Simplified Model for the Non-Linear Behaviour Representation of Reinforced Concrete Columns Under Biaxial Bending

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1 Smplfed Model for the No-Lear Behavor Represetato of Reforced Cocrete Colms Uder Baal Bedg H. Rodrges Cvl Egeerg Departmet, Uverst of Avero, Portgal aclt of Natral Sceces, Egeerg ad Techolog - Oporto Lsophoe Uverst A. Adrade-Campos Departmet of Mechacal Egeerg, Uverst of Avero, Portgal X. Romão, A. Arêde Departmet of Cvl Egeerg, aclt of Egeerg, Uverst of Oporto, Portgal H. Varm, A. Costa Departmet of Cvl Egeerg, Uverst of Avero, Portgal SUMMARY: I the preset paper a smplfed model s proposed for the force-deformato behavor of reforced cocrete members der baal loadg combed wth aal force. The startg pot for the model developmet was a estg fed-legth plastc hge elemet model that accots for the o-lear hsteretc behavor at the elemet ed-sectos, characterzed b trlear momet-crvatre laws. To take to accot the secto baal behavor, the estg model was adopted for both orthogoal lateral drectos ad a teracto fcto was trodced to cople the hsteretc respose of both drectos. To calbrate the teracto fcto t were sed mercal reslts, obtaed from fbre models, ad epermetal reslts. or the parameters detfcato, o-lear optmzato approaches were adopted, amel: the gradet based methods followed b the geetc, evoltoar ad atre-spred algorthms. all, the smplfed o-lear model proposed s valdated throgh the aaltcal smlato of baal test reslts carred ot fll-scale reforced cocrete colms. ewords: RC colms, o-lear baal behavor, smplfed o-lear model, optmzato techqes. INTRODUCTION The o-lear models shold reprodce the flece of the strctral sstem s geometr ad also of the dstrbto of mass ad stffess, partclar the effect of rreglartes terms of stffess ad/or mass, pla ad elevato. A vast mber of models have bee developed to represet the materal s o-lear behavor ad are tpcall dvded to dfferet categores, amel global models, mcroscopc models ad dscrete fte elemet (member) models (H Rodrges et al., 2; Scott et al., 28; Tacer et al., 99). The mportace of std of the trdmesoal respose of RC bldgs s recogzed, assocated wth trdmesoal earthqake actos or to bldg rreglartes, whch dces a baal bedg demad combed wth aal load the colms. Dfferet modellg strateges have bee proposed for the smlato of the baal cclc behavor of RC elemets wth aal force. However, t s recogzed that the avalable baal models are ot matre eogh to be sed practce, or to be corporated to codes/stadards as has occrred wth aal smplfed models. Detaled revews of the models avalable ca be fod the CEB Report Nº22, RC rames der Earthqake Loadg State of the Art Report (CEB, 996), ad ards (ards, 99). Besdes the theor of fbre models (Petragel et al., 999; Spacoe et al., 992; Tacer et al., 99), estg aaltcal models follow the cocepts of classcal plastct (Peckold, 974), Mroz s theor of mlt-srface plastct (ElMadooh Galal & Ghobarah, 23; Powell & Che, 986; Takzawa et al., 976), Boc-We (We, 976), hsteress modellg (Cascat, 989; ath & Rehor, 99; Romão et al., 24; C. H. Wag & We, 2), bodg srface plastct (Bosas et al., 22; M.G. Sfakaaks & M.N. ards, 99; M.G. Sfakaaks & M.N. ards, 99) or lmped damage models (M.E. Marate &

2 lórez-lópez, 22; Mara Egea Marate & lórez-lópez, 23; Mazza & Mazza, 28). The preset paper proposes a pgraded smplfed model for the represetato of the baal bedg respose of colms wth aal force, whch s based a estg aal hsteretc Costa-Costa model (Costa & Costa, 987). The geeral formlato of the proposed model s establshed the aalog ad comparso wth the baal formlato of the Boc-We smooth hsteretc model [9]. Althogh retag most of the phscal meag emboded the Boc-We model, the strctral modellg strateg adopted retas the smplct ad versatlt of the orgal pecewse lear (PWL) mercal tool. 2. COSTA AND COSTA UNIAXIAL HYSTERETIC MODEL or the developmet ad valdato of the smplfed baal model, the Costa-Costa aal hsteretc model (CEB, 996; Costa & Costa, 987) was adopted, whch s brefl descrbed the et paragraph. Ths model represets a geeralsato of the orgal Takeda model (CEB, 996; Takeda et al., 97) wth a trlear skeleto crve for mootoc loadg, defed b the crackg pot (d c ; c ) ad the eld pot (d ; ), ad cldes pchg, stffess degradato ad stregth degradato effects. Uloadg-reloadg loops pror to eldg ether drecto are blear, wth slopes eqal to those of the pre-crackg ad post-crackg braches the vrg loadg brach. After eldg, pchg s modelled b a blear reloadg crve. The frst brach of the reloadg stage has a feror slope (see gre - brach 8), whle the secod brach heads to the most etreme pot of a prevos post-eld ecrso the drecto of the reloadg. or ths the stffess s s mltpled b the factor (d /d m ), where d represets the eld dsplacemet, d m the mamm respose dsplacemet ad s a postve costat. Ths: m s d / d m (2.) d m d where m s the force at the prevos mamm respose pot ad d s the deformato at the load reversal pot. + m e d - m 8 s 4 d m + d c d m gre. Pchg effect the Costa-Costa (Costa & Costa, 987) model d c d gre 2. Uloadg stffess the Costa-Costa (Costa & Costa, 987) model d m d The loadg stffess d, after eldg, s redced relatve to the elastc stffess e b the factor (d /d m ), where s a postve costat (see gre 2). Post-eld stregth ad stffess degradato wth cclg loadg s modelled b drectg the reloadg brach, after modfcato for pchg, towards a pot at a dsplacemet eqal to d m ad at a force m=(-) m, where s the Wag ad Shah damage de (M. L. Wag & Shah, 987). After reachg ths ed pot of the reloadg brach, frther loadg takes place parallel to the post-eldg stffess of the vrg loadg crve.

3 3. THE BIAXIAL BOUC-WEN MODEL The orgal formlato of the Boc-We model, cast wth the edochroc theor framework was preseted b Boc (Boc, 97), for aal behavor represetato ( terms of orce- Dsplacemet, -), ad t was later geeralzed b We (We, 976). The geeralzed model epresses the restorg force as a combato of a elastc force ad a plastc force: ) ( (3.) where s the tal stffess, the post-eldg stffess rato, the eldg force, ad s a hsteretc parameter. Later, ths aal formlato was eteded b Park, We ad Ag (Park et al., 986; We, 976) to defe a baal force-deformato model wth copled dfferetal eqatos. Ths model was the sed ad modfed b ath ad Rehor (ath & Rehor, 99) to model the behavor of RC colms der baal loads. Later o, the model was geeralsed b Cascat (Cascat, 989) ad also b Wag ad We (C. H. Wag & We, 2), whch reslted two dfferet formlatos of the tal baal model. Sce the Wag ad We (C. H. Wag & We, 2) formlato s smpler, t was selected for to mplemetg the baal model proposed ths work. Nevertheless, the same mathematcal reasog ca be appled to the Cascat (Cascat, 989) form. The baal costrcto of the Boc-We model the Wag ad We (C. H. Wag & We, 2) form follows the same geeral dea as for the aal case. The restorg forces for both drectos are defed b: (3.2) whch the volved parameters have the same meag as for the aal case, bt are ow referred to the two orthogoal drectos X ad Y, b the sbscrpts ad respectvel. The hsteretc parameters ad are the defed b the followg copled dfferetal eqatos, where all the parameters volved have also the same meag as for the aal case ad sg() refers to the mathematcal sgm fcto. (3.3) As for the case of the aal model, Eqato 3.2 ca also be reformlated to a cremetal form: (3.4) cosderg that the global restorg forces ad reslt from: (3.5) Cosderg ow the defto of the cremetal orthogoal forces ad gve b, the frst part of ths sstem ca be wrtte, b a smple mathematcal trasformato, as: (3.6) sg sg A sg sg A sg A sg A

4 whch matches to the aal cremetal restorg forces calclated for each drecto wthot baal teracto. I order to smplf the otato, these cremetal forces wll be deoted as ad, respectvel. The remag part of the sstem correspods to the correcto factors C C f f C f ad C f accotg for the teracto betwee the two loadg drectos: sg sg (3.7) Therefore, based o Eqatos 3.6 ad 3.7, a codesed form of Eqato 4 ca be wrtte as: C f C f (3.8) Cosderg that the cremetal forces ad ca be obtaed wth a aal hsteretc model (partclarl wth well-establshed PWL models), the preseted framework trodces a smple ad fleble form to represet baal bedg colms. Ths formlato reqres the same formato eeded for the correspodg aal PWL model, ol trodcg a addtoal correcto term whch coples the two loadg drectos. Sce the aal hsteretc model that ma be cosdered to obta the cremetal forces ad cold be ver dfferet from the orgal aal Boc-We formlato, the tpe ad level of baal teracto betwee the two orthogoal drectos ca be also dfferet. Romão et al. (Romão et al., 24) proposed a addtoal parameter, whch was clded Eqato 3.8 order to scale the level of teracto betwee the two loadg drectos. The fal formlato of the proposed method s the defed as wrtte Eqato 3.9, whch states that for each cremetal dsplacemets vector ;, the cremetal forces ad as: C f C f ca be separatel calclated, (3.9) The vales of the scalg teracto factor wll be defed as for the best-fttg of the mercal reslts to the epermetal reslts obtaed wth baal tests, addtoal formato abot the preseted formlato ca be fod the lteratre (Romão et al., 24). 4. PARAMETER IDENTIICATION OR THE SCALING INTERACTION ACTOR 4. Optmzato method Nmercal o-lear smplfed models ma pla a mportat role the desg of ew strctres ad the assessmet of estg oes. More comple techqes ad models have bee developed to smlate, wth creasg accrac, the behavor of dfferet materals ad strctres. However, ma of these smlato models reqre the determato ad calbrato of a large mber of parameters adjsted to the specfc materal ad strctral problem. The detfcato of parameters for the mathematcal models adopted to descrbe the behavor of phscal sstems s a commo problem egeerg. The complet of the models, as well as the mber of parameters assocated, ormall creases wth the complet of the phscal sstem. The determato of these parameters shold be based o the comparso of the mathematcal model reslts ad epermetal reslts. However, whe the reqred mber of epermetal tests ad

5 parameters creases, t ma become mpractcal to detf the accrate parameters (Adrade-Campos et al., 27; Brhs & Adg, 999). I these cases, t ma be sed a verse formlato for the detfcato of parameters. Ths approach ofte leads to the resolto of a o-lear optmzato problem. I ths work, amg at calbratg the parameters for the hsteretc baal model based o aal models wth a teracto fcto, a gradet based-method (Leveberg-Marqardt (LM) method) was adopted, seqetall assocated wth a evoltoar algorthm method (real search space EA), groped. Wth these global cascade algorthms, t s teded to aggregate the advatages of both algorthms ad to mmse ther dsadvatages (Adrade-Campos et al., 29). or the applcato of these models, the SDL/SDoLo optmsato lab compter program was sed (Adrade-Campos, 2). The program was desged for specfc egeerg verse problems, sch as parameter detfcato ad tal shape optmzato problems. It herts the wealth of eperece gaed sch problems b the prevos SIDOLO code, ad adds the latest developmets drect search optmzato algorthms (Adrade-Campos, 2; Adrade-Campos et al., 27). 4.2 Predcto ad optmzato of the scalg factor eqato The calbrato of the teracto fcto, sg the optmzato strateges preseted the prevos secto, was performed two phases. I the frst phase, t was teded to select the aaltcal epresso-tpe most stable for the teracto scalg factor () fcto. After the epresso tpe selecto, the secod phase, cossts the calbrato of the parameters of the teracto fcto as well as of the teracto scalg factor. The parameters calbrato correspods to the best-ft of the reslts for a grop of mercal aalses. I order to defe the teracto scalg factor () whch characterzes the level of teracto, as a fcto depedet of the secto propertes ad loadg drecto, a set of mercal aalses was performed. To ths am, twet seve rectaglar RC colms were defed, varg cross-secto dmesos (33, 35 ad 36), reforcg steel rato (%,.5% ad 2%) ad aal load rato (.,.2 ad.3). or the colm, a catlever.5m hgh was cosdered. The respose of each colm was obtaed wth pshover aalss for dfferet drectos Colms were modelled wth a force-based elemet formlato ad cosderg a fbre dscretzato at the secto level. Two aal (º ad 9º) ad fve baal (at.25º, 22.5 º, 45 º, 67.5 º, º agles) pshover aalses were performed sg the compter program SesmoStrct (SesmoSoft, 24). A teracto scalg factor was determed for each baal respose of each colm, based o the Costa-Costa aal model copled wth the proposed teracto fcto prevosl preseted Eqato 3.9. or these aalses, a gradet-based optmzato algorthm was sed (Adrade-Campos et al., 27). I order to redce the mber of varables, for the shape factors (γ, ad ) ecessar to calclate C ad C were assmed eqal to the vales sggested b ath ad Rehor were f adopted (. 5 ad 2 ). f Amg at evalatg the goodess-of-ft of the gve teracto scalg factor, for each colm ad for each drecto calclato are made for, the dfferece betwee the smlated respose wth the smplfed (sm) model (wth the teracto fcto parameters) ad wth the refed mercal model (referece vales, ref). These dffereces are evalated for each drecto (X or Y) b the Relatve Global Error (RGEdrecto), as gve Eqato 4.. The combato of the error for the two drectos (RGEtotal) s calclated as preseted Eqato 4.2. [ ] ( ) (4.) where ad are, respectvel, the smlated ad referece vales of the potetal eerg assocated wth each pshover respose. (4.2)

6 Qadratc error After obtaed the optmal vales of the scalg factor () for each aalss (each colm ad each drecto), a epresso tpe was selected (see Eqato 4.3), whch depeds o the colm propertes, amel the cross-secto dmesos (h ad b), the aal load rato (ν); ad loadg drecto (α). ( ) ( ) The for costats (C, C2, C3, ad C4) are to be obtaed b optmzato for all pshover crves, as wll be show the followg Secto. Wth the adopted eqato for the scalg factor, the teracto fcto parameters were optmzed for all baal pshover crves, sg a cascade optmzato strateg. At ths stage, the Leveberg-Marqardt (LM) gradet-based method ad a evoltoar method (real search space EA) were groped seqetal/cascade strateges. Ths, as metoed before, order to combe the advatages of both algorthms ad mmse ther dsadvatages, the followg seqece LM+EA+LM was sed. A mportat aspect a cascade algorthm s the choce of the crtera to swtch from oe optmzer to aother. I the preset case, a herstc approach was adopted based o mercal epermets. The crtera, as sggested (Adrade-Campos et al., 29), were: ) Swtchg from LM to EA: f, from oe terato to aother, the relatve decrease the qadratc objectve fcto s less tha -5, or the mamm admssble terato mber (predetermed vale) s reached. ) Swtchg from EA to LM: f stagato of more tha 5 geeratos s observed or the relatve decrease the qadratc objectve fcto s less tha -5 or the mamm admssble terato mber (predetermed vale) s reached. The obtaed reslts are smmarzed Table ad the covergece evolto of the cascade optmzato strateges the parameter detfcato are preseted gre 3. (4.3) Table. Parameters acheved b the cascade optmzato strateg Parameter Vale C. C 2.23 C 3.45 C LM LM+EA+LM Iteratos/Geeratos gre 3. Covergece evolto of the cascade optmzato strateges the parameter detfcato gre 4 shows the plot of the Relatve Global Error, calclated for each drecto (sg Eqato 4.), for two statos, order to compare the referece smlato wth the smplfed model reslts, wth ad wthot cosderg the teracto fcto, represeted the fgre wth flled ad flled marks respectvel). B comparg the Relatve Global Errors for the two statos the error redcto each drecto s clear whe the teracto fcto s cosdered. gre 5 cldes a selected grop of eamples of pshover crves for dfferet colms ad dfferet pshover loadg agles. I each plot, for both drectos, the obtaed pshover crves are represeted: ) b the refed mercal fbre model (referece crves ble les wth sqare marks); ) b the smplfed model wthot the baal bedg teracto fcto (red les wth tragllar marks); ad ) b the smplfed model bt wth the teracto fcto (the optmsed solto gree les wth damod marks). Also, the eamples gre 5 cofrm the error redctos, both colm drectos, obtaed b adoptg the teracto fcto combed wth the optmzed scalg factor.

7 Base Shear (kn) Base Shear (kn) Base Shear (kn) Base Shear (kn) Relatve Global Error drecto Y (%) Wthot teracto fcto -.25º Wthot teracto fcto º Wthot teracto fcto - 45.º Wthot teracto fcto º Wthot teracto fcto º Wth teracto fcto optmzed -.25º Wth teracto fcto optmzed º Wth teracto fcto optmzed - 45.º Wth teracto fcto optmzed º Wth teracto fcto optmzed º Relatve Global Error drecto X (%) gre 4. Relatve Global Error (X ad Y colm drectos) of the smplfed model reslts, wth (flled marks) ad wthot (flled marks) teracto fcto, compared wth the refed mercal model reslts M26 36cm 2 =.2 As = 2.% Ac Pshover -.25 o bre model (X drecto) bre model (Y drecto) Orgal (X) Orgal (Y) Optmzed solto (X) Optmzed solto (Y) Horzotal Dsplacemet (m) M6 35cm 2 =. As = % Ac Pshover o bre model (X drecto) bre model (Y drecto) Orgal (X) Orgal (Y) Optmzed solto (X) Optmzed solto (Y) Horzotal Dsplacemet (m) M22 36cm 2 =. As =.5% Ac Pshover - 45 o M4 35cm 2 =.2 As =.5% Ac Pshover - 45 o bre model (X drecto) bre model (Y drecto) Orgal (X) Orgal (Y) Optmzed solto (X) Optmzed solto (Y) Horzotal Dsplacemet (m) 5 bre model (X drecto) bre model (Y drecto) Orgal (X) Orgal (Y) 5 Optmzed solto (X) Optmzed solto (Y) Horzotal Dsplacemet (m) gre 5. Eamples of pshover crves for dfferet colms ad dfferet pshover loadg agles for the refed mercal model, the smplfed model wthot the baal bedg teracto fcto ad the smplfed model wth the teracto fcto 5. VALIDATION O THE MODEL WITH RESULTS ROM CYCLIC TESTS 5. Itrodcto or the valdato of the proposed smplfed model wth teracto fctos, the epermetal reslts of cclc tests o 8 RC colms were sed of the test campag (H. Rodrges et al., 22; H. Rodrges, 22). rst, the aal tests were modelled order to obta the prmar skeleto crves for each depedet drecto. The, sg the obtaed crves, the baal tests were smlated ad the reslts of the mercal model wth the teracto fcto are compared wth the test reslts. 5.2 Aalss of the reslts or the valdato of the proposed smplfed teracto model, a trlear evelope crve was cosdered for the prmar crve of each colm drecto, b best-ft adjstmet to the aal epermetal reslts. or all aall tested colms, the epermetal reslts were well reprodced wth the adjsted

8 aal trlear crves ad wth the hsteretc rles of the orgal model, as represeted gre 6. I the fgres the followg are plotted for each colm: the mercal calclatos wth the teracto model (ble); the epermetal reslts (red); ad the trlear prmar crve (gree). As observed these fgres, the stregth degradato s dffclt to represet, partclarl for the last ccles of the epermetal respose. However, sgfcat dffereces are ol observed for demads correspodg to drfts greater tha 2.5%. The eerg dsspato evolto s also well represeted. Aga, ol for the last ccles (assocated wth the dffereces terms of stregth degradato), a overestmato of the dsspated eerg s obtaed wth the mercal model. I order to valdate the rles ad parameters of the teracto model for the smlato of the baal respose of the tested RC colms, the trlear crve adjsted to the aal test reslts was cosdered for the prmar crve each drecto. The predcto of the epermetal baal respose obtaed wth the smplfed teracto model for the tested RC colms s represeted gres 7, 8 ad 9. The same le leged s adopted as for the aal cases. As ca be observed, the mamm stregth was properl obtaed wth the smplfed baal model. or the colms strog drectos, a derestmato of the mamm stregth of 5% was observed, whle overestmato of 25% s reached the weak drecto. The loadg stffess ad the pchg effect were reasoabl reprodced most cases. The stregth degradato was also reasoabl appromatel the eamples der aalss. Ol the latter stages of the colms respose, close to the colms falre, cosderable dffereces were detected terms of stregth degradato. The evolto of the accmlated eerg dsspato s reasoable well smlated tl otceable stregth degradato s observed. I the stroger colm drecto a overestmato of arod 2% was reached ad the weak drecto the derestmato s arod 25%. I geeral, a good agreemet betwee the predcted mercal reslts ad the epermetal hsteretc respose was observed, dcatg that the proposed strateg ma be stable to smlate the respose of colms to baal loadg based o aal behavor crves assocated wth properl calbrated coplg teracto fctos. gre 6 Base-shear verss drft of colm N3 Uaal test gre 7. Base-shear verss drft of colm N4 Baal test, rhombs dsplacemet patter

9 Y Y X X X X Accmlatve hsteress dsspated eerg (kn.m) Accmlatve hsteress dsspated eerg (kn.m) gre 8. Base-shear verss drft of colm N5 Baal test, qadraglar dsplacemet patter Shear Y (kn) Shear X (kn) PB2-N6 gre 9. Base-shear verss drft of colm N6 Baal test, crclar dsplacemet patter Drft X (%) 2 Step PB2-N Drft Y (%) Step CONCLUSIONS AND INAL COMMENTS There are stll a mber of solved problems assocated wth modellg of RC elemets der baal loadg. Smplfed baal models ma be adopted f the ca adeqatel reprodce the ma characterstcs of the elemet s respose (sch as the stregth ad stffess degradato, dctlt, ad eerg dsspato capact) relatve to the colms aal respose. I the preset paper, a smplfed teracto model for the respose of RC colms to baal loadg s descrbed, based o estg aal models. The proposed model correspods to a pgrade of the estg Costa-Costa aal hsteretc model, ad adopts a teracto fcto based o the Boc- We baal hsteretc model, coplg the two loadg drectos. The model parameters were calbrated sg optmzato techqes, based o the reslts of a parametrc std o the trdmesoal respose of RC colms models wth a refed model. The valdt of the proposed model was demostrated throgh the aaltcal smlato of baal tests o RC colms. The obtaed mercal reslts were adeqate ad proved the effcec of the model, whch s a smple tool capable of reprodcg the respose of RC elemets cosderg the baal teracto. The proposed smplfed model ca be a sefl tool the desg ad assessmet of RC strctres where the respose s depedet o baal bedg of the elemets. Ths o-lear model accots for mechacal featres sch as hsteretc behavor rles, stregth ad stffess degradato, ad the pchg effect. However, addtoal research s stll ecessar to objectvel defe the teracto fcto parameters, whch establsh the coplg of the respose the two loadg drectos. Moreover, the model applcato to epermetal reslts obtaed b other athors shold be made. The mplemetato of the proposed model a strctral aalss program to obta the respose of RC mlt-store bldgs (whch strogl depeds o the baal respose of the colms) s aother mportat task that shold be archved.

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