A simple homogenized orthotropic model for in-plane analysis of regular masonry walls

International Journal of Solids and Structures 167 (2019) 156–169

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A simple homogenized orthotropic model for in-plane analysis of regular masonry walls Simona Di Nino a,b, Angelo Luongo a,b,∗ a b

International Research Center on Mathematics and Mechanics of Complex Systems, University of L’Aquila, L’Aquila, 67100, Italy Department of Civil, Construction-Architectural and Environmental Engineering, University of L’Aquila, L’Aquila, 67100, Italy

a r t i c l e

i n f o

Article history: Received 25 November 2018 Revised 7 March 2019 Available online 12 March 2019 Keywords: Masonry Homogenization Linear elasticity Orthotropy Static and dynamic analyses

a b s t r a c t Simple closed-form expressions for “equivalent” linear elastic constants of regular brick pattern masonry, solicited in their plane, are derived by means of a homogenization procedure. The stresses of the brick and mortar components are also evaluated analytically once the average stress acting on the homogeneous medium is determined. The fundamental concept of the method consists of modeling the behavior of a masonry cell using suitable designed assemblies of in-series and in-parallel springs. Thereafter, an equivalent homogeneous and orthotropic material is defined, the elementary cell of which has the same stiffness as the assembly. As a further simplified model, an equivalent isotropic material is defined. The accuracy of the theoretical results is assessed by means of comparisons with finite element (FE) analyses drawn from the literature. The proposed approach is found to match the macroscopic linear elastic laws of the literature accurately, with the advantage of taking a far simpler form. Finally, static and dynamic FE analyses are carried out on sample masonry walls, with the aim of comparing the non-homogeneous and homogeneous models. The latter is found to describe both the local and global behavior of masonry walls to a satisfactory extent. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Masonry is a composite material characterized by overall anisotropic behavior from a phenomenological perspective. This is owing to the composite and non-homogeneous structure of the geometrical arrangement of the units and mortar joints. Researchers have presented micro-modeling and macro-modeling techniques for masonry analysis. In the former approaches, discretization is carried out at the level of the components, namely units and mortar joints. In the latter approaches, the overall material is regarded as an equivalent homogeneous continuum. The micromodeling approach has experienced greater development, despite its high computational costs. Macro-modeling still appears as an interesting alternative for producing more computationally efficient models. Within this framework, masonry is considered as a homogeneous isotropic or anisotropic continuum. Accordingly, closed-form macroscopic constitutive laws, relating the average stresses to the average strains, are formulated for an equivalent homogeneous continuum. At times, these laws have only phenomenological bases or follow homogenization techniques, which ∗

Corresponding author. E-mail addresses: [email protected] (S.D. Nino), [email protected] (A. Luongo). https://doi.org/10.1016/j.ijsolstr.2019.03.013 0020-7683/© 2019 Elsevier Ltd. All rights reserved.

derive macro-constitutive laws from micro-constitutive laws. Such methodology consists of identifying an elementary cell, known as a representative volume element (RVE), which generates an entire panel by means of regular repetition, and writing a field problem onto the unit cell to achieve average values for the homogenized masonry material, starting from the knowledge of the constituent properties and elementary cell geometry. In linear elasticity, several authors have derived expressions for the effective properties of masonry, which is macroscopically orthotropic if it is made up of regularly spaced units. A brief review is provided herein of some of the most popular simplified approaches followed in the technical literature to obtain homogenized elastic moduli of the masonry (see also Lourenço et al., 2007; Taliercio, 2014). One of the first concepts was presented by Pande et al. (1989), who exploited results previously obtained by Salamon (1968) for stratified rock to obtain the five macroscopic elastic constants of masonry, which was assumed to be represented by an equivalent transversely isotropic material. In particular, in (Pande et al., 1989), so-called “two-step homogenization” was performed: in the first step, a single row of bricks and mortar head joints were homogenized as a layered system; in the second step, the homogenized material layers were further homogenized with bed joints to obtain the final material. In this manner, a simple mechanical

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system constituted by elastic springs was defined, and explicit formulae based on classical elasticity concepts were derived. The main disadvantage of the above method is that the obtained homogenized material differs if the homogenization steps are inverted. Following the concept of a multi-step approach, numerous authors proposed different approximations and ingenious assumptions. In (Pietruszczak and Niu, 1992), a two-stage homogenization procedure was employed, with the head joints considered as uniformly dispersed elastic inclusions, while the bed joints were assumed to represent a set of continuous weakness. A different approach proposed by Cecchi and Sab (2002) was based on the reduction of joints to interfaces. In (Cecchi and Sab, 2002), elastic springs with diagonal constitutive tensors were used for the joints; thus, the Poisson effect of the joint was neglected. A multi-parameter homogenization study was defined for two-dimensional (2D) and three-dimensional (3D) in-plane cases. The finite thickness of the joints was considered in an approximated manner, only in the constitutive relation of the interfaces. Therefore, the authors were able to determine “quasi-analytical” formulae. Refined finite element (FE) models, such as those proposed by Anthoine (1995) for periodic masonry, or by Cluni and Gusella (2004) for quasi-periodic masonry, are supposed to predict the macroscopic behavior more accurately. Moreover, Zucchini and Lourenço (2002) proposed approximated displacement (and stress) fields for any RVE, which were defined by a reduced number of variables, and derived the macroscopic elastic stress–strain law by prescribing approximate equilibrium and compatibility conditions at the boundaries of the different RVE parts. However, this approach provides closed-form expressions for the macroscopic inplane and out-of-plane shear moduli, while the Young’ s moduli and Poisson’ s ratios are numerically computed. More recently, Taliercio (2014) derived approximate expressions for the macroscopic in-plane elastic coefficients of brick masonry with a regular pattern in a closed form at various approximation degrees, by either using a method of he cells-type approach or minimizing the potential energy of the RVE subjected to any given macroscopic stress. A further aspect of the problem, which appears not to have been studied in depth thus far, concerns the possibility of returning to the non-homogeneous model once results for the homogeneous medium have been obtained. More specifically, by considering the stresses in the homogeneous continuum as average values for the non-homogeneous continuum, the task is the manner in which to evaluate the individual stresses of the brick and mortar components. This topic has been addressed in (Pande et al., 1989), in which the stress values in the masonry constituents were determined according to the orthorhombic layer theory developed in (Gerrard, 1982). Thus far, attention has also been devoted to describing the nonlinear behavior of the masonry. Nonlinear approaches based on homogenization can be classified as follows, according to Lourenço et al. (2007): (i) Engineered approaches, based on continuum models for components or mortar joint failure; (ii) kinematic and static limit analysis approaches; and (iii) FE nonlinear analyses. Engineered approaches have been developed, for example, in (Luciano and Sacco, 1998; Gambarotta and Lagomarsino, 1997), where masonry failure could occur as a combination of bed and head joint failure. Certain authors, such as (De Buhan and De Felice, 1997; Milani et al., 2006), have used limit analysis approaches, which are based on the assumption of perfectly plastic behavior with associated flow rules for the constituent materials. At present, FE nonlinear approaches can be used to perform analyses in the inelastic range by means of refined discretizations of the elementary cell; (Pegon and Anthoine, 1997; Massart et al., 2004) adopted a nonlinear damaging model for the constituent materials.

157

Fig. 1. Masonry regular brick pattern.

Moreover, creep effects occur in masonry, as demonstrated by the laboratory tests carried out by Shrive et al. (1997) or SayedAhmed et al. (1998). Numerical analyses were subsequently carried out on 3D FE models (Taliercio, 2013), as well as on a simplified 2D layered model (Brooks, 1990). Few authors have attempted to formulate mathematical expressions for the macroscopic creep coefficients of brick masonry. Ref. Brooks and Abdullah (1986) extended formulae previously proposed in linear elasticity to define the creep compliance of masonry mathematically under sustained vertical stress. Cecchi and Tralli (2012) proposed an analytical model based on homogenization procedures for periodic media, by confining creep phenomena to joints and reducing joints to interfaces. More recently, Taliercio (2014) described the global creep behavior of brickwork under service loads, assuming that the creep laws of units and mortar could be expressed by the Prony series. In this study, the focus is limited to the elastic properties of masonry. Although it is known that masonry exhibits nonlinear behavior even at low stress levels, the evaluation of the global homogenized material properties in the elastic field is a relevant topic. Firstly, the research area holds practical interest, because technical codes still provide directions on evaluating the elastic properties of masonry. Secondly, formulae for the equivalent elastic continuum are implemented in several commercial software packages with recognized reliability (see, for example, MIDAS®, where reference is made to the theory developed in Pande et al., 1989). Finally, several papers have appeared in international journals in the past decade concerning masonry modeling in the elastic field (see, for example, Stefanou et al., 2010; Bacigalupo and Gambarotta, 2011; Pau and Trovalusci, 2012; Milani and Cecchi, 2013; Drougkas et al., 2015). In this case, masonry is considered, characterized by a regular arrangement of units and mortar joints. In such a framework, the closed-form solutions provided by Pande et al. (1989), according to two-step homogenization, are believed to be the most simple and efficient. However, as highlighted previously, the main disadvantage of this method is that the homogenized material results are sensitive to the order of the steps. Therefore, the challenge lies in overcoming this limitation while maintaining the philosophy of the method. This paper deals with in-plane loaded masonry walls with a running/header bond pattern (Fig. 1). The objectives are to: (a) Derive analytical expressions for the (macroscopic) elastic coefficients of an equivalent homogeneous continuum; and (b) evaluate the stresses in the brick and mortar components of the wall, once the stresses in the equivalent medium have been computed. A one-step approach is proposed, which consists of modeling an RVE using springs suitably combined in series and in parallel. As explained later, such a one-step approach is not viable in identifying the shear modulus. However, a two-step homogenization process will lead to a result that is independent of the order followed. The remainder of this paper is organized as follows. Firstly, in Section 2, the RVE is defined and the analytical expressions for the macroscopic elastic constants of orthotropic homogeneous material are determined; as a particular case, the elastic coefficients of an equivalent isotropic homogeneous material are also derived. Here, the partition coefficients of the normal stresses between the brick

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Fig. 3. Scheme of homogenization procedure. Fig. 2. RVE: (a) non-homogeneous cell; (b) homogeneous orthotropic cell.

and mortar are also evaluated. In Section 3, the accuracy of the theory is assessed by means of comparisons with refined FE analyses and closed-form expressions available in the literature (collected in Taliercio, 2014); parametric analyses are also carried out. In Section 4, static and dynamic structural analyses are performed on (in-scale) masonry wall systems: non-homogeneous and homogeneous models are analyzed and compared. Finally, in Section 5, the main findings of the work are summarized, and several perspectives are mentioned. The Appendix provides further details. 2. Homogenized orthotropic model A masonry wall is considered, which is made from regularly staggered bricks, with alternating (horizontal) mortar bed and (vertical) head joints (Fig. 1). The two basic components, namely the brick and mortar, are assumed to be homogeneous and isotropic, with an elastic Young’ s modulus and a Poisson factor Eb , ν b and Em , ν m , respectively; however, owing to the geometry of the assembly, the resulting composite material (masonry) is nonhomogeneous and anisotropic. However, by exploiting the periodic local masonry structure, in principle, it is possible to substitute the non-homogeneous material with an ‘equivalent’ homogeneous material. At the macroscopic scale, the latter must preserve the mechanical properties possessed by the original material at the microscopic scale, averaged for an RVE (that is, the smallest part of the real medium that contains all of the required information). Of course, the homogenized model works if a sufficiently large scale is applied; that is, if the interest is in the behavior of a large number of assembled bricks. Owing to the geometric micro-structure, in which horizontal and vertical are the principal directions, an equivalent orthotropic model is adopted, for which the constitutive law in planar linear elasticity is:

⎡





1 Ex

⎢ εx ⎢ νyx εy = ⎢ ⎢− Ex γxy ⎣ 0

−

νxy

Ey 1 Ey 0

⎤

0

⎥ σx  ⎥ 0 ⎥ σy ⎥ 1 ⎦ τxy

(1)

Gxy

Here, x denotes an axis parallel to the bed joints, while y is an axis parallel to the head joints; Ex and Ey are the Young’ s moduli;

ν xy and ν yx are the Poisson’ s ratios (with

νxy

νyx

= for the elastic Ey Ex matrix to be symmetric); finally, Gxy is the in-plane shear modulus. The problem is stated as follows: knowing the assembly geometry (that is, the width b and height a of the brick, and the thickness t of the mortar joints), as well the elastic properties of the two elements (namely Eb , ν b , Em , ν m ), determine the five elastic constants (Ex , Ey , ν xy , ν yx , Gxy ) of the homogenized material, satisfying the symmetry condition. To solve the above problem, a single ‘unit cell’ of the periodic assembly is used as the RVE (Fig. 1); the RVE is detailed in Fig. 2(a). This has been subdivided into five regions, three of which are distinguished and named as ‘elements’ e = 1, 2, 3. Ele-

ment e = 1 is the brick, while elements e = 2 and e = 3 are pairs of half of the bed joint and half of the head joint, respectively; the cross-joints are included in elements 2 or 3, according to the mechanism under study, as discussed later. It should be noted that the selected RVE does not account for the pattern stagger, which is assumed to have a weak effect (in the elastic field) on the global masonry characteristics. This conjecture is validated numerically later (see Appendix). Moreover, the following assumptions regarding the elements e are introduced: 1. The elements are subjected to a uniform stress and strain state, representative of their average values. 2. In each element, one of the stresses prevails over the others, according to the external solicitation. Consequently, each element of the non-homogeneous cell is considered as one-dimensional (1D), and modeled by an equivalent spring of stiffness ke , depending on the element mechanical and geometrical properties. Thus, the non-homogeneous cell in Fig. 2(a) is modeled by a specific spring system, which depends on the behavior under study, namely: (i) Extension along x or y, (ii) transverse dilatation along x or y, and, (iii) shear. To this end, five spring system types are defined, each of which is composed of springs combined (on a phenomenological basis) in series and in parallel. By equating the stiffnesses of the spring assemblies and those of a homogeneous cell with dimensions of (a + t ) × (b + t ) and orthotropic directions x, y, as illustrated in Fig. 2(b), the homogenized elastic constants are finally derived. The logical flow of the proposed homogenization procedure is summarized in Fig. 3. 2.1. Identification of elastic axial moduli In order to identify the two elastic axial moduli Ex and Ey of the homogenized continuum, the behavior of the RVE under in-plane normal forces, parallel to either the bed joints, Nx , or head joints, Ny , is evaluated. Such forces are considered as resulting from the averaged stresses, σ x , σ y , acting on the cell, namely Nx = σx (a + t ), Ny = σy (b + t ). The following considerations drive the reduction of the non-homogeneous cell to a spring assembly. Let us consider, for example, that the cell is subjected to normal stresses, the resultant of which is Nx . Elements 1 and 3 undergo (by hypothesis), a uniform state of stress, viz. σ x1 , σ x3 , respectively. However, for the sake of equilibrium at the interface, σx1 = σx3 ; that is, the elements 1,3 behave as two in-series springs (known as sub-cell I), with stiffnesses kx1 and kx3 , respectively. However, for geometrical reasons (namely, to maintain the rectangular cell shape and ensure compatibility with the adjacent cells), element 2 (including the cross-joints and constituting the sub-cell II) must undergo the same total elongation as elements 1 and 3, namely I = II ; therefore, they behave as springs with stiffness kx2 , working in parallel with sub-cell I. It is worth noting that such an approximate model does not satisfy local compatibility between the brick and mortar bed, but only global compatibility (as the two elements are attached only at their ends). Similar considerations hold for stresses along the y direction, where the cross-joints are

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159

Fig. 4. Spring systems to determine: (a) Ex ; (b) Ey .

included in element 3. The two spring assemblies are represented in Fig. 4(a) and (b), respectively. By using a unitary out-of-plane thickness for the wall, the horizontal kxe and vertical kye spring stiffnesses (e = 1, 2, 3 ) are computed as follows:

a E b b t/2 = Em b+t 2a = Em t

b E a b 2b = Em t t/2 = Em a+t

kx1 =

ky1 =

kx2

ky2

kx3

ky3

kxII = 2kx2

(2)

ky1 ky2 kyI = ky2 + 2 ky1

kyII = 2ky3

and the total stiffnesses are:

ky = kyI + kyII

(4)

However, the stiffnesses of the homogeneous cell are equal to a+t b+t Ex and Ey . By equating these latter values with the b+t a+t spring stiffnesses in Eq. (4), the unknown elastic moduli Ex , Ey are finally evaluated. By introducing the nondimensional parameters:

αE

Em = Eb

αG

Gm = Gb

βa

t = a

βb

t = b

(5)

which account for material properties (α E,G ) and geometrical properties (β a,b ), we can obtain:

αE (βa (βb + αE ) + βb + 1 ) (βa + 1 )(βb + αE )

(6)

αE (βb (βa + αE ) + βa + 1 ) Ey = Eb (βb + 1 )(βa + αE )

(7)

Ex = Eb

NxII , by means of elementary computations, it follows that: t

(βa + 1 )(βb + 1 ) σ βa βb + αE βa + βb + 1 x (βa + 1 )(βb + αE ) σx 2 = σ βa βb + αE βa + βb + 1 x

(8)

where the definitions of (5) have been used. Analogously:

(3)

kx = kxI + kxII

σxII =

σx 1 = σx 3 =

By accounting for the in-series or in-parallel arrangements, the equivalent sub-cell stiffnesses are:

kx1 kx3 kxI = kx3 + 2kx1

Fig. 5. Spring systems to determine: (a) ν xy ; (b) ν yx .

2.1.1. Normal stress distribution For future purposes, it is worth investigating the manner in which the normal forces Nx = σx (a + t ) and Ny = σy (b + t ), applied to the homogeneous cell, distribute themselves between the two parts I and II of the non-homogeneous cell. By referring, for example, to Nx : for the equilibrium Nx = NxI + NxII , and as the subN N N cells are in parallel, xI = xII . Therefore, by letting σxI = xI and kxI kxII a

(βa + 1 )(βb + 1 ) σ βa βb + βa + αE βb + 1 y (βb + 1 )(βa + αE ) σy 3 = σ βa βb + βa + αE βb + 1 y

σy 1 = σy 2 =

(9)

Knowing the average normal stress acting on the homogenized medium (σ x or σ y ), these expressions allow for evaluating the (local) stress of the constituents of the non-homogeneous medium, namely on the bricks and mortar joints. It should be noted that, for small values of β a,b , the previous local stress expressions can be approximated by:

σx 1 = σx 3  σx σx2  αE σx

(10)

σy 1 = σy 2  σy σy3  αE σy

(11)

Accordingly, the normal brick stress is approximately equal to the global stress of the homogenized model, while the normal mortar stress is proportional to the former by means of the mortar-tobrick modulus ratio α E . 2.2. Identification of Poisson factor In order to identify the two Poisson factors, ν xy and ν yx , of the homogenized continuum, the transverse behavior of the nonhomogeneous cell under normal stresses, σ y or σ x , respectively, is studied. For example, let us consider the horizontal dilatations induced by vertical stresses. For (global) compatibility reasons, the brick (element 1) and mortar bed (element 2) undergo the same dilatation, so that they behave as in-parallel springs; in contrast, the mortar joints (element 3) add their dilatations to the former, thereby behaving as in-series springs. The spring system is depicted in Fig. 5(a); similar considerations for transverse vertical dilatations lead to the system in Fig. 5(b). The spring stiffnesses are

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redefined as follows (according to the exchange of the intersection regions, included in the elements e = 3 to derive kx3 and in the elements e = 2 to derive ky2 ):

a E b b t/2 = Em b a+t = Em t/2

b E a b b+t = Em t/2 t/2 = Em a

kx1 =

ky1 =

kx2

ky2

kx3

ky3

(12)

Each spring of the assembly undergoes a prescribed dilatation, εxe and εye (e = 1, 2, 3), which is proportional to the stress state described in Eqs. (8) and (9), namely:

ε x1 = −

νb

σyI Eb νm εx2 = − σyI Em ν εx3 =− m σyII

ε y1 = −

νb

σxI Eb νm ε y2 = − σxII Em ν ε y3 = − m σxI

Em

(13)

Em

As the system is statically undetermined, such strains induce a self-equilibrated stress state or self-stress state. To evaluate this, the elastic problem is solved via the displacement method. For example, reference is made to the system in Fig. 5(a). By using uxA , uxB to denote the unknown displacements at nodes A, B, respectively, and Fxe to denote the elastic force at spring e, the elastic problem is governed by:

Fx1 = kx1 (2uxA − ε x1 b) Fx2 = kx2 (2uxA − ε x2 b)



Fx3 = kx3 uxB − ε x3

t 2



(14)

which expresses the elastic law for the springs, and:

Fx1 + 2Fx2 = Fx3 Fx3 = 0

at A at B

(15)

which is the state equilibrium at the nodes. The solution to this problem reads as follows:

uxA

bEb (a + t )(aνb + t νm ) =− Ny 2(aEb + Em t )(Eb (ab + t (b + t ) ) + aEm t )

uxB

t νm (aEm + Ebt ) =− Ny 2Em (Eb (ab + t (b + t )) + aEm t )

(16)

and the total elongation of the assembly is Δ = 2uxA + 2uxB . Moreover, the transverse dilatation of the homogeneous cell is equal to Δ = −

νxy

Ny ; the unknown Poisson’ s ratio is derived by Ey equating the two elongations. By repeating the procedure for the system in Fig. 5(b), the following expressions are determined:

νxy =

νm (βb (βa +αE )(αE βa +1 )+αE βa (βa +1 ) )+αE (βa +1 )νb (βb +1 )(βa +αE )(αE βa + 1 )

σ˜ x1 =

βa (βa + 1 )(βb + 1 )(αE νb − νm ) σy (αE βa + 1 )(βa βb + βa + αE βb + 1 )

σ˜ x2 = −

(βa + 1 )(βb + 1 )(αE νb − νm ) σy α ( E βa + 1 )(βa βb + βa + αE βb + 1 )

(19)

σ˜ x3 = 0 that is, self-stress exists at the brick and mortar bed joints (in the hyperstatic sub-cell), and not at the mortar head joints. By repeating the procedure for the system in Fig. 5(b), the selfstresses are determined as:

σ˜ y1 =

βb (βa + 1 )(βb + 1 )(αE νb − νm ) σx (αE βb + 1 )(βa βb + αE βa + βb + 1 )

σ˜ y2 = 0 σ˜ y3 = −

(20)

(βa + 1 )(βb + 1 )(αE νb − νm ) σx (αE βb + 1 )(βa βb + αE βa + βb + 1 )

that is, self-stress occurs at the brick and mortar head joints (in the hyperstatic sub-cell), while it is zero at the mortar bed joints. σ˜ 1 From Eq. (19), it follows that x2 = − , from which, as β a is σ˜ x1 βa small, σ˜ x1  σ˜ x2 , that is, the self-stress at the brick is negligible. However, the self-stress at the mortar bed joint may be significant. Similar considerations follow from Eq. (20), when exchanging the bed joints and head joints. Moreover, note that, in this case, for small values of β a,b , the self-stress expressions can be approximated by:

σ˜ x1  0 σ˜ x2  −(αE νb − νm )σy σ˜ x3 = 0

(21)

σ˜ y1  0 σ˜ y2 = 0 σ˜ y3  −(αE νb − νm )σx

(22)

This means that the vertical and horizontal global normal stresses induce non-negligible transverse self-stresses in the mortar joints, which are proportional to (αE νb − νm ). The self-stress induced by the transverse elongation adds itself to the stress triggered by the axial elongation, so that the total stress is σˆ xe = σxe + σ˜ xe , σˆ ye = σye + σ˜ ye . 2.3. Identification of elastic shear modulus

(17)

νyx =

Self-stress state For future purposes, it is worth investigating the self-stress state σ˜ xe , σ˜ ye , owing to the transverse-to-load behavior of the non-homogeneous cell subjected to normal forces Ny = σy (b + t ) (Fig. 5(a) and Nx = σx (a + t ) (Fig. 5(b), respectively. For example, referring to the system in Fig. 5(a), by substituting the solutions in Eq. (16) into Eq. (14), the spring forces are evaluF F F ated; then, by letting σ˜ x1 = x1 , σ˜ x2 = x2 , σ˜ x3 = x3 , it follows a t/2 a+t that:



βa βb νm +αE 2 βa βb νm +αE νm (βa +1 )βb2 +βa + βb +αE (βb +1 )νb (βa +1 )(βb +αE )(αE βb +1 ) (18)

where the non-dimensional parameters (5) have been used. Remarkably, the Poisson factors satisfy the symmetry condition of the elastic matrix.

To identify the elastic shear modulus Gxy of the homogenized continuum, the cell behavior under in-plane horizontal shear forces Txy = τxy (b + t ) and equilibrating vertical forces Tyx = τxy (a + t ), resulting in the averaged shear stresses τ xy , is evaluated (Fig. 6(a)). Elements 1, 2, and 3 undergo (by hypothesis) a uniform stress, viz. τ xy1 , τ xy2 , τ xy3 , respectively. However, for the sake of equilibrium at the interfaces, τxy1 = τxy2 = τxy3 = τxy . Owing to the different elastic moduli of the materials, it follows that γxy1 = γxy2 = γxy3 ; however, these cannot be satisfied if a 1D displacement field,

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⎡

1 E

⎢ ⎢ ν Hiso = ⎢ ⎢− E ⎣ 0

−

161

⎤

ν

0

E 1 E

0 2 (1 + ν ) E

0

⎥ ⎥ ⎥ ⎥ ⎦

(27)

The Frobenius norm Horn and Johnson (1990) is assumed for the distance, namely: Fig. 6. Evaluation of Gxy : (a) Homogenized brick-head joint sub-cell (I) and bed joint sub-cell (II); (b) equivalent spring system.

such as horizontal, is assumed. Therefore, as opposed to the mechanisms analyzed previously, the cell shear behavior cannot be described by 1D devices. To overcome this drawback, the problem is tackled in two steps, similar to the procedure followed in (Pande et al., 1989; Zucchini and Lourenço, 2002): 1. Firstly, the subset made of the brick and mortar head joints is homogenized by equating the elastic complementary energy of the non-homogeneous and homogeneous (known as sub-cell I) sub-cells, subjected to a uniform shear stress state τ xy . This leads to the definition of the equivalent shear modulus:

G Gm ( b + t ) GI = b bGm + tGb

(23)

2. The homogenized sub-cell I and remaining sub-cell II, constituted by the mortar bed joints, are studied as 1D in-series springs (Fig. 6(b)). The sub-cell stiffnesses are:

b+t GI a 1 b+t = Gm 2 t/2

ksI = ksII

(24)

ksI ksII . As ksII + ksI b+t the shear stiffness of the homogeneous cell is equal to Gxy , a+t based on the equality, the shear modulus of the homogenized material can be determined. By using non-dimensional quantities (Eq. (5)), the following is obtained: so that the total stiffness of the spring system is ks =

Gxy =

αG (βa + 1 )(βb + 1 ) G βa βb + βa + βb + αG b

(25)

The invariance of this expression with respect to the exchange a⇔b proves the independence of the result of the homogenization order (first the head joints, as in this case, or first the bed joints). Note that the expression for the homogenized shear modulus in Eq. (25) can also be obtained by the Reuss lower bound for heterogeneous media, in which the stress is uniform everywhere. 2.4. Isotropic model To simplify the masonry modeling further, an equivalent homogeneous isotropic continuum is defined. By resorting to the notion of the ‘distance between two matrices’, the concept consists of searching the constitutive parameters E, ν of the isotropic material, so as to minimize the distance between the two flexibility matrices of the orthotropic and isotropic media; that is:

⎡

1 Ex

⎢ ⎢ νyx Hortho = ⎢ ⎢− Ex ⎣ 0

−

νxy

Ey 1 Ey 0

⎤

0

⎥ ⎥ 0 ⎥ ⎥ 1 ⎦ Gxy

(26)

Hortho − Hiso F =



tr (Hortho − Hiso )(Hortho − Hiso )T

(28)

By minimizing this value with respect to the Young’ s modulus E and Poisson’ s ratio ν , a system of two equations is drawn:

∂ Hortho − Hiso F =0 ∂E ∂ Hortho − Hiso F =0 ∂ν

(29)

from which the constants of the isotropic material can be determined in terms of those of the orthotropic model:

E=

10Ex Ey Gxy Ex (2Ey + Gxy (3 − 2νxy )) + Ey Gxy (3 − 2νyx )

ν=

Ex (2Ey + Gxy (3νxy − 2 )) + Ey Gxy (3νyx − 2 ) Ex (2Ey + Gxy (3 − 2νxy )) + Ey Gxy (3 − 2νyx )

(30)

3. Model validation In this section, the accuracy of the analytical expressions derived for the elastic constants is assessed by means of comparisons with the results of the refined FE analyses conducted by Taliercio (2014). In that study, a different RVE was used, in which the stagger was modeled. Comparisons are carried out: (i) On a specific masonry arrangement, indicated here as a benchmark case study, and (ii) through parametric investigations, in which different mechanical and geometrical properties of the elementary cell are explored. 3.1. Benchmark case study The numerical application described in (Taliercio, 2014), referring to header bond brickwork, is considered here. The cell is defined by the elastic parameters of the brick Eb = 17100 MPa, νb = 0.15 (and Gb = 7434.78 MPa) and the mortar, Em = 7700 MPa, νm = 0.2 (and Gm = 3208.3 MPa); the geometric parameters of the brick, a = 55 mm and b = 120 mm, and the thickness of the mortar bed and head joints, t = 10 mm. Table 1 compares the equivalent elastic constants, evaluated for the homogenized orthotropic model (HOM) and homogenized isotropic model (HIM), according to the closed-form expressions in Eqs. (6), (7), (17), (18), (25), and (30), with the respective FE results. To emphasize the accuracy of the innovative proposal, comparisons with other analytical results available in the literature, provided by Taliercio (2014), are also reported in Table 1. The percentage differences with respect to the numerical findings are indicated in brackets. With reference to the homogenized orthotropic model, the analytical expressions yield results that exhibit excellent agreement with the FE analyses, despite their simplicity. The errors are less than 1% for the axial and shear moduli, and only slightly higher for the Poisson’ s ratio. When the simpler isotropic model is used, the errors increase up to 5% for the axial and shear moduli, remaining acceptable; in contrast, the errors are higher for the Poisson’ s ratio. The comparisons with the other closed-form expressions demonstrate that, for the selected values of the geometric and

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FE model Literature models Reuss Voigt Salamon Brooks Pande et al. Zucchini & Lourenço Taliercio MoC Taliercio MSE HOM HIM

Ex Eb

Ey Eb

ν yx

Gxy Gb

0.8422

0.8115

0.159

0.7796

0.7891(−6.3% ) 0.8796(+4.5% ) 0.9156(+8.7% ) − 0.8406(−0.2% ) − 0.8384(−0.5% ) 0.8479(+0.7% ) 0.84279(+0.1% ) 0.80256(−4.7% )

0.7891(−2.8% ) 0.8796(+8.4% ) 0.8518(+4.9% ) 0.8136(+0.3% ) 0.8204(+1.1% ) − 0.8123(+0.1% ) 0.8170(+0.7% ) 0.81176(+0.0% ) 0.80256(−1.1% )

0.158(−0.2% ) 0.160(+1.2% ) 0.158(−0.1% ) − 0.155(−2.2% ) − 0.155(−2.2% ) 0.156(−1.8% ) 0.155(−2.5% ) 0.174(+9.7% )

0.7761(−0.4% ) 0.8755(+12.3% ) 0.8315(6.7% ) − 0.7710(−1.1% ) 0.7775(−0.3% ) 0.8194(+5.1% ) 0.7904(+1.4% ) 0.77615(−0.4% ) 0.78590(+0.8% )

elastic parameters, the present analytical results provide the best predictions in terms of the elastic moduli. Moreover, it should be noted that most of the expressions in the literature are complex (some of them very involved), while those proposed here are very simple and accurate. The generally weak anisotropy of the masonry, which emerges from the previous results, suggests that the isotropic model can also be used with strong approximation. Results similar to those obtained here are determined by the Reuss and Voigt bounds (generally used for the macroscopic elastic stiffness of any nonhomogeneous medium), specialized to periodic masonry, and simple formulae describing an equivalent isotropic material can be derived. 3.2. Parametric analyses Several parametric investigations are carried out to assess the accuracy of the homogenized model when the mechanical and geometrical properties of the elementary cell are varied. Comparisons are also conducted with the refined FE solutions, carried out in Taliercio (2014). Firstly, the influence of the material heterogeneity, quantified by the mortar-to-brick material ratios α E and α G , on the elastic constants of the homogenized continuum is investigated. The results are displayed in Fig. 7, where the curves indicate the orthotropic (black lines) and isotropic (yellow lines) models, while the dots denote the numerical results (and the red cross indicates the benchmark case study, αE = 0.45). The orthotropic model is first considered. It is observed that, in the entire examined field, the agreement between the analytical and numerical results is excellent. In contrast, for a small α E , less accurate analytical predictions are obtained for the Poisson’s ratio, which remain satisfactory. When the isotropic model is analyzed, adequate results are obtained, with the exception of the low Poisson’ s ratio at α E (similar behavior can be observed in the Reuss results reported in Taliercio, 2014). Moreover, the influence of the thickness-to-brick height ratio β a on the homogenized elastic constants is investigated, as illustrated in Fig. 8 (again, the red cross indicates the benchmark case study, βa = 0.18). The orthotropic model provides very good results up to a moderately large non-dimensional thickness, viz. βa = 0.18, beyond which the error increases. However, such a range covers the usual technical applications. The decreasing accuracy is probably owing to the fact that, when the mortar thickness is large, the hypothesis of the 1D spring behavior is no longer valid. The isotropic model provides similar qualitative results, although slightly less accurate, with a significant error in the Poisson’ s ratio (in this case, similar behavior can also be observed in the Reuss results presented in Taliercio, 2014).

Fig. 7. Elastic constants of homogenized medium vs mortar-to-brick elastic moduli Ey Gxy Ex ratio: (a) ; (b) ; (c) ν yx ; (d) . Numerical model (dashed black lines), homoEb Eb Gb geneous orthotropic model (solid black lines), and homogeneous isotropic model (yellow lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

4. Structural analyses Structural analyses regarding the in-plane behavior of masonry walls, with a header bond pattern and a standard stagger of half a brick, are carried out. The aim is to investigate the manner in which the (rough) homogenized model, in its orthotropic and isotropic variants, approximates the results of the (fine) nonhomogeneous model. Moreover, an investigation is conducted to detect the dependence of the accuracy of the homogeneous model on the number of bricks, which form a wall of given dimensions. To this end, two case studies are considered, consisting of masonry walls of nearly-equal dimensions, namely 2795 × 1430 × 250 mm3 and 2730 × 1300 × 250 mm3 , respectively. The two systems have the same mechanic characteristics as the benchmark case (Eb = 17100 MPa, νb = 0.15, Em = 7700 MPa, νm =

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Table 2 Equivalent constitutive parameters.

Homogenized orthotropic model Homogenized isotropic model

ν xy

Ex MPa

Ey MPa

–

Gxy MPa

14411 13723

13881 13723

0.149 0.174

5770 5843

Fig. 10. Load case schemes and contour plots of displacement magnitude on deformed configuration.

Fig. 8. Elastic constants of homogenized medium vs joint thickness-to-brick height Gxy Ey Ex ratios: (a) ; (b) ; (c) ν yx ; (d) . Numerical model (dashed black lines), hoEb Eb Gb mogeneous orthotropic model (solid black lines), and homogeneous isotropic model (yellow lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

2. A homogeneous model, in which the same shell elements are used, but all have the same mechanical constants, as provided by the orthotropic law determined previously (namely, Eqs. (6), (7), (17), (18), and (25)), as well the isotropic law (namely Eq. (30)), as indicated in Table 2. The equivalent mass density ϱh for the homogeneous medium is evaluated by averaging ϱb , ϱm on the cell area; that is:

h =

t (b + t ) m + a(t m + b b ) (a + t )(b + t )

(31)

from which h = 1678.1 kg/m3 . Static and dynamic analyses follow for several case studies. Fig. 9. Case studies: (a) DWW; (b) SWW.

0.2), and brick and mortar densities equal to b = 1700 kg/m3 , m = 1600 kg/m3 , respectively, but different brick and mortar joint meshes. That is: (a) One wall, known as the “dense weaving wall” (DWW), is formed by bricks with dimensions of 120 × 55 mm2 and a mortar joint thickness of 10 mm (as in the benchmark case, see Fig. 9(a); (b) a second wall, known as the “sparse weaving wall” (SWW), is constituted by bricks and a thickness of double dimensions, namely bricks of 240 × 110 mm2 and a mortar joint thickness of 20 mm (Fig. 9(b)). It should be noted that the interest in analyzing the SWW, made from bricks of large dimensions, is mainly of a speculative nature. However, in the literature, there exist experimental studies carried out on physical models, in which bricks of similar dimensions have been used (see Riddington, 1984; Cavaleri et al., 2014). For each wall, and by using a commercial code, two FE models are implemented: 1. A non-homogeneous model, in which brick, and mortar bed and head joints are finely modeled by shell elements. Any element is a Cauchy continuum body in plane stress, defined in 2D space and obeying a hyperelastic isotropic linear law. A triangular mesh with a linear displacement field is used. Adherence conditions are ensured at each element boundary. The numerical results refer to a model with 121.0 0 0 elements.

4.1. Static analysis Static analyses are carried out for the DWW and SWW described previously, under typical loading conditions, as illustrated in Fig. 10. In all cases, the wall is constrained on the bottom and free on the three remaining sides. In load case 1, a uniform distribution of horizontal forces is applied along the left vertical side. To simulate the seismic action roughly, the load intensity for a unit of length is taken as proportional to the wall weight for a unit of length, namely ϱh Lhag , where L is the wall width, h is the wall thickness, and ag = 0.26g, in which g is the gravity acceleration. In load case 2, a uniform distribution of vertical forces is applied along a portion of the top L side, with a dimension ; the load magnitude for a unit of length 5 is taken as proportional to the wall weight for a unit of length, according to 200ϱh gHh, where H is the wall height. The results of the analyses are presented in the following, in terms of the displacement and stress curves, along selected lines of the wall. Fig. 10 illustrates the contour plots of the resultant displacements obtained by the fine model. Load case 1 Load case 1 is considered and the DWW is analyzed first. The curves in Fig. 11 indicate the displacements and stresses evaluated L H at sections x = and y = . 2 2

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Fig. 11. Displacements and stresses for DWW in load condition 1: (a) u 2L , y ; (b) σx 2L , y ; (c) τxy 2L , y ; (d) v x, H2 ; (e) σy x, H2 ; (f) τxy x, H2 . Non-homogeneous model (black lines), homogeneous orthotropic model (red lines), and homogeneous isotropic model (yellow lines). Analytical normal stresses at bricks (blue lines) and mortar, excluding (green lines) and including (pink lines) self-stresses. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The results relevant to the non-homogeneous, homogeneous orthotropic, and isotropic models are reported. Moreover, the quota of normal stresses of the brick and mortar, as yielded by the analytical model (Eqs. (8) and (9) according to the normal stress distribution, summed to Eqs. (19) and (20) when the Poisson effect is considered), are plotted. It can be observed that the curves derived from the two homogeneous models are almost coincident, owing to the fact that the wall system under study is weakly orthotropic (see elastic constants in Table 2). Comparing the result with those of the fine model, it is observed that the displacement curves exhibit very strong agreement. Concerning the stress curves, it emerges that the results obtained by the homogeneous model fit those of the non-homogeneous model on average, as the true stress distribution changes rapidly when passing through the bricks and mortar. This result is confirmed by the zoomed-in images in Fig. 12, where the normal stress σ x is represented along a vertical straight line (Fig. 12(a)), and the normal stress σ y along a horizontal straight line (Fig. 12(b)). According to the comparison of the green and pink lines, it appears that the self-stresses are negligible at the mortar bed joints (the curves are superimposed), but significant at the mortar head joints. Remarkably, the analytical mortar stresses approximate the peaks of the non-homogeneous model to a satisfactory extent, as indicated in Fig. 12, despite the fact that the homogeneous model is aimed at describing global, and not local, behavior. When analyzing the results for the SWW (see Fig. 13), an additional observation can be made: the agreement between the re-

Fig. 12. Normal stress distribution between brick (blue lines) and mortar, excluding (green lines) and including (pink lines) self-stresses, along straight lines: (a) x = 2L ; (b) y = H2 . Non-homogeneous model (black lines), homogeneous orthotropic model (red lines), and mortar bed and head joints in gray. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

sults of the homogeneous and non-homogeneous models is strong, as in the DWW. Quite surprisingly, the increased cell dimensions do not appear to affect the homogenization accuracy. Load case 2 The results presented in Figs. 14 and 15 are obtained by analyzing load case 2. Fig. 14 illustrates the displacement and stress

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165



Fig. 13. Displacements and stresses for SWW in load condition 1: (a) u 2L , y ; (b) σx 2L , y ; (c) τxy 2L , y ; (d) v x, H2 ; (e) σy x, H2 ; (f) τxy x, H2 . Non-homogeneous model (black lines), homogeneous orthotropic model (red lines), and homogeneous isotropic model (yellow lines). Analytical normal stresses at bricks (blue lines) and at mortar, excluding (green lines) and including (pink lines) self-stresses. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

H for both the DWW (Fig. 14(a)) 2 and SWW (Fig. 14(b)). Once again, the homogenized model provides very good results. The green and pink lines are superimposed; that is, the self-stress is negligible at the mortar head joints. Remarkably, the analytical mortar stresses approximate the peaks of the non-homogeneous model to a satisfactory extent. Moreover, in this load case, the results indicate that the homogenization efficiency is not sensitive to the cell dimensions (in the range analyzed). Finally, the contour plots of the normal stress σ y , computed by the non-homogeneous (left side) and homogeneous orthotropic (right side) models are compared for both the DWW (Fig. 15(a)) and SWW (Fig. 15(b)). A very satisfactory description is provided by the homogenization. curves along the wall section y =

4.2. Modal dynamic analysis The natural frequencies of a wall provide a “global” measure of the accuracy of a homogenized model. To this end, modal dynamic analysis is performed for the non-homogeneous and homogeneous models of the DWW and SWW. The results relative to the first four natural modes are presented in terms of the natural frequencies, in Table 3, and modal shapes, in Fig. 16. It is observed that the homogenized (orthotropic and isotropic) models effectively approximate the in-plane behavior of this type of system, with a percentage error contained within the range of

0.5% to 2.5%. Moreover, no particular sensitivity of the results to the elementary cell dimension is revealed. 5. Conclusions and perspectives A homogenization procedure for deriving closed-form expressions for the macroscopic elastic constants of in-plane loaded masonry walls with a header/running bond pattern has been described. Firstly, an RVE (cell) was identified and modeled by springs combined in series and in parallel, based on considerations concerning the average equilibrium and compatibility at the interfaces between the bricks and mortar. In one case of shear modulus, it was necessary to resort to two-step homogenization, which, as opposed to most of results in the literature, is insensitive to the order of the steps. Thereafter, the elastic constants of the masonry were determined by equating the stiffnesses of the spring systems to those of a homogeneous and orthotropic cell. As a further simplified model, an equivalent isotropic medium was introduced, tailored at approximating the determined orthotropic model to the greatest possible extent. The spring models also allowed for determining the stress in each component, once the stresses acting at the cell boundary had been determined. In particular, the existence of a self-stress state, triggered by the Poisson effect, was highlighted. The accuracy of the closed-form expressions for the constants was assessed by means of comparisons with finite element

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Fig. 14. Displacements v x, H2 and stresses σy x, H2 , τxy x, H2 in load condition 2 for: (a) DWW; (b) SWW. Non-homogeneous model (black lines), homogeneous orthotropic model (red lines), and homogeneous isotropic model (yellow lines). Analytical normal stresses at bricks (blue lines) and at mortar, excluding (green lines) and including (pink lines) self-stresses. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 15. Contour plots of normal stress σ y in load condition 2 for non-homogeneous (left side) and homogeneous (right side) models of: (a) DWW; (b) SWW.

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Table 3 Comparison between modal frequencies (in [Hz]) of homogeneous vs non-homogeneous models for: (a) DWW; (b) SWW. Percentage errors in brackets.

(a) Mode

(b) Mode

Non-homogeneous model

HOM

HIM

1 2 3 4

269.36 505.87 543.88 551.6

267.01 (−0.87% ) 499.07 (−1.34% ) 530.25 (−2.51% ) 543.9 (−1.4% )

269.28 499.29 538.56 549.58

1 2 3 4

300.14 544.19 581.3 601.86

298.09 (−0.68% ) 538.38 (−1.07% ) 566.17 (−2.60% ) 594.08 (−1.29% )

300.56 (0.14%) 535.7 (−1.56% ) 578.73 (−0.44% ) 600.36 (−0.25% )

(−0.03% ) (−1.3% ) (−0.98% ) (−0.37% )

Fig. 17. Running/header bond DWW with stagger: (a) b/2; (b) b/3; (c) b/10.

Fig. 16. Modal shapes.

analyses and other approximated formulae provided Taliercio (2014). The following conclusion can be drawn.

by

1. As opposed to most models in the literature, in which only some of the macroscopic constants of an orthotropic medium are defined, in this study, closed-form expressions have been derived for all of them, satisfying the symmetry of the flexibility matrix. 2. The proposed approach matches the known macroscopic linear elastic laws known with higher accuracy. Moreover, it provides closed-form expressions that are the simplest formulae available to a significant extent. The second part of the work was aimed at validating the global efficiency of the homogenized models in structural analyses, also addressing the partition of stresses between the brick and mortar. Static and dynamic case studies were carried out on masonry wall systems. Non-homogeneous and homogenized models were analyzed and compared. Two systems, with dense or sparse weaving walls, were considered to test the efficiency of the procedure in relation to the cell dimension. The analyses led to the following conclusions. 1. The homogeneous model can accurately describe the local displacement field. 2. The homogeneous model fits the average stress fields of the non-homogeneous model strongly, where rapid changes occur across the constituent boundaries. 3. When the average stress is partitioned between the brick and mortar, more satisfactory results are provided. That is, the stress peaks occurring at the mortar are captured, despite the

fact that the homogeneous model is aimed at describing global, and not local, behavior. 4. The error does not appear to be sensitive to the cell dimension, at least in the range considered. 5. The modal dynamic analyses carried out demonstrate very strong agreement between the natural frequencies of the homogeneous and non-homogeneous models. Again, no sensitivity of the results to the cell dimension is apparent. 6. According to the philosophy of homogenization, the analytical model can describe on average all the cell behaviors, and therefore, of course, cannot describe local aspects (such as the brickmortar interface behavior). In spite of this, the refined FE analyses, in which such effects are captured instead, demonstrated that the analytical model provides very satisfactory results. As a possible development of the present current, the in-series and in-parallel spring models can be extended to the nonlinear field, in order to study the damage Di Nino et al. (2017) and viscosity Taliercio (2014). The concept consists of substituting the linear elastic device with nonlinear devices that have: (a) Softening elastic-brittle constitutive laws, or (b) visco-elastic laws (rheological devices). Of course, the proposed approach holds as long as the initial topology of the layout is not altered (for example, by cracks). This matter will be the object of future works. This paper has focused on the running/header bond, one of the most common layouts. However, the approach can be extended to different bonds. The concept consists of proceeding in two steps: (a) Firstly, use the RVE in Fig. 2 (which does not depend on the layout) to homogenize all types of heterogeneous cells in the wall (usually with different geometries), each made of just one brick and half a mortar bed and head joint; (b) secondly, homogenize the different cells by in-series/in-parallel springs, according to the specific layout. Note that in the case of the running/header bond

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Fig. 18. Horizontal displacement and normal stress for DWW in load condition 1: (a) u; (b) σ x . Header bond patterns with standard stagger of b/2 (black markers), stagger of b/3 (pink markers), and stagger of b/10 (gray markers). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

pattern, the cells are all identical, and therefore step (b) is trivial, so the global properties are independent of the stagger. For different patterns, substantial work has to be conducted on step (b), which will be the object of future works. Overall, the proposed homogeneous model accurately describes the linear elastic in-plane behavior of non-homogeneous masonry walls. Although the application field is limited, it is believed that it is useful for an improved understanding of the elastic interactions among the constituents, and can aid structural designers in conventional elastic calculations; moreover, it is suitable for extension to a larger class of problems. Appendix A. Stagger influence The homogenization procedure proposed in this paper is based on the definition of a cell, which does not account for the masonry pattern (Fig. 2). At first glance, it appears that the cell is not sufficiently representative of the geometric (topological) properties of the masonry, as it lacks information that is perceived as important (see, for example, the criticisms in Zucchini and Lourenço, 2002; Lourenço et al., 2007 regarding the two-step homogenization carried out in (Pande et al., 1989; Pietruszczak and Niu, 1992; Anthoine, 1995), which does not consider the stagger). Nevertheless, it is known that this simplified homogenization approach has been used by several authors, and performs very satisfactorily in the case of linear elastic analysis. However, the numerical results of Section 4, in which FE analyses were performed on nonhomogeneous header bond walls with a (standard) stagger of half a brick, (s = b/2 ), were found to be in excellent agreement with numerical results of homogeneous walls. This suggests the formulation of a conjecture, namely, when restricted to the linear elastic field, the wall response is (almost) independent of the stagger. To determine the validity of the conjecture, a numerical analysis was developed and carried out on non-homogeneous walls of the DWW type with different staggers, with s = b/3 (Fig. 17(b)) and s = b/10 (Fig. 17(c)). When considering loading condition 1, for which the results of Section 4 hold when s = b/2, the horizontal displacement u and stress σ x were evaluated at the centroids of the blocks close to the middle vertical axis (see Fig. 17(a)), and the results were compared with those relevant to s = b/2. The results are displayed in Fig. 18, which indicates small differences among the three cases: the maximum error is approximately 5%, which is reached for the

Table 4 Horizontal displacement and normal stress at top of wall (measured at point of coordinate y = 1.39 m, close to middle vertical axis) vs stagger.

Stagger

b/2 (standard) b/3 b/10

u

σx

8.46 · 10−7 8.60 · 10−7 8.84 · 10−7

−5673 −5847 −5889

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