comparative study

Daniele Baraldi a, *, Antonella Cea Department of Architecture Construction Conservationb Department of Engineering, University of Ferrara, Via

a r t i c l e i n f o

Article history:Received 31 March 2014Accepted 20 October 2014Available online 31 October 2014

Keywords:Masonry-like materialDiscrete modelMicropolar continuum

d, with constitutivepolar continua). Anus FEM is presentedn models behaviour.. All rights reserved.

1. Introduction

Masonry is a structural material obtained by composition ofblocks connected or not by mortar joints. Particularity of this

heterogeneous material is the heterogeneity size (size of block),that may be not negligible with respect to global size of structuralelement as in several composite materials. For this reason, in thelast twenty years, several researchers developed models forstudying masonry-like material adopting different approaches.

With this aim a heterogeneous FE model may be the moreappropriate procedure to investigate this material type. StaffordSmith and Rahman (1972) were the rst to adopt a rough hetero-geneous FE model for determining stresses in brickwork walls;

* Corresponding author.E-mail addresses: danielebaraldi@iuav.it (D. Baraldi), cecchi@iuav.it (A. Cecchi),

Contents lists availab

European Journal of M

.e ls

European Journal of Mechanics A/Solids 50 (2015) 39e58tra@unife.it (A. Tralli).ume (REV) chosen is discussed. For these purposes, ad hoc FE models are adoptefunctions obtained from an identication procedure (both for Cauchy and microextensive comparison between DEM, heterogeneous FEM and equivalent homogenoin some meaningful cases, taking into account also the effect of heterogeneity size o

2014 Elsevier Masson SASDEM and heterogeneous FEM are adopted to verify reliability and application eld of Cauchy andmicropolar continua. Moreover, sensitivity of micropolar model to the Representative Elementary Vol-http://dx.doi.org/10.1016/j.euromechsol.2014.10.0070997-7538/ 2014 Elsevier Masson SAS. All rights recchi a, Antonio Tralli b

, University IUAV of Venezia, Dorsoduro 2206, 30123 Venezia, ItalyG. Saragat 1, 44122 Ferrara, Italy

a b s t r a c t

The aim of this paper is to present a critical comparative review of different models that may be adoptedfor modelling the mechanical behaviour of masonry, with particular attention to microstructuredmodels.Several continuous and discrete models are discussed. Such models are based on the following as-

sumptions: i) the structure is composed of rigid blocks; ii) the mortar is modelled as an elastic materialor an elastic interface. The rigid block hypothesis is particularly suitable for historical masonry, in whichstone blocks may be assumed as rigid bodies. For this type of masonry, mortar thickness is negligible ifcompared with block size, hence it can be modelled as an interface.Masonry-like materials may be modelled taking into account their heterogeneity by adopting a het-

erogeneous Finite Element Model (FEM) or a Discrete Element Model (DEM). The former seems to bemore representative of masonry, but it is computationally onerous and results interpretation may bedifcult; the latter is limited to rigid block assumption and mortar joints modelled as interfaces. For thisreason, continuous equivalent models may be suitable to investigate masonry behaviour. Continuumequivalent models provide, in an analytical form, constitutive functions, but Cauchy model may be notsuitable to describe masonry behaviour due to not negligible size of heterogeneity (block size) withrespect to masonry panel size. For this reason, micropolar equivalent continuum may be adopted.By reference to the existing literature, a simple and effective DEM is adopted, in which masonry is

modelled as a skeleton having a behaviour depending on forces and moments transferred betweenblocks through the interfaces (mortar joints). Moreover for the micropolar equivalent continuum, an adhoc enriched homogenised FEM is formulated by means of triangular elements. The proposed numericalmodels represent two possible simple approaches for solving heterogeneous problems. Such models aredeveloped both by means of fast numerical routines and do not require specic computer codes, whereasthe heterogeneous FEM may be studied by adopting a traditional FE code.Continuous and discrete models for masonry like material: A criticaljournal homepage: wwwserved.le at ScienceDirect

echanics A/Solids

evier .com/locate/ejmsol

of Mthen, Page (Page, 1978; Ali and Page, 1988) adopted such type ofmodel for tting experimental results and taking into account thenon-linear behaviour of mortar joints. However the limit of thisapproach lies in the difculty to analyse macro-scale problems. Asexpected, the computational effort may be difcult to manage andthe interpretation of numerical results may be not easy.

Then, a discrete model (DEM), based on the assumptions of rigidblock behaviour and mortar joint modelled as interfaces, may besuitable for investigating masonry behaviour due to the smallnumber of degrees of freedom (DOFs) involved in the analysis ofmasonry panels. These assumptions seem to be appropriate forhistorical masonry, in which stone block stiffness is very large ifcompared with mortar stiffness, allowing to assume blocks as rigidbodies, and mortar joint thickness is negligible if compared withblock size, allowing to model mortar joints as interfaces. However,the assumption of rigid block imposes that boundary conditionsmust be referred only to block centres. This aspect may not berepresentative of actual mechanical behaviour. In the proposedDEM, masonry is seen as a skeleton in which the interactionsbetween rigid blocks are represented by forces and moments thatdepend on their relative displacements and rotations. Such modelwas adopted in the past by many authors for studying masonrybehaviour in linear and non-linear elds (Masiani et al., 1995;Formica et al., 2002; Casolo, 2004, 2006). In particular, Cecchi andSab (2004, 2009) dened a simple and effective DEM for studyingthe three-dimensional behaviour of masonry panels and formodelling random brickwork. Recently such model has beenextended to the viscoelastic eld by Baraldi and Cecchi (2014).

Discrete or distinct element models are widely adopted in otherscientic elds such as rock mechanics (see for example the pio-neering works of Cundall and Strack, 1979; Cundall, 1988). Limits inDEM approaches lie in the assumptions mentioned above, henceduring the last decades the original model has been modied fortaking into account the deformability of elements by introducingadditional parameters or by introducing FE discretisations (Itasca,1989). Some examples of evolution of the DEM are represented bycommercial or open source codes (Itasca, 2000; Munjiza, 2004;Mahabadi et al., 2012) that are characterised by a larger computa-tional effort with respect to the original DEM. Recently, a compari-son between suchmodels and a simple DEMhas been carried on forstudying masonry linear behaviour (Baraldi et al., 2013). Moreover,an exhaustive descriptionof discretemodels and their improvementup to recent years may be found in the work of Lemos (2007).

Although the DEM requires a small computational effort withrespect to the heterogeneous FE model at micro-scale level andpanel size level, it may be still unsuitable for studying masonrybehaviour at macro-scale level. For the above mentioned reasons,continuous material equivalent to masonry were proposed. Amongcontinuous models, homogenisation-identication proceduresrepresent a consistent part of research. Indeed, homogenisationprocedures allow to take into account different mechanical as-sumptions for blocks and mortar. Standard Cauchy continuousmodels are obtained applying periodic homogenisation techniquesand considering the elastic behaviour of both brick and mortar(Anthoine, 1995; Cecchi and Sab, 2002; 2004). In the non-lineareld some models exist in which blocks are assumed to be elasticand mortar is modelled with a coupled damage-friction behaviour(Milani et al., 2006; Sacco, 2009). In the non-linear eld, both blockand mortar may also display a non-linear behaviour (De Buhan andDe Felice, 1997; Gambarotta and Lagomarsino, 1997; Pegon andAnthoine, 1997; Luciano and Sacco, 1998; Formica et al., 2002;Massart et al., 2007; Wei and Hao, 2009).

On the other hand, micropolar or higher order continua havealso been adopted for masonry study. For micropolar continuum

D. Baraldi et al. / European Journal40see for example Masiani et al. (1995), Masiani and Trovalusci(1996), Boutin (1996), Sulem and Mhlhaus (1997), Smyshlyaevand Cherednichenko (2000), Forest et al. (2001), Casolo (2009),Salerno and De Felice (2009), Addessi et al. (2010), De Bellis andAddessi (2011) and Pau and Trovalusci (2012). For higher ordercontinuum see for example Stefanou et al. (2010), Bacigalupo andGambarotta (2012) and Trovalusci and Pau (2014).

A crucial problem with the choice of homogenisation-identication procedures is not only how kinematic, dynamic,and constitutive prescriptions of a discrete system are transferredto the continuous one, but also which continuum may be moreappropriate. Hence, constitutive functions of the continuous sys-tem may be different (Lofti and Benson Shing 1994, Loureno andRots, 1997; Del Piero, 2009).

The aim of this paper is to present a deep investigation ofdifferent models that may be adopted for modelling themechanicalbehaviour ofmasonry, with particular attention tomicro-structuredmodels. At microscopic level, blocks are assumed to be rigid andmortar joints are modelled as elastic interfaces; a Cauchy standardcontinuumand amicropolarmodele based on twodifferent REVseare considered. Then, an identication between the block structureand a plane continuum model is carried out by equating the me-chanical work in the two models for a class of regular motions. Dueto thehypotheses of the discretemodel, the identicationprocedureturns out to be simpler than a homogenisation procedure and leadsto the same results if blocks are assumed to be rigid. Hence, theconstitutive functionof the two-dimensional (2D)model is obtainedfrom actual geometry and constitutive function of the discretemodel. Such compatible identication procedure is adopted for allthe models, in order to obtain equivalent continuous macroscopicconstitutive functions, that turn out to be orthotropic starting fromisotropic constitutive behaviour of block and mortar due to thearrangement of masonry texture.

At panel size level this paper presents a comparison betweenDEM and FEM in which constitutive continuum e Cauchy andmicropolar e functions are obtained from an identication proce-dure. Furthermore, a FE heterogeneousmodel is taken into account,where constitutive functions of mortar and block are isotropic andwhere Young modulus of block is 104 time larger than Youngmodulus of mortar such as to simulate rigid block assumption. Forrepresenting the micropolar continuum and performing examplesthat can not be solved in analytical form, an enriched FE model isadopted, with triangular elements that take into account rotationsas degrees of freedom. Recently in this eld, several enriched FEmodels have been developed for studying the behaviour of genericmicropolar elastic materials (Zhang et al., 2005, 2012; Beveridgeet al., 2013). In particular, Providas and Kattis (2002) developedan enriched triangular FE model and proposed several patch tests.

The paper is organised as follows: in Section 2, a description ofthe 2Dmodel is given and themechanical power spent in its middleplane is dened. In Section 3, in a dual manner, the discrete modelis described and the mechanical power, expanded to a genericcouple of blocks at the interface, is dened. In Section 4a corre-spondence between a class of regular motions is dened for twoportions of 2D and discrete models having the same size, and theirmechanical power is equated. In this way, the stress measure in theplane is described as a function of the stress measure both forstandard Cauchy model and for micropolar model. A constitutivelinear isotropic elastic function for the mortar interface is adopted.Consequently, the abovementioned compatible identication leadsto a constitutive orthotropic function. This procedure is applied tothe case of a masonry panel with a running bond pattern and inSection 5 explicit formulas for this case are dened with referenceto two different REVs. It must be noted that this methodologicalidentication approach is embedded in linearised elasticity but

echanics A/Solids 50 (2015) 39e58may be extended to the non-linear case.

In Section 6, the enriched triangular FE is described and Section7 presents some examples of masonry panels subject to variousload and restraint conditions, together with a critical comparisonbetween the proposedmodels and taking into account the effects ofsize of heterogeneity on models behaviour. In this section theworks of Salerno and De Felice (2009), Pau and Trovalusci (2012)and Trovalusci and Pau (2014) are chosen as benchmark. In thesepapers the problem of Cauchy and micropolar equivalent continuais dealt with. The DEM proposed by Salerno and De Felice (2009) isa 2D FEMwith periodic brickworkmodelled as a Lagrangian systemof rigid bodies interacting by point elastic interfaces (Salerno et al.,2001; Formica et al., 2002), whereas Cauchy and micropolar con-tinua are modelled by rectangular high continuity FEs. Pau andTrovalusci (2012) and Trovalusci and Pau (2014) adopt commer-cial FE codes both for the continua and the DEM. In the latter case

D. Baraldi et al. / European Journal of MU 0 u3u3 0

(1)

From the above considerations, the generic displacement isdescribed by the elds:

u : S/V ; U : S/Skw V ; (2a,b)

which completely describe the V space of displacements-trans-lation and skew (Skw) or antisymmetric part i.e. rotation-of allthe code represents each block by a rectangular plane element withfour nodes, constrained so that each element behaves as a rigidbody, with mortar layers modelled as linear elastic springs. Differ-ently than these works, this paper describes an ad hoc DEM thatrequires a small computational effort with respect to other similarmodels due to the small number of DOFs (block displacements)involved in the analysis. Similarly, the enriched FEM used for thecomparison is simply obtained by adding an additional nodal DOFto traditional constant strain triangular (CST) FEs.

2. Plane continuum model

A plane model is dened, hence reference is made to a 2Dcontinuum identied by its S middle plane in a Euclidean coordi-nate system (y1, y2) of normal vector e3 along the y3 coordinatedirection (Fig. 1).

The kinematic descriptors of a generic point belonging to the 2Dcontinuum are represented by the following elds: u(y), U(y), thatare respectively the translation vector and rotation tensor of ageneric point y. u(y) is a vector with two components, namelyu1(y1,y2) and u2(y1,y2), whereas the rotation is a skew tensor U(y)with one component, dened as follows:Fig. 1. 2D Continuum model.points belonging to S. In the plane case, following the notation ofCecchi and Rizzi (2005), the static counterpart is fully described bythe eld N, collecting the in-plane actions, and by the eld M,representing the microcouple:

N : S/V ; M : S/Skw V (3a,b)

The static model envisages spatial elds of forces and couples(N, M) and eld of body forces and couples (b, B). The balanceequations for the in-plane case are:

divN b 0; divM 2SkwN B 0 (4a,b)

where div is divergence operator dened on S. For the continuum,set N and M actions, the mechanical work on S may be written as:

P N$grad uU Me3$grad U; (5)

where grad represents the gradient operator on S. If the adoptedcontinuum follows Cauchy's hypotheses, the in-plane couple isassumed equal to zero (Sulem and Mhlhaus, 1997; Stefanou et al.,2008) and the total internal work can be thus evaluated as thepower expended by membrane actions:

PCauchy N$symgrad u

: (6)

3. Discrete model

A standard running bond periodic masonry is considered andFig. 2a shows a Representative Elementary Volume (REV) having ablock Bi,j surrounded by six blocks. Block plane dimensions are: a(height) and b (width), whereas s represents block and panel thick-ness. Assuming rigid block hypothesis, the displacement of eachblock Bi,j is a rigid body motion referred to the motion of its centreand it is dened by the following expression (Cecchi and Sab, 2004):

ui;jy ui;j Ui;j

y yi;j

; (7)

where ui;j fui;j1 ui;j2 gT and Ui,j are the translation vector and rota-

tion skew tensor having one component ui; j3 of Bi,j, respectively, andyi,j is the position of its centre in the Euclidean space.

Following the procedure described by Cecchi and Sab (2009), ageneric couple of blocks Bi,j and Bik1; jk2 (Fig 2b) is considered.Considering p as the centre of the Sk1 ; k2 interface between suchblocks, the displacement of the material points y of Bi,j andBik1 ; jk2 in contact in a generic position x2Sk1 ; k2 , may be writtenas follows.

ui;jx ui;j

pUi;j

x p

ui k1j k2

x ui k1j k2

pUi k1j k2

x p

(8a,b)

The measure of deformation may be written as a function of thejump of displacement eld d(x) between Bi,j and Bik1 ; jk2 in a pointx2Sk1; k2 :

dxuik1;jk2

xui;j

x

uik1;jk2pui;j

pUik1;jk2

xp

Ui;j

xp

upUpxp

echanics A/Solids 50 (2015) 39e58 41(9)

d RE

of Mwhere up uik1 ;jk2 p ui;jp and Up Uik1 ;jk2 Ui;j. Thekinematic adopted in this discrete model will be adopted both forthe DEM (Cecchi and Sab, 2004) and for the following compatibleidentication with a micropolar continuum.

Let be ti;jx; tik1 ;jk2 x the forces respectively acting on Bi,j andBik1; jk2 blocks, the balance equation provides t

i;jx tik1 ;jk2 x. Hence, thework of the contact actions at the interfacebecomes:

P ZS

tx$dxdA

ZS

hti;jx$ui;j

y tik1 ;jk2

x$uik1;jk2

xidA; (10)

where dA is an innitesimal portion of the interface Sk1; k2 .Assuming tik1;jk2 x tx, the work of contact actions become:

P ZS

txuik1;jk2

x ui;jx

dA

tp$up UpZS

Skwtx5x pdA; (11)

where the symbol5 represents the tensor product. Hence,

Pp tp$up 12Mp$UP; (12)

where tp RS

txdA; Mp 2RS

Skwtx5x pdA.By assuming block as rigid bodies andmortar joints modelled as

linear interfaces, the constitutive function that denes interactionbetween block Bi,j and Bik1 ; jk2 is

tx Kdx: (13)

Fig. 2. Discrete model: running bon

D. Baraldi et al. / European Journal42If e is the vector orthogonal to plane of interfaces, the consti-tutive function becomes K 1=emI m le5e, where e isthe actual thickness of mortar joint and m, l are the Lames con-stants of mortar. The proposed formulation for the discretemodel iscoincident to that proposed by Salerno and De Felice (2009).Furthermore, the proposed formulation allows to dene the workof contact actions at interfaces (Eq. (10)) in terms of block degreesof freedom ui,j andUi,j. Then, forces and couples between blocks areobtained by differentiating the work at the interfaces and theequilibrium problem of the panel subject to in-plane actions issolved numerically by adopting a molecular dynamics method.More details for the DEM description and its solution may be foundin Cecchi and Sab (2004, 2009).4. Compatible identication

A portion P of a masonry panel with the same dimensions of theREV is considered (Fig. 3a). This portion is chosen so that its centreyi,j coincides with the centre of the REV (Fig. 2a). A portion of panelH, having the same edge is considered, so that the y point of Hcoincides with yi,j.

From the above considerations, it is possible to assign a corre-spondence between a class of regular displacement in P and H. Inparticular, it is assumed that the translation and rotation of thecentre of the block Bi,j in the discrete system are equal to thetranslation and rotation of the centre of the REV in the continuummodel: ui,j(y) u(y), Ui,j(y) U(y); hence

ui k1;j k2y u

y grad u

y

yi k1;j k2 y;

Ui k1;j k2y U

y grad U

y

yi k1;j k2 y:

(14)

where yi,j and yik1 ; jk2 are the centres of Bi,j and Bik1; jk22 Pgeneric couple of blocks and a rst order Taylor approximation(rst order identication) in translation and rotation is used. Ac-cording to the kinematic description adopted, the vector tp {t1pt2p}T denotes in-plane tractions (projection on S). Taking inconsideration correspondent displacement tests, from Eqs. (14) and(12) may be split into two parts for the sake of clearness (Salernoand De Felice, 2009) and re-written as follows:

tp$up tp5yik1jk2 yi;j

$grad uU

grad Untp5

hp yik1jk2

5yik1jk2 y

p yi;j

5yi;j y

io

(15a)

1Z

V (a), generic couple of blocks, (b).

echanics A/Solids 50 (2015) 39e582Mp$Up

S

tp5vpvp5tp $grad U yi k1j k2 yi;j dA

ZS

v1pt2p v2pt1p

5yi k1j k2 yi;j

grad UdA

(15b)

where the distance vector vp {v1p v2p}T can be dened asvp x p (Masiani et al., 1995). At this stage, for a chosen REV and agiven class of regular displacements, it is imposed that the me-chanical work spent by the contact actions on P and H coincides.Hence, forces and couples of the 2D continuum may be expressedin terms of the forces acting in the discrete model:

een

of MN 12A

Xntp5

yik1jk2 yi;j

(16a)

Me3 12A

Xn

ZS

v1pt2p v2pt1p

5yik1jk2 yi;j

dA

tp5hp yik1jk2

5yik1jk2 y

p yi;j

5yi;j y

i

(16b)

where A is the area of the chosen REV and the symbol Sn indicates asummation extended to the n interfaces of the chosen REV. The 1/2coefcient appearing in the above expressions for N and M isrelative only to the external interfaces of the REV, because suchinterfaces are shared by contiguous REVs. Under these assump-tions, if the Cauchy continuum is chosen, only the symmetric part ofP is considered, hence the membrane tensors N and M may beexpressed as a function of the vector tp, i.e. as a function of themeasure of the stress in the micro-mechanical model. It must benoted that the part ofP associated to Skw(gradU) is not taken intoaccount in the Cauchy continuum. In fact, in the adopted 2D modelsuch kinematic elds characterise neutral (rigid) motions. Then,applying Eq. (13), that becomes tpKup, into Eq.16a,b it is possibleto obtain constitutive equivalent functions.

5. Model for rigid blocks connected by elastic interfaces:running bond masonry

The compatible identication procedure is based on the samegeometry of discrete system shown in Fig. 2, where the texturepattern may represent a running bond brickwork; moreover, theblock Bi,j together with the six surrounding blocks, form a Repre-

Fig. 3. Identication betw

D. Baraldi et al. / European Journalsentative Elementary Volume (REV). The displacement of the blockBi,j is the rigid body displacement dened by Eq. (7). The constitutivelaw for any interface between adjoining blocks,Sk1 ; k2 , is supposed tobe a linear elastic relationship between the tractions t over the blocksurfaces and the jump in displacement d across Sk1; k2 (Eq. (13)).

Upper bounds for the strain energy of the equivalent mediummay be obtained using a suitable kinematic eld over the REV.Assuming E as the macroscopic in-plane strain tensor in theequivalent medium, the continuum equivalent in-plane tensor Aand micro-couple tensor L are obtained by solving the followingminimisation problem (Cecchi and Sab, 2009):

12$A$E 1

2grad U$L$grad U minU;U2KCE;grad Uz

(17)where z is the strain energy averaged over the REV, U and Urepresent any strain-periodic rigid body displacement of the blocks,kinematically compatible with E. The set KC of E-kinematicallycompatible (U, U) is introduced:

KCE; grad U

nhU;U

i;ui;j E$yi;j vi;j; Ui;j

ui;j;hv;u

i2L2

o(18)

where [v, u] is the in-plane rigid displacement of block Bi,j erespectively translation and rotation. Hence, uniform boundarydisplacement and rotation are applied to REV.

The approximate equivalent elastic coefcients, denoted by Aijkland Lij are such that:

12A1111E112

12A2222E222

12A1212E122

12A2121E122

L11u23;1 L22u23;2 z;(19)

where E11 u1,1, E22 u2,2, E12 u1,2 u3 and E21 u2,1 u3.Following the procedure proposed above, it is possible to dene

the equivalent micropolar continuum. As well known, for theCauchy continuum, Eq. (19) becomes:

12A1111E112

12A2222E222 2ACauchy1212

EC12

2 z (20)where EC12 EC21 1=2u1;2 u2;1.

Two different REVs may be considered. REV1 (Fig. 4a) is char-acterised by a block at the centre of the Cartesian coordinate sys-tem, with four horizontal interfaces and two vertical interfaces farfrom the centre of the cell, whereas REV2 (Fig. 4b) is characterisedby four horizontal interfaces and a vertical interface at the centre of

REV and 2D continuum.

echanics A/Solids 50 (2015) 39e58 43the Cartesian coordinate system.The constitutive functions of the components of N and C Me3

on S plane are the same obtained by Salerno and De Felice (2009)adopting a 2D discrete model and following the same compatibleidentication procedure.

N A$grad uU; C L$grad U (21a,b)The components of tensor A are the same for both REVs

considered are given by the following expressions (see AppendixA1 for more details):

A1111 h4Kv

eh.ab=a

Ghev=a

i.h4eh.a

ev=bi;

(22a)

A2222 Kh.

eh.a; (22b)

A1122 0; (22c)

A1212 Gha.eh; (22d)

A K b2.

4aeh G

b=ev

: (22e)

Stefanou et al. (2008, 2010) that compared the dispersion functionsof the discrete model with respect to those of the equivalentcontinuum.

6. Finite element formulation

An enriched Constant Strain Triangular (CST) nite element istaken into account for performing analysis of micropolar models.Nodal degrees of freedom are uj , uj , uj with j 1, 2, 3 (Fig. 5), that

tion for a triangular element. The displacement and rotational de-grees of freedom of the element may be collected in a vector qel

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e58442121 h v

where a and b represent, respectively, height and width of theblock. e represents the thickness of mortar joints and K and G arethe bulk and shear moduli of mortar. h and v superscripts andsubscripts are referred to horizontal and vertical interfaces,respectively. In the following, for simplicity, horizontal and verticalmortar joints are assumed to have the same thickness (ev eh e)and the same elastic properties (Kv Kh K, Gv Gh G).

The equivalent moduli for the Cauchy continuum do not dependon the REV chosen and they are given by Eq. (22aec), moreover itcan be demonstrated that

ACauchy1212 A1212$A2121A1212 A2121

: (23)

The components of diagonal tensor L are strictly dependent onthe REV. Expressions 24a,b are related to REV1, whereas Eq. 25a,bare related to REV2:

L11 b2.192

h16Kva2

.bev 4Khb2

.aeh

12Gha

.ehi

(24a)

L22 b2hKha

.12eh

Gha3

.4b2eh

i; (24b)

L11 b2.192

h16Kva2

.bev Khb2

.aeh

12Gha

.ehi;

(25a)

L22 b2hKha

.48eh

i: (25b)

It is worth noting that REV1 is characterised by a larger me-chanical work than REV2 (Eq. (16)). In fact, REV1 has two verticalinterfaces far from the centre of the cell, whereas REV2 has onevertical interface at the centre of the cell. Hence, coefcients Lij ofREV1 are larger than those of REV2. According to Salerno and DeFelice (2009), both REVs are centre-symmetric. However, REV1 iscentred at the centre of the block and REV2 is centred at the centreof the vertical joint. The stiffness coefcients of Eq. 24a,b are largerthan those of Eq. 25a,b due to the tangential contributions of ver-tical joints in REV1, that do not appear in REV2, in which the centreof vertical joint coincides with REV centre. The better performanceof REV2 with respect to REV1 has been also demonstrated byFig. 4. Representative Elementary Volumes (REVshaving nine components. Then, the stiffness matrix Kel of thetriangular element is determined as usual starting from the po-tential energy Pel of the triangular element and introducing theexpressions for stresses and strains of themicropolar continuum. Ina generic micropolar elastic continuum, force and couple stresseswithin the element are related to deformations by six elastic con-stants (Eringen, 1966). In this case, for modelling a running bondmasonry panel, the six constants are determined towards theidentication procedure described in section 5 and consequently,two different stiffnessmatrices are obtained, based, respectively, onREV1 and REV2. The CST element is adopted in order to follow anexistent literature dedicated to the analysis of periodic structuresmodelled as micropolar continua (Masiani et al., 1995; Masiani andTrovalusci, 1996; Providas and Kattis, 2002; Trovalusci andMasiani,2003; Wheel, 2008). Moreover, the triangular element allows toperform an exact integration for the determination of the stiffnessmatrix; further details may be found in Appendix A2.2. However, itis clear that elements better than the CST are available (Cook et al.,2001) without increasing the complexity of the problem. In thefollowing numerical tests, a symmetric discretization made oftriangular FEs is adopted for reducing the directional stiffness bias(Logan, 2012). Elements superior than the CST, such as the quad-rilateral ones, may be adopted in further developments of thepresent work.

7. Numerical tests

A numerical experimentation is performed in order to evaluatea) the opportunity of adopting a micropolar continuum instead of atraditional one, b) the effectiveness of the proposed triangular FEfor determining the behaviour of masonry-like panels, withparticular attention to the rotationsu3 of the discrete system and ofthe micropolar continuum, c) the more appropriate REV for1 2 3correspond to the independent elds u and u3, that are discretisedover the triangular area by the linear polynomials commonlyadopted in standard triangular FEs (Zienkiewicz and Taylor, 1989):

u/u1 Xj

uj1Nj; u2

Xj

uj2Nj; u3

Xj

uj3N

j; (26)

where Nj aj0 aj1y1 a

j2y2 is the common bilinear shape func-) considered for the identication procedure.

mesh for a panel with a large scale factor r may require a hugecomputational effort, hence in the following, the ne mesh isadopted for determining the behaviour of panels having a smallscale factor, whereas the rough mesh is adopted for evaluating thebehaviour of the models increasing the scale factor.

The FE models for the micropolar continuum (iii and iv) aredened by subdividing panel length and height into nel,1 and nel,2subdivisions, respectively; then 4 nel,1 nel,2 triangular elementsare dened (Fig. 7b, with nel,1 nel,2 8) in order to obtain asymmetric mesh. The elastic moduli dened in section 5 (Eqs.(22)e(25)) are assumed as elastic properties of the enriched

L 1550 mm, height H 1030 mm and thickness s 120 mm.

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e58 45representing running bond masonry behaviour by means of themicropolar continuum.

In order to evaluate also the effect of size of heterogeneity on thebehaviour of discrete and micropolar continuum models, differentscale factors r L/b are adopted, taking into account panel size Lwith respect to block size b.

The numerical experimentation is performed on several models,that are resumed in the following list:

i) Discrete Element Model (DEM);ii) Heterogeneous FE model (FEM Hetero);iii) Micro-polar FE model based on REV 1 (FEM REV1);iv) Micro-polar FE model based on REV 2 (FEM REV2);v) Cauchy orthotropic FE model (FEM Cauchy)

The DEM (i) has been briey described in section 3; differentlythan experimentations performed by other authors (Salerno and DeFelice, 2009; Pau and Trovalusci, 2012; Trovalusci and Pau, 2014),the solution for this model is simply obtained by using a standardcomputer code for matrix analysis. It is worth noting that thismodel allows to study masonry behaviour with a smaller compu-tational effort with respect to the other models considered, due to

Fig. 5. Constant Strain Triangle (CST) with three degrees of freedom per node.the small number of DOFs taken into account.A heterogeneous FE model (ii) is adopted in order to verify the

effectiveness of the DEM. This model is briey described inAppendix A2.1, together with a convergence test performed fordetermining the more appropriate mesh renement, in order toobtain accurate results and, contemporarily, avoid a large numberof DOFs involved in the analysis. Such convergence test shows thatdifferences between displacements are not evident if a rough mesh(Fig. 6a) is adopted instead of a ne mesh (Fig. 6b). Moreover, a ne

Fig. 6. Detail of the heterogeneous FE mIn order to evaluate the effect of the size of heterogeneity on thebehaviour of the models, the number of blocks along both planedirections is increased by introducing a geometry scale factor r L/b and maintaining xed block width-to-height ratio and panel di-mensions (L, H, s) as it was done by Salerno and De Felice (2009).Hence, the panel in Fig. 7a is characterised by r 6. With thisapproach, the order of magnitude of results does not vary ifdifferent scale factors are taken into account; furthermore, theelastic parameters of the Cauchy continuum are not affected bysuch scale factor, hence results obtained with this model do notvary for increasing number of blocks.

Three numerical examples are carried on varying load and re-straint conditions for each case considered. Fig 8 shows the threeexamples together with details of applied loads and restraints fortriangular FEs. In addition, a Cauchy continuum (v) is taken intoaccount. In this case and a traditional FE model is developed byadopting the FE mesh in Fig 7b and by assuming the moduli in Eq.22a,b and Eq. (23) as elastic properties of the triangular FEs. Suchmodel is obviously not able to furnish in-plane rotations, then, in-plane translations obtained by all the models are also compared.

In the equivalent continuum and heterogeneous FE models,boundary conditionse loads and restraints e are applied along theedges of the panel, whereas in the DEM such conditions areapplied at block centres. In the following, uniform load distribu-tions along the top edge of the panel are taken into account,together with load distribution over small areas, in order to eval-uate the effect of the load wavelength respect to the size of therepresentative volume.

7.1. Rectangular panel

In the rst three examples a masonry panel with a running bondtexture pattern is considered (Fig. 7a). The panel is composed byUNI bricks (b 250 mm, a 55 mm, s 120 mm), with bed andhead mortar joints having the same thickness e. The panel ischaracterised by 6 blocks in horizontal direction and 16 courses invertical direction. The mechanical characteristics of the mortar areEm 1 GPa and nm 0.2 and a standard joint thickness e 10mm isassumed. Hence, the overall dimensions of the panel are: lengthodel: (a) rough mesh, (b) ne mesh.

the panel with r 6. In rst and second example, the appliedhorizontal load distribution is q 1000/L KN/m and in the thirdexample the applied force is F 1000 KN.

In order to compare the results given by the different modelstaken into account, the following relative difference is dened:

d dDEM dFEM

DEM 100; (27)

factor r 6. Due to the asymmetric load, asymmetric horizontaltranslations and symmetric rotations are obtained.

The rst row of Fig. 9 shows horizontal translations obtainedwith the DEM and the continuous FE models (iiiev). In this case, in-plane translations determined with both micro-polar models andwith the Cauchy FE model turn out to be coincident. Moreover, themaximum horizontal displacement at the top of the panel is equalto 2.202 mm, which is coincident to the analytic solution of the

Fig. 7. Masonry panel considered for the numerical examples: (a) discrete model, (b) FE model for micropolar and Cauchy continua.

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e5846d

where d may represent a displacement or a rotation of the discretemodel or the equivalent continuum (u1, u2 or u3) and the super-script FEM may represent one of the four FE models e heteroge-neous or continuous-taken into account (iiev). Relative differencesevaluated at several points, such as ends and mid-point of sectionsAeA and BeB (Fig. 8), are collected in Table 1 and are represented inFig. 20aec for increasing scale factor r.

7.1.1. Example 1: panel subject to a horizontal shear loadIn the rst example the panel is subject to a horizontal shear

load along the top edge and it is simply supported along the otheredges (Fig. 8, rst column). In this case horizontal translations u1linearly proportional to y2 and negligible vertical translations u2 areexpected. Fig. 9 collects results relative to the panel having a scaleFig. 8. Numerical examples for thproblem (umax1 qH=ACauchy1212 ).Considering the in-plane rotations (second row of Fig. 9), the

discrete model furnishes non-uniform values over the panel,with smaller values for the blocks along the edges of the panel,where the boundary conditions are applied. Both micropolar FEmodels give coincident rotations (u3 1.450 104 rad) andtheir order of magnitude is acceptable if compared to the resultsof the DEM.

Fig. 10a,b shows the in-plane rotations along section AeA andBeB, respectively, obtained by modelling the panel with r 6 withthe different models. Along both sections, the values obtained withthe DEM and the heterogeneous FEM are not constant, whereas thevalues obtained with the micropolar FE models are constant.Considering a panel with a larger scale factor with respect to thee rectangular masonry panel.

standard case (r 30), Fig. 10c,d shows the in-plane rotations alongsection AeA and BeB, respectively. In this case the values obtainedwith the DEM and the heterogeneous FEM are almost constantalong the sections, except at section ends. Moreover with this scalefactor, results given by DEM and Heterogeneous FEM tend to becloser to equivalent continuum solutions with respect to the casecharacterised by r 6.

In order to evaluate the effect of size of heterogeneity on paneldisplacement, with particular attention to the behaviour ofdiscrete, heterogeneous and micropolar models with respect to thetraditional Cauchymodel, the horizontal displacement at themid ofthe top edge is evaluated increasing the scale factor (Fig. 11a)togetherwith the potential energy spent by themodels (Fig.11b). Inthis example, due to the simple loading condition, the

Table 1Results obtainedwith DEM and relative differences obtained by adopting other models formodelling a rectangular panel subject to three different load and restraint conditions(Fig. 8).

Ex. Displacement r DEM d [%]

FEM Hetero FEM REV1 FEM REV2 FEM Cauchy

1 u1 (L/2, H) 5 2.082 [mm] 1.55 7.5730 2.169 [mm] 0.80 2.51

u1 (L/2, H/2) 5 1.145 [mm] 0.16 3.8430 1.129 [mm] 0.35 2.48

u3 (L/2, H) 5 2.308E-04 [rad] 4.893 37.2 e30 1.462E-04 [rad] 13.81 0.82 e

2 u1 (L/2, H) 5 3.594 [mm] 0.12 5.69 8.82 11.8830 3.945 [mm] 1.64 1.23 1.57 1.93

u3 (0, H) 5 4.806E-03 [rad] 2.138 32.45 16.08 e30 6.113E-03 [rad] 13.91 14.89 9.33 e

u3 (0, H/2) 5 3.108E-03 [rad] 0.96 18.01 4.96 e30 3.946E-03 [rad] 1.77 6.33 3.03 e

3 u2 (L/2, H) 5 1.662 [mm] 2.05 16.34 32.13 118.6930 2.882 [mm] 2.34 6.23 1.36 26.12

u3 (0, H) 5 8.528E-04 [rad] 2.10 24.71 11.76 e30 7.839E-04 [rad] 2.17 1.78 1.79 e

u3 (0, H/2) 5 4.707E-04 [rad] 1.06 12.95 4.03 e30 3.606E-05 [rad] 0.88 4.39 1.75 e

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e58 47Fig. 9. Horizontal translation (rst row) and in-plane rotation (second row) for the panel sumicrop. FEM (second column).bject to a horizontal shear load, having r 6. Results given by DEM (rst column) and

D. Baraldi et al. / European Journal of M48displacement and the energy obtained with continuum models areconstant for increasing r, whereas the values obtained with DEMand heterogeneous FEM tend to converge to the results given by thecontinuum models. Hence, as can be expected, the behaviour ofmodels i and ii converge to that of a traditional continuum forincreasing number of blocks. The heterogeneous FEM appears to bemore rigid than the other models due to the rough mesh adoptedfor modelling the panel with increasing r.

Fig. 10. Rotations u3 along section AeA (a,c) and the BeB (b,d) obtained with the different mload.

Fig. 11. Horizontal displacement at the mid of the top edge (a) and expended energy(b) obtained with the different models (iev) for a panel subject to a horizontal shearload and increasing the scale factor.echanics A/Solids 50 (2015) 39e587.1.2. Example 2: panel with xed base subject to a horizontal shearload

In this example the panel is xed at the base and subject to ahorizontal shear load (Fig. 8, second column) in order to obtain aexural behaviour characterised by non uniform rotations over thepanel. Fig. 12 shows the contours of horizontal translations androtations over the panel obtained with the DEM and micropolar FEmodels, with r 6. Similarly to the previous example, asymmetrichorizontal translations and symmetric rotations are obtained.

In this case results given by models iii and iv are not coincidentand the micropolar FEM based on REV2 appears to be slightly moredeformable than that based on REV1. Fig. 13a,b shows the in-planerotations along sections AeA and BeB, respectively, for the panelhaving r 6. The values obtained with the DEM are not constantalong both sections and the values obtained with the heteroge-neous FEM are quite close to DEM results. Considering the micro-polar FE models, the results obtained with model based on REV2are quite close to those obtained with the DEM along both sections,whereas results obtained with the model based on REV1 are quitedifferent respect to those obtained with the DEM, especially at theends of each section. Considering the panel with r 30, Fig. 13c,dshows in-plane rotations along sections AeA and BeB, respectively.In this case the results given by the different models are closer toeach other with respect to the case with r 6; however, rotationsgiven by model iv (REV2) appear to be closer to DEM rotations withrespect to those given by model iii (REV1), especially at sectionends.

odels (ieiv) for a panel having r 6 (a,b) and r 30 (c,d) subject to a horizontal shear

Fig. 12. Horizontal translation (rst row) and in-plane rotation (second row) for the panel subject to a horizontal shear load, having r 6. Results given by DEM (rst column),microp. FEM based on REV1 (second column), microp. FEM based on REV2 (third column).

Fig. 13. Rotations u3 along sections AeA (a,c) and BeB (b,d) for a panel with a xed base having r 6 (a,b) and r 30 (c,d) and subject to a horizontal shear load obtained with thedifferent models (ieiv).

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e58 49

Fig. 14. Vertical stresses along section BeB obtained with different models.

Fig. 15. Horizontal displacement at the mid of the top edge (a) and expended energy (b) obtained with the different models (iev) for a panel with increasing scale factor rwith xedbase and subject to a horizontal shear load.

Fig. 16. Vertical translation (rst row) and in-plane rotation (second row) for the panel subject to a symmetric vertical load at the top edge, having r 6. Results given by DEM (rstcolumn), microp. FEM based on REV1 (second column), microp. FEM based on REV2 (third column).

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e5850

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e58 51In this example the determination of the internal developedstresses by the different models may be of particular interest.Fig. 14a,b shows vertical stress along section BeB for panel withr 6 and 30, respectively. In both cases results given by the nu-merical models considered do not show relevant differences.However for r 30 the stresses obtained with the Cauchy modelapproximate better those obtained with DEM with respect to themicropolar models.

Similarly to the previous case, the horizontal displacement at themid of the top edge and the energy spent by models are evaluatedincreasing the scale factor r (Fig. 15a,b). In this example, due to the

Fig. 17. Rotations u3 and vertical displacements u2 along sections AeA (a,c) and BeB (b,d) fora symmetric vertical load at the top edge.

Fig. 18. Vertical stresses along section Bdifferent restraint condition with respect to the previous case, onlydisplacement and energy obtained with the Cauchy continuummodel are constant for increasing r, whereas the values obtainedwithmodels ieiv tend to converge to the results given by the Cauchymodel. As it has been found in Fig 12, themicropolarmodel based onREV2 is more deformable than that based on REV1 and u1(L/2,H)given by the micropolar model based on REV2 is closer to the con-stant value given by the Cauchy model with respect to the valuesgiven by the micropolar model based on REV1. The heterogeneousFEM appear to be more rigid than other models due to the roughmesh adopted for modelling the panel with increasing r.

a panel with a simply supported base having r 6 (a,b) and r 30 (c,d) and subject to

eB obtained with different models.

relevant differences and the corresponding results obtained withr 30 are not showed for simplicity.

Similarly to the previous example, Fig. 18a,b shows verticalstress along section BeB for panel with r 6 and 30, respectively. Inboth cases micropolar models slightly underestimate stresses ob-tained with DEM, however for r 6 the stresses obtained with themicropolar model based on REV2 approximate better those ob-tained with DEM with respect to the micropolar model based onREV1.

The effect of size of heterogeneity is taken into account in thiscase by determining the vertical displacement at the mid of the topedge for increasing r (Fig. 19). Models ieiv converge to Cauchysolution more slowly with respect to the previous examples due tothe small wavelength of the applied load.

7.2. Square panel subject to a force acting on its diagonal

Fig. 19. Vertical displacement at the mid of the top edge obtained with the differentmodels (iev) for a panel with increasing scale factor r with xed base and subject to ahorizontal shear.

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e58527.1.3. Example 3: panel subject to a symmetric vertical loadIn this example the panel is simply supported at the base and

subject to a vertical force applied at the mid of the top edge (Fig. 8,third column). This problem may represent a masonry wall loadedby a beam of a wooden oor at the top edge, hence the size of theload distribution is similar to the size of the REV and it is quitesmaller than the size of the panel.

Fig. 16 shows the maps of in-plane vertical translations androtations over the panel obtained with the DEM and micropolarFE models, with r 6. In this case, due to the symmetric load,symmetric vertical translations and asymmetric rotations areobtained.

With particular attention to in-plane rotations, the contourmaps obtained with the FE model based on REV2 (third column ofFig. 16) appear to be very close to DEM results (rst column ofFig. 16). Fig. 17a,b shows in-plane rotations obtained with thedifferent models and evaluated along sections AeA and BeB of thepanel having r 6. In this case DEM and heterogeneous FEM resultsare in excellent agreement; moreover Fig. 17a,b shows clearly thatthe micropolar FE model based on REV2 is closer to the DEM thanthe micropolar FE model based on REV1 along both sectionsconsidered. If the panel with r 30 is taken into account, Fig. 17c,dconrm that the micropolar FE model based on REV2 approximatesbetter the DEM. In this case, Fig. 17d shows a detail of in-planerotations in the middle of section AeA in order to appreciate thatresults given by the micropolar model based on REV1 are far from

other results. Far from the loaded region (section BeB), in-planerotations obtained with the different models do not show

Fig. 20. Relative differences obtained by adopting continuous FEMs and Heterogeneousrectangular panel subject to three different load and restraint conditions and increasing scIn order to have a further validation of the discrete and micro-polar continuum models, part of an existing numerical experi-mentation carried out by Pau and Trovalusci (2012) and Trovalusciand Pau (2014) is taken into account as benchmark solution. Inparticular, analyses are limited to the cases a1, a2 and a3 of bothpapers, characterised by scale factors r L/b equal to 5, 10 and 20,respectively, and having block width-to-height ratio b/a 4, whichis quite close to that of UNI bricks of the previous examples. Thepanel is square, with length and height L 8000 mm and it issubject to a force F 1000

2

pKN acting along the diagonal of the

panel (Fig 21). In this example mortar is modelled accordingly toPau and Trovalusci (2012). The thickness of horizontal and verticalmortar joints is equal to 10 mm and it is kept constant and notinuenced by the scale factor, in order to have the same jointstiffness for varying blocks' size.

Fig. 22aec shows the contour maps of rotations over the panelsmodelled by DEM; as expected, the larger the scale factor, thegreater the deformability of the panel in terms of in-plane rota-tions. The order of magnitude of results is in quite good agreementwith benchmark results that may be found in Pau and Trovalusci(2012). Differently than benchmark results, contour lines inFig. 22 highlight the masonry texture pattern of each panel. Then,following the work done for the benchmark results, rotations(Fig. 23, rst column) and vertical translations (Fig. 23, secondcolumn) given by the different models ieiv are evaluated along theleft edge of the panel (y1 0) for the three scale factors considered.Considering in-plane rotations, the micropolar model based onREV2 ts DEM results better than the model based on REV1.However, differently with respect to benchmark results, the CauchyFEM in comparison with DEM for determining displacements at several points of aale factor r.

model is more deformable than other models and displacementsobtained with micropolar model converge to those obtained withthe Cauchy model for increasing scale factor.

8. Conclusions

In the present work, a deep investigation of different modelsthat may be adopted for studying the mechanical behaviour ofmasonry is presented, with particular attention to the opportunity

of using a micropolar model instead of a traditional Cauchy con-tinuum model.

The discrete model introduced by Cecchi and Sab (2004), whichconsiders masonry as a skeleton with rigid blocks and elasticmortar interfaces, is adopted for studying the in-plane behaviourof the panels and results are taken as reference solution. Moreover,an identication between the discrete model and a 2D continuumis carried out by equating the mechanical work in the two modelsfor a class of regular motions. Two different REVs are taken intoaccount for representing the micropolar continuum and a trian-gular FE in plane stress state enriched by rotational nodal degreesof freedom is adopted for modelling the micropolar continuumand representing the behaviour of in-plane loaded periodicbrickworks.

The discretemodel allows to calculate the rotations of the blocksin masonry walls and allows to compare the rotations determinedby the micropolar FE models (based on two REVs). In addition, atraditional heterogeneous FE model is taken into account in whichmortar joints are modelled as a continuous material.

The proposed examples investigate the behaviour of masonrypanels subject to both concentrated and distributed loads, in orderto evaluate the effect of the load wavelength with respect to panelsize. Moreover, the proposed examples investigate the effects ofsize of heterogeneity on masonry behaviour by introducing a scalefactor r L/b.

Fig. 21. Masonry square panel having r 5 subject to a force acting along its diagonal.

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e58 53Fig. 22. Contour maps of in-plane rotations for square panels subject to a diagonal force and with r equal to 5 (a), 10 (b) and 20 (c).

D. Baraldi et al. / European Journal of Mechanics A/Solids 50 (2015) 39e5854The DEM model turns out to be more deformable than theheterogeneous FE model, however results in terms of in-plane ro-tations and displacements appear to be in very good agreement.Then, this comparison conrms that the DEM can be the referencesolution for evaluating the effectiveness of the micropolar FEmodels.

Comparing results in terms of rotations given by the micropolarFE models with respect to the DEM, the micropolar FE model basedon REV2 turns out to be more appropriate than that based on REV1.Results given by the FE model based on REV2 turn out to begenerally closer to those given by the DEM than those given by theFE model based on REV1. In particular, considering rotations closeto the loaded regions, the micropolar FE model based on REV1completely fails to approximate DEM results.

Fig. 23. Comparison of rotations (rst column) and vertical dIt is worth noting that the micropolar FEmodel based on REV2 isquite more deformable than that based on REV1. Then, consideringresults in terms of translations, the behaviour of micropolar FEmodel based on REV1 appears to be closer to that of DEM withrespect to the model based on REV2. However the differences ofmodel based on REV2 with respect to the DEM are acceptable.

As can be expected, the FEM based on Cauchy continuum turnsout to bemore deformable than DEM, heterogeneous FEM and bothmicropolar models. It is worth noting that increasing panel sizewith respect to block size, the behaviour of micropolar modelsconverge rapidly towards that of the Cauchy model, whereasconvergence is slow for the DEM and heterogeneous FEM.

Then, the micropolar model turns out to be suitable and effec-tive in modelling the in-plane behaviour of masonry panels

isplacements (second column) along vertical line y1 0.

and grad U. According to the identication between discrete and

resenting the in-plane behaviour of masonry panels with regular

v2PREVu1;1;u2;2;u1;2;u2;1;u3;u3;1;u3;2

vu21;1

;v2PREV

u1;1;u2;2;u1;2;u2;1;u3;u3;1;u3;2

vu22;2

(A.2a,b)

v2PREVu1;1;u2;2;u1;2;u2;1;u3;u3;1;u3;2

vu21;2

;v2PREV

u1;1;u2;2;u1;2;u2;1;u3;u3;1;u3;2

vu22;1

(A.2c,d)

v2PREVu1;1;u2;2;u1;2;u2;1;u3;u3;1;u3;2

vu23;1

;v2PREV

u1;1;u2;2;u1;2;u2;1;u3;u3;1;u3;2

vu23;2

(A.2e,f)

of Mactions at the interfaces generated by simple deformation statesand then differentiating the sum of the works of the REVs withrespect to the deformation components.

The deformation states considered are: i) horizontal extensionE11, ii) vertical extension E22, iii) horizontal shear E12 u3, iv)vertical shear E21 u3. For each deformation state, the relativedisplacements and rotations of blocks are evaluated over the REVand Eq. (10) is applied for all the interfaces of the REV, taking intoaccount their constitutive relationship (Eq. (13)). It is worth notingthat REV1 is characterised by eight horizontal interfaces sharedwith contiguous REVs and two vertical interfaces, whereas REV2 ischaracterised by four horizontal interfaces shared with contiguousREVs and one vertical interface.

The work spent over a horizontal and a vertical interface may bewritten in terms of the jump of displacements (D1, D2) and rotations(d3) between adjacent blocks as follows (Cecchi and Sab, 2004):

Ph 12

ZS

dxTKhdxdA

12

8