Analysis of the structure of Gothic cathedrals: application to Barcelona cathedral

STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTI ONS 11 P. Roca , J. L. Gonúlez, E. Ona e y B. Lo u enço (Eds.) © CIMNE. Ba celona 1998 ANAL YSIS OF THE STRUCTURE OF GOTHIC CATHEDRALS APPLlCATION TO BARCELONA CATHEDRAL Pe e Roca, LaTa Pe ll eg ini, Eu genio Onn e nnd Alex Hanganu* U nil 'e s ila/ Po lil ec lli C {j de Cow/! lIl ya ·Inle na io lol Ceme o Nwne ical Me hods in Ell ginu ing (CIMNE) G a ll Capi à S/ li 08034 Ba c IO 1(J. Spai ll SU MMARY The s lldy a ancien s uc u e s composed a Slane a ches and aul s equi es h e use o speci ic echniques a analysis endowed wilh powe ul p ocedu cs a lhe madelling o geomel y, he simula ion o lh e mechanical espanse a lh e a iaus ma e iais (including ashla blacks and ma sa n y. backings and ills) and h e s im ula ion a lhe possible agenlS (g a i y, Vind , ea hquake, se lemems) w hi ch may a ec lhe s uc u e o may ha e a ec ed i h augh hi sla y. Ma e speci ically, Ih ose e chniques mu sl be able 10 epToduce lhe aclu al condi ions a h usl equilib ium be ween lhe la ge numbe o cu ed. linea o wo-dimensional membe s in ol ed. The alidi y o se e al modelling echniques wi h a e y di e enl le ei a sa phis ica ion is sludied Ih ough Ihei use in lhe s udy a a pa icula Galhic cans uc ion: Ba celona Ca hed al. The a ious lechniques used we e de eloped a lh e School a Ci il Enginee ing a lhe Tech ni cal Un i e si y o Ca alonia and ha e al eady been used in he analysis a a se! a hislo ical cons uc ions, including lhe Basi li ca o San Ma co (Onale e ai ., 19 97) and lhe C yp o lhe Colõnia Güell (Roca, 1997 ). The p ese n s udy o Ba celona Ca hed al, now unde de elop nen , is Ihus no only ai med a gaining a beue unde s anding o a ious as pec s o lhe Tesis ance o Go hic cons uc ions (and he sou hem a nd Ca alan Go hi c, in pa icula ) bu also a app aising lhe possibili ies and limi a ions o a ious analy ical echniques. 232 STRUCTURAL ANALYSIS OF HI STO RI CAL CONSTRUCTI ONS 11 1 SOME CONSIDERAT IONS REGARDING GOTHIC STRUCTURE 1.1 O e all s uc u al dcsign The esis ance c Go hic cons uc i on ga e ise o a cc ain amoun c con o e sy a e Viollcl-le-Duc's i s se ious a emp lo p opose a alienal ision c i s mechanical p incipies and pa icula s uc u aI o de ing. Hi~ cnlhusias ic in e p e a i on o GO lhic s uc u e as a ully alianal objec was la e con adic ed by olhc esca che s (see B. Bassegoda, 1974). Ra he lhan de ending Violle 's poin s a iew. hey ound ins ances o s uc u al a bi a iness and unce ain ies which we e only o bc unde s ood as composi ional mo i s, cons uc ion de ices, C pu ely a consequence c he limi ed knowl edge c in uilion c lhe his o ical dcsigne s. Violle 's heo y p esen ed Go hic s uc u al o de ing as a sys em sl iclly based on equilib ium and jus i ied by i . Fu he mo e, ali de ices o Go hic S l u clu e we e laid Oul so as lo co n ibu e o he achie emen o an o e all duc ili y. In hi s con e x , du clili y musI be unde slOod as he eapacily o lhe sl uclu e o accepl signi icanl de o ma ion wi hou sus ai ning damage o ailing. I is his duc ili y which allowed he Go hi c s uc u al componen s (pie s, a ches, au l s) o al ain hci imp essi e d im ensions and hei slende ness. Mo e ecenlly, olhe esea che s ha e ound nolable s uc u al weaknesses in Go hi c eons uc ions. Ap opos o lhe sludy o he Ca hed al o Sla. Ma ía, in Vi o ia, Spain, C oci el a I. (1 995) obse ed weak poin s such as (I) a e y Iimi ed in e connec ion be lween lhe s uclu al elemenls, p oducing a lype o isos a ic consl uc ion; (2) lhe excessi e slende ness o pie s and ela ed possible inOuence o second·o de e ec s in lhe equiJib ium, (3) lhe dc o mabili y o bu esses a nd ly in g a ches, (4) he eccen ici y c ea ed o e he pie by bu esses a ached lo pie ex ensions which usualJy do nol sland di ec 1y o e lhe pie , (5) lhe weakcning e eel causcd on suppo elemenls by lhe i o ium and possible open in gs, wi h special ega d lo hei shea e sis ance, and (6) lhe acl Ihal ib sp ingings in lhe aisle and a cade a ches may sl ongly educe lhe local esis an! sec ions o lhe uppe pie exlens ion s. The combina ion o !hese aspec s a e lhe cause o majo de onna ions and dangc ous oul· o -plumbness. Aeco ding o lhe esea che s sma l! design o cons uc ion impc eclions may lead o high de o mabili y and c acking. Fu he mo e, equen damagc pa c ns ha e been idenli ied which in some cases a e conside ed "ch onic", such as lh e so·called Sabou eL c acks lypica ll y obse ed in la ge g oined auh, (Heyman, 1983, Ba hel, 1993). The s udy o se e a1 majo GOlhic cons uc ions, including lhe case o lhe Basilica o Sl. F ancis o Assisi a e he Seplembe 1997 ea hquake (C oci, 1998 ), al50 showed he weakness o Go hic sl uc u es o special agen s such as ea hquake, inlense wind and la ge ounda i on sellle nenls. 1.2 The ole o lhe s uc u al elemen s The clea and uni ocal ole ha Viollel·de-Duc assigned lO lhe a ious sL uclu al elemenls (diagonal, l ans e se and c1e es o y a ches in aul ing, bu esses and lying P. ROCA e aI. I Analysis o Ba celona Ca hed al 233 bu esses, e c.) and lh e e y assumplion lhal a li hese elemenls a e esse n i al in lh e Go hic g amma , a e also oday dis ega ded by some specialisls. Some new unde s anding ca ne om lhe obse a ion o lhe s ales o esidual equilib ium a ained by Go hic cons uc ions a e pa ial desl u clion caused by i e, bomb exp losions (du in g Wo ld Wa 11 ) o simp ly lack o main enance. Rema kably, some aul s main ained a pe ec equi li b i um a e los in g h e ans e se o diagonal ib s. O he cons uc ions los some bu esses o lying bu l esses wi hou immedia e damage be in g obse ed in lhe sys em o aul s ( 8. Bassegoda, 1974 ). Wi h ega d o lhe ole o lhe eleme nl s in a c oss aul , some s udies e ealed ha he diagonal, longi udinal and ans e se elemen s con ibu e li le, i any hin g, 10 es is ance. The s abili y o lhe aul is due mainly o he capaci y o h e webbing o he au l s o wo k as a s heH wi h double cu a u e (Ma k, 1982, 1993). I is in e es ing o no e ha his con adic s some classical opinions wh ich assigned lO lh e diagonal a ches he main s uc u al ole and elegaled he webbing lO li lle mo e h an a simple cJosu e. The ole o he iH placed o e he a ul s is also subjecl o opinion a nd p obably has e y di e en unc ions depending on i s o ig in and ma e ial composilion. The backing -placed aI lhe haunches o lhe aul s - is o u mOs l impo ance o Slabi li se lhe au l s subjec o g a ily load. Addi ional iU placed o c lhe aul migh also p oduce a ma ginal s ab ili si ng e ee when lhe aul is only subjee o g a i y l oa d s. Howc e , such a iU may ha e e y di e en e ec s in lhe case o agen s o he Ihan g a i y. The s udies ca i ed ou on he Basilica o SI. F ancis o Assisi a e lhe ea hquakc showed he non-cohesi e illing exis ing o e lhe au l s, which was nol o iginal1y placed he e on pu pose bu me ely accumula ed o e lhe decades and had a e y un a ou able e ec du ing he occu ence o lh e ea hquake leading o he collapse o some o he au l s (C oei, 198 8). The iU w hi ch exis s in some sou hem Go hic eons uc ions (such as in Sl a. Ma ia deI Ma in Ba ce lona and Ba celona Ca hed al i sel ) consis s o a cohesi e mass o lime mo a and adi ional ce amic essels and is hus cha acle ised by conside able s eng h co mbined wi h e y li mi ed weigh (1. Bassegoda, 19 83). Supposedl y, I hi s kind o co nc e e iH makes a signi ican con ibu i on o esis ance in lhe cases men ioned, whe e a aul webbing much hinne han in mosl Go hic eons ue ions has also been obse ed. 2 TECHNIQUES APPLlED TO THE ANALYSIS OF GOTHIC STRUCTURE 2.1 Di ieul ies o he analysis Because o lh e many di icul ies which a e encounle ed in lhe sludy o GOlhic cons uc ions, ew seien i ic s udies ha e been pe onned o da e wi h a iew o gaining a deepe unde s anding o hei mechanical and esis ance p openies. The main di ieul ies a e posed by h e o llowing aspec s: Because o he agile na u e o s one mason y, analy ical s udies based on he hypo hesis o linea elas ie i y a e o e y Iimi ed a li di y. In ael, wi hou a g ea deal 234 STRUCTURAL ANALYSIS OF HI STORICAL CONSTRUCTIONS 11 a (unhe in e p e a ian by lhe analys . hey may only in o m abou lhe se icc beha iou p o ided lha! s esses keep wí hin mode a e alues. The knowledge a bo h lhe in e nai composi ion a lhe s uc u al membe s and lhe mechanical p opc ies a lhe exis ing ma e iais a e no mally e y limi ed, o se e al eascns. Fu he mo e, bOlh lhe composi ion and lhe ma e ial p ope ies may show conside able sca e ing h oughou lh e en i e cons uc ion. The mcchanical beha iou is deeply dependen on lhe his o ical cons uclion p oces s, which is mos ly unce ain, Addi ional unce ain y is p o ided by conlingencies 5uch as acciden s and pa ial ebuilding, du ing cons uc ion a immedia e cmedial mea su e s o cons ain p eeocious de o ma ion. Fina ll y, as we a e eminded by Cassinello (1998), hose co ns uc ions whieh ha e eached lhe p esen day ha e usually undc gone majo ans onna ions as a consequence o p og essi e O pennanen de onna ion, e acking, physieal o ehemieal deg ada ion o ma e iais, o possible eons uc ion al e a ions o eplacemen s. Again acco ding o Cassinello, he elabo a ion o an accu a e model shou ld hus ake in o accoun he h ee main aspeClS de ennining he s uc u al beha iou , namely he (de onned) geome y, in e nai eomposi ion. and ma e ial p ope ies. The in eg a ion o lhe his o ieal ans o ma ions should be ex ended o hese h ee aspecls. 2.2 Signi ican s udies and de elopmen s A widesp ead se o echniques has been used in he s udy o GOlhic cons uc ion, including pho oelas ic modelling, classical heo ies bascd on plas ic heo em s, con en ional ma ix calcula ion and lhe ini e elemen melhod. Pho oelas ic modellillg The esea ch ca ied ou by Ma k (1982) by means o pho oelaslic models, eXlended o a la ge numbe o majo cases such as he calhed als o Amiens, Beau ais. Bou ges, Cha es and Palma de Mallo ca, cons i u es one o lhe i s and mo e success ul a emp s o quan i a i ely analyse lhe sl uc u e o hese cons uc ions lo gain a deep unde s anding o lh ei pe onnance when subjecl o a ious agen s (wind in pa icula ). Thc s udy enables us o en isage he essen ial cha ac e is ics o lhe buildings, compa e lhei di e en design, and be e unde s and lhe ole o hei di e en s uc u al elemenls. l also p o ides an unde s anding o he o igin o some o lhe exisling damage, such as ce ain c acks and pennanen de o malions. Plas ic analysis The possibili ies o classical plas ic analysis in lhe s udy o Go hic ca hed als, based on he plas ic limi heo ems, ha e been shown by Heyman (1998). Due o lhe alidi y o lhe hypo hesis assumed by classic heo y. e y ealis ic collapsing mechanisms may be de ised h ough i s ca e ul applica ion. Howe e , classical analysis is o li le use i applied o condi ions o he han o a emo ed em ailu e. Thus, classical heo ies P. RO CA cl alo I A nal ysis o Ba celona Calhcd al 235 a e no e y use ul o in e p e lhe cause and ex en o c acks, de o ma ion o o he damage no di ec ly ela ed o he gene a ion a a co llapse. Fu he no e, hei p ac ical bu igo ous use beco nc s ex emely di icul in he case o complex s uc u es wi h mui iple elemen s (as encoun e ed by se e al au ho s, incJuding Cau in and S agni o, 1998 ). Acco ding lO Chassagnou e aI. (1998), lhe e alua ions ob ained using classical heo ies end o be oo pessimis and may lead o epai s hal a e mo e a - eaching han ac ually needed. This is so because o lhe ype o addi ional hypo hesis ha mus be assumed o make he use o hese Iheo ies easible. CO l e l ional ma ix calc ula io l Using con en ional ma ix o mula ion, Leon a nd Cassa i (1997. 1998) dc eloped a delailed pa amel ical s udy o he main geome i cal and ma e ial a iables in luencing lhe s uc u al esponse o Le 6n Ca hed al in Spain. Thc s udy was based on a ca e ul modclling o lhe main ans e se sec ion a lhe building and he na e au l s by means o an equi alen sys em a one-dimensianal elcmen s. Fini e elemen me h od The main limi a ion o mo s ini e elemen onnula ions, whcn applied o mason y elemcn s, lies in he h YPo lhesis o he con iouum, which makes il di icul o c cn impossible o simulale h e pa icula kinema ics o he mason y modes o ailu c. Elaslic analyses based 00 he ini e elcmenl me hod a e only sui ab le o cha acle ising wo king condilions subjecl o s a es o e y modc ale s esses in which comp essi e s esses a e la gely p edominan . Howe e . he adoplion o su i able cons i uli e equa ions which accoun o he main phenomena eJa ed o he ailu c o he ma e iais (c acking unde ension, c u shing unde comp ession, e c.) makes il po ssible o ep oduce mo e ad anced slages o lhe esponse and e en 10 simula e ailu e mechanisms simila 10 hose p edic ed by lhe classical Ih co ies. C oci e aI. (1995) ca ied oul a ini e elemenl analysis o lhe Ca hcd al o S a. Ma í a, in Vi o ia, Spain. The analysis, applied o he main ans e se sec ions o lh e building and o lhe na e aul s, ollowed an inc emen aI s a egy o accoun o c acking du e o ension o shea s esses, as wcll as he equilib ium second-o de e cc s. This analysis showed some o he main weaknesses o lhe building, as has al eady bcen men ioncd in Sec ion 2.1. Simila analyses we e also used o lhe s lUd y o h e co ll apse o Bcau ais Ca h ed al (C oci e al., 1998b) and he e ecIs o h e ea hquake o Sep embe 1997 on lhe Basilica o Assisi (C oci, 1998 ). The ini e elemen me hod, also combined wi h sui able cons i u i e equa ioos, ha s been success ully used o s udy Go hic c oss aul s (Ba hcl, 1993 ). Fo his pu posc, e y de ailed inile elemen models we e elabo a cd and used in combinalion wi h a ma e ial ea men enabling lhe simula ion o c acking as well as sliding be ween a ch ing join s. 236 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCT IO NS 11 Ca u in and S agn i o (1993, 19 95) ca ied oul e y in e es ing s lU dies using bo h c1assical plas ic analysis and nan-linea analysis by lhe ini c ele nc n melhad. Thei me hod was success ully applied o analyse lhe cen al na e a Reims Ca hed al. The in i o cleme n me hod has beco also u i li sed o s udy se e al Spanish Go hic ca hed als (Bu go s, Se ille) by Izquie do (1997; see al50 Cassinell o, 1998). Cha ssagnou e l aI. (1998) ha e de eloped a echnique o lhe gene ic modelling a Go hic c oss aul s, al50 based on lhe ini e elemen melhad. wi h lh e a im a making a ailable a 1001 o lhe p ac i cal e alua ion a h ei s uc u al esponsc. Ona e cl ai ( 199 7) ha e de eloped a ini e elemen damage model w hi ch was success ully used o he anal ys is o SL Ma ks Basilica in Venice. This model, b ie l y discussed in a la e sec i on, is one o lhe echniqucs chosen o he nume ical s udies o lhe Ca hed al o Ba celona p esen ed in his pape . 3 BARCELONA CAT H EDRAL 3.1 ln oduc ion o lhe s udy The choice o Ba celona Ca hed al (Figs. 1-6) o lhe pu pose o h e p esen s udy is pani a ll y due lo he acl ha abundan in o ma ion exisls on i s geome y and olhe cons uc ion aspcc s, made a a il able lo lhe au ho by lhe Diocesan A c hi e and lhe Chap e o Ba celona Ca hed al. Funhe mo e, h e e a e some publica ions wi h in e es ing his o ical and cons u c ion in o ma ion (Elías, 1926, 1. Bassegoda, 1968, 19 81, 1983). Ba celona Ca hed al also shows ce ain s uc u al singula ilies which ga e i added in e es as a subjec o s udy. • Fi s , i is a cons uc ion in lhe so -c alled Ca alan Go hic s yle, wh ic h exhibi s no able spec i i c ea u e s. Ca alan Go hic cons uc ions we e ypically designed as e y diaphanous spaces wi h la ge-spanned na es. • The dimensions o lhe aul s a e ema kable due o lhe la ge gap be ween lh e pie s nOI only in Ihei ans e sal di ec ion, bu also 10ngi udinal1 y. • The lay-ou o lhe l ans e se seclion o lhe building is cha ac le i sed by h e ai sle na es being a lm os (buI nOI quile) as high as he cen al na e. Because o lhis sl uclu a ll ay-ou , lhe aisle aul s h;we an impo an! "I nclu al unc ion in elain in g lhe Ih uSl o lhe cenl al aul s. In a celeb aled leclu e on Mallo ca Ca hed al, Rubió i Bell c (1912) dis in guished belween lh ee di e en ypes o Go hic calhed als wi h ega d lo lhe s uc u al o ganisa ion o lhei ans e se sec ion. In lhe i s ype, lhe na e and lhe ai sles a e buil o lhe same heighl (as in Sa agossa, Munich, Pe ugia and o he s). The second ype, in w hi ch lhe aisles a e almos bu nol qui e as high as h e na e, is by a lhe leas widesp ead since il only inc1udes lhe cases o Ba celona Ca hed aJ and S a . Ma ía dei Ma . The hi d ypc, in which lhe aisles a e signi ican ly lowe lhan h e na e, is by a lhe mos equen (being lhe case in Amiens, Reims. Beau ais, Cologne, Milan, Toledo, Ma ll o ca and many o he s). P. ROCA el a!. I Anal ys is o( Ba celona Ca lh ed al 237 ~''' IDlln''U:UU~iIIl l 1!U n. :~ II I!i: 'jI(U'UVl ; SllIl I:U1I1UJ 11U! Fig. I Longi udinal and ans e se sec ions o Ba celona Calhed al ( acsimile ep oduclion o plans d awn in 1864 as pa o lhe sludies p e ious o lhe cons uc ion o lhe new cimbo io). 238 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTlONS 11 . , Fig. 2 Plan o he building (1864 plans). Ui'l1i8DR1Hií I~R8 h ni'l PI."I. '",.,-.! '" I • .yp."AI. : ~~"-::' "' .'_ n. . ......................... . ~ , """"J . .., ,,,,.. : !:''' ,.... .... ,. ' ..... ; :::. ~ : Z':,; ..2 ; 'Y({E-'A~ " ~ .. . n .... :: ~::!!:.. ' n .... ~ ...... .. - ...... " ...... ....... ..... " J . ... ..,..., .. ... .,,,"'''''-- ...... ,,,,,, J'" U. J •• 1 ....... J .. .,-" ... ~ a...i ~ : :':'hL:."·- .... ~ .. ">:,, .. - " . 1" .... __ .. " ". '''-- .. J .. "...,.... ..... ",,"",-- " " ... _ .... .. . ,. ... - :~~7", .. "''''' ............ ................... , ... "J J "" .... " J -'. J ""'-, '-':: :"T,C" . ~.~. P. ROCA el alo I Anal ys is o Ba celona Ca hed al 239 Following Rubió. in lh e case o consl uclions belonging o he second ype lhe h us o he na e aul s is comple ely e ained by lhe ai sle aul s so hal no lying a ches a e needed ai ali. Whe e hey exisl. as in Ba celo na Ca lh ed al, Ihey a e non-s uc u al and Ih e i only unc ion is p obably d ainage. Th e p ominenl ole o lhe ai s le aul s as s abi li s in g eleme n s may be elaled 10 Iw o cons uclion de ices w hi ch a e consis en wi h hem and which we e p obably in co p o a ed in o de 10 sl englhen Ihem. In pa icula , lh e aislc aul s a e su cha ged w i h an amou n o co nc ClC ill g ea e Ih an ha placed o e lhe na e, as i o inc ease hei coun e ac ing h us l. ~ I O,5m Fi g. 3 T ans e se sec ion o lh e na e pie s a nd lh e pie s susl ai ning Lh e c im bo io as ep esenled in lhe 1864 plans (E 1:40 ). 3.2 B ic desc ip ion o he building Cons u c i on o lhe na es o Ba celona Ca hed al was begun in 129 8 and las ed o mo e lhan a ce n u y. As usual. lhe apse was cons uc ed i sl, being inished in 1327, while lhe cons u c li on o lhe en i e na e conlinued un il 141 7. In 1422 wo k s opped, lea ing lhe cimbo io un inished and a p o i sio nal wall cJo su e as a açade. Mosl o lhe o iginal ea u es o he building a e assumed 10 be all ibulable o he mas e builde Jaume Fab e, who wo ked on h e Calhed al om 1317 . The plan o lhe emple shows a e y uncon en ional lay-o ul in which he cimbo io is nOI e ec ed o e lhe c ossing bu o e lh e i s b ay o lhe na e elose o lhe açad e. while he c ossing is delimi ed by lhe wo majes i c elock owe s, mo e commo nl y pa o he açade. A simila dis ibulion can be obse ed in e y ew case s, such as lhe Ge man ca hed als o Ulm and F ibu g. The building in cludes h ee na es ( lh e na e and wo aisles) al hough, as a consequence a i s pa icula design, i appea s o enclose wo addilional aisles. This 246 STRUCTURAL ANALYS IS OF HI STORICAL CONSTRUCTIONS 11 Fig 10. Dc o mcd shape c a aul! quad an mode!!ed as a amed sys em, subjecI o dead loading. Fig. 11 Fo ce ajcc o ies in a GO lhic aul subjecl o dead loading (Ma k. 1982). Allalysis o lhe aul usi /g 3D solid /inile elemen s Thc FEM modeJling a lhe au l accoun s o lhe ib sys em, lhe webbing, lhe nlbble backing a1 lhe aul haunches, he conc e e iH and lhe la e al walls, ali de ined by hei co esponding mechanical p ope ies. Gi en lhe li mi ed in o ma ion a ailable on he ma e ial p ope ies, some assump ions had o be made in a de o assign mechanical pa ame e s o he a ious ma e i aIs (Table 1). The g ea es sou ce a unce ain y was he cone e c ill, composed a ce amic po e y and li me mo a , he p ope ies o which had o be oughly es ima ed. The ashla mason y and ubble we e es ima ed, on he basis o empi i ca l c i e ia, om he a e age comp essi e s eng h (74.2 MPa) o he Mon jui" c sands one used o he cons uc ion o he building. P. ROCA e aI. I Analysis o Ba celona Ca hed al 247 ig. 12 FEM model and dis ibu ion o s esses caused by o al dead load 00 lhe na e aul (maio comp essi e s esses in Pa). Fig. 12 shows he dis ibu ion o s esses which is ob aioed o he aul subjecI o o al dead load. Some o he ohse a ions allowed by lhe simpli ied ame model can be con i med, sue h as lhe signi ican ole o lhe webbing in ans e ing he o ces 10 he pie s, and lhe majo con ibu ion o lhe ans e se a ches bu small o null con ibu ion o he cle es o y ibs and diagonal in e sec ions. 248 STRUCTURAL ANALYSIS OF HI STOR ICAL CONS TRUCTI ONS 11 Table I. Mechani ca l p ope ies de ined o lhe a ious ma e iais Ma e i al D e o ma ion Poisson Comp esi e Tensile Dens i y modulus coe icien s eng h sl engh (kN/m3) ( Mp' ) (kP.) (kP a) Ashla 8000 0 .2 8000 400 2,7 mll so n y (*) Rubble (**) 2500 0.2 2500 100 2,5 Conc c c i 11 500 0.2 500 50 1,0 (*) in pi e s, ibs and aul wcbbing (**) a lhe co e o pie s and aI aul haunches The ana lysis p ediclS an un calis ica l1 y high ul ima e capacily (o e lh ee imes lhe o al dead load in g) b eca use lhe la e al displace ncn s c lhe sp ingings a e de ined as u ll y cons ain ed in lhe indi idual ca men! o lhe aul . Thus. lhe simula ed ul ima e m ec hani sm does nOI aceoun o lhe lexibili y and li mi ed capaci y c lhe e aining sys em p o ided by lhe aisle auhing and bu esses. Mo e ealis i c esu l s a e ob ai ned when, as expla in ed in Sec ion 4.3. lhe aul is in se ed inlO a mo e comple e model also including ali s uc u al elemen s in he I ans e se sec ion. 4.3 S udy o a na e bay S udy using Gene alised Ma ix Fo mula ion The pu pose o Ihis sludy was 10 analyse lhe possibili y o s udying lh e equilib ium o Go hic s uc u e up o lhe ul ima e condi ion by means o a simple model. In spi e o his simplici y, he model was elabo a ed o oblain an accu a e desc iplion o lhe geome y o lhe na e sec ion, lhe dis ibu ion o lhe dead load and lhe esponse o lhe ma e i ai s. Using he acul ies o lhe me hod, he a ious s uc u al elemenls we e modelled as s aigh o cu ed lin ea elemen s wi h a iable and co mpo si e c oss-sec ion (Fig s. 13- 14 ). l sho uld be no ed Ihal, in a e y simpli i ed way, he aul s a e also ea ed as a cu ed compos ile beam whe e each ma e ial (maso n y. ubble, conc e e i l!) is de ined by i s pa icula mechanical p opc ies. The analysis a cco un s o lhe e ccls o mas on y c acking unde ension o c ushing und e com p ession as desc ibcd in Sec ion 4.1. By g adual1y applying he g a i y load up o and beyond h e dead load alue we we e able o isualise lhe wo king condi ion o he sl uc u e, lhe p og essi e d e elopmen o c a c king and lhe inal appea ance o a numbe o c ushed zones leading o ailu e (Figs. 14-16). Th e applicd load was kep p opo ional o lhe dead load h oughou lhe p oce sso Failu e wa s eached o a o al applied g a i y load equa l o wi ce he (O a l dead load. As ca n be se cn in lhe igu es men ioned abo e, lhe sec ions a ec ed by deep c acking o c ushing de e lop a highly p onounced cu a u e and hus may be iden i ied wi h P. ROCA c aI. I Analysis o Ba celona Ca hed al 249 hinges, which would lead o an ul ima e mechanism acco ding o lhe plas ic heo y. The se o hinges leading o he ul ima e mechanism appea ed in he na e aul . lhe uppe egion o he pie sha s and he aisle au l s. In bo h lhe na e and aisle a ul s he hinges de eloped a hei c own. Howe e , he ob ained mechanism is nOI iden ical lo he heo e ical one Ihal would be ob ained by applying lhe plas ic Iheo y. because o lhe conl ibu ion o lhe gene al de o ma ion o lhe elemenls lO he ins abili y. Owing o lhis con ibu ion, lhe ailu e mechanism becomes poss ibl e e en i some hinges a e no ully de eloped, as a lhe na e aull c own, o no de eloped aI ali, as occu s in he base o lhe pie s whe e lhe mobili y needed 10 de elop lh e mechanism is gi en simply by he g ea nexibili y o he sha s. A ull lheo c ic mechanism would equi e a se o se en hinges. Fig. 13 Equi alenl ame model. Du ing lhe p ocess, lhe a e age comp essi e s ess a! he base o lhe pic s a ies om a alue o 3.5 MPa, o dead loading applied, o a alue dose o he comp essi e s eng h (8 MPa). The esponse is almos linea up o ailu e, wi h only e y sligh losses o s i ness Ih oughou lhe de elopmen o he hinges and a inal, agile ailu c. I should be menlioned ha his agile esponse may esul as a limi a ion o lhe nume ical p ocedu e o con inue lhe analysis beyond some local se e e damage. The pa ame ic s udies showed lha lhe ul ima e load is nOI dependen upon lhe comp essi e s eng h when alues hi ghe lhan 8 MPa a e conside ed, which means lha lhe ailu e is cons an ly caused by lhe gene al ul imale mechanism. This is wha should be expec ed om lhe dassical plas ic analysis. In con as!, lhe ul ima e load dec eases in p opo ion o he comp essi e s eng h when alues o less Ihan 8 MPa a e de ined because he ailu e is hen dele mined by he c ushing o lhe mason y ai lhe base o he pie s. 250 STRUc URA L ANALYSIS OF HI STORICAL CONSTRUCTI ONS " As demon s alcd by lhe pa amcl ic s udies, lhe gene a l ca pacily wus nOI signi ican ly dependem on lhe s i ness o he conc ele i!! ( o a lu cs a lhe modulus o de o ma ion a yi ng be lwecn 50 o 1000 MPa). Simi la ly. a modc a c a ia ion o i s a e age densi y h ad almosl no c eel upon lhe ul ima c load. , "" Fig. 14 Dis ibu i on o axia l s esscs and c acking o lhe s nlc u e subjccl o dead load. Fig. 15 Dis ibu ion c axial s esses and c acking aI ailu c occu ing o a dcad load ac o ised by 2. P. ROCA Cl aI. I Analysis o Ba celona Ca hed ai 251 Fig. 16 De o med shape close o ailu c. This mode! was also used o simulale lhe e ecl a an ea hquake by applying a se o ho izon al equi alen s a ic loads consis enl wi h he disl ibu ion 011 mass. The 11011- linea analysis p edicled lhe ailu e o a seismic coe icien alg eqllal o 0.12, which can bc conside ed sa e enough wi h cga d o lhe seismic illlensÍly o lhe egion o Ba celona. The applica ion o lhe Spanish seismic code would lead lO lhe conside a ioll o a coe icien equal o 0.086 o a mason y building in a simila condi ion and o a i sk pe iod o one hund ed yea s. As showll in Fig. 17, he dis ibu ion o s esses is nOI signi ican ly al e cd wi h espec o lhe dead load condi ion when a mo e mode ale ea hquake is simula ed. The co esponding de o med shape is shown in Fig. 18. ; . ... Fig. 17 Dis ibu ion o s esses caused by a se o s a ic equi alen o ces simula ing an ea hquake cha ac e ised by a coe icien alg equal o 0.086. 252 STRUc URAL ANALYSIS DF HISTDRICAL CDNSTRUCTlDNS 11 Fig. 18 Geome y de o med byea hquake. Ob iously, lhe li ni ed a li di y D lhe l ca men o lhe ea hquake as an equi alen! se D s a ic o ces should be a ken in o conside a ion and lhe ob ained alue only ega ded ao; a coa se app oxima ion o lhe possible seis mie esponse D lhe building. Fini e elemell( damage modelling In o de o keep lhe o al numbe o equa ions wi hin easonable limi s, only a qua e o lhe ans e se sec i on was disc e ised in o e ahed al solid elemen s o ca y Ou! lhe damagc analysis. Damage i s appea s a lhe c QWIl o lhe ans e se a ches D lhe na e. As add i ional load is p og essi ely app li ed, damage sp eads co e i ng la ge egions o lhe s uc u e (Figs. 19, 20). Addi ional damage ocuses a e obse ed aI lhe c own o lhe aisle aulls. ai lhe haunches o lhe aisle l ans e sc a ches and a lhe bases o lhe pie s. Failu e occu s o a o al load applied equa l lo Iwice lhe dead load. caused by lhe gene a ion o a collapsing mechanism simila o lh al ep oduced by he GMF mode . As can be seen in Figs. 19-21, bo h damage and cu alu e a e concen a ed in he egions whe e ei he se e e c acking o in ense comp essi e s esses appea ed in he GMF mode!. The main di e en ce be ween lhe wo models lies in lhe loca ion o he maximum damage in he aisle aul , which in lhe FEM model eaches i s highes! in ensi y nol aI he c own bul in he sec ion eloses o he aul haunch nea lhe pie . This is also he p obable localion o lhe heo e ieal hinges leading o lh e ul ima e mechanism. As in lhe GMF analysis, an ahnos linea ela ionship is ob ained belween he amoun o load and lhe displacemenls o lhe sl uclu e wi h only a e y sligh , g adual loss o s i ness obse ed wi h he ex ension o damage. This almos linea esponse is ob ained up o he uhima e load, wi h no u he so ening b anch being p edic ed. AIso as in he GMF analysis, lhe agile esponse migh be a pu ely nume ic al e ec! caused by insu icien mesh e inemen o by he li mila ions o lhe me hod used in he solu ion s a egy. P. ROCA c aI. I Analysis a Ba celona Ca hcd al 253 Fig. 19 Dis ibu ion o damage a loading le eis co esponding o dead load ac,o s o 0.5, 1.0, 1.5 and 2.0 (see damage seale in Fig. 20) 254 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTIONS 11 . 11 2 . 110 . 019 · "', , :)11 . UI · l.:) .2" · 172 · 8621:-1 Fig. 20 Dis ibu ion o damage a loading le eIs co csponding o dcad load ac o s o 0.5. 1.0 and 2.0. P. ROCA el aI. J Analysis o Ba celona Cn hed al 255 Fig. 21 De a med shape o g a ily load 5 CONCLUSIONS Two al e na e numc ical app oaches cha ac c ised by Ihci c y di e en deg ee o sophis ica ion we e sa is aclo ily used o ca y ou a nan-linca analysis o he sl uc u c o a Go hic cons uc ian -namely. Ba celona Ca hed al - aking in o accoun! he mo e p ominen ea u es o he mechanical esponsc o lhe s uc u al ma e ial. The wo app oachcs - he Gene alised Ma ix Fo mula ion and he 3D ini e clemen nodelling wi h a damage model -p oduced e y simila p ediclions o lhe espanse a lhe s uc u e h oughou lhe loading p ocess, lhe collapsing mechanism and lhe ul ima e capaci y in he analysis o a na e bay. Bo h app oaches. bul in pa icula he much simple GMF p ocedu e. awc / ci success in analysing mason y consl uc ions o !hei abilily o accu a ely simula e h e essen ial ac o s dCle mining Ihei s uclu al esponse. Fi sl. / ey sha e he capaci y 10 de_sc ibe com pie x geome y and 10 accu alely com pUle and disl ibu e dead loads on lhe s uC lu c. Second, hey include sui able cons i u i e equa ions lo mode! lhe essenlial cha acle is ics o lhe beha iou a he ma e ial (l he inabili y o mason y o ca y ensioll, in pa icula ). The main limi a ion o he GMF s ems om i s inadequacy o ea! Iwo-dimensional elemen s such as Go hic aul s. I an accu a e s udy o a aul is o be pe o med. he use o a de ailed ini e elemen mode ling becomes una oidablc. Fu he mo c, il equi es e y ealis ic ma e ial modelling o lhe a ious sl uc u al elemen s, ma e iais. su cha gc and suppo condi ions in ol ed (including ibs. webbing. mason y haunches. and possiblc conc e e iH).