Basd n assumpton 5, by substtuton of ths dsplamnts rlaton nto th stran-dsplamnt quatons of th lassal thory of lastty, th followng rlatons ar obtand: x u w - x x z (3a) Fgur Undformd and dformd gomtrs of an dg of a plat undr th Krhhoff assumptons y v w - y y z (3b) g xy u v w + - z y x x y (3) z (3d) g xz (3) IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6 943

5 Fnt Dffrn Enrgy Mthod for nonlnar numral analyss of rnford onrt slab usng smplfd sotrop damag modl g yz (3f) (6b) Eah lamna n th lamnat s assumd to b n plan strss stat so that th onsttutv rlaton for a typal lamna k, assumng nonlnarty of matral, an b wrttn as: és ù é x Q Q Q ù é 6 ù x ê ú ê ú ê ú s Q Q Q ê y ú ê 6 ú ê y ú êt ú ê xy Q6 Q6 Q ú ê 66 g ú ë û ë û ë xy û (4) Th lmnts of [Q ] ar rlatd to th ngnrng onstants E, G and u, aordng assumpton 6, as follows. Transformaton of lamna strss and stran from th - axs to th rotatd x - y axs (orntaton θ ) s shown hr n matrx notaton. Q Q os q + Q + Q sn q os q + Q sn q 4 66 ( 4 ) sn q os q ( sn 4 q os 4 q) Q Q + Q - Q + Q + 66 (5a) (5b) Q G - D 66 (6) (6d) whr: E s th longtudnal last modulus n drton x ; E s th longtudnal last modulus n drton y ; u u s th major Posson s rato of layr assoats to drtons x and y ; G s th shar modulus assoats to drtons x and y ; D rprsnts th damag of matral. In ths formulaton, th loads and momnts an b xprssd n matrx form as (Fgur ): (7a) Q Q sn q + Q + Q sn q os q + Q os q 4 66 (5) (7b) Q Q - Q - Q snq os q + Q - Q + Q sn q osq Q Q - Q - Q sn q osq + Q - Q + Q snq os q) (5d) (5) whr Nxx, N yy and N xy ar n plan mmbran fors pr unt lngth (du to strthng of th plat md surfa), M xx and M yy Fgur Coordnat systm and layr numbrng usd for a lamnatd plat ( ) sn q os q ( sn 4 q os 4 q) Q Q + Q - Q - Q + Q (5f) (6a) 944 IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6

6 M. V. A. LIMA J. M. F. LIMA P. R. L. LIMA ar bndng momnts pr unt lngth about th y and x axs, rsptvly, and M ar twstng momnt pr unt lngth (smlar xy to torson n a bam). Ths matrs ar trmd: A j xtnsonal stffnss matrx; D j bndng stffnss matrx; B j xtnson-bndng ouplng matrx. A onvnnt form for th [ A ], [ B ] and [ D ] matrs an b th stablshd by xamnng th poston of th k lamna n Fgur : Fgur 3 Fors and momnt rsultants on a plat lmnt n å ( k ) A Q z - z dz j j k + k k (8a) (8b) 3 3 ( + ) n D Q z - z dz å ( k ) j j k k 3 k (8) Can b obsrvd n quaton 7 that ntrnal fors and momnts ar dpndnt of dsplamnts, layr poston and matral proprts. Thrfor, th applaton of nonlnar modls (damag modl of onrt and plastty of stl) afft drtly th mhanal bhavor of lamnatd plats. Th vrtual work don by ntrnal fors s xprssd by: ì ï æ u ö æ v ö é æ u ö æ v ö ù æ w ö dwnt ò ín xxd ç + N yyd + N xy d + d - M xxd - è ê ç ú x ø è ç y ø è ç y ø è x ø ç x r ï è ø î ë û æ w ö æ w ö ïü -M yyd ç - M xyd ý dxdy è y ø ç è x y ø ïþ (9) whr r s th ara of th md-surfa of th plat. Th vrtual work don by xtrnal fors an b xprssd by Eq. () onsdrng loads atng on th doman and on th boundary of th plat (Fgur 3 shows postv drtons of loads) and aordng assumpton 7: () whh s th transvrsal for pr unt ara atng on md-surfa of th plat; F, F and F ar th fors pr lngth unt along th dgs xx xy xz x and x a, sond to th drtons x, y z, rsptvly; M xx s th bndng momnt pr unt lngth, appld along th dgs x and x a; F, yx F and yy F yz ar th fors pr lngth unt along th dgs y and y b, aordng to th drtons x, y z, rsptvly; s th Bndng momnt pr unt lngth appld along th dgs y and y b. Th momnt of torson was not onsdrd n th xtrnal work yy sn t s not usually appld n slabs, dspt t appars th ntrnal fforts. 3. Numral formulaton Aftr xprssng th work don by ntrnal fors as a funton of dsplamnt, by rplang th xprssons of ntrnal fforts (7a) and (7b) n (9), th numral tratmnt aordng to EFDM an b appld. Frst, t s nssary to ntrodu n quatons of th δ Wnt and δ W xt th rprsntatons usd n fnt dffrns for th drvatvs of th dsplamnts. It s assumd, for valuaton ths vrtual works, that th plat s dvdd nto stons of ntgraton (ara lmnts) n whh all magntuds nvolvd ar supposd onstant. Aftr dtrmnaton of δ Wnt δ Wxt, by th summaton of th ontrbutons of th varous stons of ntgraton, a systm of algbra quatons of qulbrum s obtand by applaton of prnpl of vrtual work. Th rsoluton of ths systm prmts to obtan th numral soluton basd n unknowns nodal dsplamnts u, v w. To dtrmnaton of th systm of algbra quatons t s onsdrd th knmat ondton of th boundary and othrs varaton of dsplamnts, arbtrary and ndpndnt of ah othr. In th numral formulaton dvlopd n ths work ar usd two typs of rprsntatons for th drvatvs of th dsplamnts: ntrd rprsntaton and rdud rprsntaton. For ths, onsdr th Fgur 4, whr f ( x ) rprsnts th funton u, v w and m s th pont whr th drvatvs ar valuatd (pvotal pont). Th nodal spang s λ. Frst and sond drvatv of th funton f ( x ) valuatd at pont m ar f ' ( f f ) m λ m + and " m fm ( fm+ fm + fm ). λ IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6 945

7 Fnt Dffrn Enrgy Mthod for nonlnar numral analyss of rnford onrt slab usng smplfd sotrop damag modl Fgura 4 Funton f (x) usd to rprsnt th drvatv of dsplamnt Fgur 5 Fnt dffrn dsrtzaton of plat: nn stons of ntgraton; vrtual nods and ral nods In aras loatd along th dgs of th board, th us of rprsntaton ntrd at th frst drvatv of u and v an aus sngularty n th offnt matrx. Thrfor, for thos drvatvs s assumd to rdud rprsntaton, as suggstd by Graça [4]. Ths rprsntaton an b dfnd by f ' m ( f m + f m ). λ For th numral dsrtzaton of th doman La Lb of plat, th mthod provds a gnraton of stons of ntgraton obtand from subdvsons n th x and y drtons: n x and n y, rsptvly. Thus, an gnr ston of ntgraton prsnts th rtangular dmnsons λ x La / nx and λ y Lb / ny, wth a total of ( nx + 3) ( ny + 3) nods and ( nx + ) ( ny + ) ara lmnts for ntgraton, wth nn dffrnt typs of ston, as shown n Fgur 5. Th ara lmnt typ, 3, 7 and 9 ar loatd n th ornrs of λ th plat and dsplay ara x λy, whl th lmnt typ, 4, 6 4 λ and 8, th ara s x λy. Th rmanng stons of ntgraton (typ 5) hav ara λx λy, as llustratd n Fgur 5. Th ston of ntgraton (s Fgur 5) s omposd of nn nodal ponts and for ah of ths ponts ar assoatd wth thr dgrs of frdom u, v and w, whh norporats, for ah ara lmnt a total of 7 dsplamnts. Not that th sond-ordr drvatvs of th dsplamnts of th nods, assoatd wth th dgs of th board, rqur xtrnal nods, alld vrtual nods. Mor nformaton about th squn of dvlopmnt of th numral tratmnt, aordng to EFDM, an b found n [3-5]. 4. Comparsons wth xprmntal rsults Two slabs solvd n th ltratur was rsolvd by th ad of th nwly dvlopd omputr program: a two-way slab supportd by ts four dgs and submttd to dstrbutd loadng [7]; a two-way slab supportd by ts four ornr and submttd to Tabl Matral proprts Slab supportd by ts four dgs Conrt stl E(GPa) v f (MPa) A B f t (MPa) ε d A t B t f y (MPa) 3,4,8 35,4, 6 3,6, x -4,8 6,9 375,9 Slab supportd by ts four ornrs Conrt stl E(GPa) v f (MPa) A B f t (MPa) ε d A t B t f y (MPa) 8,6,5 37,9, 65 3,8,33 x -4,8,3 345,4 946 IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6

8 M. V. A. LIMA J. M. F. LIMA P. R. L. LIMA Fgur 6 Analytal strss-stran urvs usng Damag Modl of Mazars onntratd load [8]. Th numral dsrtzaton usd th gomtry and loadng symmtry, analyzng thus, a quartr of plat. Th numral rsults wr ompard wth xprmntal load-dflton urvs of slabs and ruptur load. 4. Computr smulaton of strss stran urvs Tabl shows th xprmntal dats of usd onrt ( E, f, f t u ) and stl ( E s f ) and th paramtr valus assoatd y wth damag modl. Basd on dats, th thortal strss-stran urvs (Fgur 6) wr obtand usng quatons () and (). f Intally, th stran valu t ε do was dtrmnd. Aftr ths, E approprat onstants A t, B t, A and B.ar tstd untl to obtan a good agrmnt btwn xprmntal and thortal valus. Th obtand tnsl paramtr A t shown smlarty wth valus ndatd by Lmatr Mazars [34] and Challaml [35]. How th xprmntal strss-stran urvs wr not obtand by authors, th omprssv paramtrs A and B wr hangd untl to obtan a sam xprmntal omprssv strngth f and a pak stran btwn and 3. Smlar produr was usd by many rsarhrs [8, 9, 3, 5, 34, 36]. Rnforng stl was modld as an unaxal layr onsdrd as an on-dmnsonal matral n th rnformnt drton. Th ntr of stl layr onds wth rnformnt ntr of xprmntal slabs and has an quvalnt ara. An lastoplast modl was usd and thr paramtrs ar nssary as nput data to dfn th stl modl: th ntal Young s modulus E s, th yld strss f and th ultmat stran ε su (adoptd as ). y Fgur 7 Gomtry, strutur and matral paramtrs of RC-slab aordng to [6] IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6 947

9 Fnt Dffrn Enrgy Mthod for nonlnar numral analyss of rnford onrt slab usng smplfd sotrop damag modl Fgur 8 Efft of damag of transvrsal last modulus on load-dsplamnt bhavor 4. Two-way slab supportd by ts four dgs Fgur 7 shows th gomtry, strutur and rnformnt haratrsts of RC-slab xprmntd by Taylor t al [7]. Th slab, supportd by ts four dgs and subjtd to a dstrbutd load at th top, has a squar shap wth 95 mm sd lngth and 5 mm thknss. On vry mportant fatur of non-lnar modls appld to th analyss of strutural rnford onrt lmnts s th problm of msh sz. In ths xampl, two numral paramtrs wr tstd. Th dal dsrtzaton was obtand aftr svral tsts and rsultd n a plat wth 8 x 8 subdvsons. Th ross ston was dvdd nto, 5, and 36 onrt and stl layrs suprmposd. Fgur 9 Influn of quantty of layrs on numral modlng at slab supportd by ts four dgs Th load dflton urv, alulatd wth 8 x 8 subdvsons and 5 layrs s dptd n Fg. 8, togthr wth th xprmntal rsults. Can b obsrvd th prsn of thr numral urvs rsultant of applaton, or not, of dffrnt damag paramtr to shar modulus G : E ) G, whr E E and u u ; + u ) G 3) G E - D ( + u) E - atdt ( + u), whr E E, u u and D atdt + ad ;, whr E E and u u. It was obsrvd a grat nflun of varaton of transvrsal modulus on load-dflton bhavour of slabs. Th adopton of undamagd modulus (stuaton ) rsults n th mor stffnss soluton whr th loss of load aftr th onrt rakng annot to b adquatly modld. Howvr, th applaton of damag offnt Th rsults llustratd n Fg. 9 ndat n a numral urv wth lowr stffnss, aftr rakng, than xprmntal urv. In fat, th man fator that afft th loss of transvrsal stffnss of onrt s th rakng of slab n rgon undr traton. In stuaton 3 th sotrop damag of G s assoatd wth th D t paramtr and a bst agrmnt btwn xprmntal and numral rsults s obsrvd. Th fal of slab was haratrzd by yldng of stl of rnformnt ( s > ). In ordr to study th nflun of layr, th xampl was analyzd by thr othr mshs, omposd of, and 36 layrs for th ross ston. Th rsults llustratd n Fg. ndat, for th prsntd xampl, that th nrasng of th numbr of layrs from 5 to 36 produd almost th sam rsults for th load-dsplamnt urv. So, t s mportant to pont out that 5 layrs for th ross ston s rally nough, vn whn th non-lnarty ffts ar vry sgnfant. Th load dflton rlatonshp obtand by ths modl s agan shown n Fg. n addton to rsults prsntd by Jang Mrza Fgur Comparson btwn xprmntal, proposd modl and FEM modl 948 IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6

10 M. V. A. LIMA J. M. F. LIMA P. R. L. LIMA Fgur Gomtry and strutur of slab supportd by ts four ornrs aordng to [37] [6]. Ths last modl s basd n Fnt Elmnt Mthod, Plast Modl to onrt and Class Thory of Plats. Th obtand rsults ar vry los to th numral ons, showng th good auray of th prsnt modl wth a smlar dgr of frdom. 4.3 Two-way slab supportd by ts four ornrs Th sond xampl xamnd (tstd by MN [8]) onsstd of squar ornr-supportd slab subjtd to a pont load appld at th ntr. Th two-way slab was 95x95 mm squar and 44 mm thk, and rnford wth an orthogonal msh gvng a rnformnt rato of.85%. Gomtr proprts, rnformnt and matral paramtrs ar shown n Fg.. Matral paramtrs ar shown n Tabl. Th transvrsal modulus was damag aordng stuaton 3 ndatd n th last tm. In rprsntng ths spmn, a 4 x 4 subdvson (5 x 5 ara lmnts) was usd to modl on-quartr of th slab. Ths xampl also was analyzd by mshs omposd of, 5, and 36 layrs for th ross ston. Th rsults llustratd n Fg. ndat rasonably aurat prdtons of load-dflton urv and thy onfrm th us of 5 layrs as th bttr soluton. Th modfaton of stffnss, aftr rakng of onrt, s rsultant of numral applaton of damag to onrt. In ultmat load t s vrfd a larg onrt stran ( ε > 5.7 ) and a stl stran n last rgm ( ε s < ) that ndat a ruptur of slab by rushng of onrt undr omprsson. Th MN slab s on oftn usd as a bnhmark for albratng nonlnar analyss. Comparson of prsnt modl wth anothr numral modl ar shown n Fgur 3. Kratz Pollng [37] usd lastoplast damag modl to rprsnt th strss-stran bhavor of onrt undr tnson and omprsson n a Fnt Elmnt Mthod (FEM) program. Crsfld [38] valuatd th ombnaton of FEM, usng squar lmnts, wth lnar softnng bhavor of onrt undr traton. Th rtron of Von Mss was adoptd as th falur rtron for th onrt undr omprsson. Doulah Kabr [39] usd FEM (squar lmnt wth 4 dgrs of frdom) and Class Thory of Plats of Mndln. To onrt bhavor was adoptd th lnar softnng n traton and b-ln- Fgur Influn of quantty of layrs on numral modllng at slab supportd by ts four ornrs Fgur 3 Comparson btwn proposd modl and FEM modls to slab supportd by ts four ornrs IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6 949

11 Fnt Dffrn Enrgy Mthod for nonlnar numral analyss of rnford onrt slab usng smplfd sotrop damag modl ar urv n omprsson. Zhang t al [7] usd FEM (squar lmnt wth 4 dgrs of frdom) assoatd wth Class Thory of Plats of Mndln and Rssnr. Conrt was modld as last lnar undr omprsson and to th tnsl bhavor undr traton was usd a nw modl that onsdr th rakng of onrt n orthogonal drton wth man drton. It s obsrvd a bttr agrmnt wth Krazt and Pollng modl whl anothr modls shown a mor stffnss soluton aftr frst rak and ruptur load vry dffrnt of th xprmntal rsults. Thus th damag modl appars as a good altrnatv for modlng onrt bhavor. 5. Conluson A nw modl was prsntd n ths work to prdt th flxural bhavor of rnford onrt slabs. Combnng th Fnt Dffrn Enrgy Mthod wth Classal Thory of Lamnats t was possbl to prform a nonlnar analyss by applaton of Damag Modl, proposd by Mazars, to rprsnt th onrt bhavor undr traon and omprsson. Th rsults ndat th nssty of th applaton of tnsl damag paramtr ( D t ) to rdus th transvrsal last modulus aftr rakng of onrt. In fat, how th valuaton of rnford onrt slab s a bdmnsonal problm, th rakng afft th proprts n all drtons. Th omparson wth xprmntal rsults dmonstrat th ffny of mthod to dsrb th load-dflton bhavor and to dtrmn th ruptur load. Comparng wth othr numral solutons, usng FEM, t s obsrvd a hghr auray of proposd modl, manly onsdrng th low dsrtzaton nssary. 6. Aknowldgmnts Th authors would lk to thank th CNPq and FAPESB (PPP /) and CAPES for fnanal support. 7. Rfrns [] KURRER K-E., RAMM, E. Th Hstory of th Thory of Struturs. From Ar Analyss to Computatonal Mhans. John Wly & Sons, 8, 848pp. [] THOMÉE, V.. From fnt dffrns to fnt lmnts: A short hstory of numral analyss of partal dfrntal quatons. Journal of Computatonal and Appld Mathmats, vol. 8,, p. 54. [3] HAGIYA, H., MORISHITA, M., ANDO, T.; TANAKA, H. MAT- UO, K. Damag valuaton of rnford onrt slabs subjtd to ontat dtonaton loads wth numral smulaton. Sn and tnhnology of nrgt matrals, vol. 64, n. 5, 3, p.9-. [4] FERRETI, D. ;SAVOIA, M. Non-lnar modl for R/C tnsl mmbrs strngthnd by FRP-plats. Engnrng Fratur Mhans, vol. 7, 3, p [5] HABIB, A.; MOHARRAMI,H. Nonlnar snstvty analyss of rnford onrt frams, Fnt Elmnts n Analyss and Dsgn, vol. 46,, p [6] VIRDI, K. S. Fnt dffrn mthod for nonlnar analyss of struturs, Journal of Construtonal Stl Rsarh, vol. 6, 6, p. 8. [7] JONES, J., C. WUA,, D.J. OEHLERS, A.S. WHITTAKER, W. SUN, S. MARKS, R. COPPOLA. Fnt dffrn analyss of smply supportd RC slabs for blast loadngs. Engnrng Struturs, vol. 3, 9, p [8] HOUBOLT, J.C. A Study of Svral Arothrmolast Problms of Arraft Struturs n Hgh-Spd Flght. Zurh, 958, Dotoral Thss - Th Swss Fdral Insttut of Thnology. [9] GRIFFIN, D. S.; VARGA, R.S. Numral soluton of plan lastty problms. Journal of Soty of Industral Appld Mathmat, vol., n. 4, 963. [] BUSHNELL, D. Fnt-dffrn nrgy modls vrsus fnt-lmnt modls- Two varatonal approahs n on omputr program. Numral and omputr mthods n strutural mhans.(a ) Nw York, Aadm Prss, In., 973, p [] PLETZ, E. Análs d strutura lamnars plo método da nrga dsrtzada. Ro d Janro, 983, Ts (Mstrado) - Unvrsdad Fdral do Ro d Janro. [] NISHIKAVA, L. K. Análs numéra d plaas om antos rntrants. Ro d Janro, 99, Ts (Mstrado) - Unvrsdad Fdral do Ro d Janro. [3] GRAÇA, M. S. B. A. Método das dfrnças fntas nrgétas na análs státa dnâma d plaas dlgadas spssas, Ro d Janro,, Ts (Doutorado) - Unvrsdad Fdral do Ro d Janro. [4] LIMA, J. M. F. Estudo omparatvo ntr formulaçõs gomtramnt não lnars para a flxo-torção d hasts d pards dlgadas d sção abrta, Ro d Janro, 4, Ts (Doutorado) - Unvrsdad Fdral do Ro d Janro. [5] NEVES, J. B; LIMA, J. M. F.; LIMA, P. R. L. Método das Dfrnças Fntas Enrgétas para smulação do omportamnto sob flxão do onrto rforçado om fbras urtas d aço. Rvsta Sul-amrana d Engnhara Estrutural, vol., n., 3, p [6] JIANG, J.; MIRZA, F. A. Nonlnar analyss of rnford onrt slabs by a dsrt fnt lmnt approah. Computrs & Struturs, vol.65, n.4, 997, p [7] ZHANG, Y. X.; BRADFORD, M. A.; GILBERT, R. I. A layrd shar-flxural plat/shll lmnt usng Tmoshnko bam funtons for nonlnar analyss of rnford onrt plats. Fnt lmnts n analyss and dsgn, vol.43, 7; p [8] MAZARS, J. A Dsrpton of mro and marosal damag of onrt struturs, Engnrng Fratur Mhans, vol.5, 986; p [9] FERNANDES, G. R. Método dos lmntos d ontorno aplado à análs não lnar d plaas, São Paulo, 998, Dssrtação (Mstrado) - Esola d Engnhara d São Carlos - Unvrsdad d São Paulo. [] CRESCE, S. H. Análs não-lnar d pavmnto d onrto armado plo método d lmntos d ontorno, São Paulo, 3, Ts (Doutorado) - Unvrsdad d São Paulo. [] BANDEIRA M. S. Análs não-lnar d lajs d onrto armado plo método dos lmntos fntos, Goás, 6, Dssrtação (Mstrado) - Unvrsdad Fdral d Goás. [] WANG, T.; HSU, T. Nonlnar fnt lmnt analyss of onrt struturs usng a nw onsttutv modls. Computr and struturs, vol. 9,, p IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6

12 M. V. A. LIMA J. M. F. LIMA P. R. L. LIMA [3] ASSAN, A.E., Nonlnar analyss of rnford onrt ylndral shlls. Computr and struturs, vol. 8,, p [4] HAIDO, J. BAKAR, B.H.; ABDUL-RAZZAK, A.A.; JAY- APRAKASH, J. Dynam rspons smulaton for rnford onrt slabs. Smulaton Modllng Prat and Thory, vol. 8,, p [5] SANCHES, F. J.; VENTURINI, W. S. Damag modlng of rnford onrt bams. Advans n Engnrng Softwar, vol.38, 7, p [6] LIMA, J.M.F.; NAGAHAMA. K.J.; LIMA, P.R.L.; SANTOS, G. J. B dos. Um modlo para análs não lnar físa da flxão d vgas d onrto armado. Mâna Computaonal, vol. XXIX,, p [7] TAYLOR, R.; MATHER, D. R. H.; HAYES, B. Efft of th arrangmnt of rnformnt on th bhavour of rnford onrt slabs. Magazn Conrt Rsarh, vol.8, 966, p [8] MCNEICE, G. M., Elast-Plast Bndng of Plats and Slabs by th Fnt Elmnt Mthod, Ph. D. Thss, London Unvrsty, 967. [9] NOH, H.C.. Nonlnar bhavor and ultmat load barng apaty of rnford onrt natural draught oolng towr shll. Engnrng Struturs, vol. 8, 6, p [3] OLIVEIRA, R.S.; RAMALHO, M.A.; CORREA, M.R.S. A layrd fnt lmnt for rnford onrt bams wth bond slp ffts. Cmnt & Conrt Composts, vol. 3, 8, p [3] GUAN, H.; COOPER, C.; LEE,, D-J. Ultmat strngth analyss of normal and hgh strngth onrt wall panls wth varyng opnng onfguratons. Engnrng Struturs, vol. 3,, p [3] JONES, R. M. Mhans of ompost matrals. Taylor & Frans Group, 999. [33] REDDY, J. N. Mhans of Lamnatd Compost Plats and Shlls: Thory and analyss. CRC, 4. [34] LEMAITRE, J.; MAZARS, J. Applaton d la théor d l ndommagmnt au omportmnt non lnéar t à la ruptur du béton d strutur. D I nsttut thnqu Du batmnt t dês travaux publs, vol.4, 98, p [35] CHALLAMEL, N. A varatonally basd nonloal damag modl to prdt dffus mrorakng voluton. Intrnatonal Journal of Mhanal Sns, vol.5,, p [36] PITUBA, J. J. C. Avalação d um modlo ansótropo d dano onsdrando a prda d rsstêna dvdo ao salhamnto do onrto. Asoaón Argntna d Mána Computaonal, vol.9, ; p [37] KRATZIG, W.B., AND POLLING, R. An lasto-plast damag modl for rnford onrt wth mnmum numbr of matral paramtrs, Computrs and struturs, vol.8, n.5-6, 4, p.-5. [38] CRISFIELD, M. A. Alratd soluton thnqus and onrt rakng. Hr Majsty s Statonry Off, London, v. 33, pp , 98. [39] DOULAH, N.; KABIR, A. Non lnar Fnt Elmnt analyss of rnford onrt rtangular and skw slabs. Journal of Cvl Engnrng, Insttutons of Engnrs, Bangladsh, v. 9,. IBRACON Struturs and Matrals Journal 4 vol. 7 nº 6 95