Arch Comput Methods Eng (2010) 17: 299325DOI 10.1007/s11831-010-9046-1

O R I G I NA L PA P E R

Structural Analysis of Masonry Historical Constructions.Classical and Advanced Approaches

Pere Roca Miguel Cervera Giuseppe Gariup Luca Pela

Received: 15 January 2010 / Accepted: 15 January 2010 / Published online: 22 July 2010 CIMNE, Barcelona, Spain 2010

Abstract A review of methods applicable to the study ofmasonry historical construction, encompassing both classi-cal and advanced ones, is presented. Firstly, the paper of-fers a discussion on the main challenges posed by historicalstructures and the desirable conditions that approaches ori-ented to the modeling and analysis of this type of structuresshould accomplish. Secondly, the main available methodswhich are actually used for study masonry historical struc-tures are referred to and discussed.

The main available strategies, including limit analysis,simplified methods, FEM macro- or micro-modeling anddiscrete element methods (DEM) are considered with regardto their realism, computer efficiency, data availability andreal applicability to large structures. A set of final considera-tions are offered on the real possibility of carrying out realis-tic analysis of complex historic masonry structures. In spiteof the modern developments, the study of historical build-ings is still facing significant difficulties linked to computa-tional effort, possibility of input data acquisition and limitedrealism of methods.

P. Roca () M. Cervera G. GariupTechnical University of Catalonia, Jordi Girona 1-3, 08034Barcelona, Spaine-mail: pere.roca.fabregat@upc.edu

M. Cerverae-mail: miguel.cervera@upc.edu

G. Gariupe-mail: giuseppe.gariup@upc.edu

L. PelaUniversity of Bologna, Bologna, Italye-mail: luca.pela@unife.it

1 Introduction. Purpose and Challenges

Studies oriented to conservation and restoration of histori-cal structures have recourse to structural analysis as a wayto better understand the genuine structural features of thebuilding, to characterize its present condition and actualcauses of existing damage, to determine the true structuralsafety for a variety of actions (such as gravity, soil settle-ments, wind and earthquake) and to conclude on necessaryremedial measures. In short, structural analysis contributesto all the phases and activities (including diagnosis, reliabil-ity assessment and design of intervention) oriented to grantan efficient and respectful conservation of monuments andhistorical buildings. Accurate structural analysis is neededto avoid erroneous or defective conclusions leading to ei-ther over-strengthen the structure, causing unnecessary lossin terms of original material and cultural value, or to insuffi-ciently intervene on it, and hence generate inadmissible riskson people and heritage. Unsurprisingly, ancient structureshave been studied, since long time ago, using the most ad-vanced tools available for structural assessment.

The application of advanced computer methods to theanalysis of historical structures was pioneered by the stud-ies of the Brunelleschi Dome by Chiarugi et al. [30], thePisa Tower by Macchi et al. [79], the Colosseo in Rome byCroci [34], see also Croci and Viscovik [36], Mexico Cathe-dral by Meli and Snchez-Ramrez [88] and San MarcosBasilica in Venice by Mola and Vitaliani [93], among oth-ers (Figs. 1 and 2). By then, the development of methodsfor accurate analysis of steel and concrete structures, includ-ing non-linear applications, was already at a very advancedstage thanks to the work of Zienkiewicz and Taylor [136],Ngo and Scordelis [97] and many others. Notwithstanding,analysts attempting to use computer tools for the study his-torical structures were by then facing overwhelming chal-lenges. Methods then available were not yet prepared to

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tackle the specific problems of ancient constructions con-cerning materials, structural arrangements and real preser-vation condition. In fact, the difficulties posed by histori-cal structures are still very challenging, and still reminiscentof those encountered by the pioneers, in spite of significantprogress during the last decades.

Some of difficulties encountered are related to the de-scription of geometry, materials and actions, all of whichacquire remarkable singularity in the case of historical con-struction. Additional important difficulties are related to theacquisition of data on material properties, internal morphol-ogy and damage, as well as to the adequate interpretationof structural arrangements, overall organization and histori-cal facts. Because of all these difficulties, it is generally ac-cepted (Icomos/Iscarsah Committee [60]) that the study ofa historical structure should not only base on calculations,but should integrate as well a variety of complementary ac-tivities involving detailed historical investigation, deep in-spection by means of non destructive techniques (NDT)and monitoring, among other. Structural analysis of histor-ical structures constitutes in fact a multidisciplinary, mul-tifaceted activity requiring a clever integration of differentapproaches and sources of evidence. These difficulties arediscussed into more detail in the following paragraphs.

1.1 Material

Historical or traditional materials such as earth, brick orstone masonry and wood are characterized by very com-plex mechanical and strength phenomena still challengingour modeling abilities. In particular, masonry is character-ized by its composite character (it includes stone or brickin combination with mortar or day joints), a brittle responsein tension (with almost null tensile strength), a frictional re-sponse in shear (once the limited bond between units andmortar is lost) and anisotropy (for the response is highly sen-sitive to the orientation of loads). In spite of the very signif-icant effort invested to characterize and mathematically de-scribe masonry mechanics and strength, the accurate and ef-ficient simulation of masonry response is still a challenge inneed of further experimental and theoretical developments.Important results by Ali and Page [1], Loureno [70], Bindaet al. [12] and many others have yielded a very significantlevel of understanding.

Historical materials, including brick or stone masonry,are normally very heterogeneous even in a single buildingor construction member. Moreover, historical structures of-ten show many additions and repairs done with different ma-terials. Material characterization is constrained by due re-spect to the monument and original material. Non- destruc-tive, indirect, tests (NDT) and minor destructive tests (MDT)should be preferred. If any, only a very limited number ofpits or cores allowing direct observation and laboratory test-ing are normally acceptable. In practice, only limited and

partial information can be collected. Additional assumptionson morphology and material properties may be needed in or-der to elaborate a model.

1.2 Geometry

Historical structures are often characterized by a very com-plex geometry. They often include straight or curved mem-bers. They combine curved 1D members (arches, flyingarches) with 2D members (vaults, domes) and 3D ones(fillings, pendentives . . .). They combine slender memberswith massive ones (massive piers, walls buttresses, foun-dations . . .). However, today numerical methods (such asFEM) do afford a realistic and accurate description of geom-etry. Due to it, geometry is perhaps one of the least (althoughstill meaningful) challenges to be faced by the analysis.

1.3 Morphology

A more significant problem lays in the characterization anddescription of the internal morphology of structural mem-bers and their connections. Structural members are oftennon-homogeneous and show complex internal structures in-cluding several layers, filling, material, cavities, metal inser-tions and other possible singularities. Connections are sin-gular regions featuring specific geometric and morphologi-cal treats. The transference of forces may activate specificresisting phenomena (contact problems, friction, eccentricloading). Modeling morphology and connections in detailmay be extremely demanding from a computational point ofview. Nevertheless, the main difficulty is found in physicallycharacterizing them by means of minor- or non-destructiveprocedures.

1.4 Actions

Historical structures may have experienced (and keep on ex-periencing) actions of very different nature, including theeffects of gravity forces in the long term, earthquake, envi-ronmental effects (thermal effects, chemical or physical at-tack), and anthropogenic actions such as architectural alter-ations, intentional destruction, inadequate restorations . . . .Many of these actions are to be characterized in historicaltime. Some are cyclic and repetitive (and accumulate signifi-cant effect in the long term), some develop gradually in verylong time periods, and some are associated to long returnperiods. In many cases, they may be influenced by historicalcontingency and uncertain (or at least, insufficiently known)historical facts.

1.5 Damage and Alterations

Existing and general alterations may affect very significantlythe response of the structure. Damage and deformation are to

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 301

(a)

(b)

Fig. 1 Some pioneering FEM studies on historical structures.(a) Tower of Pisa: FEM model and substructuring of the colonnade sys-tem (Macchi et al. [79]). (b) Mexico City Cathedral (Meli and SnchezRamrez [88]). (c) The Colosseum in Rome. Tensile horizontal stressesdue to the seismic action (Croci [34])

be modeled, as present features of any existing structure, togrant adequate realism and accuracy in the prediction of theactual performance and capacity. Damage encompasses me-chanical cracking, material decay (due to chemical or phys-ical attack) or whatever phenomena influencing on the orig-inal capacity of materials and structural members.

(c)

Fig. 1 (Continued)

1.6 History

History is an essential dimension of the building and mustbe considered and integrated in the model. The following ef-fects linked to history may have had influence on the struc-tural response and existing damage: Construction process,architectural alterations, additions, destruction in occasionof conflicts (wars . . .), natural disasters (earthquake, floods,fires . . .) and long-term decay or damage phenomena. His-tory constitutes a source for knowledge. In many occasions,the historical performance of the building can be engineeredto obtain conclusions on the structural performance andstrength. For instance, the performance shown during pastearthquakes can be considered to improve the understandingon the seismic capacity. The history of the building consti-tutes a unique experiment occurred in true scale of space andtime. In a way, knowledge of historical performance makesup for the mentioned data insufficiency.

2 Desirable Features of Methods Applied to HistoricalStructures

Because of the aforementioned challenges, attempts tomodel and simulate the response of a historical structureshould try to satisfy some basic requirements. Firstly, anymodelling technique should be able to adequately describethe geometry and morphology of the real construction, in-cluding the structural form, internal composition, connec-tions and support conditions. An accurate description of thedistribution of mass and external forces is essential for bothgravity and seismic analyses.

Secondly, constitutive equations should be adopted al-lowing an adequate description of the essential mechanical

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and strength features of the different materials existing in thebuilding. It is important to highlight that simple linear elas-tic analysis fails to simulate essential features of non-tensionresisting materials such as stone and masonry. More sophis-ticated, non-linear constitutive equations will normally be

Fig. 2 The finite element model of the St. Marks Basilica: Top: globaldiscretization; Bottom: soil foundation discretization including defor-mation (Mola and Vitaliani [93])

necessary. In turn, the use of such constitutive equationswill require the availability of non-linear properties to be ob-tained by means of different laboratory or in-situ mechanicaltests.

Actions (mechanical, physical, chemical . . .) are also tobe modelled by means of mathematical formulations de-scribing their mechanical effect in terms for forces on thestructure, imposed movements or deformations, or possiblevariations of the material properties.

An accurate model of the structure should also afford thedescription of damage and alterations existing in the struc-ture, including cracks, disconnections, crushing, deforma-tion and out-of-plumb, and construction defects. Some dam-age types can be modelled indirectly as a disconnection be-tween elements or a local reduction of material properties. Inorder to characterize the actual capacity in the present con-dition of a building, the analysis should be carried out on themodel accounting for its real damaged and deformed state.

As the analysis of historical structures will normally beoriented to identify needs for restoration and strengthening,analysis methods should able to incorporate and model pos-sible stabilization, repair or strengthening measures. In somecases, these can be taken into account in an indirect wayby adequately modifying material properties, modifying thesectional dimensions or configuration, or by adding forcesto represent their mechanical effect.

The interaction of the structure with the soil is also to betaken into account except in cases it is judged to be irrele-vant. Taking it into account will often require the inclusionof a large portion of foundation soil as part of the entire FEMmodel as done in the analysis of San Marcos Basilica byMola and Vitaliani [93] or the modal analysis of a masonrytower by Fanelli [44] (Figs. 2 and 3).

Fig. 3 Examples of graphical rendering for a FE mathematical model of a masonry tower: discretized geometry, including the soil foundation,and first two vibration modal shapes (Fanelli [44])

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Fig. 4 Different stages of themodel (a) to (d) and successiveboundary conditions variation(Casarin [21]). The surroundingbuildings are taken into accountby directly modelling their wallsor by simulating them withtranslational springs

Fig. 5 Deformed meshes andmaximum displacements (ingrey scale) for seismic loadacting along the longitudinaldirection of the building.Monastery of Jernimos(Loureno and Mouro [72])

Fig. 6 Detailed modelling of Mallorca cathedral, in Spain, including the different parts of the structure which can influence on the modal analysis(tower, faade, choir) and first modal shapes (Roca et al. [126])

Certain types of analyses, as in particular dynamic one,may require the inclusion of neighboring buildings into themodel with an adequate description of existing connections.This is so because of their possible effect on the modalshapes and overall dynamic response. Modelling accuratelythe dynamic response will often require to construct a globalmodel incorporating all the distinct parts of a complex struc-ture as in the analysis of Reggio Emilia Cathedral (Casarin[21], Casarin and Modena [22]), of the Monastery of Jern-imos (Loureno and Mouro [72]) or of Mallorca Cathedral(Roca et al. [126]) (Figs. 46). The connection between thedifferent parts should be modelled accurately by taking into

account the real contact conditions, which requires a pre-vious detailed inspection and investigation. The agreementbetween numerical and experimental vibration modes andfrequencies may be considered as a way to validate the de-scription of connection among different parts in the struc-ture.

Studies on different historical structures (Roca [121],Roca et al. [123]) have shown that real deformations are nor-mally much larger (one or even two orders of magnitude)than those predicted by conventional instantaneous calcu-lations. This is due to the fact that these analysis neglecthistory-related aspects such as (1) deformations occurred

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Fig. 7 (Color online) Analysis of Tarazona Cathedral for different his-torical configurations (Roca [121]) by FEM isotropic damage model.(a) Distribution of tensile damage at the initial condition after con-struction (12th13th c.); (b) compression damage after overloading

of lateral vaults during late Middle Age; (c) tensile stresses (N/m2)after thinning of piers (16th c.); tensile damage during momentarydismantling of flying arches for restoration purposes (20th c.)

during the construction process, as those due to the deflec-tion of centerings and forms and the deformation of struc-tural members during intermediate and incomplete config-urations of the structure, (2) initial and historical soil set-tlements, (3) architectural alterations, (4) the non-reversibleeffect of multiple thermal and hygrometric cycles, and (4)long term damage of mechanical, physical or chemical na-ture, among other phenomena.

A significant part of the damage and deformation todayvisible in ancient construction may have been experiencedduring the construction process or at very early stages of thestructures life. Because of it, it is hardly possible to deter-mine the amount of deformation mechanically connected tothe gravity forces or other possible actions. In any case, agood approach to the prediction of deformation and, possi-bly, damage, can hardly be derived from a purely instanta-neous analysis which does not take into account, to some ex-tent, the changes and historical events or physical phenom-ena affecting the structure. Accurate studies should ideallyafford the simulation of the following aspects:

(1) the subsequent historical stages experienced by thebuilding (in particular, the construction process) througha sequential analysis,

(2) the actions occurring in historical periods, such as majorearthquakes or the repeated effect of minor earthquakesor thermal cycles, and

(3) long-term damage processes (such as those related tolong-term creep) developed across the life of the con-struction.

As observed apropos of the study of the collapse of the CivicTower of Pavia and Noto Cathedral in Italy (Papa and Tal-

iercio [108], Binda et al. [11, 13, 14]), the effect of creepunder constant stress, at the long term, may induce signifi-cant, cumulative damage in rock-like materials. The investi-gation of specimens cut from the walls of the Pavia Towerafter its collapse allowed to Anzani et al. [4] the formula-tion, for the first time on ancient masonry, of the hypothesisof a collapse due to the long-term behavior of the material.The identification of the problem fostered some research ef-fort to better characterize the phenomenon, and some mod-els have been already proposed for its description (Lourenoand Pina-Henriques [73], Taliercio and Papa [133]).

It must be noted, however, that a realistic simulation ofhistorical actions or long-term damage processes is still re-quiring additional experimental studies and numerical de-velopments. In fact, limitations linked to the capacity of thecomputers and data available make it very difficult to inte-grate the aforementioned capabilities in the analyses. A re-alistic modelling including both structural history and time-dependent effects is, at the moment, still too demanding withregard to the possibilities of conventional calculations ap-proaches. However, these aspects may be taken into accountin an approximate way or may at least be considered to ob-tain a better interpretation the results.

As an attempt to examine the influence of the architec-tural alterations experienced by the structure of TarazonaCathedral in Spain (Roca [120], Fig. 7), the analysis wasindependently carried out on a series of structural configu-rations corresponding to different historical moments. Forthe analysis of Mallorca cathedral, whose final conditionis believed to have been very much affected by a delicateconstruction process, a true sequential analysis was carriedout involving the superposition of two consecutive construc-

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 305

Fig. 8 Sequential analysis of Mallorca Cathedral taking into account the construction process (Clemente [31]) using a FEM isotropic damagemodel. Phases considered (left) and corresponding distribution of deformation and damage (right)

tion stages (Clemente [31], Fig. 8). The construction processwas reconstructed thanks to a detailed research on historicaldocuments. The deformation and damage predicted by thesequential analysis was significantly larger than those ob-tained by instantaneous analysis on the final structural con-figuration.

3 Review of Classical Methods

It was Robert Hooke who discovered that the ideal shape ofa masonry arch in equilibrium is that of the inverted cate-nary curve drawn by a chain subjected to the same weightdistribution. At the end of a printed lecture on Helioscopesand some other instruments in 1676 he inserted the fol-lowing problem: The true mathematical and mechanicalform of all manner of arches for building, with the true but-ment necessary to each of them. A problem which no ar-chitectonick writer hath ever yet attemted, much less per-formed. He then provided the solution in the form of ananagram whose decipherment was only revealed after hisdeath in 1705. The solution read: Ut pendet continuumflexile, sic stabit contiguum rigidum inversumas hangsa flexible cable, so inverted, stand the touching pieces of anarch (Heyman [58]). Meanwhile, the equation for the curveof a hanging flexible line or catenary had been already de-rived by David Gregory, who by 1698 had independentlyreached and extended Hookes assertion to the case of ma-terial arches with finite thickness. According to Gregory,arches are stable when some catenary can be fitted withinits thickness.

The possibility of analysing masonry arches by the anal-ogy between the equilibrium of compressed members withthat of funicular models was used during 18th and 19th c.,and even at the beginning of 20th c., for the design and as-sessment of masonry bridges and buildings. A well known

example is the study of the dome of St. Peter in the Vaticanby Poleni [116], Fig. 9. At the beginning of 20th c., architectA. Gaud used the same principle to design complex struc-tures, such as the Church of Colnia Gell near Barcelona,by means of 3D hanging models (Fig. 10).

In France, during the 18th c., La Hire, Couplet andCoulomb undertook the problem from a different approach.Their understanding resulted from viewing the arch as a con-junction of rigid bodies which could experience relative dis-placements. According to Couplet, the collapse occurs whenthe arch develops enough hinges (or sections experiencing arelative rotation) as to become a mechanism (Heyman [57]).The first general theory on the stability of arches was pub-lished by Coulomb in his 1773 essay [33]. Coulomb devel-oped a consistent and general theory providing the math-ematical base for the description of the different possiblemodes of collapse, taking into account both relative rotationsand sliding between parts. He also stated that the failure dueto sliding is rare and suggested to consider only overturning(rotational) failures for practical purposes. He proposed theuse of a theory of maxima and minima (from our mod-ern point of view, and optimization method) to determinethe position of the more unfavourable hinges or sections ofrupture.

A further development arrived with the thrust line theoryand graphic statics during 19th c. Graphic statics supplied apractical method consistently based on the catenary princi-ple. Graphic statics was actually used for the assessment ofa large amount of masonry bridges and large buildings upto the beginning of 20th c. An example is given by Rubiosanalysis of the structure of Mallorca Cathedral (Rubi [127],Fig. 11). Huerta [59] provides a comprehensive review of theapplication of the thrust line theory to the analysis of vaultsand domes.

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Fig. 9 Polenis solution todescribing the equilibrium ofSaint Peters dome as theinverted shape of a catenary(Poleni [116])

Fig. 10 Gaudis hanging model used to design the church of ColniaGell c. 1900 (Rfols [118])

4 Modern Theories to Support Classical Approaches.Plastic Analysis

Heymans [56] formulation for the plastic (or limit) analy-sis of masonry arches synthesizes all the mentioned histori-cal insights and provides a powerful and theoretically soundtool for the study of this type of constructions. Accordingto this formulation, the limit theorems of plasticity can beapplied to masonry structures provided the following condi-tions are verified: (1) The compression strength of the ma-terial is infinite; (2) Sliding between parts is impossible;(3) The tensile strength of masonry is null. These conditionsenable the application of the well known limit theorems ofplasticity.

In particular, these conditions enable the following for-mulation of the lower-bound (or safe) theorem: The struc-ture is safe, meaning that the collapse will not occur, if astatically admissible state of equilibrium can be found. Thisoccurs when a thrust line can be determined, in equilibriumwith the external loads, which falls within the boundariesof the structure. The load applied is a lower-bound of theactual ultimate load (causing failure). The lower-bound the-orem supports the so-called static approach (or static limitanalysis) for safety assessment of masonry structures.

According to the upper-bound theorem, If a kinemati-cally admissible mechanism can be found, for which the

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 307

Fig. 11 Rubis [127]application of graphic statics toanalyze the stability of MallorcaCathedral (detail)

work developed by external forces is positive or zero, thenthe arch will collapse. In other works, if a mechanism is as-sumed (by arbitrarily placing a sufficient number hinges),the load which results from equating the work of the exter-nal forces to zero is an upper-bound of the actual ultimateload. The application of the upper bound theorem leads tothe so-called kinematic approach (or kinematic limit analy-sis) for the study of masonry buildings.

A corollary of the safe theorem, the so-called unique-ness theorem, is also applicable: A limit condition will bereached (i.e., the structure will be about to collapse) if a bothstatically and kinematically admissible collapsing mecha-nism can be found. In other words, the collapsing config-uration will be reached if a thrust line can be found causingas many hinges as needed to develop a mechanism. Hingesare caused by the thrust line becoming tangent to the bound-aries. When this occurs, the load is the true ultimate load, themechanism is the true ultimate mechanism, and the thrustline is the only possible one.

In spite of its ancient origin, limit analysis is regardedtoday as a powerful tool realistically describing the safetyand collapse of structures composed by blocks (includingnot only arches and structures composed of arches, but also

towers, faades and entire buildings). It must be remarked,however, that it can hardly be used to describe the responseand predict damage for moderate or service load levels notleading to a limit condition. Strictly speaking, limit analysiscan only be used to assess the stability or safety of structures.

Limit analysis entails a very deep and conspicuous realityand should be always considered as a complementary tool,or at least as a guiding intuition, when performing alterna-tive computer analyses. Experience shows that, no matterthe level of sophistication of any computer method, it willproduce, at ultimate condition, results foreseeable by meansof limit analysis.

Based on the observation of real seismic failure modes ofhistorical and traditional buildings in Italy, Giuffr [51, 52],see also Giuffr and Carocci [53] and Carocci [23], pro-posed an approach for the study of the seismic vulnerabil-ity of masonry buildings based on their decomposition intorigid blocks (Figs. 12 and 13). The collapse mechanisms arethen analyzed by applying kinematic limit analysis. This ap-proach is particularly interesting as a tool for seismic analy-sis of buildings which do not conform to box behavior be-cause of lack of stiff floor slabs or because of weaker partialcollapses affecting the faade or inner walls.

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More recently, Giuffrs proposal has experienced re-newed interest thanks to the possibility of combining blockanalysis with the capacity spectrum method (Fajfar [43],Lagomarsino et al. [64], Lagomarsino [63]) for the seismicassessment of masonry structures. The method is applied tobuildings, churches and towers (Fig. 14). The resulting ver-ification methodology has been adopted by the seismic Ital-ian code OPCM 3274 [101].

Research on the possibilities of classical limit analysis isstill being carried out. Recently, improved graphic orientedtechniques for analysis of masonry arch and vault structures,based on the combination of the static and kinematic ap-proaches, have been presented by Block et al. [17]. Limitanalysis is currently exploited as a useful tool to analyzeancient structures (Ochsendorf [98], De Luca et al. [42],Block [15]). Roca et al. [125] have produced a method forgraphically-oriented analysis of reinforced masonry struc-tures based on the same principles.

Fig. 12 Failure modes for buildings with no ties (a) and with ties an-coring the faade to lateral walls (Carocci [23])

5 Advanced Computer Developments Based on LimitAnalysis

5.1 Analysis of Blocky Structures

Most modern computer developments based on limit analy-sis exploit the potential of the kinematic approach for theanalysis of masonry structures composed of block assem-blages. The following hypotheses are normally adopted:(1) Limit load occurs at small overall displacements. (2) Ma-sonry has zero tensile strength. (3) Shear failure at the jointsis perfectly plastic. (4) Hinging failure mode at a joint oc-curs for a compressive load independent from the rotation.Hypothesis (1) is true for most cases. Assumption (3) is fullysupported by experimental results. In the case of masonrycrushing, hypothesis (4) might be questionable, but crushingbehavior has minor importance in the response of masonrystructures except for very shallow arches, pillars, towers andmassive vertical structures.

The use of the aforementioned principles in combinationwith modern computers and advanced numerical methodshas provided powerful tools for the analysis of masonry con-structions. Livesley [67], Melbourne and Gilbert [86] (seealso Gilbert and Melbourne [49], and Gilbert [48]), Baggioand Trovalusci [6], Ferris and Tin-Loi [45], Casapulla anddAyala [24], Ordua and Loureno [102104] and Gilbertet al. [50], among others, have proposed different methodsfor the assessment of masonry structures by limit analysis.Modern developments afford the analysis of blocky struc-tures for both in-plane or out-of plane loading.

Ordua and Loureno [102104] have proposed a CapModel for limit analysis for both plane or spatial structures

Fig. 13 Failure mechanisms for buildings embedded within urban texture (based on DAyala and Speranza [41])

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 309

Fig. 14 Kinematic analysisapplied to possible collapsemodels of the faade of SantaMaria del Mar church inBarcelona (Roca et al. [126])

Fig. 15 Out of plane loadedwall, supported at one edge;(a) model; (b) FEM failuremechanism ( = 0.210);(c) limit analysis failuremechanism ( = 0.216)(Ordua and Loureno [104])

made of rigid blocks which takes into account the non-associated flow rules and limited compressive strength ofmasonry. Results obtained with the model have been satis-factorily compared with available experimental results. Anexample concerning a simple wall subjected to seismic loadsis given in Fig. 15, where the amount of horizontal forces re-sisted is measured as a multiplier over the gravity forces.

Limit analysis formulation for blocky structures placessome difficulties regarding the treatment of the normalitycondition. Standard formulations adopt a simple frictionalCoulomb law characterized by a friction angle at the con-tact interfaces. Applying the normality condition (or associ-ated flow rule) leads, in this case, to a fixed dilatancy (nor-mal separation between blocks) characterized by an angle necessarily equal to , where tan is the ratio between nor-mal and tangent deformation (Fig. 16). In reality, no physicalcondition leads to this value, real dilatancy of masonry being

variable and almost null in many cases. However, the limittheorems of plasticity, previously enunciated, are only ap-plicable when associated flow rules are adopted (i.e., whenthe normality condition stands). Adopting a non-associatedflow rule by assuming, for instance, null dilatancy, leadsto non-standard limit analysis for which the limit theoremsare not strictly applicable. Considering finite compressionstrength (for instance, by assuming a contact region sub-jected to uniform yielding compression stresses) leads aswell to non-standard limit analysis. As shown by Orduaand Loureno [103], looking to the safest solution by min-imizing the multipliers obtained by the kinematic approach(upper-bound theorem) may, for non associate flow rules,cause severe underestimation of the collapse load and incor-rect failure mechanisms. Ordua and Loureno [103] haveovercome this problem by means of a load-path followingprocedure in which equilibrium and yield conditions are

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Fig. 16 Associated flow rule (top) with = tan = tan , andnon-associated flow rule (below), with null dilatancy (tan = 0). Theyield criterion is represented as relationship between the normal (N)and shear forces (V ). The corresponding displacements are the normal(n) and tangent (s) ones

first applied for only constant loads. Once the constant loadshave been applied, the variable ones are steadily increased.A solution for the entire set of equations is obtained for asmall compressive effective stress (i.e., a reduced compres-sion strength taking into account transverse cracking) at theinterfaces and the load factor is minimized. Then, solutionsare computed for successively raising compressive effectivestresses until the assessed value is reached.

Gilbert et al. [50], in turn, use a non-associative fric-tional joint model (a specially modified Mohr-Coulomb fail-ure surface) which is continuously updated, within a itera-tive procedure, until a converged solution is obtained. Theprocedure provides reasonable estimates of the ultimate ca-pacity for a wide range of problems, including entire ma-sonry faades (Fig. 17).

5.2 Limit Analysis of Domes, Vaults and Complex SpatialStructures

Recent proposals are exploring the potential of the lower-bound theorem (static approach) for the analysis of vaultedand 3D spatial systems. A formulation for the analysis ofcurved shell masonry members was presented by ODwyer[99], consisting of the decomposition of the shell elementinto a system of arches in equilibrium. By applying thesafe theorem and maximizing the ultimate load, via an op-timization process, ODwyer establishes a procedure whichallows the general application of Heymans limit analysisto vaults and domes. Block and Ochsendorf [16] have pre-sented a Thrust-Network Analysis method for generatingcompression-only surfaces and spatial systems based on aduality between geometry and in-plane forces in networks.

The method is applicable to the analysis of vaulted historicalstructures and for the design of new vaulted ones. Lucchesiet al. [77] have generalized the static approach for arches tothe case of masonry vaults through a maximum modulus ec-centricities surface, a concept analogous to the line of thrustin the case of arches.

Andreu et al. [2, 3] have developed a computer techniquefor the assessment of complex masonry constructions, in-cluding 3D framed structures and shells, inspired on Gaudishanging models (Fig. 18). Skeletal masonry constructionsare modelled as 3D catenary nets composed of numerousvirtual strings subjected to arbitrary loading. Based on thisdescription, limit analysis is applied according to the staticapproach. Cable net solutions complying with the limit the-orems of plasticityin particular, the safe (or lower-bound)and the uniqueness theoremsare generated by means ofconvenient optimization techniques.

6 Elastic Linear Analysis. Possibilities and Limitations

Linear elastic analysis is commonly used in the calculationof steel and reinforced concrete structures. However, its ap-plication to masonry structures is, in principle, inadequatebecause it does not take into account the non-tension re-sponse and other essential features of masonry behavior. Itmust be noted that, due to its very limited capacity in ten-sion, masonry shows a complex non-linear response even atlow or moderate stress levels. Moreover, simple linear elas-tic analysis cannot be used to simulate masonry strength re-sponses, typically observed in arches and vaults, character-ized by the development of partialized subsystems workingin compression. Attempts to use linear elastic analysis to di-mension arches may result in very conservative or inaccurateapproaches. Linear elastic analysis is not useful, in particu-lar, to estimate the ultimate response of masonry structuresand should not be used to conclude on their strength andstructural safety.

Notwithstanding, linear elastic analysis has been used,with partial success, as an auxiliary tool assisting in the di-agnosis of large masonry structures. Easy availability andreduced computer costs have promoted its use, in spite ofthe mentioned limitations, before the development and pop-ularization of more powerful computer applications.

Some examples are the studies of San Marco in Venice(Mola and Vitaliani [93]), Fig. 3, the Metropolitan Cathedralof Mexico (Meli and Snchez-Ramrez [88]), the Tower ofPisa (Macchi et al. [79]), the Colosseum of Rome (Croci[34]) (Fig. 2) and the Church of the Gell Colony inBarcelona, by Gonzalez et al. [54], see also and Roca [119],Fig. 19, among many other. The case of Hagia Sophia hasdeserved much attention and has been analyzed by differ-ent authors using similar modeling techniques (Mark et al.

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 311

Fig. 17 Failure modes for wallssubjected to vertical andhorizontal loading obtained byGilbert et al. [50]

[82, 83], Croci et al. [37], and others). In all these cases, thelimitations of the method were counterbalanced by the verylarge expertise and deep insight of the analysts.

During the last years, non-linear analysis is becomingmore popular thanks to larger software availability and in-creasing computer capacity. However, linear analysis is al-ways performed, prior to the application of more sophisti-cated approaches, to allow a quick and first assessment ofthe adequacy of the structural models regarding the defini-tion of meshes, the values and distribution of loads and re-actions, and the likelihood of the overall results.

Meli and Pea [87] have discussed the possibilitiesof elastic-linear models in providing preliminary infor-mation for the seismic study of masonry churches andthe possibility of using information obtained from themfor constructing more detailed models of critical partsto be then studied separately with more complex analy-ses.

7 Simplified Modelling

7.1 Extensions of Matrix Calculation for Linear Members

The limitations of linear elastic analysis, on the one hand,and limit analysis, on the other hand, can be partly over-come by means of simple generalizations of matrix calcula-tion of frame structures, extended with (1) Improved tech-niques for the description of complex geometries (curved

members with variable sections . . .) and (2) Improved de-scription of the material (for instance, including simple con-stitutive equations yet affording the consideration of crack-ing in tension and yielding / crushing in compression, yield-ing in shear).

These tools are, in principle, only applicable to 2D or3D systems composed of linear members (namely, skeletalstructures). However, there are some proposals to treat 2Dmembers (vaults, walls) as equivalent systems composed ofbeams.

In fact, the application of conventional frame discretiza-tion yields inaccurate results when dealing with shear wallsystems (Karantoni and Fardis [61]). However, these resultscan be improved through the definition of a set of specialdevices to represent more realistically the shear deforma-tion of the walls. Among some different proposals, Kwans[62] technique, in which the shear deformation of the wallsis transferred to rigid arms connecting wall panels or wallswith lintels, has shown to be very accurate for both solid orhollow walls and faades.

Molins and Roca [94] developed a method for the analy-sis of masonry skeletal structures as an extension of con-ventional matrix calculation to systems composed of curvedmembers with variable cross-section. The method includesa set of partial models for the description of the non-linearresponse of masonry taking into account cracking in ten-sion and yielding or crushing in compression. Roca et al.[124] extended the method to analyse 3D systems including

312 P. Roca et al.

Fig. 18 Examples of inverted multiple nets (top) and application to describe the equilibrium condition of a tower of the faade of BarcelonaCathedral (Andreu et al. [2])

masonry load bearing walls using Kwans proposal for the

modelling of wall systems as equivalent frames. The method

has been successfully used in the assessment of faades and

entire buildings.

7.2 Use of Rigid and Deformable Macroelements

Important research efforts have been devoted to the devel-opment of computational approaches based on rigid anddeformable macroelements. Each macroelement models an

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 313

entire wall or masonry panel, reducing drastically the num-ber of degrees of freedom of the structure. Brencich, Gam-barotta and Lagomarsino ([18], Fig. 20) use two nodesmacroelements taking into account the overturning, damageand frictional shear mechanisms experimentally observed inmasonry panels. The overall response of buildings to hori-zontal forces superimposed to the vertical loads is obtainedby assembling shear walls and flexible floor diaphragms.The former are made up of both macroelements, representa-

Fig. 19 (Color online) Linear elastic analysis of the Crypt of the GellColony near Barcelona (Roca [119]). Regions subjected to tensile nor-mal stresses are represented in red colour

tive of piers and spandrels, and rigid elements, representingthe undamaged parts of the walls.

More recent developments, as those due to Casolo andPea [26] and Chen et al. [29], show the permanent inter-est of this type of simplified approaches and their ability tocombine satisfactory accuracy with computer efficiency.

Casolo and Pea [26] have developed a specific rigid ele-ment approach for the in-plane dynamic analysis of masonrywalls. A rigid body spring model (RBSM) has been adoptedconsisting of a collection of plane quadrilateral rigid ele-ments connected to each other by two normal springs andone shear spring at each side. Specific separate hystereticlaws are assigned to the axial and shear deformation be-tween elements. A Coulomb-like law is adopted to relatethe strength of the shear springs to the vertical axial load-ing. The satisfactory performance of the approach has beenproven by comparison with available experimental and nu-merical results on pier walls and faades. The technique hasbeen also successfully applied to the study of a large realmasonry construction, namely the Maniace Castle in Syra-cuse (Casolo and Sanjust [25]).

8 FEM Based Approaches. Macro-Modelling

The finite element method offers a widespread variety ofpossibilities concerning the description of the masonrystructures within the frame of detailed non-linear analysis.

Fig. 20 Analysis withmacro-blocks of a faade walltested by Calvi and Magenes[20]. Damage distribution undermonotonic loading (Brencich,Gambarotta and Lagomarsino[18]) at peak load (A) and at theend of the monotonic loadhistory (B)

314 P. Roca et al.

Fig. 21 Analysis of a shearwall with the plasticity model ofLoureno et al. [75]: deformedmesh (a) and cracks (b)

An example of a pioneering application is found in Markset al. [82, 83] analysis of Hagia Sophia allowing differentportions of the structure to weaken at different levels of ten-sion.

Most of modern possibilities based on FEM fall withintwo main approaches referred to as macro-modelling andmicro-modelling.

Macro-modelling is probably the most popular and com-mon approach due to its lesser calculation demands. Inpractice-oriented analyses on large structural members orfull structures, a detailed description of the interaction be-tween units and mortar may not be necessary. In these cases,macro-modelling, which does not make any distinction be-tween units and joints, may offer an adequate approach tothe characterization of the structural response. The macro-modelling strategy regards the material as a fictitious homo-geneous orthotropic continuum.

An appropriate relationship is established between aver-age masonry strains and average masonry stresses. A com-plete macro-model must account for different tensile andcompressive strengths along the material axes as well as dif-ferent inelastic properties along each material axis. The con-tinuum parameters must be determined by means of testson specimens of sufficiently large size subjected to homo-geneous states of stress. As an alternative to difficult ex-perimental tests, it is possible to assess experimentally theindividual components (or simple wallets and cores, seeBenedetti et al. [8]) and consider the obtained data as inputparameters of a numerical homogenization technique. Com-pared to more detailed approaches affording the descriptionof discontinuities, macro-modelling shows significant prac-tical advantages. In particular, FE meshes are simpler sincethey do not have to accurately describe the internal struc-ture of masonry and the finite elements can have dimensionsgreater than the single brick units. This type of modellingis most valuable when a compromise between accuracy andefficiency is needed.

The macro-models, also termed Continuum Mechanics fi-nite element models, can be related to plasticity or damageconstitutive laws. An example of the former approach is thework of Loureno [70] and Loureno et al. [75] which pro-posed a non-linear constitutive model for in-plane loadedwalls based on the plasticity theory, for which the materialadmissible field is bounded by a Hill-type yield criterion for

compression and a Rankine-type yield criterion for tension(Fig. 21).

In the case of Continuum Damage finite element mod-els, isotropic criteria have been usually preferred because oftheir mathematical simplicity and the need for only few ma-terial parameters. However, a number of orthotropic mod-els have been also proposed. Papas [107] orthotropic modelconsists of a unilateral damage model for masonry includinga homogenization technique to keep into account the tex-ture of brick and mortar. Berto et al. [9] developed a specificdamage model for orthotropic brittle materials with differentelastic and inelastic properties along the two material direc-tions. The basic assumption of the model is the acceptanceof the natural axes of the masonry (i.e. the bed joints and thehead joints directions) also as principal axes of the damage(Fig. 22).

The macro-models have been extensively used with theaim of analyzing the seismic response of complex masonrystructures, such as arch bridges (Pela et al. [114]), historicalbuildings (Mallardo et al. [80]), and mosques and cathedrals(Roca et al. [123], Martnez et al. [84]; Murcia-Delso et al.[96], Figs. 23 and 24).

A drawback of the macro-modelling approach lays in itsdescription of damage as a smeared property spreading overa large volume of the structure. In real unreinforced ma-sonry structures, damage appears normally localized in iso-late large cracks or similar concentrated lesions. A smearedmodelling of damage provides a rather unrealistic descrip-tion of damage and may result in predictions either inaccu-rate or difficult to associate with real observations.

An interesting enhancement to smeared damage repre-sentation was recently proposed by Clemente et al. [32]. Themethod is based on the so-called smeared-crack scalar dam-age model, modified in such a way that it can reproducelocalized individual (discrete) cracks. This is achieved bymeans of a local crack-tracking algorithm. The crack track-ing model enables the simulation of more realistic damagedistributions than the original smeared-crack model. The lo-calized cracks predicted by the crack tracking model be-have, in consistency with limit analysis, as a set o hingesdeveloping gradually and finally leading to a full collaps-ing mechanism. The model has been used to analyze the re-sponse of the structure of Mallorca Cathedral under gravityand seismic forces (Fig. 24). More recently, Clementes et

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 315

Fig. 22 Analysis of acyclically-loaded wall withopenings (from Berto et al. [9]):(a) dx, (b) dy, (c) dx+ and(d) dy+ numerical damagecontours

al. [32] isotropic damage has been modified to account formasonrys orthotropy by Pela et al. ([113], see also Pela[112]).

Very few attempts to combine non-linear constitutiveequations with solid 3D element meshes have been formu-lated. Oate et al. [100]), see also Hanganu [55], proposed a3D continuum damage model for masonry and concrete un-der physical and environmental effects and applied it to thestudy of the Domes of the San Marcos Basilica in Venice.The model was later used to study Barcelona Cathedral(Roca et al. [122]).

9 FEM Based Approaches. Discontinuous Models

Macro-models encounter a significant limitation in their in-ability to simulate strong discontinuities between differentblocks or parts of the masonry construction. Such disconti-nuities, corresponding either to physical joints or individualcracks formed later in the structure, may experience phe-nomena such as block separation, rotation or frictional slid-ing which are not easily describable by means of a FEMapproach strictly based on continuum mechanics.

A possible way of overcoming these limitations consistsof the inclusion within the FEM mesh of joint interface-elements to model the response of discontinuities. A pio-neering application of this possibility is found in Marks etal. [82] study of the buttresses of Hagia Sophia in Istambul

by means of a model combining large blocks with interfaceelements allowing the sliding along the large blocks shapingthe structure.

A more sophisticated set of discontinuous models wasused by Pegon et al. [110, 111] to study the seismic re-sponse of the cloisters of the So Vicente de Fora Monasteryin Lisbon. A full-scale model of part of the cloisters wasbuilt in laboratory and was subjected to intensive experi-mentation to, among other purposes, validate the models.The models included joint models to simulate the block-to-block and block-to-masonry interfaces. Elastic-perfectly-plastic and material softening joint models were used. Theplastic range was defined by means of a Mohr-Coulom lawcharacterized by a friction angle and acohesion. However,the softening model allowed the initial friction angle to de-crease towards a residual value, whereas the cohesion re-mained constant. The study included a combination of 2Danalyses involving the entire faade with more detailed 3Danalysis on selected parts of the structure (Fig. 25). The ma-terial between the joints (stone or brick masonry) was de-scribed as either linear elastic or non-linear plastic homoge-neous material.

10 FEM Based Approaches. Micro-Modelling

In the micro-modelling strategy, the different components,namely the units, mortar and the unit/mortar interface are

316 P. Roca et al.

Fig. 23 (Color online) Analysisof Kk Ayasofya Mosque inIstanbul by a damagemechanicsbasedmacro-model. Distribution oftensile damage parameter (inchromatic scale) for thestructure subjected to deadloading (Roca et al. [123])

Fig. 24 Seismic analysis of Mallorca Cathedral. Smeared damage approach (a) versus localized damage approach (b), for an earthquake of 475years of return period (Clemente et al. [31]). Both diagrams represent the tensile damage scalar parameter in chromatic scale

distinctly described. The so-called detailed micro-models

describe the units and the mortar at joints using continuum

finite elements, whereas the unit-mortar interface is repre-

sented by discontinuous elements accounting for potential

crack or slip planes (Fig. 26). Detailed micro-modelling is

probably the more accurate tool available to simulate the real

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 317

Fig. 25 3D failure mechanismof part of the cloister of SoVicente de Fora Monastery inLisbon as obtained by Pegon etal. [111]

Fig. 26 Modelling strategiesfor masonry structures(Loureno, [70]): masonrysample (a); detailed (b) andsimplified (c) micro-modelling;macro-modelling (d)

behavior of masonry. It is particularly adequate to describethe local response of the material. Elastic and inelastic prop-erties of both unit and mortar can be realistically taken intoaccount.

The detailed macro-modelling strategy leads to very ac-curate results, but requires an intensive computational effort.This drawback is partially overcome by the simplified micro-models (Lotfi and Shing [69]; Tzamtzis [134]; Lourenoand Rots [74], Fig. 27; Gambarotta and Lagomarsino [46],Fig. 28; Sutcliffe et al. [132]), where expanded units, repre-sented by continuum elements, are used to model both unitsand mortar material, while the behavior of the mortar jointsand unit-mortar interfaces is lumped to the discontinuouselements (Fig. 26). Masonry is thus considered as a set ofelastic blocks bonded by potential fracture/slip lines at thejoints.

The micro-modelling approaches are suitable for smallstructural elements with particular interest in strongly het-erogeneous states of stress and strain. The primary aim isto closely represent masonry based on the knowledge of theproperties of each constituent and the interface. The nec-essary experimental data must be obtained from laboratory

tests on the constituents and small masonry samples. Nev-ertheless, the high level of refinement required means an in-tensive computational effort (i.e. great number of degreesof freedom of the numerical model), which limits micro-models applicability to the analysis of small elements (aslaboratory specimens) or small structural details.

11 Homogenization

Midway between micro-modelling and macro-modellingstands the so-called homogenized modelling. If the structureis composed by a finite repetition of an elementary cell, ma-sonry is seen as a continuum whose constitutive relationsare derived from the characteristics of its individual com-ponents, namely blocks and mortar, and from the geometryof the elementary cell. Most of the methods of homogeniza-tion simplify the geometry of the basic unit with a 2 stepintroduction of vertical and horizontal joints and thus with-out taking into account the regular offset of vertical mortarjoints. However, this kind of approach results in significanterrors when applied to non-linear analysis

318 P. Roca et al.

Micromechanical homogenization, derived independentlyby van der Pluijm [135], Lopez et al. [68] and Zucchiniand Loureno [137], is based on the detailed finite elementanalysis of the elementary cell and overcomes the approx-imations introduced by the 2 steps simplified method. Re-cent advances in terms of sophisticated analysis homoge-nization tools are discussed in Loureno et al. [76] and in-clude the polynomial stress field expansion approach of Mi-lani et al. [89] and the mesoscopic approach of Massart etal. [85], Calderini and Lagomarsino [19], and Shieh-Beygiaand Pietruszczak [130].

A micro-mechanical model for the homogenized limitanalysis of in-plane loaded masonry has been proposed byMilani et al. [8992]. It is developed with the aim to obtainthe homogenized failure surfaces for masonry. The strengthdomains are implemented in finite element limit analysis

Fig. 27 Micro-modelling of masonry shear walls (Loureno [70]).Top: Load-displacement diagram. Bottom: Results of the analysis ata lateral displacement of 2.0 mm: (a) deformed mesh; (b) damange

codes and numerically treated both with a lower and an up-per bound approach (Fig. 29). The main advantages of thismethod, compared with classical micro-modelling, are thefollowing: (1) The finite element mesh does not have to re-produce the exact pattern of the masonry units nor it hasto be so fine. The structure can be meshed automatically.(2) Once the homogeneous properties have been calculatedfrom the micro-mechanical model, standard finite elementmethod can be used to perform the analysis avoiding thecomplications introduced by elements interfaces.

The micro-mechanical approach has been successfullyapplied to both linear and non-linear problems. In a firststep, Zucchini and Loureno [137] introduced homogeniza-tion in the elastic field by deriving the mechanical propertiesof masonry from a suitable micro-mechanical model tak-ing into account the staggered alignment of the units. Thehomogenization model has been subsequently extended tonon-linear problems in the case of a masonry cell failure un-der tensile loading parallel to the bed joint, Zucchini andLoureno [138], or under compressive loading perpendic-ular to the bed joint, Zucchini and Loureno [139]. Theseresults are accomplished by coupling the elastic micro-mechanical model with a damage model in tension anda plasticity model in compression. To extend the modelto mixed loading conditions Zucchini and Loureno [140]used a full periodic cell with an antisymmetric deformationmechanism. The internal structure of the cell is representedby five different components, namely units, two antisym-metric bed joints, head joints and cross joints.

12 Non-linear FEM Analysis of Vaulted Structures

Few non-linear developments have been produced aiming tothe analysis of vaulted structures. Domes and vaults posesignificant difficulties due to their curved, two-dimensionaland spatial character. In fact, most of the applications fornon-linear analysis of masonry structures so far mentionedare oriented to two-dimensional planar problems.

A pioneering work is found in the studies by Oate et al.[100] of the domes of San Marco Basilica in Venice. Theset of vaults was modelled by means of a continuum dam-age model for masonry and concrete under mechanical and

Fig. 28 Micro-modelling ofmasonry, from Gambarotta andLagomarsino [46]

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 319

Fig. 29 3D homogenized limitanalysis of a masonry building(Milani et al. [92])

Fig. 30 Study of Saint Marcosdomes in Venice by a continuumdamage model (Oate et al.[100])

other (physical, chemical, biological) deterioration effects.The analysis attained a characterization of the safety condi-tion of the system of vaults (Fig. 30).

Croci et al. [38] carried out a finite element analysis ofthe Cathedral of Sta. Mara, in Vitoria, Spain. The analysis,applied to the main transverse sections of the building andto the nave vaults, followed an incremental strategy to ac-count for cracking due to tension or shear stresses, as well as

the equilibrium second-order effects. Similar analyses were

also used for the study of the collapse of Beauvais Cathe-

dral (Croci et al. [39]) and the effects of the earthquake of

September 1997 on the Basilica of Assisi (Croci [35]).

Barthel [7] elaborated very detailed finite element mod-

els to analyze Gothic cross-vaults. The models were used

in combination with partial constitutive models enabling the

320 P. Roca et al.

simulation of masonry cracking as well as sliding betweenarch ring joints.

Cauvin and Stagnitto [27, 28] carried out studies onGothic cross vaults using both limit analysis and FEM non-linear analysis. Their method was successfully applied to thestudy of the central nave of Reims Cathedral.

Loureno [71] proposed a formulation for the study ofmasonry spatial and curved shells. To our knowledge, it isthe only one based on the micro-modelling technique. It in-cludes constitutive equations, stemming from plasticity, tosimulate the response of the material in combination withjoint elements to describe the sliding of blocks.

13 Discrete Element Method

The Discrete element method (DEM) is characterized bythe modeling of the material as an assemblage of distinctblocks interacting along the boundaries. According to its pi-oneer proposers, Cundall and Hart [40], the name discreteelement applies to a computer approach only if (1) it al-lows finite displacements and rotations of discrete bodies,including the complete detachment and (2) it can recognizenew contacts between blocks automatically as the calcula-tion progresses.

The formulation, initially oriented to the study of jointedrock, was later extended to other engineering applicationsalso requiring a detailed study of contact between blocksor particles, such as soils and other granular materials(Ghaboussi and Barbosa [47]). Finally, it has also been ap-plied to the modeling of masonry structures (Pagnoni [105];Lemos [66]; Sincraian [131]). The common idea in the dif-ferent applications of the discrete element method to ma-sonry is the idealization of the material as a discontinuumwhere joints are modeled as contact surfaces between dif-ferent blocks. This approach affords the modeling of vari-ous sources of non linear behavior, including large displace-ments, and suits the study of failures in both the quasi staticand dynamic ranges.

Distinct element methods, discrete-finite elements anddiscontinuous deformation analysis are different formula-tions of the discrete element method with important appli-cations to masonry structures.

Distinct element methods are direct derivations of thefirst work by Cundall and Hart. They involve soft contactformulations where a normal interpenetration is needed torecognize contact between two different bodies. The mainfeatures of these methods are: (1) No restriction of blockshapes and no limitation to the magnitudes of translationaland rotational displacements. The approximation that all thedeformations occur at the surfaces of blocks is made. It isalso assumed that forces arise only at contacts between acorner and an edge. (2) Forces arise due to deformation.

A change in displacement results in a change in force whichis added to the existing force stored for the contact. (3) Ac-celerations are computed from the forces and moments foreach block. The accelerations are further integrated into ve-locity and displacements. (4) Contact updating is performedwhen the sum of the displacements of all of the elementshas exceeded a certain value. To increase the efficiency, onlyblocks within a certain distance range are checked for newcontacts.

Discrete-finite element methods recollect different at-tempts of combining FEM with multi-body dynamics.Munjiza et al. [95] developed a method for the simulationof fracturing problems considering deformable blocks thatmay split and separate during the analysis. Mamaghani et al.[81] used a fixed contact system with a small deformationframework and finite deformations concentrated in contactelements. Contacts, discontinuities and interfaces were con-sidered as bands with a finite thickness. The contact elementwas a two-nodded element having normal and shear stiff-nesses. The method was applied to the stability analysis ofdifferent masonry structures. The failure mode of a masonryarch is shown in Fig. 31.

The discontinuous deformation analysis (DDA) is a 2Dmethod developed by Shi and Goodman [129] for rock engi-neering analysis subsequently applied to masonry (Ma et al.[78]). Blocks are considered deformable but with a uniformstrain and stress distribution. Contact is considered rigid andno interpenetration is permitted. This condition is enforcednumerically by an iterative procedure at each time step.

The natural field of application of DEMs is composedby structures formed by regurarly shaped masonry or stoneblocks. Rocking motion of stone blocks (Pea et al. [115]),static and dynamic analysis of load bearing walls (Pagnoni[105], Baggio and Trovalusci [5], Schlegel and Rauten-strauch [128]), stone bridges (Lemos [65], Bicanic et al.[10]), columns and architrave (Papastamatiou and Psy-charis [109], Psycharis et al. [117], Fig. 32), arch and pillar(Pagnoni [105], Pagnoni and Vanzi [106], Lemos [66]) aretypical examples of DEM analysis. The analysis of complexstructures is still a controversial topic in DEM. Computa-tional viability of analysis may limit severily the numberof block elements that can be included in a model. Modelsprepared to simulate the response of real structures may re-sult in too coarse or unrealistic discretizations or 2D, andspecially, 3D real masonry structures.

14 Conclusions

In spite of the important advances experienced by structuralanalysis methods, the study of masonry historical structuresis still a challenging activity due to the significant difficultiesencountered in the description of their complex geometry,

Structural Analysis of Masonry Historical Constructions. Classical and Advanced Approaches 321

Fig. 31 Failure mode of a masonry arch (Mamaghani et al. [81])

Fig. 32 Final position of the column-architrave model of theParthenon Pronaos, without reinforcement, for a simulated earthquake(Psycharis et al. [117])

materials, morphology (member composition and connec-tions), and present condition, including damage and alter-ations. In particular, the modelling of the mechanical behav-

ior of masonry is very demanding due to its fragile nature intension and composite character at the macro-scale.

A wide set of possibilities have been developed to de-scribe masonry structures to different levels of accuracy,from rough (but useful) engineering approaches to very de-tailed modelling taking into account the distinct responseof the individual components. A very significant effort isundertaken at present to produce homogenization criteriaallowing a better interconnection between the micro- andmacro-analysis levels.

Detailed methods, such as micro-models, are still requir-ing high computer effort and, in practice, can only be usedto analyze small individual members such as solid or hol-low walls. Moreover, sophisticated methods often requirecomplex material properties which can only be determinedthrough costly and sophisticate laboratory experiments. Inthe case of the study of real buildings, such properties are notnormally available and need to be indirectly estimated basedon more common and available evidence. However, inade-quate assumptions on these properties may compromise thereal gain in accuracy provided by the sophisticated methodsand even lead to inadequate results.

With further developments in computer technology andnumerical methods, the analysis of entire complex histori-cal structures (including for instance, Gothic cathedrals) us-ing very accurate approaches, may become possible in thenear future. Further developments will make very demand-ing analyses, such as the nonlinear time-domain dynamicone, possible even if used in combination with complexstructural and material models. However, many of the dif-ficulties mentioned, as those related to the adequate surveyand description of the structure, including material featuresand damage, will still compromise the accuracy and realismof the numerical predictions. In this context, the judgmentof the analyst is essential in order to conclude on the accept-ability of the results.

Sufficient validation or calibration of numerical models,based on the comparison with empirical information, willbe always necessary to grant the reliability of the numer-ical models and their capacity to predict on the structuralresponse and safety. In the case of historical structures, thisempirical information can be obtained from historical inves-tigation on the past performance, inspection and monitoring(Icomos/Iscarsah Comittee [60]). In the study of a historicalstructure, structural analysis is a key activity to be developedin combination with other complementary, but also impor-tant activities oriented to both gathering the input informa-tion and allowing later calibration or validation of results.

Acknowledgements Part of the studies presented here were devel-oped within the research projects SEDUREC and BIA2004-04127,funded by DGE of the Spanish Ministry of Science and Technology,whose assistance is gratefully acknowledged.

322 P. Roca et al.

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