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

STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTI ONS 11 P. Roca , J. L. Gonúlez, E. Onate y rB. Lo urenço (Eds.) © CIMNE. Ba rcelona 1998 ANAL YSIS OF THE STRUCTURE OF GOTHIC CATHEDRALS APPLlCATION TO BARCELONA CATHEDRAL Pere Roca, LaTa Pe ll egr ini, Eu genio Onnte nnd Alex Hanganu* U nil 'e rs ila/ Po lil ec lli C {j de Cow/! lIl ya ·Inlernatioflol Cemer for Nwnerical Methods in Ell ginu ring (CIMNE) Gra ll Capità S/ li 08034 Bar ct IO f1(J. Spai ll SU MMARY The stlldy af ancient structu re s composed af Slane arches and vaults requires th e use of specific techniques af analysis endowed wilh powerful procedurcs far lhe madelling of geomelr y, the simulation of lh e mechanical respanse af lh e variaus materiais (including ashlar blacks and ma sa nr y. backings and fills) and th e s im ulation af lhe possible agenlS (gravity, \Vind , earthquake, settlemems) w hi ch may affect lhe structure or may have affected it thraugh hi slar y. Mare specifically, Ih ose te chniques mu sl be able 10 repToduce lhe aclu al conditions af thrusl equilibrium between lhe large number of curved. linear or two-dimensional members involved. The validity of several modelling techniques wi th a ve ry differenl levei af sa phistication is sludied Ihrough Iheir use in lhe study af a particular Galhic canstruction: Barcelona Cathedral. The various lechniques used were developed at lh e School af Civil Engineering af lhe Tech ni cal Un iversity of Catalonia and have already been used in the analysis af a se! af hislorical constructions, including lhe Basi li ca of San Marco (Onale et ai ., 19 97) and lhe Crypt of lhe Colõnia Güell (Roca, 1997 ). The prese nt study of Barcelona Cathedral, now under developrnent, is Ihus not only ai med at gaining a beuer understanding of various as pects of lhe Tesistance of Gothic constructions (and the southem a nd Catalan Got hi c, in particular) but also at appraising lhe possibilities and limitations of various analytical techniques. 232 STRUCTURAL ANALYSIS OF HI STO RI CAL CONSTRUCTI ONS 11 1 SOME CONSIDERAT IONS REGARDING GOTHIC STRUCTURE 1.1 Overall structural dcsign The resistance cf Gothic constructi on gave rise to a ccrtain amount cf controversy after Viollcl-le-Duc's fi rst serious attempt lo propose a ralienal vision cf its mechanical principies and particular structuraI ordering. Hi~ cnlhusiastic interpretati on of GO lhic structure as a fully ralianal object was later contradicted by olhcr rescarchers (see B. Bassegoda, 1974). Rather lhan defending Viollet's points af view. they found instances of structural arbitrariness and uncertainties which were only to bc understood as compositional motifs, construction devices, Cf purely a consequence cf the limited knowl edge cr intuilion cf lhe historical dcsigners. Viollet's theory presented Gothic structural ordering as a system slriclly based on equilibrium and justified by ir. Furthermore, ali devices of Gothic S lru clure were laid Oul so as lo co ntribute to the achievement of an overall ductility. In thi s conte xt , du clility musI be underslOod as the eapacily of lhe slruclure to accepl significanl deformation without sust ai ning damage or failing. It is this ductility which allowed the Got hi c structural components (piers, arches, vau lt s) to altain thcir impressive d im ensions and their slenderness. More recenlly, olher researchers have found nolable structural weaknesses in Got hi c eonstructions. Apropos of lhe sludy of the Cathedral of Sla. María, in Vitoria, Spain, Croci el a I. (1 995) observed weak points such as (I) a very Iimited interconnection be lween lhe struclural elemenls, producing a lype of isostatic conslruction; (2) lhe excessi ve slenderness of piers and related possible inOuence of second·order effects in lhe equiJibrium, (3) lhe dcformability of buttresses a nd fly in g arches, (4) the eccentricity created over the pier by buttresses attached lo pier extensions which usualJy do nol sland direct1y over lhe pier, (5) lhe weakcning effeel causcd on support elemenls by lhe triforium and possible open in gs, with special regard lo their shear re sistance, and (6) lhe facl Ihal rib springings in lhe aisle and arcade arches may slrongly reduce lhe local resistan! sections of lhe upper pier exlens ion s. The combination of !hese aspects are lhe cause of major defonnations and dangcrous oul· of -plumbness. Aecording to lhe researchers sma l! design or construction impcrfeclions may lead to high deformability and cracking. Furthermore, frequent damagc pattcrns have been idenlified which in some cases are considered "chronic", such as lh e so·called SaboureL cracks lypica ll y observed in large groined vauh, (Heyman, 1983, Barthel, 1993). The study of severa1 major GOlhic constructions, including lhe case of lhe Basilica of Sl. Francis of Assisi after the Seplember 1997 earthquake (Croci, 1998 ), al50 showed the weakness of Gothic slructures to special agents such as earthquake, inlense wind and large foundati on selllernenls. 1.2 The role of lhe structural elements The clear and univocal role that Viollel·de-Duc assigned lO lhe various sLruclural elemenls (diagonal, lransverse and c1erestory arches in vaulting, buttresses and flying P. ROCA et aI. I Analysis of Barcelona Cathedral 233 buttresses, etc.) and lh e very assumplion lhal a li these elemenls are esse nti al in lh e Gothic grammar, are also today disregarded by some specialisls. Some new understanding carne from lhe observation of lhe stales of residual equilibrium attained by Gothic constructions after partial desl ru clion caused by fi re, bomb exp losions (dur in g World War 11 ) or simp ly lack of maintenance. Remarkably, some vaults maintained a perfect equi li b ri um after los in g th e transverse or diagonal rib s. Other constructions lost some buttresses or flying butlresses without immediate damage be in g observed in lhe system of vaults ( 8. Bassegoda, 1974 ). With regard to lhe role of lhe eleme nl s in a cross vault, some studies revealed that the diagonal, longitudinal and transverse elements contribute little, if anythin g, 10 res istance. The stability of lhe vault is due mainly to the capacity of th e webbing of the vau lt s to work as a s heH with double curvature (Mark, 1982, 1993). It is interesting to note that this contradicts some classical opinions wh ich assigned lO lh e diagonal arches the main structural role and relegaled the webbing lO litlle more th an a simple cJosu re. The role of the fiH placed over the va ult s is also subjecl to opinion a nd probably has ve ry different functions depending on its orig in and material composilion. The backing -placed aI lhe haunches of lhe vaults - is of utmOs l importance to Slabi li se lhe vau lt s subject to gravily load. Additional fiU placed ovcr lhe vault might also produce a marginal stab ili si ng effeet when lhe vault is only subjeet to gravity l oa d s. Howcver, such a fiU may have very different effects in lhe case of agents other Ihan gravity. The studies ca rri ed out on the Basilica of SI. Francis of Assisi after lhe earthquakc showed the non-cohesi ve filling existing over lhe vau lt s, which was nol original1y placed there on purpose but merely accumulated over lhe decades and had a very unfavourable effect during the occurrence of lh e earthquake leading to the collapse of some of the vau lt s (Croei, 198 8). The f iU w hi ch exists in some southem Gothic eonstructions (such as in Sl a. Maria deI Mar in Bar ce lona and Barcelona Cathedral itself) consists of a cohesive mass of lime mortar and traditional ceramic vessels and is thus characlerised by considerable strength co mbined with very li mi ted weight (1. Bassegoda, 19 83). Supposedl y, I hi s kind of co ncrete fiH makes a significant contributi on to resistance in lhe cases mentioned, where a vault webbing much thinner than in mosl Gothic eonstruetions has also been observed. 2 TECHNIQUES APPLlED TO THE ANALYSIS OF GOTHIC STRUCTURE 2.1 Diffieulties of the analysis Because of lh e many difficulties which are encounlered in lhe sludy of GOlhic constructions, few seientific studies have been perfonned to date with a view to gaining a deeper understanding of their mechanical and resistance propenies. The main diffieulties are posed by th e fo llowing aspects: Because of the fragile nature of stone masonry, analytical studies based on the hypothesis of linear elastie it y are of very Iimited va li dity. In fael, without a great deal 234 STRUCTURAL ANALYSIS OF HI STORICAL CONSTRUCTIONS 11 af (unher interpretatian by lhe analyst. they may only inform about lhe servicc behaviour provided lha! stresses keep wíthin moderate values. The knowledge af both lhe internai composition af lhe structural members and lhe mechanical propcrties af lhe existing materiais are normally very limited, for several reascns. Furthermore, bOlh lhe composition and lhe material properties may show considerable sca ttering throughout lh e entire construction. The mcchanical behaviour is deeply dependent on lhe historical construclion proces s, which is mostly uncertain, Additional uncertainty is provided by conlingencies 5uch as accidents and partial rebuilding, during construction ar immediate rcmedial mea su re s to constrain preeocious deformation. Fina ll y, as we are reminded by Cassinello (1998), those co nstructions whieh have reached lhe present day have usually undcrgone major transfonnations as a consequence of progressive Of pennanent defonnation, eracking, physieal or ehemieal degradation of materiais, or possible eonstruction alterations or replacements. Again according to Cassinello, the elaboration of an accurate model shou ld thus take into account the three main aspeClS detennining the structural behaviour, namely the (defonned) geometry, internai eomposition. and material properties. The integration of lhe historieal transformations should be extended to these three aspecls. 2.2 Significant studies and developments A widespread set of techniques has been used in the study of GOlhic construction, including photoelastic modelling, classical theories bascd on plastic theorem s, conventional matrix calculation and lhe finite element melhod. Photoelastic modellillg The research carried out by Mark (1982) by means of photoelaslic models, eXlended to a large number of major cases such as the calhedrals of Amiens, Beauvais. Bourges, Chartres and Palma de Mallorca, constitutes one of lhe first and more successful attempts to quantitatively analyse lhe slructure of these constructions lo gain a deep understanding of lh eir perfonnance when subjecl to various agents (wind in particular). Thc study enables us to envisage the essential characteristics of lhe buildings, compare lheir different design, and better understand lhe role of their different structural elemenls. lt also provides an understanding of the origin of some of lhe exisling damage, such as certain cracks and pennanent deformalions. Plastic analysis The possibilities of classical plastic analysis in lhe study of Gothic cathedrals, based on the plastic limit theorems, have been shown by Heyman (1998). Due to lhe validity of lhe hypothesis assumed by classic theory. very realistic collapsing mechanisms may be devised through its careful application. However, classical analysis is of little use if applied to conditions other than or far removed frem failure. Thus, classical theories P. RO CA cl alo I A nal ysis or Barcelona Calhcdral 235 are not very useful to interpret lhe cause and extent of cracks, deformation or other damage not directly related to the generation af a co llapse. Furtherrnore, their practical but rigorous use beco rnc s extremely difficult in the case of complex structures with muitiple elements (as encountered by several authors, incJuding Cauvin and Stagnitto, 1998 ). According lO Chassagnou et aI. (1998), lhe evaluations obtained using classical theories tend to be too pessimist and may lead to repairs thal are more far-reaching than actually needed. This is so because of lhe type of additional hypothesis that must be assumed to make the use of these Iheories feasible. COtlvetlrional marrix calc ulariotl Using conventional matrix formulation, Leon a nd Cassati (1997. 1998) dcveloped a delailed paramelrical study of the main geome tri cal and material variables influencing lhe structural response of Le 6n Cathedral in Spain. Thc study was based on a careful modclling of lhe main transverse section af lhe building and the na ve vau lt s by means of an equivalent system af one-dimensianal elcments. Finire element merh od The main limitation of mo st finite element fonnulations, whcn applied to masonry elemcnts, lies in the h YPo lhesis of the contiouum, which makes il difficult or cvcn impossible to simulale th e particular kinematics of the masonry modes of failurc. Elaslic analyses based 00 the finite elcmenl method are only suitab le for characlerising working condilions subjecl to states of very modcrale stresses in which compressive stresses are largely predominant. However. the adoplion of su itable constitulive equations which account for the main phenomena reJated to the failurc of the materiais (cracking under tension, c ru shing under compression, etc.) makes il po ssible to reproduce more advanced slages of lhe response and even 10 simulate failure mechanisms similar 10 those predicted by lhe classical Ih cories. Croci et aI. (1995) carried oul a finite elemenl analysis of lhe Cathcdral of Sta. Marí a, in Vitoria, Spain. The analysis, applied to the main transverse sections of lh e building and to lhe nave vaults, followed an incrementaI strategy to account for cracking du e to tension or shear stresses, as wcll as the equilibrium second-order effccts. This analysis showed some of the main weaknesses of lhe building, as has already bcen mentioncd in Section 2.1. Similar analyses were also used for lhe s lUd y of th e co ll apse of Bcauvais Ca th edr al (Croci et al., 1998b) and the effecIs of th e earthquake of September 1997 on lhe Basilica of Assisi (Croci, 1998 ). The finite element method, also combined with suitable constitutive equatioos, ha s been successfully used to study Gothic cross vaults (Barthcl, 1993 ). For this purposc, very detailed finile element models were elaboratcd and used in combinalion with a material treatment enabling lhe simulation of cracking as well as sliding between arch ring joints. 236 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCT IO NS 11 Ca uvin and Stagn it o (1993, 19 95) carried oul very in teresting s lU dies using both c1assical plastic analysis and nan-linear analysis by lhe finitc elernc nt melhad. Their method was successfully applied to analyse lhe central nave af Reims Cathedral. The fin it o cleme nt method has beco also uti li sed to study several Spanish Gothic cathedrals (Burgo s, Seville) by Izquierdo (1997; see al50 Cassinell o, 1998). Cha ssagnou e l aI. (1998) have developed a technique for lhe generic modelling af Gothic cross vaults, al50 based on lhe finite element melhad. with lh e a im af making available a 1001 for lhe prac ti cal evaluation af th eir structural responsc. Onate cl ai ( 199 7) have developed a finite element damage model w hi ch was successfully used for the anal ys is of SL Marks Basilica in Venice. This model, brie fl y discussed in a later sec ti on, is one of lhe techniqucs chosen for the numerical studies of lhe Cat hedral of Barcelona presented in this paper. 3 BARCELONA CAT H EDRAL 3.1 lntroduction to lhe st udy The choice of Barcelona Cathedral (Figs. 1-6) for lhe purpose of th e present study is pani a ll y due lo the facl that abundant informarion exisls on its geome tr y and olher construction aspccts, made ava il able lo lhe author by lhe Diocesan Arc hi ve and lhe Chapter of Barcelona Cathedral. Funhermore, th ere are some publications with interesting historical and const ru ction information (Elías, 1926, 1. Bassegoda, 1968, 19 81, 1983). Barcelona Cathedral also shows certain structur al singularilies which gave it added interest as a subject of study. • First, it is a construction in lhe so -c alled Catalan Gothic style, wh ic h exhibits notable spec ifi c feature s. Catalan Gothic constructions were typically designed as very diaphanous spaces with large-spanned naves. • The dimensions of lhe vaults are remarkable due to lhe large gap between lh e piers nOI only in Iheir transversal direction, but also 10ngitudinal1 y. • The lay-out of lhe lransver se seclion of lhe building is char ac le ri sed by th e ai sle naves being a lm ost (buI nOI quile) as high as the central nave. Because of lhis slruclura ll ay-out, lhe aisle vaults h;we an importan! "Irnclural function in relain in g lhe IhruSl of lhe cenlral vault s. In a celebraled leclure on Mallorca Cathedral, Rubió i Bellvcr (1912) dist in guished belween lhree different types of Gothic calhedrals with regard lo lhe structural organisation of lheir transverse section. In lhe first type, lhe nave and lhe ai sles are built to lhe same heighl (as in Saragossa, Munich, Perugia and others). The second type, in w hi ch lhe aisles are almost but nol quite as high as th e nave, is by far lhe least widespread since il only inc1udes lhe cases of Barcelona CathedraJ and Sta . María dei Mar. The third typc, in which lhe aisles are significantly lower lhan th e nave, is by far lhe most frequent (being lhe case in Amiens, Reims. Beauvais, Cologne, Milan, Toledo, Ma ll orca and many others). P. ROCA el a!. I Anal ys is o( Barcelona Ca lh edral 237 ~''' IDlln''U:UU~iIIltl t1!U n. :~ II I!i: 'jI(U'UVl ; SllIltI:U1I1UJr11U! Fig. I Longitudinal and transverse sections of Barcelona Calhedral (facsimile reproduclion of plans drawn in 1864 as part of lhe sludies previous to lhe construction of lhe new cimborio). 238 STRUCTURAL ANALYSIS OF HISTORICAL CONSTRUCTlONS 11 . , Fig. 2 Plan of the building (1864 plans). Ui'l1i8DR1Hií I~R8fhtni'l PI."I. t'",.,-.! '" 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 f-'. J ""'-, '-':: :"T,C" . ~.~. P. ROCA el alo I Anal ys is of Barcelona Cathedral 239 Following Rubió. in lh e case of conslruclions belonging to the second type lhe thrust of the nave vaults is completely retained by lhe ai sle vaults so thal no flying arches are needed ai ali. Where they exisl. as in Barcelo na Ca lh edral, Ihey are non-structural and Ih e ir only function is probably drainage. Th e prominenl role of lhe ai s le vaults as stabi li s in g eleme nt s may be relaled 10 Iw o construclion devices w hi ch are consistent with them and which were probably in co rp orated in order 10 slrenglhen Ihem. In particular, lh e aislc vaults are surcharged w ith an amou nt of co ncrClC fill greater Ih an that placed over lhe nave, as if to increase their counteracting thrus l. ~ I O,5m Fi g. 3 Transverse section of lh e na ve piers a nd lh e piers susl ai ning Lh e c im borio as represenled in lhe 1864 plans (E 1:40 ). 3.2 Bricf description of the building Const ru c ti on of lhe naves of Barcelona Cathedral was begun in 129 8 and lasted for more lhan a ce ntur y. As usual. lhe apse was constructed firsl, being finished in 1327, while lhe const ru c li on of lhe entire nave conlinued until 141 7. In 1422 wo rk stopped, leaving lhe cimbor io unfinished and a pro vi sio nal wall cJo sure as a façade. Mosl of lhe original reatures of the building are assumed 10 be allribulable to the master builder Jaume Fabre, who worked on th e Calhedr al from 1317 . The plan of lhe temple shows a ve ry unconventional lay-o ul in which the cimborio is nOI erected over lhe crossing but over lh e first b ay of lhe nave elose to lhe façad e. while the crossing is delimited by lhe two majes ti c elock towers, more commo nl y part of the façade. A similar distribulion can be observed in very few case s, such as lhe German cathedrals of Ulm and Friburg. The building in cludes three naves ( lh e nave and two aisles) although, as a consequence af its particular design, it appears to enclose two addilional aisles. This 246 STRUCTURAL ANALYS IS OF HI STORICAL CONSTRUCTIONS 11 Fig 10. Dcformcd shape cf a vaul! quadrant mode!!ed as a framed system, subjecI to dead loading. Fig. 11 Force trajcctories in a GO lhic vault subjecl to dead loading (Mark. 1982). Allalysis of lhe vault usif/g 3D solid /inile elements Thc FEM modeJling af lhe vau lt accounts for lhe rib system, lhe webbing, lhe nlbble backing a1 lhe vault haunches, the concr ete fiH and lhe lateral walls, ali def ined by their co rresponding mechanical properties. Given lhe li mited infor ma tion available on the material properties, some assumptions had to be made in arde r to assign mechanical parameters to the various mate ri aIs (Table 1). The greatest source af uncertainty was the coneretc fill, composed af ceramic pottery and li me mortar, the properties of which had to be roughly estimated. The ashlar masonry and rubble were estimated, on the basis of empiri ca l criteria, from the average compressive strength (74.2 MPa) of the Monr jui" c sandstone used for the construction of the building. P. ROCA et aI. I Analysis of Barcelona Cathedral 247 fig. 12 FEM model and distribution of stresses caused by total dead load 00 lhe nave vault (maio compressive stresses in Pa). Fig. 12 shows the distribution of stresses which is obtaioed for the vault subjecI to total dead load. Some of the ohservations allowed by lhe simplified frame model can be confirmed, sue h as lhe significant role of lhe webbing in transferring the forces 10 the piers, and lhe major contribution of lhe transverse arches but small or null contribution of the clerestory ribs and diagonal intersections. 248 STRUCTURAL ANALYSIS OF HI STOR ICAL CONS TRUCTI ONS 11 Table I. Mechani ca l properties defined for lhe various materiais Materi al D efo rmation Poisson Compresive Tensile Dens it y modulus coefficient strength slrenght (kN/m3) ( Mp' ) (kP.) (kP a) Ashlar 8000 0 .2 8000 400 2,7 mll so nry (*) Rubble (**) 2500 0.2 2500 100 2,5 Concrctc fi 11 500 0.2 500 50 1,0 (*) in pi er s, ribs and vault wcbbing (**) at lhe core of piers and aI vault haunches The ana lysis prediclS an unrcalistica l1 y high ultimate capacily (ov er lhree times lhe total dead load in g) b eca use lhe lateral displacerncnts cf lhe springings are defined as fu ll y constrain ed in lhe individual trcatmen! of lhe vaul t. Thus. lhe simulated ultimate m ec hani sm does nOI aceount f or lhe flexibility and li mit ed capacity cf lhe retaining system pr ovided by lhe aisle vauhing and buttresses. Mor e realis ti c resu lt s are obtai ned when, as expla in ed in Section 4.3. lhe vault is in sert ed inlO a more complete model also including ali structural elements in the Iransverse section. 4.3 Study of a nave bay Study using Generalised Marrix Formularion The purpose of Ihis sludy was 10 analyse lhe possibility of studying lh e equilibrium of Gothic structure up to lhe ultimate condition by means of a simple model. In spite of this simplicity, the model was elaborated to oblain an accurate descriplion of lhe geometry of lhe nave section, lhe distribution of lhe dead load and lhe response of lhe materi ai s. Using the faculties of lhe method, the various structural elemenls were modelled as straight or curved lin ea r elements with variable and co mpo site cross-section (Fig s. 1314 ). lt sho uld be not ed Ihal, in a very simplif i ed way, the vaults are also treated as a curved compos ile beam where each mate rial (maso nr y. rubble, concrete fi l!) is defined by its particular mechanical propcrties. The analysis a cco unts for lhe effccls of mas onry cracking under tension or crushing und er com pr ession as describcd in Section 4.1. By gradual1y applying the gravity load up to and beyond th e dead load value we were able to visualise lhe working condition of the slructure, lhe progressive d ev elopment of cra c king and lhe final appearance of a number of crushed zones leading to failure (Figs. 14-16). Th e applicd load was kept proportional to lhe dead load throughout lhe proce sso Failure wa s reached for a total applied gravity load equa l to twi ce the (Ota l dead load. As ca n be se cn in lhe figur es mentioned above, lhe sections affected by deep cracking or crushing deve lop a highly pronounced curvature and thus may be identified with P. ROCA ct aI. I Analysis of Barcelona Cathedral 249 hinges, which would lead to an ultimate mechanism according to lhe plastic theory. The set of hinges leading to the ultimate mechanism appeared in the nave vault. lhe upper region of the pier shafts and the aisle vau lt s. In both lhe nave and aisle va ults the hinges developed at their crown. Howeve r, the obtained mechanism is nOI identical lo the theoretical one Ihal would be obtained by applying lhe plastic Iheory. because of lhe conlribution of lhe general deformation of lhe elemenls lO the instability. Owing to lhis contribution, lhe failure mechanism becomes poss ibl e even if some hinges are not fully developed, as at lhe nave vaull crown, or not developed aI ali, as occurs in the base of lhe piers where lhe mobility needed 10 develop lh e mechanism is given simply by the great nexibility of the shaft s. A full lheorctic mechanism would require a set of seven hinges. Fig. 13 Equivalenl frame model. During lhe process, lhe average compressi ve stress a! the base of lhe picrs varies from a value of 3.5 MPa, for dead loading applied, to a value dose to the compressive strength (8 MPa). The response is almost linear up to failure, with only very slight losses of st iffness Ihroughout lhe development of the hinges and a final, fragile failurc. It should be menlioned that this fragile response may result as a limitarion of lhe numerical procedure to continue lhe analysis beyond some local severe damage. The parametric studies showed lhar lhe ultimate load is nOI dependent upon lhe compressive strength when values hi gher lhan 8 MPa are considered, which means lhat lhe failure is consrantly caused by lhe general ultimale mechanism. This is what should be expected from lhe dassical plastic analysis. In contras!, lhe ultimate load decreases in proportion to the compressive strength when values of less Ihan 8 MPa are defined because the failure is then delermined by the crushing of lhe masonry ai lhe base of the piers. 250 STRUcrURA L ANALYSIS OF HI STORICAL CONSTRUCTI ONS " As demon stralcd by lhe paramclric studies, lhe genera l ca pacily wus nOI signiricantly dependem on lhe st iffness of the concrele fi!! (for va lu cs af lhe modulus of deformat ion varyi ng be lwecn 50 to 1000 MPa). Simi larly. a modc ratc variation of its average density h ad almosl no cffeel upon lhe ultimatc load. , "" Fig. 14 Distributi on ofaxia l stresscs and cracking for lhe stnlcture subjccl to dead load. Fig. 15 Distribution cf axial stresses and cracking aI failurc occurring for a dcad load factorised by 2. P. ROCA Cl aI. I Analysis of Barcelona Cathedrai 251 Fig. 16 Deformed shape close to failurc. This mode! was also used to simulale lhe effecl af an earthquake by applying a set of horizontal equivalent static loads consistenl wirh the dislribution 011 mass. The 11011linear analysis predicled lhe failure for a seismic coefficient alg eqllal to 0.12, which can bc considered safe enough with rcgard to lhe seismic illlensÍly of lhe region of Barcelona. The application of lhe Spanish seismic code would lead lO lhe consideratioll of a coefficient equal to 0.086 for a masonry building in a similar condition and for a ri sk period of one hundred years. As showll in Fig. 17, the distribution of stresses is nOI significantly altercd with respect to lhe dead load condition when a more moderale earthquake is simulated. The corresponding deformed shape is shown in Fig. 18. ; . ... Fig. 17 Distribution of stresses caused by a set of static equivalent forces simulating an earthquake characterised by a coefficient alg equal to 0.086. 252 STRUcrURAL ANALYSIS DF HISTDRICAL CDNSTRUCTlDNS 11 Fig. 18 Geometry deformed byearthquake. Obviously, lhe lirnited va li dity Df lhe lrcatment of lhe earthquake as an equivalen! set Df static forces should be ta ken into consideration and lhe obtained value only regarded ao; a coarse approximation to lhe possible seis mie response Df lhe building. Finite elemell( damage modelling In order to keep lhe total number of equations within reasonable limits, only a quarter of lhe transverse secti on was discretised into tetrahedral solid elements to carry Ou! lhe damagc analysis. Damage first appears at lhe c r QWIl of lhe transverse arches Df lhe nave. As add it ional load is progressively app li ed, damage spreads cove ri ng larger regions of lhe structure (Figs. 19, 20). Additional damage focuses are observed aI lhe crown of lhe aisle vaulls. ai lhe haunches of lhe aisle lransversc arches and at lhe bases of lhe piers. Failure occurs for a total load applied equa l lo Iwice lhe dead load. caused by lhe generation of a collapsing mechanism similar to lh al reproduced by the GMF mode\. As can be seen in Figs. 19-21, both damage and curvalure are concentrated in the regions where either severe cracking or intense compressive stresses appeared in the GMF mode!. The main differen ce between lhe two models lies in lhe location of the maximum damage in the aisle vault, which in lhe FEM model reaches its highes! intensity nol aI the crown bul in the section elosest to the vault haunch near lhe pier. This is also the probable localion of lhe theoretieal hinges leading to lh e ultimate mechanism. As in lhe GMF analysis, an ahnost linear relationship is obtained belween the amount of load and lhe displacemenls of lhe slruclure with only a very slight, gradual loss of stiffness observed with the extension of damage. This almost linear response is obtained up to the uhimate load, with no further softening branch being predicted. AIso as in the GMF analysis, lhe fragile response might be a purely numeric al effec! caused by insufficient mesh refinement or by the li milations of lhe method used in the solution strategy. P. ROCA ct aI. I Analysis af Barcelona Cathcdral 253 Fig. 19 Distribution of damage at loading leveis corresponding to dead load fac,ors of 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 Distribution of damage at loading leveIs corrcsponding to dcad load factors of 0.5. 1.0 and 2.0. P. ROCA el aI. J Analysis of Barcelona Cnthedral 255 Fig. 21 Def armed shape for gravily load 5 CONCLUSIONS Two alternate numcrical approaches charactcrised by Ihcir vcry different degree of sophistication were satisfaclorily used to car ry out a nan-lincar analysis of the slructurc of a Gothic constructian -namely. Barcelona Cathedral -taking into accoun! the more prominent features of the mechanical responsc of lhe structural material. The two approachcs -the Generalised Matrix Formulation and the 3D finite clement rnodelling with a damage model -produced very similar prediclions for lhe respanse af lhe structure throughout lhe loading process, lhe collapsing mechanism and lhe ultimate capacity in the analysis of a nave bay. Both approaches. bul in particular the much simpler GMF procedure. awc t/tcir success in analysing masonry conslructions to !heir abilily to accurately simulate th e essential factors dClermining Iheir struclural response. Firsl. t/tey share the capacity 10 de_scribe com pie x geometry and 10 accuralely com pUle and dislribute dead loads on lhe struC lurc. Second, they include suitable constitutive equations lo mode! lhe essenlial characleristics of lhe behaviour af the material (l he inability of masonry to carry tensioll, in particular). The main limitation of the GMF stems from its inadequacy to trea! Iwo-dimensional elements such as Gothic vaults. If an accurate study of a vault is to be performed. the use of a detailed finite element modetling becomes unavoidablc. Furthermorc, il requires very realistic material modelling of lhe various slructu ral elements, materiais. su rchargc and support conditions involved (including ribs. webbing. masonry haunches. and possiblc concrete fiH).