Engineering Structures 56 (2013) 1787–1799

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Engineering Structures

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TREMURI program: An equivalent frame model for the nonlinear seismicanalysis of masonry buildings

0141-0296/$ - see front matter � 2013 Published by Elsevier Ltd.http://dx.doi.org/10.1016/j.engstruct.2013.08.002

⇑ Corresponding author. Tel.: +39 0103532521; fax: +39 0103532546.E-mail address: sergio.lagomarsino@unige.it (S. Lagomarsino).

Sergio Lagomarsino a,⇑, Andrea Penna b, Alessandro Galasco b, Serena Cattari a

a Dept. of Civil, Environmental and Chemical Engineering, University of Genoa, Italyb Dept. of Civil Engineering and Architecture, University of Pavia, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 18 February 2013Revised 30 July 2013Accepted 2 August 2013Available online xxxx

Keywords:Masonry buildingsSeismic assessmentEquivalent frame modelPushover analysisMixed masonry–r.c. constructions

The seismic analysis of masonry buildings requires reliable nonlinear models as effective tools for bothdesign of new buildings and assessment and retrofitting of existing ones. Performance based assessmentis now mainly oriented to the use of nonlinear analysis methods, thus their capability to simulate thenonlinear response is crucial, in particular in case of masonry buildings. Among the different modellingstrategies proposed in literature, the equivalent frame approach seems particularly attractive since itallows the analysis of complete 3D buildings with a reasonable computational effort, suitable also forpractice engineering aims. Moreover, it is also expressly recommended in several national and interna-tional codes. Within this context, the paper presents the solutions adopted for the implementation ofthe equivalent frame model in the TREMURI program for the nonlinear seismic analysis of masonrybuildings.

� 2013 Published by Elsevier Ltd.

1. Introduction

The large population of existing and historical unreinforced ma-sonry buildings all over the world and their potential high vulner-ability to earthquake require to improve the knowledge of theirseismic behaviour, setting up analytical and numerical modelsfor their structural assessment. Actually, the reliability of modelsrepresents one of the most important issues involved in both thedesign of new buildings and, in particular, in the assessment andstrengthening of existing ones.

In this paper, the attention is focused only on the global re-sponse of masonry buildings, that is assuming that proper connec-tions prevent the activation of local failure modes mainlyassociated with the out-of-plane response of walls. Within thiscontext, the global seismic response is strictly related both to thein-plane capacity of walls and to the connection and load transfereffects due to the floor and roof diaphragms. Thus, in most of thecases, it is necessary to refer to methods of global analysis andthree-dimensional models. As regards the analysis methods forthe seismic assessment, in the last decades, the performance-basedearthquake engineering concepts have led to an increasing use ofnonlinear static analyses (pushover). As it is well known, thesesimplified procedures based on pushover analysis [1,2] result inthe comparison between displacement capacity of the structure(identified for different performance limit states) and the displace-

ment demand, which depends both on structure and earthquakecharacteristics. The definition of the displacement capacity for sig-nificant limit states requires the evaluation of a force–displace-ment curve (‘‘pushover’’ curve), able to describe the overallinelastic response of the structure under horizontal seismic load-ings and to provide essential information to idealize its behaviourin terms of stiffness, overall strength and ultimate displacementcapacity. This curve can be obtained by a nonlinear incrementalstatic (pushover) analysis, i.e. by subjecting the structure, idealizedthrough an adequate model, to a static lateral load pattern (simu-lating seismic inertial forces), increasing the total force and/or thedisplacements, with possible updating of the force distribution(adaptive pushover).

Among the possible modelling strategies proposed in literatureand codes, this work is focused on the equivalent frame modellingstrategy [3,4]. According to this approach, each resistant masonrywall is subdivided into a set of deformable masonry panels, inwhich the deformation and the nonlinear response are concen-trated, and rigid portions, which connect the deformable ones. Thisapproach, which is also suggested in some seismic codes [5,6], re-quires a limited number of degrees of freedom, with a reasonablecomputational effort, allowing the analysis of complex three-dimensional models of URM structures, obtained by assemblingwalls and floors, mainly referring to their in-plane strength andstiffness contributions. Moreover the idealization as an equivalentframe easily allows to introduce other structural elements, such asreinforced concrete beams or columns, together with the masonryones. Thus, it appears particularly versatile to model also mixed

Fig. 2. URM wall idealization according to simplified and equivalent frame models.

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structures (e.g. mixed masonry and reinforced concrete structureswhich are quite common in existing buildings).

The solutions adopted for the implementation of the equivalentframe model in the TREMURI computer program are presented anddiscussed in the following. TREMURI was originally developed andgradually improved at the University of Genoa, starting from 2001[7,8], and subsequently also implemented in the commercialsoftware 3Muri [9]. Starting from the presentation of some generalissues on the possible strategies for the idealization of the masonrywall in an equivalent frame (Sections 2 and 3), the solutionsadopted in TREMURI for the different structural elements(Section 4), the assembling of complete 3D models and the seismicanalysis by means of nonlinear static analysis procedures(Section 5) are illustrated in the following sections. Regarding theformulations of structural elements (both masonry and reinforcedconcrete ones) the attention is only focused on the nonlinear beammodel with lumped inelasticity idealizations. Finally, someexamples of applications are illustrated in Section 6 to show thecapability of the proposed model of assessing the seismic responseof masonry buildings.

2. Equivalent frame modelling of URM walls

Structural element modelling strategies are based on the iden-tification of macroscopic structural elements, defined from a geo-metrical and kinematic point of view through finite elements(solid, shell or frame) and described from a static point of viewthrough their internal generalized forces. In the field of structuralelement models, the ‘‘equivalent frame’’ ones are the most widelydiffused. They consider the walls as an idealized frame, in whichdeformable elements (where the nonlinear response is concen-trated) connect rigid nodes (parts of the wall which are not usuallysubjected to damage). Focusing on the in-plane response of com-plex masonry walls with openings, usually two main structuralcomponents may be identified: piers and spandrels. This idealiza-tion starts from the earthquake damage observation that showsas usually cracks and failure modes are concentrated in such ele-ments (Fig. 1). Piers are the main vertical resistant elements carry-ing both vertical and lateral loads; spandrel elements, which areintended to be those parts of walls between two vertically alignedopenings, are secondary horizontal elements (for what concernsvertical loads), which couple the response of adjacent piers in thecase of lateral loads. It is worth noting that, although ‘‘secondaryelements’’, spandrels significantly affect the boundary conditionsof piers (by allowing or restraining end rotations) with significantinfluence on the wall lateral capacity.

Fig. 2 reports a sketch aiming at representing the idealization ofa wall with openings as an assemblage of structural elements. Dif-ferent schemes are illustrated according to very simplified models,for which the actual modelling of spandrels behaviour is notrequested, and the Equivalent Frame (EF) discretisation that

Fig. 1. Examples of in-plane failure modes with damage concentration in piers and spandright).

considers both pier and spandrel elements. In particular, the ideal-ization of a ‘‘strong spandrels-weak piers’’ model (SSWP in Fig. 2) isbased on the assumption piers crack first, thus preventing the fail-ure of spandrels which can be then assumed as infinitely stiff por-tions, assuring a perfect coupling between piers. This correspondsto assuming a fixed-rotation boundary condition at the piersextremities and it is also known as ‘‘storey mechanism’’ [10]. Onthe contrary, in case of the ‘‘weak spandrels-strong piers’’ (WSSPin Fig. 2), the hypothesis of both null strength and null stiffnessof spandrels is adopted then assuming the piers as uncoupled (thiscorresponds to the cantilever idealization). However, it is worthnoting that in most cases it is correct to assume that horizontal dis-placement of the vertical structural elements are at least coupledat the floor levels by the presence of horizontal diaphragms.

Once the choice has been made, according to the assumptions ofthese simplified models, since only pier elements are modelled, thedefinition of both their effective height and boundary conditionsplays a crucial role for the reliable assessment of the overall capac-ity of the wall. Usually only preliminary evaluations on the effec-tiveness of spandrels are requested in order to properly orientatethe choice between these two extreme idealizations. Both SSWPand WSSP models are expressly suggested by FEMA guidelines[11,12] and model SSWP is consistent with the POR method [13],which was largely adopted in Italy after the 1980 Irpinia earth-quake [14]. In the Italian Building Code [6] the WSSP hypothesis

rels (examples from L’Aquila 2009 – left and centre – and Emilia 2012 earthquakes –

S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799 1789

is assumed for the simplest allowed modelling technique (cantile-ver models), whilst the SSWP hypothesis (storey mechanism) is nomore allowed for the assessment of multi-storey masonrybuildings.

Despite the advantage of adopting very simplified and manage-able models, since they are based on an aprioristic choice, the fol-lowing troublesome issues arise. First of all, it is conceivable thatboth of these limiting cases are inappropriate for certain walls,which may display both types of response in different regions orwhich can be involved in a different behaviour with the increaseof nonlinear response. Moreover, it is not at all a foregone conclu-sion that the presence of certain constructive details (e.g. r.c.beams coupled to spandrels), not supported by a quantitative eval-uation of their effectiveness, is sufficient to assure the achievementof the hypotheses which these simplified models are based on [15].On the contrary in case of the EF Model, since both pier and span-drel elements are modelled, the transition through differentboundary conditions is directly a consequence of the progressivedamage of the elements. Actually, in some cases, the use of theEF Model is regulated in codes, by defining the cases in which ma-sonry spandrels may be taken into account as coupling elements inthe structural model [5]; these provisions mainly concern thebonding to the adjoining walls, the connection both to the floortie beam and to the lintel.

Once having idealised the masonry wall as an assemblage ofstructural elements, the reliable prediction of its overall behaviourmainly depends on the proper representation of characteristics ofeach single structural member. In this paper the attention is fo-cused only on the nonlinear beam and lumped inelasticity idealiza-tions. Within this context, several applications may be found in theliterature. Some of them focus on the formulation of nonlinearbeams [16] or programs specifically oriented to the analysis of ma-sonry buildings [17,18]. Others are based on the use of general pur-pose software packages [19–22].

In the following the attention is focused on the solutionsadopted in TREMURI program developed at the University of Genoa[7], starting from the formulation of a more refined nonlinearmacro-element model [23–26]. A more simplified nonlinear for-mulation similar to the one suggested in [17] was then introducedmainly addressing to engineering practice aims and to performingpushover analysis (this is also implemented in the commercial re-lease of the program, 3Muri [9]).

The main distinctive features of the TREMURI program, whencompared to the other models mentioned above, are: (a) as it isspecifically oriented to the seismic analysis of masonry structures,the possiblity to easily implement different formulations for ma-sonry panels (Section 4.1.1) and alternative algorithms for thepushover analysis (Section 6.1); (b) the explicit modelling of flexi-ble horizontal diaphragms (Section 5.2), which are very common,particularly in ancient existing buildings; (c) the 3D assemblingof masonry walls, which behave in-plane, and floor/roof dia-phragms, drastically reducing the number of degrees of freedom(Section 5.1).

3. Idealization of the masonry wall in an equivalent framemodel

The first step for modelling of the masonry wall as an equiva-lent frame is the identification of the main structural components,previously introduced as piers and spandrels.

For the identification of the geometry of pier and spandrelelements, conventional criteria are often assumed in literature,supported by the damage survey after earthquakes and experimen-tal campaigns. However, a systematic parametrical analysis eithernumerical or experimental has never been performed in order to

define rigorous criteria. Despite this, although the identificationof masonry piers and spandrels may result rather trivial and easilyautomated in case of perforated walls with regularly distributedopenings, it becomes more difficult and ambiguous when openingsare irregularly arranged (Fig. 3). In the following some possible cri-teria are examined.

Usually the criteria for the definition of the height of masonrypiers are defined as a function of that of adjacent openings. A com-monly adopted criterion conventionally assumes a maximum 30�inclination of the cracks starting from the opening corners andconsistently provides an increased height for the external piers.This is also the initial hypothesis proposed in [27] for the definitionof the equivalent height of masonry panels in models based on thestorey mechanism. In [28] it is proposed to define it as the heightover which a compression strut is likely to develop at the steepestpossible angle (i.e. assuming that cracks can develop either hori-zontally or at 45�). In case of existing buildings, the pattern ofpre-existing cracks should be taken into account in order to prop-erly define the geometry of spandrels and piers.

In the following some criteria that can be easily automated (andactually already implemented in the 3Muri software) are dis-cussed. Fig. 4 summarizes the main steps of the frame idealizationprocedure in a regularly perforated masonry wall: from the identi-fication of spandrels and piers (steps 1 and 2) to that of nodes (step3). Spandrel elements (step 1) are defined on basis of the verticalalignment and overlap of openings: the length and the height areassumed equal to the distance and width (in case of full alignment)of the adjacent openings, respectively. Pier elements (step 2) aredefined starting from the height of adjacent openings: when theselatter are perfectly aligned, as the case of the internal pier shown inFig. 4, the height is assumed equal to that of openings. For the def-inition of height of the external piers the possible development ofinclined cracks from the opening corners (and/or from the linteledges) has to be considered, as previously discussed. As possibleapproximated criteria, it can be assumed equal to the height ofthe adjacent opening or as the average of the interstorey heightand the height of the opening. The geometry of the rigid nodes(step 3) comes out directly from the previously defined elementsthat are connected to them.

To complete the frame idealization for the whole wall, such acalculation is done separately for each storey and each wall. It isworth noting that the application of such a criterion without anylimitation to the cone diffusion angle may induce a significantoverestimation of the effective in-plane aspect ratio of externalpiers in case of adjacent openings with a limited height and closeto the wall edge. Actually, in these situations flexural failuremodes are likely predicted for such slender piers, with possibleunderestimation of the lateral strength and overestimation ofthe deformation capacity. The presence of other structural ele-ments, such as lintels and r.c. tie-beams, can influence the effec-tive height of masonry piers and, in principle, for irregularlydistributed openings it should also vary depending on the direc-tion of analysis.

In case of not perfectly aligned openings, a possible choice is toconventionally assume a mean value for the height of spandrel ele-ments as a function of the overlapping part between the openingsat the two levels (Fig. 5); when no overlap is present or the openinglacks at all, it seems more appropriate to assume the portion of ma-sonry as a rigid area (Fig. 5). Further studies, based on both exper-imental testing and numerical research, should be performed inorder to validate the capability of the presented procedure for dif-ferent types of opening layout.

Finally, the actual efficiency of masonry panels must be care-fully assessed and considered in the equivalent frame modellingof the wall. For example, infilled openings (as shown in Fig. 3)are sometimes weak and badly connected and, in this case,

Fig. 3. Examples of façades with regularly and irregularly distributed openings.

Step 2- Identification of piers Step 1- Identification of spandrels

b s

b s/2

Step 3- Identification of nodes Equivalent frame

(Hin

t. + H

door

) /2

Fig. 4. Example of equivalent frame idealization in case of regularly distributed openings.

Pier

Spandrel

Rigid node

Barycentric Axis of element

Non-linear beam / macro-element

Equivalent Frame Idealisation

Fig. 5. Example of equivalent frame idealization in a case of irregularly distributedopenings.

1790 S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799

they could be idealized, on the safe side, as openings, henceneglecting the contribution of added masonry. This can be justi-fied by the difficulty to guarantee a full interlocking with theadjacent pre-existing masonry portions and the stress redistribu-tion effects, which hardly may reproduce the original configura-tion without opening. As an alternative, reduced mechanicalproperties could be assigned to the corresponding infilled ma-sonry portions.

4. Modelling of structural elements

Once having idealised the masonry wall into an assemblage ofstructural elements, the reliable prediction of its overall behaviourmainly depends on the proper interpretation of the single elementresponse. As mentioned above, several formulations, characterizedby different degrees of accuracy, may be adopted either formasonry panels and other structural types. The possibility ofmodelling the nonlinear response of structural elements other than

masonry ones, such as reinforced concrete (r.c.), steel or woodenbeams, is particularly useful for the analysis of new and existingbuildings. As an example, from the beginning of the twentieth cen-tury, the spreading of r.c. technology has caused the birth of mixedstructural solutions inspired by practical aspects and higher archi-tectural freedom: (a) new mixed masonry–r.c. buildings (e.g. build-ings with perimeter masonry walls and internal r.c. frames); (b)mixed buildings resulting from interventions carried out on exist-ing masonry structures (e.g. replacement of internal masonry wallsby r.c. frames, r.c. walls inserted for supporting lifts and staircases,additional storeys made of r.c. structure). Indeed, these structuralmodifications may turn out in a potential high increase of the seis-mic vulnerability, as discussed in [29].

In the following, the attention is focused on a simplified formu-lation based on nonlinear beam elements with lumped inelasticityidealization (bilinear elastic perfectly plastic behaviour). The ele-ment response is directly faced in terms of global stiffness,strength and ultimate displacement capacity by assuming a properforce–displacement relationship and appropriate drift limits (orchord rotation limits in the case of r.c. elements). Despite someunavoidable approximations of the actual behaviour (e.g. relatedto the mechanical description of damage and dissipation mecha-nisms), this simplified formulation implies the following mainadvantages:

– It allows performing nonlinear static analyses with a reasonablecomputational effort, suitable also in engineering practice;

– It is based on few mechanical parameters that may be quitesimply defined and related to results of standard tests.

Moreover, it has to be stressed this formulation is consistentwith the recommendations included in several seismic codes[30,31,6], since strength criteria defined for both bending andshear failure modes can be easily implemented and adopted todefine the lateral strength of the different structural elements.

S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799 1791

4.1. Masonry elements

The specific characterization of the force–displacement rela-tionship, aiming to describe the masonry panels behaviour, startsfrom the knowledge and interpretation of the different failuremodes which may occur.

Observation of seismic damage to complex masonry walls, aswell as laboratory experimental tests, have shown that a masonrypanel subjected to in-plane loading may show two typical types ofbehaviour: flexural behaviour, that may be associated to the failuremodes of Rocking (panel starts to behave as a nearly rigid bodyrotating about the toe) and Crushing (panel is progressively charac-terized by a widespread damage pattern, with sub-vertical cracksoriented towards the compressed corners); shear behaviour, thatmay be associated with the failure modes of Diagonal Cracking (pa-nel usually develops cracks at its centre, that after propagate to-wards the corners) and Shear Sliding (failure is attained withsliding on a horizontal bed joint plane). Despite this classification,it is evident that also mixed modes are possible and quite common.Actually it is worth noting that this classification usually is implic-itly referred to the pier element type. In fact, while many experi-mental researches related to the behaviour of piers have beencarried out in the last decades, tests on spandrels are very limitedand quite recent [32–35]. Indeed, the boundary conditions thatcharacterize spandrel elements and the orientation of main mortarjoints activated are very different from those of piers: as a conse-quence, relevant differences may be noticed. In particular in caseof flexural behaviour, due to low values of axial load, which usuallycharacterize spandrel elements (especially in case of lack oftie-rods or r.c. beams), Crushing represents a very rare instance.Moreover, in case of the shear behaviour, due to the interlockingphenomena, Sliding failure (meant as sliding on a vertical jointplane at the end-sections) usually cannot occur.

As well known, the occurrence of these different failure modesdepends on several parameters. In case of piers, they may be sum-marized as follows: the geometry; the boundary conditions; theaxial load; the mechanical characteristics of the masonry constitu-ents (mortar, blocks and interfaces); the masonry characteristics(block aspect ratio, in-plane and cross-section masonry pattern).In case of spandrels, as shown in the referenced experimental cam-paigns, some additional variables can play an important role, likethe interlocking phenomena which can be originated at end-sec-tions with the contiguous masonry portions, the type of lintels(in particular masonry arches or architraves in stone, timber, steelor r.c.), the interaction with other structural elements coupled to it(in particular if tensile resistant such as r.c. beams or steel tie-rods).

The above introduced failure modes may be interpreted, interms of resultant maximum shear, by some simplified strengthcriteria, based on mechanical or phenomenological hypotheses,which are proposed in literature and codes. Usually, they are basedon the approximate evaluation of the local/mean stress state pro-duced by the applied forces on predefined points/sections of thepanel, assessing then its admissibility with reference to the limitstrength domain of the constituent material, usually idealisedthrough oversimplifications based on few mechanical parameters.As a function of the current value of the axial force (N) acting onthe element, the minimum value – as predicted by the criteriaadopted to model the flexural and shear responses, respectively –is usually assumed as reference. In addition, it is worth noting that,due to the application of horizontal load patterns, aimed to simu-late seismic actions, the acting axial load changes from the initialvalue consequent to the vertical dead loads; moreover, due toredistribution phenomena associated with the progressing of non-linear response, further variations may occur. As a consequence, itis evident how also the value of the corresponding shear strength

varies in each panel during the nonlinear static analysis. Then, fail-ure of the panel is usually defined through the definition of a max-imum drift (du) based on the prevailing failure mechanismoccurred in the panel (e.g. as proposed in both national and inter-national codes [5,6,31,30]). Fig. 6 schematically shows the above-mentioned issues.

Further details on the specific criteria and formulations imple-mented in TREMURI program are illustrated in Section 4.1.1.

4.1.1. Modelling of masonry piers and spandrelMasonry panels (piers and spandrels) are modelled as 2D ele-

ments by assuming a bi-linear relation with cut-off in strength(without hardening) and stiffness decay in the nonlinear phase(for non-monotonic action). The kinematic variables and general-ized forces aimed to describe the 2D element are summarized inFig. 6. It is important to stress that loads are applied only on nodes,thus no loads act along the element. The initial elastic branch is di-rectly determined by the shear and flexural stiffness, computed onthe basis of the geometric and mechanical properties of panel, assummarized in the stiffness matrix (Ke, in squared brackets), asfollows:

Vi

Ni

Mi

Vj

Nj

Mj

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;¼

12EJgh3ð1þwÞ

0 � 6EJgh2ð1þwÞ

� 12EJgh3ð1þwÞ

0 � 6EJgh2ð1þwÞ

0 EAh 0 0 � EA

h 0

� 6EJgh2ð1þwÞ

0 EJgð4þwÞhð1þwÞ

6EJgh2ð1þwÞ

0 EJgð2�wÞhð1þwÞ

� 12EJgh3ð1þwÞ

0 6EJgh2ð1þwÞ

12EJgh3ð1þwÞ

0 6EJgh2ð1þwÞ

0 � EAh 0 0 EA

h 0

� 6EJgh2ð1þwÞ

0 EJgð2�wÞhð1þwÞ

6EJgh2ð1þwÞ

0 EJgð4þwÞhð1þwÞ

26666666666664

37777777777775

ui

wi

/i

uj

wj

/j

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;ð1Þ

where the w coefficient is computed as 1.2El2/(Gh2); E and G are theYoung and shear moduli, respectively; A and J are the cross-sectionand the moment of inertia of the panel, respectively; l and h arelength and height of the panel; g is a stiffness reduction coefficientaiming at accounting for the panel ‘‘cracked’’ conditions. As regardsthe g coefficient, since the progressive degradation of the stiffness isnot actually modelled, a calibration of the initial mechanical prop-erties is necessary. Concerning this point, codes and recommenda-tions [5,6] provide only rough information. Usually, it is proposedto adopt reduced values of the elastic stiffness properties: unlessmore detailed information are available, a reduction of 50% is pro-posed. Indeed, the results of parametric nonlinear FEM analysesperformed by the authors on panels subjected to static in-planeloading with different levels of axial loads and slenderness[36,37], showed that the reduction factor depends on the actingcompressive state (likewise to that proposed in [38] for r.c. ele-ments). Fig. 7 shows the reduction factor g as a function of the com-pressive stress state (ry), normalized to the masonry compressivestrength (fu), for two different levels of the nonlinear masonrybehaviour.

Rigid end offsets are then used to transfer static and kinematicvariables between element ends and nodes. A nonlinear correctionprocedure of the elastic prediction is carried out based on compar-ison with the limit strength values as defined hereafter; the redis-tribution of the internal forces is made according to the elementequilibrium.

The ultimate shear and bending strength is computed accordingto some simplified criteria that are consistent with the most com-mon ones proposed in the literature and codes for the prediction ofthe masonry panel’s strength as a function of the different above-mentioned failure modes. Table 1 summarizes the criteria imple-mented in TREMURI program for URM piers and spandrels,respectively; as debated in Section 4.1, the program updates, at

ViNi

Nj Vj

Mj

Mi

δ N

V

V u,flexural

V u,shear

N k δu

Failure criteria Bi-linear relationship

N k+1

Influence of the current axial load acting on the panel

V u =min (V u,shear; V u,flexural)

N k-1

VShear strength domainFlexural strength domain

Idealization of the single panel

K

(ui , wi , ϕi )

(uj , wj , ϕj )

Kinematic variables, generalized forces and geometrical properties

h

t

l

Fig. 6. Sketch of the idealization of masonry pier response by adopting simplified strength criteria based on applied axial compression force.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1

For increasing values of axialand shear compliance

η

σy/ fu

Fig. 7. Reduction factor of the elastic stiffness properties (as results from nonlinearFEM analyses in [36]).

1792 S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799

each step of the nonlinear analysis, the current ultimate strengthtaking into account the axial load variation.

A check for the ultimate compressive strength is also imple-mented in the TREMURI nonlinear analysis procedure. The maxi-mum element capacity in compression is limited to Nu = 0.85ltfu,

Table 1Strength criteria for URM panels implemented in TREMURI program.

Failure mode and element type Strength domain Notes

Rocking/crushing Piers Mu ¼ Nl2 ð1� N

0:85fu ltÞ fu maso

Spandrels Mu ¼dH0p

2 1� H0p0:85f hudt

h iH0p is asas proppresentthe spastrengt

Mu ¼ f N; ftufhu;lc ;lt

� �ftu ¼ min fbt

2 ; c þ lrsu� � As prop

constituequivalcoefficiof com

Shear Bed jointsliding

Piers Vu;bjs ¼ l0tc þ lN 6 Vu;blocks Coulom(Vu,block

Spandrels Vu = htc h heighmechan

Diagonalcracking

Piers/spandrels

Vu;dc 1 ¼ lt 1:5sob

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ N

1:5so lt

qso maso

Vu;dc 2 ¼ 1b ðlt~c þ lNÞ 6 Vu;blocks Coulom

relatedb = 1). Tcomme

where fu is the masonry compressive strength, l is the length ofthe cross-section, t the wall thickness.

When different strength domains are implemented for the samefailure mode (e.g. in case of the shear mode), a choice has to bemade by the user. Recently, a critical review of the use and thechoice among these criteria as a function of the masonry typehas been discussed in [37]. In particular, it has been noticed thattwo main parameters may address this choice: the regularity ofthe masonry pattern and the ratio between the strength/stiffnessparameters of mortar and blocks.

Moreover, it has to be highlighted how, due to the rather lim-ited and quite recent attention specifically addressed on spandrelelements, most of these strength criteria have been formulatedand validated only by comparison with experimental results onpier elements. Thus, common practice is to adopt the same failurecriteria for both element types, assuming spandrel behaviour asthat of a pier rotated by 90�. Indeed, very few specific formulationsare proposed in both literature and codes, as recently discussed in[44]. For example, the Italian Building Code [6] makes a distinctionin the strength criteria to be adopted for spandrels as a function ofthe acting axial load. If it is known from the analysis, the same cri-teria assumed for piers are adopted. If it is not known (that is the

nry compressive strength, l length of section, t thickness

sumed as the maximum value between the axial load N acting on spandrel and Hp

osed in [6] by assuming a strut-and-tie mechanism (if a tension member is). Hp is the minimum value between the tensile strength of elements coupled tondrel (such as r.c. beam or tie-rod) and 0.4fhudt, where fhu is the compressionh of masonry in horizontal directionosed in [39] the limit domain is obtained by assuming an elasto-perfectly plastictive law with limited ductility both in tension (lt) and compression (lc) and an

ent tensile strength for spandrel ftu (fbt tensile strength of bricks; l and c frictionent and cohesion of mortar joint, respectively; / interlocking parameter; rs entitypressive stresses acting at the end-sections of the spandrel)

b criterion with: l0 length of compressed part of cross section. A limit values) is imposed to take into account in approximate way the failure modes of blockst of spandrel transversal section (assumed only in case of a strut-and-tieism may develop)nry shear strength, b stress distribution factor as function of slenderness [40,41]

b-type criterion with: l and c equivalent cohesion and friction parameters,to the interlocking due to mortar head and bed joints, as proposed in [42] (withhe introduction of b, proposed in this paper, is implicitly justified in [42] by soments on the shear stress distribution; a similar corrective factor is proposed in [43]

S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799 1793

case when floors are modelled as infinitely stiff), if the spandrel iscoupled to another tensile resistant element (e.g. steel tie rod or r.c.beam), a strut-and-tie mechanism is assumed to be developed,with a maximum compression force in the spandrel equal to thetension strength in the coupled element. However, as introducedin Section 4.1, several factors differentiate spandrels from piers.In particular, regarding the flexural response, due to both the lowvalues of axial load or the lack of other tensile resistant elementscoupled to spandrel (as usual in case of existing buildings), the cri-teria proposed in codes lead to very conservative predictions of thespandrel strength: as a consequence in many cases flexural failuretends to prevail over shear much more frequently than that ob-served by earthquake damage assessment in existing buildings orin experimental campaigns. To overcome this result and, in gen-eral, to take into account for the additional factors which mayinfluence the spandrel behaviour (as discussed in Section 4.1), itturns out the pressing need of referring to more reliable and cor-roborated strength criteria for them. As regards to this and withparticular reference to the flexural failure mode, the formulationproposed in [39], as shown by the comparison with experimentalresults discussed in [44], seems to provide rather good results, atleast in the case of clay brick masonry. The formulation is basedon the assumption that, in case of spandrels, masonry may exhibitan ‘‘equivalent’’ tensile strength ftu in the horizontal direction (par-allel to mortar bed joints) in virtue of the interlocking phenomenaoccurring with the contiguous masonry portions.

In TREMURI program, the user can choose between the flexuralstrength criterion proposed by Cattari and Lagomarsino [39], if onewants to consider the contribution of ftu, and the criteria proposedin [6]. In this latter case, the maximum value provided by twoabovementioned hypotheses on the acting axial load is assumedas reference. This assumption is justified by the observation thatthe axial force computed by the program for spandrels representsan underestimation of the actual one (usually very low, apart fromthe case of the presence on tensioned tie-rods): in fact, some phe-nomena (e.g. the effect of interaction with floors) are modelledonly in an approximate way.

It is worth stressing that expressions summarized in Table 1only refer to the case of URM masonry, while the TREMURI pro-gram also allows modelling reinforced masonry structures. To thisaim, proper strength criteria, consistent with those usually pro-posed in codes, have been implemented to predict the panel’sstrength.

The panel collapse, is checked by assuming a limit value for thedrift (d), computed as:

d ¼ ðuj � uiÞh

þðuj þuiÞ

26 du ð2Þ

The limit value assumed (du) varies as a function of the prevail-ing failure mode that occurs in the panel. According to some rec-ommendations proposed in codes [6,30,31], in case of URMmasonry piers, it usually ranges from 0.4% to 0.8%; indeed, in caseof spandrels, the recent experimental campaigns showed generallygreater values. Once collapse is reached, the element becomes astrut; this assumption is on the safe side, because no residual shearand bending strengths are considered, while the axial load is stillsupported, checking it does not exceed the axial strength Nu.

4.2. Reinforced concrete elements

Nonlinear r.c. elements, modelled as 2D or 3D elements in thecase of beams or columns and walls, respectively, are idealizedby assuming elastic-perfectly plastic hinges concentrated at theends of the element. The choice of this simplified concentratedplasticity model, with respect to more accurate ones like as the fi-bre approach, is justified by the will to assume a computational

burden comparable to that of masonry elements and a similar levelof accuracy.

The initial elastic branch, similarly to masonry elements, is di-rectly determined by the stiffness contributions in terms of shearand flexural behaviour by neglecting that offered by reinforcement.It is computed by means of a stiffness matrix analogous to thatintroduced in Eq. (1) (with some necessary modifications in caseof 3D elements). The reduction of stiffness due to cracking phe-nomena may be taken into account, analogously to masonry ele-ments, by the g coefficient (e.g. assumed as proposed in [38]),kept constant during the analysis.

Shear and compressive/tensile failures are assumed as brittlefailures while combined axial-bending moment, modelled by plas-tic hinges at the end of element, are regarded as ductile failure.

Shear strength is computed according to the criteria proposedin [6,45] in the case of low-medium ductility classes, for differentelement types (beam, column and r.c.-wall). Both cases of trans-verse shear reinforcements present or not are considered; if pres-ent, the shear strength criteria adopted are based on anequivalent truss with the variable strut inclination method.

In the case of combined axial force (N) and bending moment(M), the interaction M–N domain is computed on the commonhypotheses of: plane-sections; perfect bond between concreteand steel bars; rectangular stress block distribution.

In case of columns, only the case of symmetrical reinforcementsis considered; in case of r.c. walls, the domain is computed takinginto account the contribution of longitudinal bars in their actualposition. In order to determine the formation of a plastic hinge,the comparison between the elastic prediction and the limit valuesobtained from the M–N interaction domain, is carried out. The caseof r.c. walls and columns is more complex since these elements canbe affected by a biaxial bending–compression behaviour. In thislatter case, the Mx–My–N domain is traced by computing, on thebasis of the axial force acting on the element, the resistant bendingmoments separately in each plane (Mx,Rd and My,Rd, respectively)and, then, by assuming a proper interaction domain (linear or withmore accurate formulations, such as elliptic). It is necessary topoint out that the plastic hinge, once activated, involves both Xand Y planes at the same time.

The ultimate limit state of the section, in the case of ductilemechanisms, is identified when the chord rotation (computedreferring to the shear span LV) reaches its ultimate value (hu), cal-culated by widely used expressions [30,46,47], based on an empir-ical approach starting from a number of experimental data [46].

Once failure is reached, for both ductile and brittle failuremodes, the beam element is converted to a strut, as in the caseof masonry elements. Instability phenomena and second order ef-fects are not considered. This modelling approach has been re-cently adopted for the assessment of mixed masonry–r.c.buildings [29].

4.3. Steel and wooden elements

Also steel and wooden beams or tie-rods may be modelled.Similarly to r.c. elements, steel and wooden beams are idealized

by assuming elastic-perfectly plastic hinges, concentrated at theends of the element. Obviously the strength criteria adopted as ref-erence are modified according to these different materials andchecks on the ultimate deformation capacity are not included.Tie-rods are idealized as non-compressive spar elements with thepossibility to assign also an initial strain e0 (thus to impose a cor-responding pre-stress action equal to EAe0, with E Young modulusof material and A transversal section of tie-rods). In this case, thestiffness matrix of the element Ke (Eq. (1)) is updated by resettingall terms containing the J contribution; thus, the nonlinearity iskept into account by updating at each step of the analysis the

1794 S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799

global stiffness K (obtained by assembling those of all elements)accounting only for the active tie-rods (i.e. those in tension).

5. 3-Dimensional model

Starting from the equivalent frame modelling discussed in theprevious paragraphs for a single wall, complete 3D models maybe assembled.

The 3-dimensional modelling of whole URM buildings startsfrom the following basic hypotheses: (a) the construction bearingstructure, both referring to vertical and horizontal loads, is identi-fied with walls and horizontal diaphragms (roofs, floors or vaults);(b) the walls are the bearing elements, while diaphragms are theelements governing the sharing of horizontal actions among thewalls; (c) the flexural behaviour of the diaphragms and the wallout-of-plane response are not computed because they are consid-ered negligible with respect to the global building response, whichis governed by their in-plane behaviour. The global seismic re-sponse is possible only if vertical and horizontal elements areproperly connected; then, if necessary, ‘‘local’’ out-of-plane mech-anisms have to be verified separately through suitable analyticalmethods.

Within this general context, two main issues have to be solved,in particular related to: (i) the strategy for assembling 2D masonrywalls (discussed in Section 5.1); (ii) the modelling of floors(Section 5.2).

5.1. 3D assembling of masonry walls

In order to assemble a 3D model, a global Cartesian coordinatesystem (X,Y,Z) is defined. The wall vertical planes are identified bythe coordinates of one point and the angle formed with the global Xaxis (Fig. 8). In this way, the walls can be modelled as plane framesin the local coordinate system and internal nodes can still be 2-dimensional nodes with 3 d.o.f. At corners and where two or morewalls intersect 3-dimensional nodes are used. They are character-ized by 5 degrees of freedom (d.o.f.) in the global coordinate sys-tem (uX, uY, uZ, /X, /Y). In fact, the rotational degree of freedomaround vertical Z axis can be neglected because of the membranebehaviour adopted for walls and floors (Fig. 8). These nodes canbe obtained assembling 2D rigid nodes acting in each wall planeand projecting the local d.o.f. along global axes. The assemblage

Fig. 8. 3D assembling of masonry walls: classification of 3D and 2D rigid nodes andout-of-plane mass sharing.

is then obtained condensing the degrees of freedom of two2-dimensional nodes by assuming the full coupling among theconnected walls. This solution is particularly efficient to reducethe total number of d.o.f. and perform nonlinear analyses with areasonable computational effort also in case of large and complexbuilding models.

Since the 2D nodes have no d.o.f. along the direction orthogonalto the wall plane, the nodal mass component related to out-of-plane degrees of freedom is shared to the corresponding d.o.f. ofthe two nearest 3D nodes of the same wall and floor according tothe following relations:

MIx ¼ MI

x þmð1� j cos ajÞ l� xl

MIy ¼ MI

y þmð1� j sinajÞ l� xl

ð3Þ

where the meaning of the terms is shown in Fig. 8. This solutionpermitted to maintain the adopted simplification hypotheses inthe implementation of static analyses with 3 components of accel-eration along the 3 principal directions and 3D dynamic analyseswith 3 simultaneous input components.

5.2. Modelling of diaphragms

A proper assumption on the diaphragm stiffness may signifi-cantly affect the overall response. In fact, in the limit case of ‘‘infi-nitely’’ flexible floors, there would be no load transfer from heavilydamaged walls to still efficient structural elements. On the con-trary, in the other limit case of floors assumed as ‘‘infinitely’’ stiff,this contribution could be overestimated. Although it represents acrucial feature to be considered, the floor behaviour in 3D model-ling is frequently assumed (with a rough approximation) as com-pletely rigid. This hypothesis may be completely unrealistic incase of existing buildings (e.g. historical masonry structures),where various ancient constructive technologies (i.e. timber floorsand roofs, structural brick or stone vaults) can be found for floorand roofing systems; moreover, this is also a major issue in newmasonry buildings with wooden floors and roofs.

In order to simulate the presence of flexible diaphragms, spe-cific floor elements were introduced in the TREMURI model. Theyare modelled as 3- or 4-nodes orthotropic membrane finite (planestress) elements, with two displacement degrees of freedom ateach node (ux, uy) in the global coordinate system. They are identi-fied by a principal direction (floor spanning orientation), withYoung modulus E1, while E2 is the Young modulus along the per-pendicular direction, m is the Poisson ratio and G12 the shear mod-ulus. The moduli of elasticity E1 and E2 represent the normalstiffness of the membrane along two perpendicular directions, alsoaccounting for the effect of the degree of connection between wallsand horizontal diaphragm and providing a link between the in-plane horizontal displacements of the nodes belonging to the samewall-to-floor intersection, hence influencing the axial force com-puted in the spandrels. The most important parameter is G12,which influences the tangential stiffness of the diaphragm andthe horizontal force transferred among the walls, both in linearand nonlinear phases. Starting from these entities, the orthotropicmatrix bD, aiming at correlating strain and stress (in case of 3 nodesmembranes) may be computed as follows:

D ¼

E11�et2

etE11�et2 0

etE11�et2

eE11�et2 0

0 0 G12

264

375 ð4Þ

where e is the ratio E2/E1. Thus, the matrix bD may be modified (in D)through a proper rotation matrix R in order to take into account forthe actual orientation of the diaphragm. Finally, the stiffness matrix

Fig. 9. 4-Node membrane element as average of 3-node element meshes.

S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799 1795

is assembled starting from D by adopting linear shape functions. Foreach node i of the 3-node element, the matrix Bi can be defined as:

Bi ¼1

2A

yj � yk 00 xk � xj

xk � xj yj � yk

264

375 ð5Þ

where xj, yj, xk, and yk are the coordinates of nodes j and k, and A isthe area of the triangle. Starting from the matrices Bi and D, thestiffness matrix of the 3-node membrane element can be assembledas

K ¼kii kij kik

kji kjj kjk

kki kkj kkk

264

375 ð6Þ

where kij ¼ BTi DBjAs, with s equivalent thickness assumed for the

membrane element. In the case of 4-nodes elements, the stiffnessmatrix is obtained as the averaged contribution of the two possiblemeshes of 3-node elements (Fig. 9).

The evaluation of the abovementioned quantities may be rathersimple in case of some floor typologies, ascribing it to the struc-tural role shown by some specific elements. For example, in thecase of a r.c. floor with beams and slab the shear stiffness is mainlygiven by the slab whereas the beam axial stiffness leads to the def-inition of E1. On the contrary, in case of various ancient floor tech-nologies, as the case of vaults, beside thickness and materialproperties, the stiffening contribution strongly depends on shapeand geometrical proportion (e.g., rise-to-span ratio). In [48], start-ing from the results of both linear and nonlinear FEM numerical

Fig. 10. Front view and 3D building model (on the

simulations, the definition of equivalent stiffness properties hasbeen proposed for some types of vaults (barrel, cross and cloistervaults) as a function of different thickness-to-span and rise-to-span ratios, constraints conditions and masonry texture pattern(parallel, orthogonal and oblique). In case of timber floors androofs, some recent experimental works suggest a critical reviewof the formulae proposed in the literature and codes [31,49–51].

6. Seismic analysis procedures

In order to perform nonlinear seismic analyses of masonrybuildings a set of procedures has been implemented in the TREMU-RI program [8]: incremental static with force or displacement con-trol; 3D pushover analysis with fixed load pattern; 3D time-historydynamic analysis (Newmark integration method; Rayleigh viscousdamping). In the following, the attention is focused on the numer-ical algorithm implemented for pushover analyses, which becamemore and more popular for seismic structural assessment in thelast decades, in particular in conjunction with the spreading of per-formance-based earthquake engineering concepts.

6.1. The pushover analysis

The pushover procedure implemented [7,18] transforms theproblem of pushing a structure maintaining constant ratios be-tween the applied forces into an equivalent incremental staticanalysis with displacement control at only one d.o.f. In this sensethis procedure is conceptually similar to the one proposed in [52].

In particular, the general formulation of the pushover problemcan be represented by equations:

KFF KFm KFC

KTFm kmm kmC

KTFC kT

mC KCC

264

375

xF

xm

xc

8><>:

9>=>; ¼

kfF

kfm

rc

8><>:

9>=>; ð7Þ

where m is the control degree of freedom and fF is the coefficientvector of the applied load pattern.

The system of equations can be transformed subtracting themth row, multiplied by a proper factor, from the first m � 1 rows;the ith equation then becomes:

top); modal analysis results (on the bottom).

Fig. 11. Comparison between the real and numerically simulated damage pattern (from [54]): the experimental one (on the left); that simulated through TREMURI program(at the centre); that simulated by the finite element mode (on the right, in terms of principal inelastic strain).

0

40

80

120

160

200

0 5 10 15 20 25

V [

kN]

u [mm]

FEM

Experimental test

EF-full stiffness

EF-reduced stiffness

Fig. 12. Simulation of the in-plane response of Door Wall: comparison in terms ofpushover curve (from [54]).

1796 S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799

Ki1 �fi

fmkm1

� �x1 þ � � � þ kim �

fi

fmkmm

� �xm þ � � �

þ kin �fi

fmkmn

� �xn

¼ 0 ð8Þ

The new system of equations, with a modified stiffness matrix:

eK FFeK Fm

eK FC

KTFm kmm kmC

KTFC kT

mC KCC

264

375

xF

xm

xc

8><>:

9>=>; ¼

kfF

kfm

rc

8><>:

9>=>; ð9Þ

is then equivalent to a displacement control one, in which the mthd.o.f. xm is the imposed one. This formulation was obviously rewrit-ten by introducing the nonlinear contribution in incremental form,in order to be implemented in the nonlinear procedure.

The algorithm results quite effective and robust, so allowingpushover analyses on very complex models.

Finally, it has to be stressed that horizontal forces, proportionalto tributary masses and the assumed mode shape, are applied toeach node at the level of each floor. Actually, the application of no-dal forces in the pushover analysis represents a crucial issue, inparticular, for a reliable computation of the axial load acting onspandrel elements. As an example, if in case of flexible floors forcesare applied only on the corner nodes, a wrong axial load on span-drel elements is obtained.

7. Validation of the model and examples

In the following, some examples of applications are illustrated,showing the capability of TREMURI program (and more in generalof the equivalent frame approach) in describing the seismic re-sponse of masonry buildings.

The first example shows how, despite of some unavoidableapproximations consequent to the idealization of a real structurein an equivalent frame model (e.g. in the proposed model thedeformability of portions idealized as rigid nodes is neglected), itis able to provide a good simulation of the structural behaviourin linear range. In particular, the application concerns the casestudy of the Hall of the Giuncugnano village in Tuscany, an instru-mented building included in the Structure Seismic Observatory(OSS) program of the Italian Department of Civil Protection.Fig. 10 shows the front view of the building and the 3D model. ThisURM building, like several others all around Italy, now hosts a setof accelerometers that permanently monitor its dynamic response.Based on both in situ and laboratory tests, a characterisation of the

structural behaviour both for linear and nonlinear response hasbeen carried out, as described in [18]. In particular, modal tests al-lowed good data for dynamic identification of the 3D model(Fig. 10): numerical modal analysis results, accordingly to experi-mental ones, are shown in Fig. 6. MAC index value (Modal Assur-ance Criteria), which quantify the agreement between modalshapes from numerical model and experimental data, is 0.96 and0.94 for the first two modes.

The second example focuses on the model ability to reproducethe seismic response in nonlinear range. In particular, the in-planeresponse of the ‘‘Door-Wall’’ of the full scale two storey brick ma-sonry building prototype, tested at the University of Pavia by Mag-enes et al. [53], has been analyzed. Experimental results have beensimulated by the TREMURI program (as described in more detail in[54]) and compared with those obtained from a finite elementmodel by shell elements, adopting the nonlinear continuum dam-age model proposed in [55]. The parameters of the models havebeen defined on the basis of those directly obtained by experimen-tal tests on the components (mortar and brick) and their assem-blies (triplets, masonry prisms, in-plane cyclic shear tests onpiers) carried out on the same brick masonry adopted for the fullscale prototype [56]. No additional calibration of parameters hasbeen performed. In fact, the will was to simulate a real conditionof seismic assessment on an existing building for which, in

Flexural plastic phase

Flexural collapse

Shear collapse

Shear plastic phase

Elastic phase

Rigid Node

Flexural plastic phase

Flexural collapse

Shear collapse

Shear plastic phase

Elastic phase

1 2

3 4

5 6

78

9

10 11 12

13 14 15

n17

n18

n19

n20

N1

N2

N3

N4

N5

N6

N7

N8

1 2

3 4

5 6

78

9

10 11 12

13 14 15

n17

n18

n19

n20

N1

N2

N3

N4

N5

N6

N7

N8

1 2

3 4

5 6

78

9

10 11 12

13 14 15

n17

n18

n19

n20

N1

N2

N3

N4

N5

N6

N7

N8

19 20

21 22

23 24

2526

27

28 29 30

31 32 33

n25

n26

n27

n28

N9

N10

N11

N12

N13

N14

N15

N16

3 3

CA

SE B

CA

SE A

1 m 1 m 2 m 1 m 1 m

3 m

3 m

3 m

0

0,05

0,1

0,15

0,2

0,25

0,3

0 2 4 6 8

V/ W

u/uSSWP

WSSP

SSWP

EF Model - Case A

EF Model - Case B

EF Model - Case C

Fig. 13. Comparison among simplified models (SSWP and WSSP) and the Equivalent Frame model as a function of different hypotheses assumed for spandrels (from A to C)for a three-storey URM wall (adapted from [15]).

S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799 1797

addition, results of standard tests are available. According to themasonry type (brick masonry) and configuration (two steel beamsintroduced to apply the horizontal forces that worked as ‘‘tie-rods’’at level of spandrels) of the ‘‘Door Wall‘‘, nonlinear beams in theTREMURI program have been modelled by adopting the criterionproposed in [42] for the shear response of masonry elementsand, in particular for spandrels, the criteria proposed in [6] (see Ta-ble 1). Figs. 11 and 12 show the comparison of results in terms ofdamage pattern and pushover curve (global shear forces V at thebase of the wall versus mean displacement u of the second floor),respectively. It is worth noting that, differently from the experi-mental tests, in the numerical simulations the nonlinear staticanalysis has been performed monotonically. Results are in fairagreement and substantially confirm the reliability of the EF mod-elling approach, in particular for masonry walls characterized byregular opening patterns. As regards the pushover curve, the mod-erate overestimation of the global strength given by the FE and EFmodels with respect to the experimental one may be explained byconsidering that:

– numerical simulation has been performed monotonically;– a scattering of the material parameters (in particular those of

joints) should characterize the real building;– parameters have been calibrated on those obtained on tests per-

formed on single panels (for which a higher building care isexpected than the whole structure leading to betterperformance).

In the case of the TREMURI model, two hypotheses for the stiff-ness properties have been adopted (by assuming ftu/fhu – see Ta-ble 1 – equal to 1 and 0.5, respectively). The case of ftu/fhu = 0.5highlights how this reduction of the stiffness parameters is rathercoarse. Actually, in the case examined, the response is driven byfew structural elements: thus, the incorrect assignment of thedeformability parameters strongly influences the overall responseof the structure. Finally, as regards the damage pattern numericallysimulated, it can be observed that different failure modes occur inthe two lateral piers at the ground storey. Although these piers are

characterized by the same slenderness, the flexural behaviour pre-vails in the left one, while diagonal cracking prevails in the rightone. This is due to the different levels of normal forces they aresubjected to, associated with the global overturning of the wall.In the central pier, which is the more squat, diagonal cracking oc-curs. The more symmetric experimental damage pattern is due tothe application of a cyclic load history.

Finally, the third example aims to showing the higher versatilityof the equivalent frame model to simulate the actual behaviour ofmasonry building with respect to other more simplified ap-proaches. Fig. 13 illustrates the response of a three-storey URMwall with two lines of vertically aligned openings [54] in which,respectively: SSWP and WSSP represent the two extreme idealiza-tion proposed in [11,12]; Case A refers to the EF model in which thesame strength criteria have been assumed for piers and spandrels;Case B to the EF Model in which for the flexural behaviour the cri-terion proposed in [39] has been assumed for the spandrel byassuming g as 0.05; Case C to the EF Model in which reinforcedconcrete beams have been modelled coupled to spandrel elements.Results are illustrated in terms of V/W (ratio between base shearand total weight of the structure) versus u/uSSWP (displacementof control node located on top of the wall, normalized to the ulti-mate value obtained in the case of SSWP). SSWP and WSSP definethe range of the possible pushover curves of the structure. It ap-pears too wide in terms of strength, stiffness and ductility defini-tion, all three aspects that play a fundamental role by referringto the adoption of nonlinear static procedures as tools of verifica-tion. Moreover, even if the systematic adoption of WSSP in caseof existing buildings of course should lead to results which areon the safe side, such a severe underestimation of the actual capac-ity would not be acceptable, in particular if an unsuccessful assess-ment would lead to unrealistically heavy retrofitting interventions.The comparison between SSWP and Case C stresses how the pres-ence of certain constructive details is not in general sufficient to as-sure the satisfaction of some simplified hypotheses: this is the caseof the presence of reinforced concrete beams (usually associatedwith the presence of rigid floors), characterized by a finite stiffness,which does not correspond with the assumption of fixed rotations

1798 S. Lagomarsino et al. / Engineering Structures 56 (2013) 1787–1799

at the level of each floor. As a consequence, SSWP provides anupper bound which operates on the unsafe side. Case A providesresults similar to those of WSSP because resistance criteria as-sumed for spandrel elements are too cautionary. By assuming thecriterion proposed in [39] for spandrel elements (Case B), both asignificant increase in the overall resistance and a decrease in theglobal ductility can be observed with respect to case A. The latterresult can be explained by both the different pattern and sequenceof damage that occur in cases A and B (Fig. 13). In fact in Case A, dueto the moderate axial load acting on the spandrel elements, sincethe initial steps of the analysis a Rocking mechanism occurs in al-most all spandrels which thus supply a weak coupling for piers.On the contrary, in case B the following phases are recorded: a firstphase in which only the spandrels located on the top floor showthe activation of a Rocking mechanism (in fact, due to the moderatecompressive stresses acting on the contiguous masonry portions,they cannot rely much on the interlocking phenomena); a finalphase, in which damage increases in spandrels and also spreadsto piers located on the ground floor. Analogous results and com-ments on the role of spandrel elements are presented in [3,17].

8. Conclusions

The theoretical bases of TREMURI program are explained in thepaper, with particular reference to its distinctive features:

� the creation of an equivalent 3-dimensional frame based onassembling 2-dimensional (plane) structures (masonry wallsand floor/roof diaphragms), which allows for an effective con-densation of the degrees of freedom of the global model, soreducing the computational burden for the global analysis ofthe building;� the implementation of specific elements allowing for the repre-

sentation of the main characteristics of nonlinear response ofmasonry piers and spandrels, as well as other structural mem-bers (r.c. members, steel tie-rods, etc.);� the implementation of an orthotropic membrane element, for

the simulation of the in-plane behaviour of flexible diaphragms(e.g. timber roofs and floors), which can be commonly found innew and existing masonry structures;� an original and versatile algorithm for the pushover analysis,

suitable for assessing the nonlinear evolution of the lateralresponse of 3-dimensional masonry buildings, including thedeterioration of the base shear for increasing lateral displace-ments after the attainment of peak strength.

Most of the TREMURI features were developed and imple-mented with the aim of filling some of the existing gaps in the seis-mic analysis of masonry structures, with a specific target on theassessment of existing buildings with flexible diaphragms. Thecomputer program that has been developed in the last decade al-lows, with a reasonable computational effort, a reliable simulationof the actual behaviour of traditional masonry structures, providedthat proper connections prevent the activation of local out-of-plane mechanisms and favour the development of a globalresponse, governed by the in-plane behaviour of the differentstructural components.

This versatile computer program for the analysis of the seismicresponse of historical masonry structures represents a useful toolfor supporting their conservation. In fact, a correct identificationof the most critical walls and related collapse mechanisms (softstorey/failure in spandrel elements) may suggest differentstrengthening strategies, addressed to increase strength or dis-placement capacity of masonry panels, as well as stiffness of hori-zontal diaphragms. TREMURI is a framework in continuous

progress for the development and implementation of new ad-vanced nonlinear elements and analysis procedures (static/dy-namic), both at research level and engineering practice.

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[5] EN 1998-1. Eurocode 8. Design provisions for earthquake resistance ofstructures. Part 1-1: General rules – seismic actions and generalrequirements for structures. CEN, Brussels, Belgium; 2004.

[6] NTC 2008. Decreto Ministeriale 14/1/2008. Norme tecniche per le costruzioni.Ministry of Infrastructures and Transportations. G.U. S.O. n.30 on 4/2/2008;2008 [in Italian].

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