Damage Model with Crack Localization Application to Historical Buildings

Transcription

1 Structural Analysis of Historical Constructions, Nw Dlhi 2006 P.B. Lournço, P. Roca, C. Modna, S. Agrawal (Eds.) Damag Modl with Crack Localization Application to Historical Buildings Robrto Clmnt, Pr Roca Tchnical Univrsity of Catalonia, Dpartmnt of Construction Enginring, Barclona, Spain Migul Crvra Tchnical Univrsity of Catalonia, Dpartmnt of Strngth of Matrials and Structural Enginring, Barclona, Spain ABSTRACT: A modl for th analysis of larg structurs, abl to rproduc th mchanical dgradation of th matrial and to prdict th potntial collaps undr crtain load conditions, is prsntd. Th modl is basd on th so-calld smard-crack scalar damag modl, modifid in such a way that it can rproduc localizd individual (discrt) cracks. This is achivd by mans of a local crack tracking algorithm. Th modl has bn usd to analyz th rspons of th structur of Mallorca Cathdral undr gravity and sismic forcs. Compard with th traditional smard cracking approach, th tracking mthod has shown a bttr capacity to prdict ralistic collapsing mchanisms; th rsulting damag in th ultimat condition appars localizd in individual cracks, which bhav similarly to ral plastic hings; th computd ultimat loads bcom lss snsitiv to th variation of th tnsil strngth and othr matrial paramtr. 1 INTRODUCTION Modls dscribing damag as a smard variabl distributd ovr th volum of th structur ar normally usd for th analysis of rinforcd concrt structurs and thy provid accptabl rsults. Howvr, th analysis of unrinforcd structurs, and particularly, th study of unrinforcd masonry ons rquirs mor sophisticatd approachs abl to dscrib th localizd damag which can b normally obsrvd in ths structurs both in srvic and at th ultimat condition. In th so-calld smard-crack scalar damag modl (Crvra, 2003), th damag in a point is dfind by mans of a scalar valu (calld damag variabl) that affcts th lastic constitutiv matrix and rprsnts th lvl of dgradation of th matrial, ranging from intact matrial (lastic) to compltly damagd matrial with no stiffnss. Th first rsults obtaind by th application of this modl to th finit lmnt analysis of masonry buildings showd that damag is simulatd in an unralistic way involving significant volums sprading ovr larg rgions of th structur. As a rsult of this, a modification was implmntd to dscrib localizd damag in th form of discrt cracks. This is achivd by mans of a crack tracking algorithm. This algorithm is abl to dtct th point of th boundary of th structur whr a crack is originatd and thn it lts th crack dvlop as a function of th dirction of th main tnsil strss. Th algorithm marks a track of finit lmnts prtaining to th crack path which can xprinc potntial damag. As part of a largr projct involving diffrnt approachs, th structur of Mallorca Cathdral has bn studid using both th smard-crack scalar damag modl and th crack tracking algorithms, for both gravitational and sismic loads. Also, som studis hav bn carrid out using both 2D and 3D structural modls. Th comparison of th rsults obtaind from th diffrnt

2 1126 Structural Analysis of Historical Constructions approachs considrd illustrats th capacity of th crack tracing approach to provid a mor ralistic simulation of th strngth rspons of masonry historical buildings. 2 TENSION COMPRESION DAMAGE MODEL Th modl is basd on th ffctiv strss concpt ( σ ), dfind as th strss associatd to lastic strains ε ; or σ = D : ε, whr D is th lastic constitutiv tnsor. Th tnsor σ is dividd into + a positiv or tnsil part σ and a ngativ or comprssiv part σ, in th following way: 3 = σ i i i = + i= 1 σ p p σ σ σ (1) whr σ i is th i-th principal strss of σ ; p i rprsnts th unit vctor associatd with th principal dirction i and ar th Macaulay brackts. Two intrnal variabls of damag, ach on associatd with a sign of th strsss ar also dfind: d + for tnsil damag and d for comprssiv damag. Undr ths considrations, th constitutiv quation is writtn as: ( 1 d ) ( 1 d ) + + σ = σ + σ (2) Ths variabls stat th lvl of damag rachd at ach intgration point, in such a way that d ± = 0 mans that th matrial is intact and d ± = 1 indicats th total matrial failur. Th us of a diffrnt variabl for ach sign of th strsss implis that, for xampl, a matrial prviously damagd to traction would rcovr its original bhaviour if it is put undr comprssion, and vic vrsa. Nxt, th quivalnt ffctiv strss norm is dfind. This is a scalar positiv valu that is usd to compar diffrnt strss stats in thr dimnsions, and it is usful to unify load, unload and rload concpts. Th quivalnt norms for tnsil ffctiv strss (τ + ) and for comprssiv ffctiv strss (τ ) hav th form: ( ) 1 2 : : τ ± = σ ± C ± σ ± [U3] (3) + whr th two non-dimnsional fourth ordr mtric tnsors C and C ar idntical and qual to th invrs of tnsor ( D E), bing E th Young s modulus of th matrial.. Starting from th prvious dfinitions, two diffrnt critria of damag can b introducd, namly, g + and g, dfind as ( τ ) g ± ±, r ± = τ ± r ± 0 (4) whr r + and r ar intrnal variabls that control th siz of th damag surfac in th ± ± strsss spac in vry tim stp. Thir initial valus ar r 0 = f, whr f + and f ar th strngth of matrial th strsss at which th matrial fails and damag appars. In this sns, th xplicit dfinition of ths intrnal variabls has th form: ( ) r ± = max r ± 0, max τ ± (5) Th critria statd abov imply that damag volution occurs whn condition (4) is not satisfid. In this cas, r ± is updatd using (5), until damag critrion is satisfid again. Finally, th damag variabls d + and d ar dfind xplicitly as a function of thir rspctiv intrnal variabls r ±. Thy ar monotonic incrasing functions of th form ± ± 0 d r 1. Th post-pak bhaviour is dfind by mans of th tnsil and comprssiv ( )

3 R Clmnt, P Roca and M Crvr 1127 fractur nrgy of th matrial G ± f. This paramtr is normalizd rspct to th charactristic lngth of th finit lmnts, to nsur objctivity rspct to th msh siz. 3 TRACKING ALGORITHM Th damag modl formulatd abov is not abl to simulat individual discrt cracks undr tnsil strsss. Th modl is basd on continuous displacmnts and strain filds, which can not dscrib th discontinuity inhrnt to th crack. For th prsnt application, this limitation is ovrcom by forcing th crack to dvlop along a singl row of finit lmnts by mans of a tracking mthod. Th fractur nrgy normalization rspct to th charactristic lngth nsurs that dissipation will b lmnt-siz indpndnt. Th proposd mthod is applid at vry tim stp during th analysis, just bfor th strss valuation. Th mthod works with a flag systm, whr finit lmnts ar lablld to dlimit th zons whr cracks will appar or dvlop. Th critria usd to dfin ths zons dpnd on th magnitud and dirction of th principal strsss at ach lmnt. Th algorithm has bn implmntd for 2D problms using thr-nodd triangular lmnts. Th procdur is dividd into two stps. First, nw cracks ar dtctd by chcking th strss valus at vry finit lmnt locatd on th boundary of th structur. Thos lmnts whr th tnsil strngth of th matrial is rachd ar lablld as a crack root. Whn svral nighbour lmnts rach th tnsil strngth at th sam tim stp, th initial radius critrion is applid. This radius (dfind by th usr) is th minimum distanc btwn two crack root lmnts, and is usd to guarant th cration of sparatd discrt cracks. At th bginning of th scond stp, xisting cracks from prvious tim stps ar chckd to dtrmin th lmnts locatd at th crack tips. Thn, th following procdur is applid on ach crack tip lmnt and ach nw crack root lmnt: 1. Dtrmin th xit point coordinats. A vctor is drawn from th ntranc point coordinats (dfind blow), using th dirction prpndicular to th principal tnsil dirction of th lmnt. Th xit point is dfind as th intrsction of that vctor with th corrsponding fac of th lmnt (s Fig. 1.a). principal strss lmnt I K J nxt lmnt nxt lmnt K crack dirction crack xit point lmnt J I crack xit point lmnt I I J principal strss lmnt J crack dirction crack ntranc point lmnt I crack ntranc point lmnt J (a) Figur 1 : (a) Exit point of lmnt I (b) Nw potntial lmnt (J) and its ntranc point 2. Dtrmin th nxt (potntial) lmnt on th currnt crack. It would b th nighbouring lmnt whos fac in common with th currnt lmnt corrsponds with th fac whr its xit point is locatd. Th lmnt is markd as a potntial lmnt blonging to this crack. 3. Dtrmin th ntranc point coordinats. Th ntranc point of th nw lmnt on th crack is locatd at th sam coordinats of th xit point of th prvious lmnt (s Fig. 1.b). 4. Rpat from stp 1, taking th nw potntial lmnt as th crack tip lmnt. (b)

4 1128 Structural Analysis of Historical Constructions For ach crack, th prvious procdur is rpatd until on of th following critria is satisfid: Strss thrshold critrion. Elmnt tracking and lablling is stoppd whn th principal tnsil strss is lowr than a thrshold dfind by th usr. Exprinc has dmonstratd that 75% of tnsil strngth usually works wll. Crack mting critrion. Th procdur stops whn a prviously damagd lmnt, or an lmnt markd as a potntially cracking on, is found along th currnt crack. This mans that two cracks hav mt, and from thn on thy will b considrd as a singl on. Boundary critrion. Whn th xit point of an lmnt is on th boundary of th structur, th cracking procss is considrd finishd. Onc any of th prvious critria is rachd, th currnt crack is considrd totally dvlopd and th nxt on is studid by rstarting th cycl. Finally, aftr applying this procdur to all th cracks, ach lmnt will hav on of th thr following labls: - Intact lmnt, not abl to damag (out of potntial crack track; it will kp lastic bhaviour during th currnt tim stp) - Intact lmnt, abl to damag (in a potntial crack track; it will initiat inlastic bhaviour if th matrial strngth is rachd) - Damagd lmnt (blonging to a crack consolidatd in prvious tim stps; it will dvlop inlastic bhaviour during th rst of th calculations) Th analysis procdur rcognizs ths labls and activats th corrsponding constitutiv rspons in ach lmnt for th currnt tim stp. Also, onc th strsss hav bn updatd (and thrfor th damagd indxs ar known), th lmnts with potntial cracking that rally suffr damag ar rlablld as includd in a consolidatd crack for th rst of calculations. Finally, lmnts potntially cracking that do not suffr damag ar rstord to thir original status, unlablld. 4 NUMERICAL ANALYSIS OF MALLORCA CATHEDRAL Th Cathdral of Palma d Mallorca, built during th 13th to th 15th cnturis, is a thr-nav building composd of a cntral nav spanning 19.9 m, 43.9 m high to th vaults kyston, and two collatral nav spanning 8.72 m ach and 29.4 m high. Th cntral nav is sustaind on octagonal pirs with a circumscribd diamtr of 1.6 or 1.7 m and a hight of 22.7 m to th springing of th vaults. Th cntral vaults transfr th latral thrust towards th buttrsss by mans of a doubl battry of flying archs (s Fig. 2). Th main distrsss obsrvd on th building ar larg latral dformation in th pirs and vrtical cracks at thir bas. Mor information on th building and prvious analytical or numrical analyss can b found in Rubió (1912), Mark (1982) and Salas (2002). Figur 2 : 3D and 2D gomtrical modls of a typical bay

5 Robrto Clmnt, Pr Roca and Migul Crvra Numrical modl Th analyss ar carrid out on th typical bay of th building using both 2D (plan strss) and 3D modls (Fig. 2). As statd bfor, th localizd damag modl can only b applid on th 2D modl, its xtnsion to 3D problms bing considrd as a futur dvlopmnt. Th 3D finit lmnt msh is composd of four-nod ttrahdral lmnts and nods. Th 2D finit lmnt msh is composd of thr-nod triangular lmnts and 8824 nods. Two main matrials ar dfind. Th first typ of matrial is considrd to modl th soft limston masonry found in most of th structur: buttrsss, vaults and walls. Flying archs and pirs ar dfind with a scond typ of matrial rprsnting th hardr limston masonry found in pirs, archs and flying archs. Th matrial proprtis ar stimatd basd on xprimnts carrid out on cor spcimns takn from th building. Th paramtrs usd ar shown in Tabl 1. In all cass a larg valu of th matrial fractur nrgy is considrd to simulat th formation of plastic hings with considrabl ductility. Prvious paramtric studis hav provd that this valu has influnc on th collaps load factor, but its study scaps of th rach of th prsnt papr. Tabl 1 : Matrial paramtrs Structural lmnts Young s modulus Comp. strngth Tns. strngth MPa MPa MPa Matrial 1 buttrsss, vaults, walls Matrial 2 flying archs, archs, pirs Analysis of th structur subjctd to gravity loads Analyss undr gravitational loads ar carrid out using both th smard crack and th localizd damag modls. Th formr is prformd on th two and thr-dimnsional modls; th rsults obtaind can b usd to compar th accuracy of th 2D plan strss approximation. It is important to highlight that th rsults from th 2D analysis wr obtaind aftr an intricat calibration procss of th thicknss of th diffrnt structural Thlmnts. Th localizd damag analysis lattr is prformd only on th 2D modl. Thr diffrnt initial radii ar dfind: 1 m, 2 m and 3 m. In all analyss, gravity is incrasd until a total collaps of th structur is rachd. Th damag distribution and th collaps mchanisms obtaind for vry cas studid ar showd in Figs All cass show similar failur mods. Failur is controlld by damag on th buttrss, at th bottom lft sid of th window. Also, a dislocation occurs in th main vault, and two hings ar clarly formd on ach flying arch. It is obsrvd that th 2D approximation is accptabl du to its similaritis with th original 3D modl. 2D 3D Figur 3 : Collaps mchanism and distribution of th tnsil damag indx undr gravity loads with th smard crack modl

6 1130 Structural Analysis of Historical Constructions With rgard to th rsults of th localizd damag modl (Fig. 4), it can b obsrvd that varying th initial radius dos nithr influnc th collaps mchanism, nor th failur load. Howvr, it is intrsting to highlight that th gnration of localizd cracks, acting as plastic hings, rprsnsrprsnts mor ralistically th bhaviour of th structur in th ultimat condition. Fig. 5 shows th comparison of th curvs rlating th load factor with th horizontal displacmnt on top of pirs throughout th loading obtaind for th diffrnt analyss carrid out. r = 3 m r = 2 m r = 1 m Figur 4 : Collaps mchanism and distribution of th tnsil damag indx undr gravity loads with th localizd damag modl and diffrnt initial radius (r) valus Figur 5 : Collaps load multiplir undr gravity loads (SC: smard crack modl; LD: localizd damag modl; r: initial radius) Fig. 5 shows th comparison of th curvs rlating th load factor with th horizontal displacmnt on top of pirs throughout th loading obtaind for th diffrnt analyss carrid out. A collaps gravity multiplir load of 2.0 and 2.05 is obtaind for th 2D and th 3D analyss rspctivly. In th localizd damag analysis, th failur load factor is 2.15[U12]. Thus, th computd ultimat load is clos in all cass. 4.3 Snsitivity analysis Givn th difficulty found in th xprimntal dtrmination of th avrag tnsil strngth of masonry, numrical analysis normally adopt mpirical valus basd on th xprinc or on vry gnral corrlations. In th prsnt study, th tnsil strngth f + is takn as 5% of th comprssiv strngth. A paramtric study is carrid out to study th influnc of this paramtr

7 Robrto Clmnt, Pr Roca and Migul Crvra 1131 on th ultimat capacity of th structur. For that purpos, both th 2D localizd damag modl with r = 3 m and th 2D smard crack modl ar analyzd with four diffrnt tnsil strngth valus ( f +, f + 2, f + 4, f + 10 ). Fig. 6 shows th curvs rlating th collaps load factor with th tnsil strngth valu usd. Th smard crack modl shows a significant snsitivity to th valu adoptd for th tnsil strngth, showing a variation of th ultimat load of about 30% with rspct to th valu calculatd for th initial valu f +. Th variation found with th localizd damag modl is only 5%. Figur 6 : Variation of th collaps gravity load multiplir in function of th tnsil strngth f Analysis of th structur subjctd to sismic loads Th sismic prformanc of th building is assssd by mans of a push-ovr analysis consisting of th gradual application of a systm of latral quivalnt static forcs on th structur. Th analysis is dividd into two stps. Th gravity load is applid in th first stp. In th scond stp, th latral sismic forcs ar applid and incrasd gradually until raching failur. In Fig. 7 th sismic load factor (dfind as th ratio btwn th latral unit forc and gravity) is plottd against th horizontal displacmnt on th top of pirs. Th smard crack modl causs failur at a load factor of 0.1. This valu is th sam for both modls (2D and 3D). Th localizd cracking modl producs a highr failur load factor as should b xpctd du to th rstrictions that th modl imposs to th propagation of damag. In that cas, th building fails at load multiplirs 20% highr for r = 1 m and r = 2 m. Th diffrnc is 40% (load factor 0.14) for an initial radius of r = 3 m. Figur 7 : Collaps load multiplir undr sismic loads (SC: smard crack modl; LD: localizd damag modl; r: initial radius)

8 1132 Structural Analysis of Historical Constructions Fig. 8 dpicts th dformations and damag distribution for th smard crack 2D modl and localizd damag modls (r = 1 m, r = 2 m and r = 3 m). As xpctd, th modl with highr initial radius shows a lssr numbr of cracks. Togthr with th larg incrassincrass producd in th load multiplir, this fact illustrats that th us of larg xclusion radii is not suitabl and may lad to an ovrstimation of th strngth of th structur. Convrsly, th load multiplir and th damag distribution convrgd for xclusion ratios btwn 1-2 m, showing that th rsult is not dpndnt on th valu of th xclusion ration providd that this is sufficintly small. SC LD r = 1 m LD r = 2 m LD r = 3 m Figur 8 : Collaps mchanism and distribution of th tnsil damag indx undr sismic loads for 2D modls (SC: smard crack modl; LD: localizd damag modl; r: initial radius) 5 CONCLUSIONS Th crack tracking modl nabls th simulation of mor ralistic damag distributions than th original smard-crack modl. Th localizd cracks prdictd by th crack tracking modl rproduc consistntly a st of xpctabl plastic hings dvloping gradually in th structur and lading to th full collapsing mchanism. Morovr, a smallr dpndncy with som matrial paramtrs (spcially with th tnsil strngth) has bn found. In turn, th smard-crack modl dscribs damag in a widly distributd, unralistic way. Th tracking modl rprsnts a mor suitabl mthod to prdict th structural bhaviour of masonry buildings without rquiring any significant additional computation cost. Furthr dvlopmnts ar now bing into considration. In particular, it is intndd to combin th crack tracking approach with an anisotropic constitutiv quation for masonry. Th rsulting modl would provid improvd ralism. In particular, it would nabl th modlling of th influnc of th actual orintation of th mortar bd joins on th rspons of th structur. Also, th xtnsion of th tracking algorithm to thr dimnsions is bing considratd.

9 Robrto Clmnt, Pr Roca and Migul Crvra 1133 REFERENCES Crvra, M Viscolasticity and rat-dpndnt continuum damag modls. Monograph M79. Barclona: CIMNE. Mark, R Exprimnts in gothic structur. Cambridg: MIT Prss Rubió i Bllvr, J Confrnc about th organical, mchanical and constructiv concpts of th Mallorca Cathdral (in Spanish). Anuario d la Asociación d Arquitctos d Cataluña. Barclona. Salas, A Structural study of a tipicaltypical bay of Mallorca Cathdral (in Spanish). Dgr dissrtation. Dpt. of Construction Enginring. Barclona: Univrsitat Politècnica d Catalunya

10 1134 Structural Analysis of Historical Constructions.