A Model of URM for Seismic Evaluation With Diagonals_ok Ok Ok

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A new discrete element model for the evaluation of the seismic behaviourof unreinforced masonry buildings

Ivo Caliò ⇑, Massimo Marletta, Bartolomeo Pantò

Department of Civil and Environmental Engineering, University of Catania, Italy

a r t i c l e i n f o

Article history:

Received 27 May 2011Revised 20 October 2011

Accepted 5 February 2012

Available online 31 March 2012

Keywords:

Masonry buildingUnreinforced Masonry structures

In-plane masonry response

Macro-element approach

Discrete element model

Seismic vulnerability

Nonlinear masonry analysis

a b s t r a c t

The evaluation of the nonlinear seismic response of masonry buildings represents a subject of consider-able importance whose resolution is nowadays a main research topic in earthquake engineering. Refined

nonlinear finite element models require a huge computational cost that makes these methods unsuitablefor practical application. In this paper an innovative discrete-element model, conceived for the simulation

of the in-plane behaviour of masonry buildings, is presented. The basic idea of the proposed approach isto approximate the in-plane nonlinear response of masonry walls by an equivalent discrete element. Thiselement is able to reproduce the typical in-plane collapse behaviour of a masonry wall subjected to earth-

quake loading. The reliability of the proposed approach has been evaluated by means of nonlinear incre-

mental static analyses performed on masonry structures, for which theoretical and/or experimentalresults are available in the literature. The proposed computational strategy provides a relatively simpleand practical tool which could be of significant value for the design and the vulnerability assessment

of unreinforced masonry structures in seismic areas. 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The assessment of the nonlinear seismic behaviour of UnRein-

forced Masonry buildings (URM) represents a subject of greatimportance, however it is rather difficult to solve. In order to esti-mate the seismic vulnerability of an existing building and to ascer-tain if the structure requires a seismic upgrade, a structural

engineer needs simple and efficient numerical tools whose com-plexity and computational demand must be appropriate for thepractical engineering purposes. However, the simulation of thenonlinear dynamic behaviour of a masonry building represents a

challenging problem which rigorously requires the use of compu-tationally expensive nonlinear finite element models and, aboveall, expert judgment. For this reason, the seismic assessment andsubsequent rehabilitation of URM constitutes a contemporary is-

sue in most seismic regions in which the historical masonry build-ings represents a significant number of the existing structures.Even in the USA, the need of introducing practical approaches forthe evaluation of the seismic behaviour of URM has been recently

highlighted [1–3].The different behaviour of masonry structures, compared to or-

dinary concrete and steel buildings, requires ad hoc algorithms

capable of reproducing the nonlinear behaviour of masonry mediaand providing reliable numerical simulations. Refined finite ele-ment numerical models, such as the smeared cracked and discrete

crack finite element models [4,5], able to predict the complex non-linear dynamic mechanical behaviour and the degradation of themasonry media, require sophisticated constitutive laws and a hugecomputational cost. As a consequence these methods are nowa-

days not suitable for practical application and extremely difficultto apply to large structures. An alternative approach to the nonlin-ear FEM is represented by the rigid-body spring models (RBSM),specifically formulated with the aim of approximating the macro-

scopic behaviour of masonry walls with reduced degrees of free-dom. Some valuable application of this approach are relative tohistorical masonry buildings [6,7]. However, the difficulties inthe model implementation make the RBSM rarely used in the engi-

neering practice. A comprehensive review in the current develop-ment on numerical issues on masonry mechanics is reported in[8]. In the latter work it is also highlighted that a complex analysistool does not necessarily provide better results than a simple tool.

In the last three decades, many authors have developed simpli-fied or alternative methodologies that, with a reduced computa-tional effort, should be able to predict the nonlinear seismic

behaviour of masonry buildings and to provide reliable numericalresults for engineering practice purposes. The most commonlyused practical approach for the analysis of URM is the so called‘equivalent frame model’, in which the masonry building is repre-

sented by an equivalent nonlinear frame structure constituted by

0141-0296/$ - see front matter 2012 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.engstruct.2012.02.039

⇑ Corresponding author. Address: Department of Civil and Environmental Engi-neering, 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: [email protected] (I. Caliò).

Engineering Structures 40 (2012) 327–338

Contents lists available at SciVerse ScienceDirect

Engineering Structures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e n g s t r u c t

http://dx.doi.org/10.1016/j.engstruct.2012.02.039mailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2012.02.039http://www.sciencedirect.com/science/journal/01410296http://www.elsevier.com/locate/engstructhttp://www.elsevier.com/locate/engstructhttp://www.sciencedirect.com/science/journal/01410296http://dx.doi.org/10.1016/j.engstruct.2012.02.039mailto:[email protected]://dx.doi.org/10.1016/j.engstruct.2012.02.039


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nonlinear beam elements and rigid offsets [9,10]. According to the

equivalent frame model approach, each wall of the building is sub-divided into macro-elements representing piers, spandrels and ri-

gid zones. The nonlinear behaviour of piers and spandrels aresimulated by nonlinear frame element, while the rigid zones, in

which the damage cannot occur, are substituted by rigid offsets.The substitution of a masonry portion with a frame element hassome restrictions due to the inaccurate simulation of the interac-tion between macro-elements, to the difficulties that arise for com-

plex geometry and to the weak modelling of the cracked conditionof panels. For these reasons some authors proposed the use of two-dimensional macro-elements, D’Asdia and Viskovic [11] model the

masonry by linear elastic macro-finite elements. Namely, the non-linearity is simulated by updating the geometry of the elementexcluding the volumes undergoing tensile stresses, or cracked, orgenerally no more co-operating. A similar approach has been pro-

posed by Braga et al. [12] that introduced the so-called ‘multi-fan

panel element’ in which the stress field of the panel is assumedto follow a multi-fan pattern, the material behaviour is assumedlinear elastic in compression and non reacting in tension. More re-

cently Vanin and Foraboschi [13] proposed a new approach basedon modelling masonry panels by means a strut and tie model.Comparative evaluations of commonly used techniques for practi-cal analysis of realistic URM structures subjected to seismic load-

ings are reported in the studies of Lourenço [8], Seible andKingsley [5], Magenes and La Fontana [9] and Kappos et al. [10].An overview on recent code developments and state-of-the-artmethods of earthquake resistant design of masonry buildings is

reported in [14] where the experimental results are also used inorder to justify the analysed numerical approaches.

In this paper a new modelling approach for the simulation of the

seismic behaviour of masonry buildings, suitable for current engi-neering practice applications, is presented. The proposed approach

is based on the concept of macro-element discretization [8] and has

been conceived with the aim of capturing the nonlinear behaviourof an entire masonry wall and of the entire building, as an assem-

blage of several walls. The model is based on a plane nonlinear dis-crete element, able to simulate the behaviour of masonry wall in its

own plane. The basic macro-element consists of an articulatedquadrilateral with rigid edges in which two diagonal springs governthe shear behaviour. The flexural and sliding shear behaviour isgoverned by discrete distributions of springs in the sides of the

quadrilateral that preside over the interaction with the adjacentmacro-elements. The calibration of the model require only a fewparameters to define the masonry material based on results from

current experimental tests. The computational cost of the proposednumerical approach is greatly reduced, compared to a traditionalnonlinear finite element modelling. Since the equivalence betweenthe masonry portion and the macro-element is based on very

simple physical considerations, the interpretation of the numerical

results is simple and straightforward. This novelty approach is in-tended as a tool, which requires low computational resources, forinvestigating the nonlinear behaviour of URM buildings.

In the paper the efficiency of the computational tool is shown bymeans of nonlinear static analyses, carried out on case studies of masonry walls and structures that have been object of theoreticaland experimental researches. The influence of the discretization

of the model associated to the size of the mesh of macro-elementand the fine-tuning of the nonlinear links have been alsoinvestigated.

2. The basic discrete-element

The basic element of the proposed approach has a simplemechanical scheme, Fig. 1. It is represented by an articulated

(a) (b)

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

(a) (b) (c)F

q

F

q

F

q

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

328 I. Caliò et al. / Engineering Structures 40 (2012) 327–338


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quadrilateral constituted by four rigid edges connected by four

hinges and two diagonal nonlinear springs. Each side of the panelcan interact with other panels or elements or supports by meansof a discrete distribution of nonlinear springs, denoted as interface.Each interface is constituted by n nonlinear springs, orthogonal to

the panel side, and an additional longitudinal spring which controls

the relative motion in the direction of the panel edge. In spite of itsgreat simplicity, such a basic mechanical scheme is able to simu-

late the main in-plane failures of a portion of masonry wall subjected to horizontal and vertical loads. These well-knowncollapse mechanisms, namely the flexural failure, the diagonal shear failure and sliding shear failure, are approximately represented in

Fig. 2 where the typical crack patterns together with the qualita-tive kinematics of masonry portion are also sketched. Fig. 3 showshow the proposed macro-element allows a simple and realisticmechanical simulation of the corresponding failure mechanisms.

The flexural failure mode, sketched in Fig. 2a, is associated to the

rocking of the masonry portion in its own plane. The loss of thebearing capacity is related to the progressive rupture of the panelin the tensile zone and/or to the crushing of the panel in the com-

pressive zone. The latter mechanism is reproduced by the consid-ered macro-element as shown in Fig. 3a. The axial and flexuraldeformability of the masonry panel is concentrated in the orthog-onal nonlinear links according to a simple fiber model discretiza-tion, as is usually adopted for reinforced concrete structures.

The diagonal-shear failure mode, drafted in Fig. 2b, is associated

to the loss of the bearing capacity of the masonry panel due toexcessive shear and to the consequent formation of diagonal cracksalong the directions of the principal compression stresses. Throughthe proposed discrete-element this collapse mechanism can be

governed by means of the two diagonal nonlinear springs whichhave the role to simulate and predict the nonlinear shear responseof the modelled masonry portion, Fig. 3b.

The sliding-shear failure mode is associated to the sliding of the

masonry panel in its own plane. In this case the loss of the bearingcapacity is associated to the formation of cracks parallel to the bed-

joints (Fig. 2c). This failure mode, which generally occurs for lowlevels of vertical loads and low values of friction coefficients, can

be controlled by the longitudinal nonlinear springs of the inter-faces, Fig. 3c.

In the following paragraphs, the kinematics and the mechanical

characteristics of the basic element are outlined. Consequently, asimple procedure, based on a fiber modelling approach, for thecalibration of the mechanical parameters of the nonlinear links of the discrete element, is presented.

2.1. The kinematics

According to the proposed discrete element approach, a ma-sonry macro-portion is assimilated to an equivalent mechanical

scheme in which the physical role of each component is simple

and unambiguous. This feature is not common to all the simplifiedmodels proposed in the literature and is noteworthy since it makesthe direct implementation of the model in many general-purposefinite element platforms possible. As mentioned in the sequel,

the model can be also improved by using a refined mesh of ele-

ments in order to obtain a more detailed description of the kine-matics of the structure and the collapse mechanism.

The degrees-of-freedom of the structural scheme are thoseassociated to the in-plane motion of the basic elements. Each ele-

ment exhibits three degrees-of-freedom, associated to the in-planerigid-body motion, plus a further degree-of-freedom needed forthe description of the shear deformability. In Fig. 4 the chosenLagrangian parameters for the panel and the corresponding

deformed configurations are reported. The deformations of theinterfaces are associated to the relative motion between corre-sponding panels, therefore no further Lagrangian parameters mustbe introduced in order to describe their kinematics. Each interface

can be identified by two nodes, i and j. Each node actually corre-sponds to the joints of the model belonging to the panels which

are connected by the interface.Since the proposed modelling approach is suitable for describ-

ing the behaviour of a plane wall loaded in its own plane, athree-dimensional masonry building can be modelled as an assem-blage of plane walls. Therefore, in this simplified approach the

behaviour of the wall in the out-of-plane direction is not consid-ered. The latter limitation is common to all the proposed simplifiedapproaches. The inclusion of the out-of-plane behaviour of the ma-sonry wall in the proposed model constitutes a further develop-ment which, at present time, is a work in progress.

In Fig. 5 it is shown how a simple masonry wall can be modelledby means of the proposed modelling approach. Namely, Fig. 5arefers to a basic scheme composed by 12 panels and is character-ised by 48 degrees of freedom, while in Fig. 5b the same wall is

(a) (b) (c)F

FF

qq q

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.

u3

u1

2uu4

1u u3

u2 u4

(a) (b)

Fig. 4. (a) Chosen Lagrangian parameters for the quadrilater; and (b) deformedconfigurations corresponding to the activation of each degree-of-freedom.

I. Caliò et al. / Engineering Structures 40 (2012) 327–338 329


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modelled through a more refined mesh composed by 48 quadrilat-erals which require 192 degrees of freedoms. The use of a more re-fined mesh is not mandatory however in some cases can providemore accurate results and a better description of the collapse

mechanism.

2.2. Calibration of the nonlinear springs

It has been shown that the kinematics of the proposed model isable to simulate the main failure mechanisms of a masonry ele-ment. However the effectiveness of the simulation of the nonlinear

behaviour relies on a suitable choice of the mechanical parametersof the model inferred by an equivalence between the masonry walland a reference continuous model characterised by simple but reli-able constitutive laws. This equivalence relies on a straightforwardfiber calibration procedure, and is based only on the main mechan-

ical parameters which characterise the masonry according to anorthotropic homogeneous medium.

2.2.1. Calibration of the interface orthogonal springs

Since the masonry is considered as a homogeneous medium itsglobal 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 simulated

by the interface orthogonal springs connecting the panel to adja-cent elements. Each spring is here calibrated by adopting a specificconstitutive law for the masonry media, according to a simplifiedfiber model approach. In the description of the calibration proce-

dure an elasto-plastic behaviour with limited deformability is as-sumed. The orthotropic nature of the masonry can be simplyconsidered by calibrating separately the horizontal and vertical

interfaces according to the mechanical properties of the corre-sponding directions.

The case regarding a horizontal interface connecting two gener-ic panels, k and l, characterised by different mechanical and

geometrical properties, is represented in Fig. 6. The vertical orthog-onal springs have the role of simulating the deformability of thetwo panels in the vertical direction. The stiffness calibration of

void

void

void

void

(a) (b)

Fig. 5. Masonry wall and corresponding macro-element discretizations with different mesh resolutions.

F

uu

spring 1Panel k

Panel l

Panel k

Panel l

spring 2

single springequivalent to springs1 and 2 in series

L 1

L /21

L /22

L 2

ty1

ty1 tu1

F cy1

cy1u cy1u

ucy2 cy2

F ty2

F cy2

ty2 tu2uu

u

F

u

F

u

influence area

λ

zerothickness

influence area

Fig. 6. Calibration of the interface springs.



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each panel is simply obtained by assigning to each link the axialrigidity of the corresponding masonry strip. The masonry strip is

identified by the influence area of the spring and the half-dimen-sion of the panel in the direction perpendicular to the interface.This procedure provides couples of springs in series. Each coupleof springs is then replaced by a single resulting nonlinear elasto-

plastic spring as described in what follows.

With reference to a single panel, according to the above-de-scribed procedure, the initial stiffness K p, the compression and ten-sile yielding strengths F cy, F ty and the corresponding ultimate

displacements ucu, utu are derived by the knowledge of themechanical material properties of the masonry as follows:

K p ¼ 2E ks

L ð1Þ

F cy ¼ skrc ; F ty ¼ skrt ð2a;bÞ

ucu ¼ L

2ecu; utu ¼

L

2etu ð3a;bÞ

where E is the Young’s modulus in the direction orthogonal to theinterface; rc and rt are the compressive and tensile yielding stres-

ses, ecu and etu are the ultimate compressive and tensile strains; s isthe thickness of the represented masonry and k is the distance be-

tween two nonlinear link.Each couple of elasto-plastic springs in series, once the mechan-

ical properties have been defined, can be replaced by a singleresulting nonlinear spring, Fig. 6, defined by appropriate stiffness,yielding strengths and ultimate displacements. The substitution

of two nonlinear springs in series with a nonlinear single springleads to a reduction of the computational cost.

In the initial elastic range the resulting stiffness of the twosprings in series, of stiffness K p1 and K p2 is as follows:

K ¼ K p1 K p2K p1 þ K p2

; ð4Þ

while the yielding strengths of the resultant spring are given by theminima between the yielding strengths of the two springs in series.Considering two springs corresponding to panels of length L1 and L2and characterised by ultimate deformation ecu1


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where f v is the average shear stress in the wall attained at the max-imum resistance, f t is the tensile strength of masonry, b is the shearstress distribution factor (depending on the geometry of the wall

and on the value of the ratio between the vertical N and horizontalH load), ro is the average compression stress due to vertical load N .

The calibration of the diagonal springs in the initial linearelastic range is simply obtained by enforcing an elastic equivalencebetween the panel and the corresponding masonry wall, consid-

ered as a pure shear deformable homogeneous plate with tangen-tial modulus G, transversal area At and height h, Fig. 7. In the

homogeneous media the relationship between the shear force V ,and the top horizontal displacement d is given by

d ¼ V

G At h ð9Þ

while in the macro-element the corresponding force–displacementrelationship is:

d ¼ V

2 cos2 a K diag ð10Þ

where K diag represents the initial elastic stiffness of each diagonallink and a = arctan (b/h), being b the base length. By equatingexpressions (9) and (10) the initial elastic stiffness of each diagonalspring is derived:

K diag ¼ G At 2h cos2 a

ð11Þ

The ultimate shear load V u and the corresponding displacementuu of each diagonal spring are therefore derived by the knowledge

of the mechanical material properties of the masonry by means of the adopted shear resistance criteria. Since the macro-elementmust incorporate the mechanical property of the finite portion of the masonry wall that represents, the ultimate shear displacement

of the element is directly associated to the ultimate shear general-ised deformability of the masonry media, whose value can be eval-uated experimentally or can be suggested by technical codes. If

experimental data are available, a better prediction of experimen-tal results can be obtained by considering different values of ulti-

mate shear loads and displacements for piers and spandrels.According to the proposal of Magenes and Calvi [15], the ultimate

displacement of the wall can be associated to a specific value of theultimate angular deformation cu = du/h that is dependent on theparticular masonry media. The post-yielding behaviour of eachdiagonal spring can be characterised according to different consti-

tutive plastic models as shown in the numerical applicationsreported in the following section.

It is worth to notice that in the proposed approach each macro-element inherits the plane geometrical properties of the modelled

masonry portion, as a consequence, contrary to the simplifiedmodels based on equivalent frame element approach, there is noneed to define an effective dimension of the structural element.

Once the constitutive laws have been defined both force and

displacement controlled load processes can be performed accord-ing to procedures currently used in finite element analyses

[20,21]. In the applications reported in the following displacementcontrolled nonlinear static analyses have been performed. Further

details on the numerical procedures are reported in the subsequentparagraph.

3. Simulation of experimental results

In this section the proposed discrete model is employed to sim-ulate the nonlinear response of real masonry walls and structuresfor which numerical and/or experimental results are available fromprevious studies. In particular, the results of an extensive experi-mental and numerical research on the behaviour of unreinforced

brick masonry buildings [22,23] have been considered. The firstapplications are relative to numerical simulations of experimentalresults of simple masonry panels characterised by various geomet-rical ratios [24]. Secondly, a case study relative to a large-scale

masonry model built and tested at the University of Pavia, in theNorth of Italy, is considered [23].

3.1. Case study 1: masonry panels

The first case study considers the results of cyclic shear tests onbrick masonry walls [24] that have been considered by severalauthors aiming at validating different numerical approaches [25–28]. Namely the experimental behaviour of simple piers under

fixed–fixed end conditions with the aim to reproduce flexuraland diagonal shear cracking behaviours have been analysed. Thedimensions of the squat wall, that collapsed according to a diago-nal shear behaviour failure mechanism, are 100 135 cm2. The

dimension of the slender wall, which exhibited a collapse responsedominated by the rocking, are 100 200 cm2. Both the walls arecharacterised by a thickness of 25 cm and have been subjected toan initial mean compressive stress of 0.6 MPa, associated to a

constant vertical load at the top, in addition to their own weights.Further details on the specimens and test procedures are reported

on the referenced paper [24].The experimental results of the considered panels are reported

in Fig. 8. In particular Fig. 8a reports the flexural response charac-terised by a moderate hysteretic energy dissipation and an almostnonlinear elastic behaviour with negligible strength degradation.Fig. 8b reports the shear-diagonal typical force–displacement

curve dominated by shear cracking with high energy dissipationand a significant strength and stiffness degradation.

For the simulation of the cyclic degrading hysteretic behaviourof a masonry wall, subjected to a combination of constant vertical

load and to a sequence of lateral load reversals, the use of an idea-lised bi- or tri-linear resistance envelope is recommended [14].Here the cyclic shear-diagonal behaviour of the considered wallshas been modelled with the proposed approach by means of a

single macro-element. Furthermore, the shear-diagonal behaviourhas been modelled by means of the idealised bi-linear envelopereported in Fig. 9. The relevant parameters of the bilinear envelopehave been identified according to the following simple rules:

– The initial slope of the idealised envelope is evaluated accordingto a secant stiffness, that can be associated to the formation of the cracks, called effective stiffness of the wall K e. The latterhas been obtained by assuming the resistance at the crack limit

as a percentage (60%) of the experimental maximum resistance,as follows:

K e ¼ H cr dcr

ð12Þ

– The maximum resistance of the idealised curve is set equal tothe maximum resistance attained during the test H max;

Fig. 7. Calibration of the diagonal springs in the linear elastic range.



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– The idealised ultimate states, determined by the maximum dis-placement attained during the test dmax and the correspondingresistance H dmax are set to be coincident to the experimental

ones.

The ratio between the resistance H dmax at the ultimate displace-ment and the maximum resistance H max defines a strength

degradation factor C sd [14].

C ds ¼ H dmaxH max

ð13Þ

The slope of the envelope curve in the post-yielding behaviour is gi-ven by the kinematic hardening (or softening) parameter.

a ¼ H dmax H maxdmax dH max

ð14Þ

In order to calibrate the shear diagonal nonlinear link of the macro-

element, that governs the pure shear behaviour, it is necessary to

observe that a fixed-ended masonry wall exhibits a deformationthat is partly associated to bending and partly to shear. Denotingthe shear and the flexural stiffness as K V and K F , respectively, theoverall stiffness exhibited by the wall during the experiment, K ,

can be considered as the result of two springs in series, from whichthe shear stiffness can easily be obtained

K V

¼ K K F

K F K ð15Þ

where the flexural stiffness K F is provided, in simplified form, by the

following stiffness of an equivalent beam

K F ¼ 12 EI

h3

ð16Þ

where E is the Young modulus, I is the moment of inertia of the wallcross-section and h is the height of the wall. The unloading stiffnesshas been expressed according to the following simple expression

K u ¼ K oK I K o

bð17Þ

where K o is the initial stiffness, K I is the value of the stiffness that is

obtained by connecting the point corresponding to the current plas-tic deformation with the origin of the axes and b is a real number

that can assume value in the range 0–1. In the simulation the value0.8 has been set. The re-loading stiffness K r , in the cyclic behaviour,has been set by pointing 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 and an unloading stiffness expressed according to theEq. (17) with b = 0.8, the ductility of the orthogonal springs hasbeen set equal to 5. The other parameters, assumed in the simula-

tion according to the above described criteria, are reported inTable 1.

The results of the numerical simulation are reported in Fig. 10.Even though the adopted constitutive models are very simple,

there is a satisfactory agreement between the experimental andthe numerical results.

3.2. Case study 2: masonry model (two-storey prototype brick

masonry building)

This case study is relative to a two-storey prototype brick

masonry building tested by Magenes and Calvi [22,23] at theUniversity of Pavia and considered in [29] for comparing and vali-dating different simplified numerical approaches. The rectangularplan of the building is 6 4.4 m while the total height is equal to6.4 m. The building consists of four two-wythe solid brick walls

with a total wall thickness of 250 mm. The prototype has been

tested by means of cyclic horizontal forces acting along the longi-tudinal direction. The building is composed by two masonry wallswith openings, in the longitudinal direction, and two masonry

walls with no openings in the transversal direction. These forces,of equal value, have been applied directly to the two longitudinalwalls at the floor levels. As a result of the performed cyclic tests,the cyclic force–displacement relationships for both the longitudi-nal walls have been obtained. The spatial configuration of the aper-

tures in the walls with openings are different: the main entrancewall of the building, denoted as ‘door wall’, possesses two windowsand two doors, while the wall opposite to this, denoted as ‘windowwall’, is characterised by six regularly located windows. The door

wall was disconnected from the adjacent transversal walls, whilethe other (window wall) was connected to the adjacent walls with

an interlocking brick pattern around the corner. The floor consisted

-80

-60

-40

-20

0

20

40

60

80

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 f o r c e ( K N )

displacement (cm)

-100

-80-60

-40

-20

0

20

40

60

80

100

-1,0 -0,5 0,0 0,5 1,0 f o r c e ( K N )

displacement (cm)

(a)

(b)

Fig. 8. Experimental behaviour of simple piers, from Ref. [12]: (a) example of

flexural response and (b) example of behaviour with diagonal shear cracking.

Fig. 9. Idealisation of experimental resistance envelope with bi-linear relationship.



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of a series of isolated steel beams designed to simulate a veryflexible diaphragm. The constant vertical forces acting on the wallsare due to the self-weight of the masonry and to the gravity andaccidental loads of the floor slabs (248.4 kN for the first storeyand 263.8 kN for the second storey). The seismic forces were sim-

ulated by the application of equal concentrated horizontal forcesdirectly applied at the longitudinal walls at the floor levels bymeans of displacement controlled screw jacks.

The results of the numerical simulations reported in the follow-

ing aim to predict the response of the door wall. In Fig. 11 the geo-metric layout of the wall and its basic representation by means of the proposed macro-element approach is reported.

In Table 2 the mechanical masonry characteristic that havebeen used for the numerical simulations are summarised. The re-sults of the numerical simulation are relative to a displacement-controlled push-over analysis. Since the experimental test hasbeen performed by applying equal concentrated forces at the floor

levels, the nonlinear numerical analyses have been conducted byimposing, at each step, different floor displacement values leadingto equal resisting forces. These latter values have been obtained bymeans of a Newton–Raphson iterative procedure imposing a global

equilibrium condition.The diagonal springs of the panels have been modelled by con-

sidering the well known Turnsek Cacovic criterion [16,17] in which

Table 1

Case study 1. Mechanical characteristics of the masonry.

Flexural Shear-diagonal

E (MPa) rc (MPa) rt (MPa) G (MPa) f t (MPa) b b

2100 6.20 0.1 500 0.275 1.5 0.8

-80

-60

-40

-20

0

20

40

60

80

-1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 f o r c e ( K N )

displacement (cm)

-100

-80

-60

-40

-20

0

20

40

60

80

100

-1,0 -0,5 0,0 0,5 1,0 f o r c e ( K N )

displacement (cm)

(a)

(b)

Fig. 10. Numerical simulation of experimental tests on piers: (a) example flexural

response and (b) example of behaviour with diagonal shear cracking.

void void

void void

(a) (b)

Fig. 11. The door wall (a) geometric layout; and (b) equivalent mechanical scheme.

Table 2

Case study 2. Mechanical characteristics of the masonry.

Flexural Shear-diagonal Sliding

E

(MPa)

rc (MPa)

rt (MPa)

G

(MPa)

f t (MPa)

b f o(MPa)

l

Piers 2100 6.2 0.05 500 0.18 1.5 0.2 0.5

Spandrels 2100 6.2 0.05 500 0.225 1.5 0.2 0.5

-150

-100

-50

0

50

100

150

-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0

B a s e S h e a r ( K N )

Top displacement (cm)

-150

-100

-50

0

50

100

150

-3,0 -2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5 3,0



(a)

(b)

Fig. 12. ‘‘Door’’ prototype wall. Comparison between (a) experimental and (b)

numerical cyclic tests.



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the damage has been controlled by a strength degradation param-eter a = 0.08. The hysteretic behaviour has been defined accord-ing to the bi-linear idealisation described in the previoussubsection.

In Fig. 12 a comparison between the experimental and numer-ical results in terms of base shear versus the absolute value of the

top displacement is reported. By observing the maximum reached

forces and displacements as well as the exhibited hysteretic behav-iour and keeping in mind that the shear-diagonal behaviour hasbeen calibrated according to a simple bi-linear representation,

the agreement between the experimental and numerical resultscan be considered satisfactory. Fig. 13 reports a comparison be-tween the experimental and numerical results in terms of baseshear of the wall versus the first inter-storey drift; also in this case

the numerical prediction appears to be acceptable.In Fig. 14 a simplified representation of the damage scenarios

predicted by the model, corresponding to the value of drift of 3%

and to the ultimate value of 4.3%, are compared with the corre-sponding crack patterns obtained experimentally. It is worth to no-tice how the proposed approach is able to grasp the progressivedistribution of damage in the wall. The used representation in

the interface allows to distinguish the reactive compressive zonefrom the cracked one due to tensile forces; diagonal bars inside apanel indicate the yielding of the diagonal springs.

The further representation of the ultimate state, reported in

Fig. 15, clearly shows the composite shear-flexural failure of thepiers and the diagonal shear failures of spandrels as well as thecrack patterns due to the partialisation of the interfaces.

-150

-100

-50

0

50

100

150

-2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5


First interstory displacement (cm)

-150

-100

-50

0

50

100

150

-2,5 -2,0 -1,5 -1,0 -0,5 0,0 0,5 1,0 1,5 2,0 2,5


First interstory displacement (cm)

Fig. 13. ‘‘Door’’ prototype wall. Comparison between (a) experimental and (b)

numerical cyclic tests.

Fig. 14. Simplified drafts of damage scenarios corresponding to different values of the applied drifts: (a) 0.3% and (b) 0.43% (maximum drift).



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In Fig. 16, the resultants of the compressive and shear forces in

the piers are reported as functions of the top displacement. It isworth to notice how the proposed approach is able to model thevariation of the coupled normal and shear forces during the nonlin-

ear response of the structure.

3.3. Influence of the model discretization

In the considered approach the macro-element inherits thegeometry of the masonry portion that is modelled, this aspect

constitutes a great advantage that is not common to all the simpli-

fied approaches based on a macro-element discretization. Further-more the consistent geometry of the element makes possible theimplementation of models with an irregular distribution of the

openings and allows the implementation of models characterisedby different level of discretization associated to the mesh resolu-tion and to the fine-tuning of nonlinear links in the interfaces. InFig. 17 the results of push-over analyses, performed on the doorwall of Pavia prototype, associated to different mesh resolutions

are reported. Three different mesh resolutions have been

Fig. 15. Damage scenarios corresponding to a maximum applied drift 0.43%.

0

40

80

120

160

200

0,0 0,5 1,0 1,5 2,0 2,5

P a n

e l C o m p r e s s i o n ( K N )


rigth pier

-40

-20

0

20

40

0,0 0,5 1,0 1,5 2,0 2,5

S h e a r ( K N )


rigth pier

0

40

80

120

160

200

240

0,0 0,5 1,0 1,5 2,0 2,5

P a n e l C o m p r e s s i o n ( K N )


central pier

-100

-60

-20

20

60

100

0,0 0,5 1,0 1,5 2,0 2,5

S h e a r ( K N )


central pier

0

40

80

120

160

200

0,0 0,5 1,0 1,5 2,0 2,5

P a n e l C o m p r e s s i o n ( K N )


left pier

-40

0

40

80

0,0 0,5 1,0 1,5 2,0 2,5 S h e a r ( K N )


left pier

(a) (b)

Fig. 16. ‘‘Door’’ prototype wall: the resultants of compressive (a) and shear forces (b) in the piers of the door wall as a function of the absolute top displacement.



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considered whose increasing computational cost is summarised in

Table 3. Mesh A correspond to the basic mesh size in which piersand spandrels are modelled by a single macro-element. Mesh Band C represent more refined mesh resolutions in which piersand spandrels are represented by several macro-elements. It is well

known that, in the case of the finite element approach, convention-ally conducted within the framework of the local continuum

theory [20,21], the numerical solution is sensitive to the mesh size

when strain softening models are used. For this reason with the

aim to investigate the influence of the mesh in the numericalsimulations, in Fig. 17, for the Pavia prototype, two different com-parisons have been considered according to a strain-softening anda strain hardening shear-behaviour characterised by a = 0.08 and

a = 0.08, respectively. All the other parameters are reported in Ta-ble 2. It can be observed as, in presence of strain softening plastic-ity (Fig. 17a), the localisation of plastic deformation causes a meshdependence in the softening branch of the push-over curve, while

in presence of strain-hardening behaviour (Fig. 17b) there is a goodagreement between the results obtained by different meshresolutions.

With reference to the mesh A and an elasto-perfectly plasticbehaviour, in Fig. 18 the influence of the fine-tuning of the Nonlin-ear Links in investigated. Namely, the results of push-over analysesassociated to different numbers of Nlinks for unit length are com-

pared. The comparison shows that, in the analysed case, all the con-sidered Nlinks distributions provide results in very good agreement.

3.4. Influence of the tensile and shear strength

In the following a sensitivity analysis on the tensile strength,

associated to the flexural and shear-diagonal behaviours, has beenperformed in terms of monotonic push-over curves on the doorwall of the Pavia prototype. The discretization of the model is con-sistent with the mesh A and an elasto-perfectly plastic behaviour,according to the parameters reported in Table 2. Namely, in

Fig. 19a the push-over curves for three different value of the tensilestrength rt , that governs the flexural behaviour of the panel, arerepresented. In Fig. 19b the sensitivity analysis has been conducted

by varying the tensile strength f t associated to the Turnsek andCacovic criterion. It is worth to notice, that the present modellingapproach allows to calibrate each fundamental collapse behaviour

(flexural failure, diagonal shear failure and sliding shear failure)

0

25

50

75

100

125

150

175

0.00 0.50 1.00 1.50



mesh A

mesh B

mesh C

0

25

50

75

100

125

150

175

0.00 0.50 1.00 1.50

B

a s e S h e a r ( K N )

displacement (cm)

mesh A

mesh B

mesh C

(a)

(b)

Fig. 17. ‘‘Door’’ prototype wall: Push-over analyses associated to different mesh-

resolutions: (a) strain-softening behaviour; and (b) strain-hardening behaviour.

Table 3

mesh resolutions.

Label Number of macro-elements

Number of degrees of freedom

Maximum Nlinkdistance (cm)

A 16 64 10B 64 256 10

C 144 576 10

020

40

60

80

100

120

140

160

0.00 0.50 1.00 1.50 2.00 2.50



10 cm

20 cm

40 cm

Fig. 18. ‘‘Door’’ prototype wall: Push-over analyses associated to different Nlinks

distributions in the interfaces.

0

25

50

75

100

125

150

175

0.00 0.50 1.00 1.50 2.00

B a

s e S h e a r ( K N )


0

25

50

75

100

125

150

175

0.00 0.50 1.00 1.50 2.00

B

a s e S h e a r ( K N )


f t= 0 .12 MPa

f t= 0 .075 MPa

f t= 0 .05 MPa

t= 0 .05 MPaσ

t= 0 .10 MPaσ

t= 0 .20 MPaσ

(a)

(b)

Fig. 19. ‘‘Door’’ prototype wall: Push-over analyses associated to different values of

tensile strength associated to the flexural (a) and to the shear-diagonal (b)

behaviours.



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according to ad hoc adopted collapse criteria, this constitutes anovelty with respect to the other approaches presented in the

literature. The parameters associated to each failure mechanismare based on results from current experimental tests.

4. Conclusions

The problem of the evaluation of the seismic vulnerability of unreinforced masonry buildings represents a subject of high practi-cal relevance but, at the same time, a very difficult task. For theassessment of URM building seismic vulnerability and its possibleseismic retrofitting, the structuralanalyst needssimple and efficient

numerical tools whose complexity and computational demandmust be appropriate. In the present work an advanced model, basedon the definitionof a plane macro-element, forthe simulation of theseismic behaviour and the evaluation of the seismic vulnerability of

unreinforced masonry buildings has been introduced. The proposedmodel has been conceived to provide a simulation method capableof predicting the seismic behaviour of a masonry structure at alow computational cost, compared to a nonlinear finite element

simulation. A considerable reduction of the computational cost isachieved since the nonlinear behaviour of a masonry portion is de-scribed by very few degrees of freedom and by means of uniaxialnonlinear springs. A suitable definition of the parameters of the

model that leads to a reasonable equivalence between the masonrywall and its simplified nonlinear mechanical representation hasbeen proposed. This method possesses many advantages with re-spect to other existing simplified approaches. The ability of the

macro-element to model a portion a masonry which inherits thesame plane configuration, and without distinguishing betweenpiers, spandrelsand immune zones, makes the modellingof the glo-

bal in-plane response of masonry façades with irregular openingpattern possible. The presence of the interfaces between macro-elements allows also the modellingof the interactionwith nonlinearframe elements. As a result infilled frame structures could also be

modelled. Furthermore the introduction of ad hoc interface ele-

ments could lead to a simple evaluation of the global behaviour of three dimensional masonry building in presence of concrete ringbeams at the floor levels. This latter aspects constitute a research

in progress that will be the subject of a paper to come.Although simple to comprehend, the proposed approach for

wide spread practical application requires the implementation of ad hoc user interfaces able to simplify the application by third

party or practicing engineers. The effectiveness of the proposedmodelling has been evaluated by means of nonlinear incrementalstatic analyses performed on masonry walls and structures whichhave been the object of theoretical and experimental research in

the past. The numerical results describe fairly well the experimen-tal results, in terms of the cyclic response, stiffness, strength anddissipative behaviour, as well as damage distribution.

Acknowledgement

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

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