Computers and Structures 143 (2014) 91–107

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Computers and Structures

journal homepage: www.elsevier .com/locate /compstruc

A macro-element modelling approach of Infilled Frame Structures

http://dx.doi.org/10.1016/j.compstruc.2014.07.0080045-7949/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Address: Dipartimento Ingegneria civile e Architettura –DICAR, University of Catania, Viale Andrea Doria, 6, 95125 Catania, Italy.Tel.: +39 (0)95 738 2255; fax: +39 (0)95 738 2249.

E-mail address: icalio@dica.unict.it (I. Caliò).

Ivo Caliò ⇑, Bartolomeo PantòDipartimento Ingegneria Civile e Architettura, University of Catania, Italy

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

Article history:Received 23 October 2013Accepted 9 July 2014Available online 12 August 2014

Keywords:Infilled Frame Structures (IFS)Masonry Infilled Reinforced Concrete frame(MIRC)Macro-element approachDiscrete element approachSeismic vulnerabilityMicro-models

In this paper a macro-modelling approach for the seismic assessment of Infilled Frame Structures (IFS) ispresented. The interaction between frame and infill is simulated through an original approach in whichthe frame members are modelled by means of lumped plasticity beam–column elements while the infillsare described by plane macro-elements. The reliability of the approach is evaluated by means of nonlin-ear analyses, performed on infilled masonry reinforced concrete structures, for which experimentalresults are available in literature. The proposed computational strategy is intended to provide anumerical tool suitable for the design and the vulnerability assessment of IFS.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Infilled Frame Structures (IFS) represent a high percentage ofexisting and new buildings in many seismically prone areas aroundthe world. These mixed structural systems are governed by theinteraction between frame and infill wall. The frame can be builtwith different materials: reinforced-concrete, steel or wood, whilethe infill wall can be of unreinforced masonry or concrete. MasonryInfilled Reinforced Concrete frames (MIRC) are, at present, widelyadopted for the construction of new buildings as well as structuralretrofitting strategy of medium and low-rise buildings. Existingmasonry infilled frame buildings are often difficult to be classifiedsince, in many cases they are the result of low-engineered struc-tures built before the emanation of seismic codes and conceivedto resist only gravity loads [1]. Currently, new infilled frame build-ings are mainly concentrated in several regions around the worldwhich are often characterised by a high-density population, sincethis structural typology represents a rapid and low cost buildingstrategy [2]. A large number of buildings are built with masonryinfill walls for non-structural reasons, since the role of infill isassociated with architectural needs. In these cases the structuralcontribution of masonry infill panels is generally neglected in thestructural analyses, leading to a significant inaccuracy of theprediction of lateral stiffness, strength and ductility capabilities

of the structure. As highlighted by many authors [1,3,4], ignoringthe role of frame-infill panel interaction is not always safe, result-ing in a possible change of the seismic demand due to the modifi-cation of the fundamental period of the mixed structural system.Furthermore, the presence of infill walls can modify the stiffnessand ductility distribution along the structure causing local col-lapses, during seismic events, ignored at the structural designphase.

The ability to assess the seismic behaviour of IFS is of crucialimportance and several contributions have been provided in thelast five decades by many researchers by means of both numericaland experimental investigations. The highly nonlinear masonry-infill response and the ever-changing contact condition along theframe-infill interfaces make the simulation of the nonlinear behav-iour of an entire infilled frame building a challenging problem. Adetailed simulation of the complex nonlinear behaviour of infilledframes requires the rigours use of expensive computationally non-linear finite element models, capable to reproduce the nonlineardegrading behaviour of the infill masonry and the complex interac-tion between the frame and the embraced infill. Refined finite ele-ment numerical models [2,5,6], such as the smeared cracked anddiscrete crack finite element models, able to predict the complexnon-linear dynamic mechanical behaviour and the degradation ofthe masonry media, require sophisticated constitutive laws andhuge computational resources. As a consequence these numericalapproaches are, at present, unsuitable for practical applicationand extremely difficult to apply to large structures. On the otherhand, in order to estimate the seismic vulnerability of an existingbuilding and to assess whether the structure requires a seismic

92 I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107

upgrade, a structural engineer needs simple and efficient numeri-cal tools, whose complexity and computational demand must beappropriate for practical engineering purposes. For these reasons,in the last six decades, many authors have developed simplifiedor alternative methodologies for predicting the nonlinear seismicbehaviour of IFS. According to the classification proposed by Aster-is et al., in a recent comprehensive review of existing mathematicalmacro-models of infilled frames [7], the current numericalapproaches can be classified in two main categories, macro- andmicro-models. The macro-models try to grasp the global behaviourof the Infilled Frame Structures without claiming to obtain adetailed nonlinear response and to describe all the possible modesof local failure. On the contrary, the micro-models are conceivedfor a detailed behaviour simulation of the infilled frame trying toencompass all the possible modes of collapse due to differentdamage scenarios in the masonry infill and the surrounding frame.Refined micro-models can also be useful for validating andcalibrating simplified approaches in those cases in which experi-mental results are unavailable. The most commonly used macro-model practical approach is the so called ‘diagonal strut model’,according to this approach the infilled masonry is represented bya diagonal bar under compression. The first who suggested thepossibility of considering the effect of the infill as an equivalentdiagonal bracing was Polyakov [8]. This suggestion was taken upby Holmes [9], who modelled the infill by an equivalent pin-jointed diagonal strut made of the same material with the samethickness as the infill panel and a width equal to one-third of theinfill diagonal length. This so called ‘one-third’ rule was suggestedas being applicable irrespective of the relative rigidities of theframe and the infill. Many authors [10–20] suggested alternativeproposals, for the evaluation of the equivalent strut width. Someof these studies provide analytical expressions, which encompassthe presence of openings in the infill or analyse special layoutssuch as the case in which infill and frame are not bonded together[17].

In the last two decades some authors highlighted the limit ofusing a single strut-element for modelling the complex nonlinearbehaviour of IFS and proposed more complex macro-element withthe aim to obtain a better description of the effect of infill on thesurrounding frame. Thiruvengadam [21], with the aim to evaluatefrequencies and modes of vibrations of IFS, proposed a ‘multiplestrut model’, wherein the infills are represented by a set of equiv-alent multiple struts. The model is able to account for the frame-infill separation and infill openings and was also included inFEMA-356 [22]. Subsequently, a number of researchers proposedthe multi-strut approach for modelling both the linear and nonlin-ear behaviour of IFS [23–30]. The main advantage of the multiple-strut models, despite the increase in complexity, is the ability torepresent the actions in the frame more accurately; a comprehen-sive description of the multiple strut models is reported in thereview paper [7].

In this paper an alternative innovative approach for the simula-tion of the seismic behaviour of Infilled Frame Structures, suitableboth for research and current engineering practice applications, ispresented. In this approach, the infilled wall is modelled by meansof a discrete element, originally conceived for the simulation of thenonlinear response of masonry building [31,32], while the rein-forced concrete frame is modelled by means of inelastic beam–col-umn elements, in which the plastic hinges can originate indifferent positions along the beam-span. The computational costof the proposed numerical approach is greatly reduced in compar-ison to that involved in nonlinear finite element simulations,which require finite element modelling of both the frame andthe infill. The basic macro-element, adopted for the simulation ofthe infilled masonry, consists of an articulated quadrilateral,with rigid edges, in which two diagonal springs govern the

shear-diagonal behaviour. The flexural and shear-sliding behaviouris governed by discrete distributions of springs on the sides of thequadrilateral that govern the interaction with the adjacentelements. The calibration of the model requires few parametersto define the masonry material based on results from currentexperimental tests. The equivalence between the masonry portionand the macro-element is based on very simple physical consider-ations [32] and the interpretation of the numerical results isstraightforward and unambiguous. This novel approach has beenrecently successfully applied to mixed reinforced concrete-masonry structures in the academic context [33–38,52] thusconfirming that it can be considered as a low cost computationaltool suitable for the investigation of the nonlinear structuralbehaviour where the seismic capacity results from the interactionbetween masonry and reinforced concrete.

In Section 2 a detailed description of the proposed macro-modelapproach is provided and the required fiber calibration proceduresare described in detail. The capability of the model to simulate thecomplex interaction between the infill and the surrounding frameis outlined in Section 3, where the typical failure mechanism ofinfilled frames and its macro-element modelling are considered.

In the numerical applications (Section 4), the proposedapproach is adopted for the simulation of the nonlinear responseof masonry infilled frame for which experimental results are avail-able from previous studies listed in the literature. In the same sec-tion the influence of the model discretization and the ability of themodel of taking into account the presence of openings through asimple a macro-element discretization is highlighted.

2. The infilled frame model

In the present study, the complex nonlinear behaviour ofInfilled Frame Structures is analysed according to a hybridapproach in which the surrounding frame is modelled using con-centrated plasticity beam–column elements while the nonlinearresponse of the infill is simulated by means of a plane discrete ele-ment recently proposed by the authors [32] which has been imple-mented in a software used both for research and engineeringpractice [39]. In the following it is clarified how the beam–columnelement and the discrete-element contribute to provide a reliablesimulation of the nonlinear behaviour of Infilled Frame Structures.

2.1. The discrete-element for modelling masonry infill

Masonry infill is modelled by means of a macro-element,recently introduced by Caliò et al. [32], conceived for the simula-tion of the nonlinear in-plane behaviour of unreinforced masonrywalls. This element is characterised by a simple mechanicalscheme, Fig. 1, constituted by an articulated quadrilateral withrigid edges connected by four hinges and two diagonal nonlinearsprings. Each side of the quadrilateral can interact with other ele-ments or supports by means of a discrete distribution of nonlinearsprings, denoted as interface. Each interface is constituted by nnonlinear orthogonal springs, perpendicular to the panel side, andan additional longitudinal spring, parallel to the panel edge. In spiteof its simplicity, such a basic mechanical scheme is able to simulatethe main in-plane failures of a masonry wall portion subjected toin-plane horizontal and vertical loads.

These well-known collapse mechanisms, namely the flexuralfailure, the diagonal shear failure and sliding shear failure, areroughly represented in Fig. 2 where the typical crack patterns,together with the qualitative kinematics of the masonry portion,are also sketched.

Fig. 3 shows how the proposed plane macro-element allows asimple and realistic mechanical simulation of the corresponding

(a) (b)

Fig. 1. The basic macro-element for the masonry infill: (a) undeformed configuration; and (b) deformed configuration.

qq qF F F

(a) (c)(b)

Fig. 2. Main in-plane failure mechanisms of a masonry portion. (a) flexural failure; (b) shear-diagonal failure; and (c) shear-sliding failure.

qq q

F

F F

(a) (c)(b)

Fig. 3. Simulation of the main in-plane failure mechanisms of a masonry portion by means of the macro-element. (a) flexural failure; (b) shear-diagonal failure; and (c) shear-sliding failure.

I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107 93

failure mechanisms. The flexural failure mode, sketched in Fig. 2a.,is associated with the rocking of the masonry panel in its ownplane; the activation of this collapse mode is governed by the inter-face orthogonal springs, Fig. 3a. The diagonal-shear failure mode,shown in Fig. 2b, is related to the loss of bearing capacity of themasonry panel due to excessive shear and to the consequent for-mation of diagonal cracks along the directions of the principalcompression stresses; this mechanism is controlled by the diago-nal nonlinear links, Fig. 3b.

The shear-sliding failure mode is characterised by the sliding ofthe masonry panel in its own plane. In this case the loss of bearing

capacity is associated to the formation of cracks parallel to the bed-joints, Fig. 2c; this mechanism is governed by the sliding spring ofthe interface, Fig. 3c.

According to the proposed discrete element approach, amasonry macro-element is modelled by an equivalent mechanicalscheme in which the physical role of each component is simple andunambiguous [32].

Each discrete element exhibits three degrees-of-freedom, asso-ciated with the in-plane rigid-body motion, plus a further degree-of-freedom, needed for the description of the shear deformability.The deformations of the interfaces are associated to the relative

94 I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107

motion between corresponding panels, therefore no furtherLagrangian parameters have to be introduced in order to describetheir kinematics.

In Fig. 4 a typical discretization of an infilled frame with andwithout a central door opening is shown. Namely Fig. 4(a, d) referto the geometrical layouts of the infilled frames, Fig. 4(b, e) reportthe corresponding discretization according to the basic neededmesh while the representations of Fig. 4(c, f) are relative to a morerefined mesh. The use of a more refined mesh is not mandatoryhowever, in some cases, can provide more accurate results and abetter description of the collapse mechanism. This aspect is partic-ularly important in the modelling of infilled frames since it allowsa better description of the infilled masonry also in the presence ofwindow or door openings and in some cases can lead to more accu-rate modelling of the infilled frame, as described in the subsequentparagraphs.

It is worth to notice that each macro-element inherits the planegeometrical properties of the corresponding modelled masonryportion avoiding any effective dimension definition of the element,as required by the simplified models based on equivalent strut ele-ment approach.

The efficacy of the masonry infill model relies on a suitablechoice of the mechanical parameters of the macro-elementinferred by an equivalence between the masonry media and a ref-erence continuous model characterised by simple but reliable con-stitutive laws. This equivalence is based on a straightforwardcalibration procedure that exploits the main mechanical parame-ters of the masonry only, according to an orthotropic homogeneousmedium, as highlighted in the following.

2.1.1. Calibration of the interface orthogonal springsSince the masonry is considered as a homogeneous medium its

global behaviour should be ascribed to the flexural and shearingcharacteristics of a finite portion of an orthotropic inelastic con-tinua. As mentioned before, the flexural behaviour is governed bythe interface orthogonal springs connecting the panel to adjacentelements. Each spring is calibrated by adopting a specific constitu-tive law for the masonry media, according to a fiber modellingapproach. The orthotropic nature of the masonry is simply consid-ered by calibrating separately the horizontal and vertical interfacesaccording to the mechanical properties of the correspondingdirections.

(a) (b)

(d) (e)

Fig. 4. Modelling of infilled frame with and without a central door opening. (a, d) thcorresponding to a more refined mesh resolution.

In reference [32] a detailed description of the calibration of themodel for unreinforced masonry structures is reported, here thecalibration procedure is specialised for a typical masonry-framelayouts, as reported in Figs. 4 and 5. The deformability of the infillalong the horizontal and vertical directions are concentrated in thenonlinear links of the vertical and horizontal interfaces. Consis-tently with a fiber calibration approach, the stiffness calibrationof the panel is simply obtained by assigning to each link the axialrigidity of the corresponding masonry strip. Each masonry strip isidentified by its influence area and the half dimension of the panelin the direction perpendicular to the interface, Fig. 5.

With reference to a single orthotropic panel, the initial stiffness,the compression and tensile yielding strengths and the corre-sponding ultimate displacements are derived by the mechanicalmaterial properties of the masonry in horizontal and vertical direc-tions as reported in the Table 1, with reference to a simple elasto-plastic behaviour with limited deformability.

In Table 1, Eh and Ev are the Young’s modulus in horizontal andvertical direction of the homogenised orthotropic masonry media;rhc, rvc and rht, rvt are the corresponding compressive and tensileyielding stresses, ehcu, evcu and ehtu, evtu are the ultimate compres-sive and tensile strains; s is the thickness of the masonry infilland kh and kv are the distance between two nonlinear links inthe vertical and horizontal interfaces.

Also in presence of openings the calibration of the macro-ele-ments is simple and straightforward. In Fig. 4d–f it is shown howa central opening can be represented through two different meshdiscretizations. It is worth noticing that only the link associatedto interfaces between elements have to be calibrated. As an exam-ple, in Fig. 5d–f a simple case of a partially infilled frame, in whichthe opening is adjacent both to a column and to a portion of thebeam, is represented. In this case the expressions reported inTable 1 have to be referred to the actual size of the masonrymacro-element, furthermore in the horizontal direction only thenonlinear links of the column-panel interface have to be consid-ered and calibrated.

The collapse behaviour of the panel is dealt with by differentcriteria in compression and in tension. Precisely, once the compres-sive ultimate displacement is reached, the spring is removed fromthe model and the relevant reaction is applied as an external forceloading the corresponding elements. On the other hand, when thetensile limit displacement is attained, although the reaction is

(c)

(f)

e geometrical layout; (b, e) model corresponding to the basic mesh; (c, f) model

H

L

λhL/2

v

λh

λv

H/2

(a) (b) (c)

H

L'

λh

λv

λh

λv

H/2

L'/2

'

'

(d) (e) (f)

Fig. 5. Fiber calibration of the orthogonal springs of masonry-frame interfaces.

Table 1calibration of the orthogonal spring of masonry-frame interfaces.

Initial elastic stiffness Compression yielding force Tensile yielding force Compression ultimate displacement Tensile ultimate displacement

Orthogonal springs in the vertical interfaces (Fig. 5b)

Khp ¼ 2 EhkhsL

Fhcy ¼ skhrhc Fhty ¼ skhrht uhcu ¼ L2 ehcu uhtu ¼ L

2 ehtu

Orthogonal springs in the horizontal interfaces (Fig. 5a)

Kvp ¼ 2 Ev kv sH

Fvcy ¼ skvrvc Fvty ¼ skvrvt uvcu ¼ H2 evcu uvtu ¼ H

2 evtu

I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107 95

again re-distributed to the corresponding elements, the link is notremoved from the model, since it will be able to bear further com-pressive loads once the contact with the corresponding panel willbe restored.

2.1.2. Calibration of the sliding springs of the interfaceThe longitudinal spring governs the sliding-shear failure by

considering the potential sliding between two adjacent elements.The characteristics of this spring depend on the effective contactsurface between the adjacent elements. When the contact zonebetween the elements is zero the sliding spring is no longer active.The sliding springs have been modelled by means of a rigid-plasticconstitutive behaviour which is governed by a Mohr–Coulombyielding surface, sliding occurs in particular when the force inthe nonlinear link reaches its limit value, Flim, which is given bythe expression:

F lim ¼ ðc þ l rmÞAo ð1Þ

in which c is a cohesion parameter and l is the friction coefficient;rm is the current average value of the compressive stresses actingon the interface and Ao is the effective contact area of the two adja-cent elements. The sliding behaviour that can occur between theinfill and the adjacent frame has to be characterised by specific val-ues of the cohesion and the friction coefficient which, in general, arenot coincident with the values associated to the masonry media, inwhich the brick texture plays an important role.

2.1.3. Calibration of the diagonal springsShear failure is the most common type of masonry collapse

when the infilled frame is subjected to action in its own plane, thisis aided by the kinematic constraint exerted by the surroundingframe. In the macro-element model the shear failure collapse iscontrolled by the diagonal springs [32] according to a suitableyielding criteria. As highlighted in reference [40], two fundamen-tally different hypotheses, which lead to similar results, have beendeveloped in order to model the shear failure mechanism. In thefirst case, which has been accepted by Eurocode 6 (Design ofmasonry structures) [41], the shear strength is defined accordingto a Mohr Coulomb law

Fm ¼ fmo þ lc rn ð2Þ

where fmo is the shear strength associated to a zero value of com-pression strength, lc is a friction coefficient, defining the contribu-tion of compressive stresses, rn is the value of the averagecompressive stress. Values for fmo and lc should be determined byexperimental test [42–44]. It is worth highlighting that this crite-rion is the same suggested for the description of the sliding-shearfailure, although characterised by appropriate values of fmo and lc

that in general do not coincide with the corresponding values thatgovern the sliding-shear failure.

Alternative theories associate the diagonal shear failure to theprinciple tensile stresses that develop in the wall when subjectedto vertical loads and increasing horizontal forces. The mostadopted criterion based on this assumption is the well knownTurnsek and Cacovic criterion [42] in its modified form [44] that

u1

v1

up3

φ1

u2

φ2

v2

up2

rigid edge

um

ks

vonvo1

φo1 φon

k1 knkn-1

vo2

φo2

k2

up4

up1

beam/column internal degrees of freedombeam/column external degrees of freedom

Fig. 6. Qualitative representation of an interface between a beam column and theadjacent macro-elements and the corresponding degrees of freedom.

96 I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107

takes into account the influence of the geometry of the wall andthe distribution of action at maximum resistance. The latter crite-rion can be expressed as

fv ¼f t

b

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ rn

f t

rð3Þ

where fv is the average shear stress in the wall attained at the max-imum resistance, ft is the tensile strength of masonry, b is the shearstress distribution factor (depending on the geometry of the walland on the value of the ratio between the vertical N and horizontalH load), rn is the average compression stress due to vertical load N.

In the initial linear elastic range, the calibration of the diagonalsprings is simply obtained by enforcing an elastic equivalencebetween the panel and the corresponding masonry wall, which isconsidered as a pure shear deformable homogeneous plate, asreported in reference [32]. The ultimate shear load and the corre-sponding displacement of each diagonal spring are thereforederived by the knowledge of the mechanical material propertiesof the masonry by means of the adopted shear resistance criteria[32]. Since the macro-element must incorporate the mechanicalproperty of the finite portion of the modelled masonry wall, theultimate shear displacement of the element is directly associatedto the ultimate shear generalized deformability of the homoge-nised masonry media. The latter value can be evaluated experi-mentally or can be obtained from technical codes. The ultimatedisplacement of the wall can be associated with a specific valueof the ultimate angular deformation which is dependent on theparticular masonry media.

The post-yielding behaviour of each nonlinear link can be char-acterised according to different constitutive models, as shown inthe numerical applications reported in Section 4.

2.2. The interacting beam column element

For mixed masonry-reinforced concrete structures, beam–col-umn lumped plasticity elements are included in the model. Theinteraction between the frame elements, along the entire length,with the adjacent masonry is modelled by means of the interfacesof the macro-elements.

The frame element interacts with the masonry panels by meansof nonlinear-links distribution along the macro-element interfaces.Each interface, as those between masonry panels, is constituted byn orthogonal and a single longitudinal nonlinear links. In Fig. 6 thedegrees of freedom which govern the interaction between a paneland an adjacent beam element are indicated; upk (k = 1. . .4) are thefour degrees of freedom that describe the kinematic of the macro-element, u1, v1, /1, u2, v2, /2 are the degrees of freedom of the beamends, while voj, /oj (j = 1. . .n) and um are the degrees of freedomassociated to the nonlinear links of the interface. For the evaluationof the nonlinear behaviour of the frame element it has beenassumed that plastic hinges can occur in each sub-beam elementbetween two nonlinear links. This latter assumption provides areliable frame element model since it is able to embed the occur-rence of plastic hinges at different positions and it is consistentwith the adopted level of infill discretization. The inelastic behav-iour of the frame element, concentrated at plastic hinges is gov-erned by the interaction of the axial force and two flexuralmoments consistent with the yield surfaces of the concrete crosssections. In the application developed in this work the yieldsurfaces have been evaluated according to a standard approach,modelling the inelastic behaviour of beam cross sections consistentwith an inelastic-perfectly plastic behaviour [45]. Once the consti-tutive laws have been defined, both force and displacement con-trolled load processes can be performed according to procedurescurrently used in finite element analysis [45,46].

3. The typical failure mechanism of infilled frames and itsmacro-element modelling

Infilled frames exhibit a highly nonlinear inelastic behaviour asa result of the interaction between the masonry infill panel and thesurrounding frame. The most important factors contributing to thenonlinear behaviour are: cracking and crushing of the masonry,cracking of the concrete, yielding of the reinforcing bar, local bondslip, degradation of the bond friction mechanism at the panel-frame interface associated to a variation along the contact length.Macro-models, due to their simplicity, cannot accurately representall the various failure mechanisms of infilled frame.

Numerous experimental and numerical studies have identifiedmany complicated failure mechanisms due to the frame–panelinteraction [3,7]. According to the classification reported in therecent state of art review by Asteris et al. [7], these different failuremechanisms can be classified as an out-of plane buckling mode orone of four distinct in-plane modes.

The Diagonal Compression Buckling mode (DCB) is associatedwith the crushing of the infill within its central region due toout-of-plane buckling of the infill. This failure mode mainly hap-pens in slender infills and cannot be directly controlled throughthe in-plane response of the proposed macro-element approach.Several analytical models have been introduced to represent theout-of-plane behaviour of infill panels during seismic events. Themajority of experimental data suggest that after the initial crackingof the URM infill wall, the out-of-plane strength depends upon thecompressive strength of the masonry, not upon its tensile strength,due to the arching action of the infill wall. Recently Kadyiewski andMosalam [47] proposed an alternative model that considers boththe in-plane and out-of-plane behaviour consisting of two beamcolumn elements with a node at mid-span used to account forout-of-plane inertial forces.

The in-plane modes are qualitatively represented in Fig. 7together with their equivalent macro-element representations,according to two different levels of discretization. The CornerCrushing mode (CC), Fig. 7a1 is associated with the crushing ofthe infill in at least one of its loaded corners. This failure mecha-nism generally occurs in presence of weak masonry infill panel sur-rounded by a frame with weak joints and strong members [7,29].In Fig. 7a2 and a3 simplified representations of the kinematic

(d1)(c1)(b1)(a1)

(d2)(c2)(b2)(a2)

(d3)(c3)(b3)(a3)

Fig. 7. Qualitative representation of the in-plane collapse mode and its equivalent macro-element discretization. (a) Corner Crushing (CC) mode, (b) Diagonal cracKing (DK)mode, (c) Sliding Shear (SS) mode, and (d) Frame Failure (FF) mode.

I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107 97

associated to the proposed formulation, corresponding to the twodifferent mesh resolutions, are reported. The same figure high-lights how the Corner Crushing of the infill can be controlled bythe deformation of the nonlinear links of frame-panel interfaces.

The Diagonal cracKing mode (DK) corresponds to a shear diag-onal collapse and manifests by cracking along the compresseddiagonal of the infill, Fig. 7b1. The activation of this failure modeis governed by the diagonal nonlinear links of the macro-elementsas qualitative reported in Fig. 7b2 and b3. This is a common mech-anism that is often associated with damage within the frame.

The Sliding Shear (SS) mode is associated to the sliding shearfailure through bed joints, this mechanism generally happens inthe case of infill with weak mortar joints surrounded by a strongframe, Fig. 7c1. If the infill is modelled by only one element, thisbehaviour can also be captured by an appropriate calibration ofthe diagonal links of the macro-element which aims to controlboth the DK and SS failure modes at the macro-scale as qualita-tively reported in Fig. 7c2. When the infill is modelled as a meshof macro-elements, this mechanism is mainly controlled by thenonlinear links, along the horizontal interfaces between adjacentmasonry elements [32]. In Fig. 7c3 a simplified qualitative repre-sentation of the kinematic associated to this failure mechanism isreported.

The Frame Failure mode (FF) is seen in the form of a distributionof plastic hinges producing a mechanism in the frame, as inFig. 7d1; this mode occurs in weak frame and strong infill. Fig. 7d2

and d3 report a qualitative representation of the collapse mecha-nism associated to the two considered mesh discretizations, itcan be observed as a refined mesh can allow a better representa-tion of the collapse mechanism. Further details on the capabilityof the present approach to provide a satisfactory simulation ofthe collapse behaviour of infilled frames are reported in Section 4.

It is worth noting that all the considered basic mechanisms canoccur under different combinations and can involve low or heavydamage in the surrounding frame. The proposed approach is ableto identify combined mechanisms and the simultaneous presenceof damage corresponding to different failure modes.

4. Numerical applications

In this section the proposed model is employed to simulate thenonlinear response of masonry infilled frame for which experimen-tal results are available from previous studies reported in the liter-ature. In particular, the results of the following two differentexperimental programs have been taken into account:

– A research program, performed by the Construction EngineeringResearch Laboratory (Champaign USA), aimed to investigate theseismic vulnerability of masonry-infilled non-ductile reinforcedconcrete frames (designed to resist gravity loads) [48].

– An experimental campaign, carried out at the University of Col-orado, devoted to the seismic assessment of reinforced concreteframes designed in accordance with the Uniform Building Code[3,49,50].

4.1. Simulation of experimental results of non-ductile RC infilledframes

The first experimental program, under consideration, was per-formed at the Construction Engineering Research Laboratory, atChampaign (Illinois) to determine the seismic vulnerability ofexisting dormitory-type buildings, constructed during the 1950sand 1960s in the USA. The research investigated the structural

98 I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107

behaviour of non-ductile RC frames fully infilled with masonrypanels. The term non-ductile, as used in the reference paper [48],refers to the RC bare frames based on reinforcement details typicalof structures designed to resist gravity loads without taking intoaccount seismic actions. In the experimental campaign, five half-scale models were tested, all subjected to lateral, in-plane mono-tonic loading. The five models were all single-storey RC frames ofsingle-, double- or triple-bay construction. In the following, theresults of a single-bay infilled frame are considered, the detailedmodel configurations and the material properties of the consideredspecimen are reported in the reference paper [48].

Fig. 8 shows the layout corresponding to the geometrical char-acteristics and the reinforcement of the infilled frame. The inputdata, assumed in the numerical simulations, refer to a homoge-nised representation of the infill. The stress/strain relationshipfor concrete in compression is assumed to be of parabolic typeup to the strain eco and of rectangular type up to the ultimate strainecu. The stress/strain relationship for steel has been taken as that ofelastic-perfectly plastic.

The properties of the nonlinear links that govern the behaviourof the macro-element discretization of the masonry infill havebeen evaluated according to the straightforward calibration proce-dure reported in Section 2.

The proposed model has been implemented in the software3DMacro [39], and used for the numerical simulations presentedin this section. The reliability of the latter implementation, forthe seismic assessment of Unreinforced Masonry Building (URM),has been recently investigated, also by other authors [51] wheredifferent structural component models for URM have been com-pared. Further validations both for unreinforced and confinedmasonry structures are reported in [32–38,52].

The input parameters required for calibrating the reinforcedconcrete frame and the discrete element are reported in Tables 2and 3 respectively, according to the failure criteria considered inSection 2.

The first numerical application analyses the behaviour of thebasic single-bay frame without considering masonry infill contri-bution. Fig. 9 reports the comparison between the experimentaland numerical results relative to the bare RC frame. A good agree-ment can be observed between experiment and simulation both interms of the capacity curve and the collapse mechanism, the latteris associated to the activation of plastic hinges in column ends.

The results relative to the single-bay RC infilled frame arereported in Fig. 9b, where the base shear versus the top horizontaldisplacement is reported. The continuous line is relative to theexperimental results while the dashed lines refer to the numericalsimulation; the latter has been performed by considering a 3 � 3macro-elements discretization. The agreement between experi-mental and numerical results, at least in terms of maximum baseshear force and top displacements, can be considered satisfactory.

1829 203 203

76

381 381

L

1327

76

381

H

Fig. 8. Layout corresponding to (a) the geometrical characteristics a

A further investigation of the role of the frame and the infill dur-ing the interaction is conducted in Fig. 10. In Fig. 10a the contribu-tions of the infill and the interacting frame to the base shear arereported together with the total base shear, already displayed inFig. 9b. Furthermore, in Fig. 10b a comparison of the monotonicnonlinear response of the frame in bare and interacting conditionsis reported. It is interesting to observe how the contribution of theframe increases when it interacts with the infill.

The shear sliding properties, reported in Table 3, have been cho-sen by assuming typical values of brick masonry media. However,with the aim to investigate the role of the cohesion and the frictionangle, which control the shear sliding properties, in Fig. 11 the baseshear versus the top displacement has been evaluated for differentvalue of the cohesion c and friction angle l, it can be observed avery small sensitivity to the friction angle and a low dependenceon the cohesion.

In Fig. 12b the damage scenario predicted by the model corre-sponding to the ultimate value of drift of 2.86%, is compared tothe corresponding damage crack patterns obtained experimentally[48], as shown in Fig. 12a. It is worth to notice that, due to theirsimplicity, macro-elements provide a simplified representation offailure mechanisms for infilled frame. With reference to the com-parison reported in Fig. 12a and b it has to be considered that atthe base of the column there is the overlapping of steel reinforce-ment, as detailed in [48]. This overstrength can partly justify thedifference between numerical and experimental collapse simula-tions, since the experiment does not show any hinge at the baseof the left column, provided by the numerical simulation, whichis instead collocated in the middle.

The used representation in the interface allows the distinctionof the reactive compressive zone from the cracked one due to ten-sile forces; diagonal bars inside a panel indicate the yielding of thediagonal springs. It is worth noting how the proposed approach isable to grasp the distribution of damage on both the infill and thesurrounding frame.

In Fig. 12c and d the flexural moment distribution in the frame,corresponding to the values of drift 0.55% and 2.86%, are reported.These further representations show how the proposed approachprovides a simulation of the complex interaction between frameand infill, characterised by continuous variation of the contact zonewith redistributions of internal forces both in the infill and the sur-rounding frame.

4.2. Simulation of experimental results of RC infilled frames designed inaccordance with the UBC

To investigate the performance of masonry-infilled RC framessubjected to in-plane lateral loads, a comprehensive study wascarried out at the University of Colorado. The results and majorconclusions of this study, obtained from the experimental

610

S4S2

S8S5

457

S6

457

S7

197

127

203

5 #3 127

S3S1 203

6 ga./12.7mm

6 ga./12.7mm 4 #3

nd (b) the typical geometrical reinforcing, from reference [48].

Table 2Case study 1. Mechanical characteristics of concrete and steel.

Concrete Steel

E (MPa) rc (MPa) rt (MPa) ec0 (%) ecu (%) w (kN/m3) E (GPa) fy (MPa) eu (%)

29,992 38.50 1.50 0.20 0.35 25 200 377 10

Table 3Case study 1. Mechanical characteristics of masonry infill.

Flexural

w(kN/m3)

E(MPa)

rc

(MPa)rt

(MPa)k (cm)

18 2500 5.00 0.15 10

Shear diagonal

G (MPa) fv0 (MPa) lc

1000 0.30 0.15

Shear sliding

(horizontal) (vertical)

c (MPa) l c (MPa) l

0.30 0.4 0.7 0.5

I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107 99

investigation conducted on twelve one-half-scale frame speci-mens, are summarised in the paper by Mehrabi et al. [50] andreported in more details in [49]. The study focused on RC framesdesigned in accordance with code provisions, with and withoutthe consideration of strong earthquake loadings. A six-storey,three-bay, reinforced concrete moment-resisting frame wasselected as a prototype structure. The design loads complied withthe specifications of the Uniform Building Code (UBC) (1991).Two types of frames were considered: (i) a ‘‘weak’’ frame, repre-sentative of existing reinforced concrete structures not designedto resist to earthquake loadings; (ii) a ‘‘strong’’ frame, designedfor Seismic Zone 4 according to the UBC. In the present study theresults of a loaded specimen designed for Seismic Zone 4 are

base

she

ar [

KN

]

0

10

20

30

40

0 10 20 30 40

top displacement [mm]

(a)

Fig. 9. Numerical simulation of experimental tes

considered. All the details of the considered specimen, identifiedin the reference works by the number 7, are reported in the studies[3,49,50]. Here, for the sake of comprehensiveness, the layout ofthe frame, together with IFS geometrical and reinforcing character-istics, are displayed in Fig. 13.

The considered specimen is characterised by a strong framewith a strong solid type masonry infill and possesses an aspectratio, h/L, equal to 2/3, where h is the height and L is the widthof the infill. The ultimate resistance and failure were dominatedby the Corner Crushing of the infill at a displacement of approxi-mately 130 mm, and internal crushing at around 180 mm. In spiteof the presence of a strong panel, no shear failure was observed inthe columns. The aim of simulating the nonlinear cyclic experi-mental behaviour, by means of the adoption of simple constitutivelaws, has been pursued by means of the following assumptions forthe reinforced concrete frame and the infilled masonry. The nonlin-ear cyclic behaviour of the reinforced concrete has been modelledaccording to a bilinear-envelope Takeda scheme [53] as reported inTable 4.

For the simulation of the cyclic degrading hysteretic behaviourof the masonry infill, subjected to a combination of vertical loadsand to a sequence of lateral load reversals, the use of an idealisedbi-or tri-linear resistance envelope is recommended [40]. Here,the shear-diagonal behaviour of the masonry infill has been mod-elled by means of an idealised bi-linear envelope, consistent withthe approach reported in [32] for unreinforced masonry, whose rel-evant parameters are summarised in Table 5, these have been cho-sen by considering the experimental data reported in [3], whichalso contain the results of shear sliding experimental tests. Theparameter Ko represents the initial slope of the idealised envelope,in the post-yielding behaviour, the kinematic hardening (or soften-ing) is governed by the parameter a, that is given by

a ¼ Hd max � Hmax

dmax � dH maxð4Þ

where Hmax is the resistance at the elastic limit, Hdmax is the resis-tance at the ultimate displacement and dHmax and dmax are the cor-responding displacements of the envelope curve.

The unloading stiffness has been expressed according to the fol-lowing simple expression

base

she

ar [

KN

]

0

25

50

75

100

0 10 20 30 40

top displacement [mm]

(b)

ts on the (a) bare frame and (b) infill frame.

0

20

40

60

80

100

0 10 20 30 40

base

she

ar [

KN

]

top displacement [mm]

0

20

40

60

0 10 20 30 40

base

she

ar [

KN

]

top displacement [mm]

(a) (b)

Fig. 10. Masonry and frame contribution to base shear.

0

20

40

60

80

0 2 4 6 8 10

bas

e sh

ear

[KN

]

top displacement [mm]

μ

μ

μ

0

20

40

60

80

0 2 4 6 8 10

bas

e sh

ear

[KN

]

top displacement [mm]

μ

μ

μ

Fig. 11. Base shear versus top displacement for different values of the shear sliding paramenters.

(b)(a)

(d)(c)

Fig. 12. Collapse behaviour. Damage scenario: (a) experimental test, (b) numerical model; Flexural moment distribution in the frame for different values of drift: (c) 7.5 mm,and (d) 40 mm.

100 I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107

254

203

420

F2 F1 F1 F2

8 #5

229

2 #5

92

92

254

203

280 280 430

152

4 #6

203

203

420

Lateral force

1422

230

360

254 178 178 254 2133

6 #5 4 #4

Fig. 13. Layout corresponding to (a) the geometrical characteristics and (b) the typical geometrical reinforcing, from reference [3].

Table 4Case study 2. Mechanical characteristics of concrete and steel.

E (MPa) rc (MPa) rt (MPa) ec0 (%) ecu (%) w (KN/m3) Cyclic behaviour

Concrete18,670 25.46 1.00 0.20 0.35 25

E (GPa) fy (MPa) eu (%)

Steel200 420 10

I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107 101

Ku ¼ KoKI

Ko

� �b

ð5Þ

where Ko is the initial stiffness, KI is the value of the stiffness that isobtained by connecting the point corresponding to the current

Table 5Case study 2. Mechanical characteristics of masonry infill.

Flexural w (kN/m3)

E(MPa)

rc

(MPa)rt

(MPa)k(cm)

18 2100 5.00 0.15 10

Shear diagonal G(MPa)

fv0

(MPa)lc a

(%)b

750 0.70 0.40 7.5 0.8

Shear sliding

(horizontal) (vertical)

c (MPa) l c (MPa) l

0.2 0.8 0.4 0.8

plastic deformation with the origin of the axes and b is a real num-ber that can assume a value in the range 0–1. The re-loading stiff-ness Kr, in the cyclic behaviour, has been set by returning to thepoint which corresponds to the maximum reached plasticdeformation.

The flexural behaviour is governed by the orthogonal interfacesprings. These have been calibrated according to an elastic-per-fectly-plastic constitutive law, with different tensile and compres-sive limits, as specified in Table 5, and an unloading stiffnessexpressed according to Eq. (5) with b = 0.8. Furthermore, the duc-tility of the orthogonal springs has been considered infinite in com-pression and equal to 1.5 under tensile actions.

Fig. 14 reports the comparisons between the experimental andnumerical results in terms of base shear versus the top displacement.In Fig. 15 a comparison between the experimental observed failuremechanism and the simplified numerical prediction, is reported.

Fig. 16 reports, separately, the contributions of the interactingframe and of the infill obtained by the numerical simulation. Itcan be observed how the infill is characterised by a strong degrad-ing behaviour, that has been controlled simply by setting theparameters of the assumed constitutive law, consistent with a bi-linear degrading envelope.

The low contribution of the infill, after several cyclic loads, andits brittle behaviour is consistent with the actual failure mecha-nism, reported in the Fig. 15a, in which the masonry infill appearsseverely damaged.

Keeping in mind that both the reinforced concrete and themasonry infill have been calibrated according to very simple con-stitutive laws, by observing the maximum reached forces and theexhibited hysteretic behaviour, the agreement between the exper-imental and numerical results can be considered satisfactory.

base

she

ar [

KN

]

displacement [mm]

base

she

ar [

KN

]

displacement [mm]

(b)(a)

Fig. 14. Base-shear versus top displacement: (a) experimental results; and (b) numerical simulations.

Fig. 15. Collapse behaviour. Damage scenario: (a) experimental test, and (b) numerical model.

base

she

ar [

KN

]

top displacement [mm]

base

she

ar [

KN

]

top displacement [mm]

(a) (b)

Fig. 16. Contribution of interacting frame (a) and infill masonry panel (b).

102 I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107

The choice of a simple constitutive law is justified by the goalof proposing a method suitable for practical applications andbased on a limited number of parameters, however the proposedmodel is able to account for more complex constitutivebehaviours.

4.3. Influence of the model discretization: from macro- to micro-modeling

In the considered approach each macro-element inherits thegeometry of the masonry portion that represents, this aspect con-stitutes a great advantage that is not common to all the simplifiedapproaches based on a macro-element discretization.

Furthermore the consistent geometry of the element allows aneasy implementation of openings in the infills and permits theimplementation of models characterised by different level of dis-cretization according to different mesh resolutions and to thefine-tuning of nonlinear links of the interfaces, this latter associ-ated to the distance between the NLinks k.

In Fig. 18 the results of push-over-analyses, performed on thenon-ductile model considered in Section 4.2, associated to differentmesh resolutions are reported. Further four different mesh resolu-tions have been considered whose increasing computational cost issummarised in Table 6.

Mesh A corresponds to the basic mesh size in which the infill ismodelled by a single macro-element. Mesh B and D represent more

Table 6Computational resources associated to the considered discretizations.

Mesh Number of elements Number of degrees of freedom

Panels Frames Panels Frames Total

A 1 3 4 6 10B 4 6 16 15 31C 9 9 36 24 60D 16 12 64 33 97M 143 40 572 117 689

Table 7Mechanical characteristics of micro model.

Flexural w (kN/m3) E (MPa) rc (MPa) rt (MPa)

18 2500 5.00 0.15

Shear diagonal G (MPa) fv0 (MPa) lc

1500 Linear elastic Linear elastic

Shear sliding (horizontal) (vertical)

c (MPa) l c (MPa) l

0.30 0.4 0.7 0.5

0

25

50

75

100

0 10 20 30 40

base

she

ar [

KN

]

top displacement [mm]

Fig. 18. Influence of the mesh discretization of single-bay RC infilled frame; baseshear as a function of the top horizontal displacement.

I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107 103

refined mesh resolutions in which the infill is represented by 2 � 2and 4 � 4 macro-elements respectively while the 3 � 3 mesh, con-sidered in the simulation reported in Section 4.1, has been identi-fied as C.

Mesh M is relative to a detailed model in which each discrete-element corresponds to a single brick and is assigned to representboth the brick and the mortar joints properties according to thecorrespondence reported in Fig. 17a and b. In Fig. 17c–f a shrinkrepresentation of the macro-element is reported, although the ele-ment-interfaces have zero thickness, with the aim to show thenonlinear links of the interfaces. In the same figures it is high-lighted how the nonlinear links are delegated to represent themortar joints and the deformability of bricks by indicating the cor-responding influence area.

The parameters used in the micro-model refined simulation aresummarized in Table 7.

This micro-element application shows the capability of the dis-crete element to be used on a different scale, versus a micro-modelrepresentation.

From Fig. 18 it can be observed a small sensitivity to the meshsize and very good agreement between the micro and macro-model numerical simulations. It is worth highlighting that, to theauthor knowledge, this is the first simplified approach that canbe used both at macro- and micro-scale.

Micro Model discrMasonry portion

Longitudinal SDiagonal Springs

(a) (b)

(d) (e)

Fig. 17. The micro-model discretazation. (a) The geometrical layout; (b) the discretizatio(e) the influence areas of the longitudinal springs, and (f) the influence area of orthogon

Furthermore the micro-model approach can also be used forvalidating and better calibrating the macro-model parameters,which is a great advantage.

In Fig. 19 simplified representations of the damage scenarioscorresponding to the mesh A, B, D, M are reported. All the simula-tions show collapse mechanisms which produce a shear-diagonal

etization Mechanical scheme

prings Orthogonal Springs

(c)

(f)

n; (c) the mechanical representation; (d) the influence area of the diagonal springs;al springs.

Fig. 19. Influence of the mesh discretization of single-bay RC infilled frame; Simplified representations of the damage scenario predicted numerically.

0

20

40

60

80

0 10 20 30 40

base

she

ar [

KN

]

top displacement [mm]

0

20

40

60

80

0 10 20 30 40

base

she

ar [

KN

]

top displacement [mm]

(1)

(2)

d1 d2 d3

(3)

w1 w2 w3

Fig. 20. Influence of openings in the macro-model numerical simulation of single-bay RC infilled frame; (1) base shear as a function of top displacement; (2) damage scenarioin the presence of door openings; and (3) damage scenario in the presence of windows openings.

104 I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107

Table 8Geometrical characteristics of the openings.

d1 d2 d3 w1 w2 w3

ao = bo/bi 1/3 1/2 3/5 1/3 1/2 3/5ko = ho/bo 2 1.5 1 1 2/3 1/2

I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107 105

failure with damage on the frame. In particular, the micro-modelsimulation provides results in better agreement with the experi-mental damage scenario reported in Fig. 12a.

4.4. Modelling of openings in the masonry infills

As highlighted in the previous paragraph, since each discrete-element is assigned to represent the nonlinear behaviour of thecorresponding macro-portion, as in a finite element simulation,this approach allows a simple modelling in the presence ofopenings.

0

20

40

60

80

0 10 20 30 40

base

she

ar [

KN

]

top displacement [mm]

(a)

Fig. 21. Comparison between micro- and macro-model simulations in terms of base

Fig. 22. Comparison between micro- and macro-model simulations in terms

Although this important feature should require an extensiveinvestigation and a possible validation with experimental results,in this context, for the sake of conciseness, it is shown how the pro-posed approach leads to a straightforward and reliable modellingof openings in the Infilled Frame Structures.

With reference to the RC masonry infilled frame analysed inSection 4.1, the influence of a central door and window openingsin the masonry infill are examined.

The first investigation, reported in Fig. 20, analyses the reduc-tion of the base shear due to the presence of central door or win-dow openings, of various dimensions, characterised by a width bo

and a height ho. Each geometrical layout is identified by the ratioao = bo/bi, between the width bo of the opening and the width bi

of the infill, and the aspect ratio ko = ho/bo, as reported in Table 8.The results are expressed in terms of base shears as function of

the top displacements and are compared with the correspondingvalues of the bare frame and of the infilled frame withoutopenings.

0

20

40

60

80

0 10 20 30 40

base

she

ar [

KN

]

top displacement [mm]

(b)

shear versus top displacement; (a) door openings; and (b) windows openings.

of collapse mechanisms; (a) door openings; and (b) windows openings.

106 I. Caliò, B. Pantò / Computers and Structures 143 (2014) 91–107

With the aim to obtain a numerical validation of the obtainedresults, Fig. 21 reports a comparison, in terms of base shear as afunction of top displacement, between the macro- and the micro-model simulations with reference to the cases identified as d2and w2. Furthermore, Fig. 22 reports a simplified representationof the corresponding collapse mechanisms. A strong correlationbetween the basic macro-model and the refined micro-model pre-dictions can be observed.

5. Conclusions

Infilled Frame Structures represent a high percentage of existingand new buildings in many seismically prone areas around theworld. These composite structural systems are governed by theinteraction between frame and infill walls. The high nonlinearresponse of the masonry infill and the ever-changing contact con-ditions along the frame-infill interfaces make the simulation of thenonlinear behaviour of an infilled frame building a challengingproblem currently involving many research groups worldwide.

In this context, an alternative innovative approach for the simu-lation of the seismic behaviour of Infilled Frame Structures, suitableboth for research and current engineering practice applications, ispresented in this paper. In this approach, the infilled wall is mod-elled by means of an innovative discrete element, originally con-ceived for the simulation of the nonlinear response ofUnreinforced Masonry Building [32], while the reinforced concreteframe is modelled by means of inelastic beam–column elements inwhich the plastic hinges can form at different positions along thebeam-span. The computational cost of the proposed numericalapproach is significantly lower in comparison to the nonlinear finiteelement modelling, which requires both discretization of the frameand the infill. Furthermore, the adopted strategy, based on a combi-nation of finite element and macro-element discretization, offersmany advantages with respect to the existing simplified methodsin which the contribution of the infill is represented by one or morediagonal struts. The equivalence between the masonry portion andthe macro-element is based on very simple physical considerationsfounded on a fiber element calibration [32] and the interpretationof the numerical results is simple, straightforward and unambigu-ous. Since each discrete-element is assigned to represent the non-linear behaviour of the corresponding macro-portion, as in a finiteelement simulation, this approach allows a simple modellingincluding the presence of openings and can be used on differentscales, from macro- to micro-modelling, although its use has beenconceived in the context of macro-elements. The effectiveness ofthe proposed modelling strategy has been evaluated by means ofnonlinear monotonic and cyclic static analyses performed on RCmasonry infilled frame which have been the object of theoreticaland experimental research. The results seem to indicate that theproposed approach can be adequate for seismic assessments andthe design of Infilled Frame Structures since it requires very lowcomputational resources, allows easy interpretation of results andprovides satisfactory accuracy.

Acknowledgement

This research has been supported by the Italian Network ofSeismic Engineering University Laboratories (ReLUIS).

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