DISCRETE ELEMENT MODEL FOR IN-PLANE LOADED VISCOELASTIC MASONRY

Journal for Multiscale Computational Engineering, 12 (2): 155175 (2014)

DISCRETE ELEMENTMODEL FOR IN-PLANE LOADEDVISCOELASTIC MASONRY

Daniele Baraldi & Antonella Cecchi

Dipartimento di Architettura Costruzione Conservazione, Universita` IUAVdi Venezia, Italy

Address all correspondence to Daniele Baraldi, E-mail: [email protected]

A viscoelastic constitutive model is proposed to evaluate the evolution in time of historical masonry behavior. Masonrystructures may be subject, over time, to damage due to creep phenomena, accompanied by a consequent redistributionof stresses and strains. Two models are presented and compared. A discrete element model and a continuous modelbased on analytical homogenization procedures. Both models are based on the following assumptions: (i) the structure iscomposed of rigid blocks; (ii) the time dependence of masonry behavior is concentrated in mortar joints, modelled as vis-coelastic interfaces. The rigid block hypothesis is particularly suitable for historical masonry, in which stone blocks maybe assumed as rigid bodies; the hypothesis of viscoelastic mortar is based on the observation that nonlinear phenomenamay be concentrated in mortar joints. The continuum homogenized model provides, in an analytical form, constitutiveequivalent viscous functions; the discrete model describes masonry as a rigid skeleton such as to evaluate both its globaland local behavior. A parametric analysis is carried out to investigate the effect of (i) mortar-to-brick thickness ratio;(ii) masonry texture (running versus header bond); and (iii) size of heterogeneity (block dimensions) with respect topanel dimensions. Elementary cases are proposed to directly compare constitutive functions of continuum and discretemodels. In addition, a meaningful case is proposed: a masonry panel in which the principal stresses are both of compres-sion and the no-tension assumption may therefore be discounted. A further investigation pointed out the sensitivity toheterogeneity size such as to verify model reliability and applicability field.

KEYWORDS: discrete model, masonry-like materials, viscoelastic, homogenization

1. INTRODUCTION

A wide set of models exists in the literature apt to investigate masonry behavior. The difficulty in modelling masonryis due to its heterogeneity and its nonlinear behavior.

Under service loads, strain usually stabilizes after a certain time and the masonry components, bricks and mortar,can be assumed to have a linear viscoelastic behavior. Because of the difference in viscoelastic properties of thecomponents, however, the long-term stresses and strains over any masonry element can differ, even considerably,from the short-term ones.

Several rheological models have been proposed by Choi et al. (2007) based on experimental tests, studies that havefurther emphasized the importance of the issue. An attempt to predict the creep coefficients of masonry accordingto the properties of the individual constituents was made by Brooks and Abdullah (1986) and Brooks (1990). Somerecent damage models have been successfully applied to the analysis of masonry. The definition of damage by suitablenonscalar criteria and the introduction of the orthotropy typical of the masonry structures into the model still remaintroublesome issues (Luciano and Sacco, 1997; Berto et al., 2004; Lourenco and Rots, 1997; Taliercio, 1991; Papa andTaliercio, 2005).

As far as creep is concerned, it should be noted that the existing technical literature relies heavily on empiricaland semiempirical approaches derived from a limited number of experimental results (Choi et al., 2007; Binda et al.,1991; Brooks and Abdullah, 1986; Brooks, 1990), with the result that many of the proposed models seem to featureunacceptable levels of accuracy in the validation of their experimental results. Moreover, these models also sufferfrom a lack of generality, having been derived for specific situations of creep in historical masonry.

15431649/14/$35.00 c 2014 by Begell House, Inc. 155

156 Baraldi & Cecchi

Here, a hypothesis of rigid blocks connected by interfaces (mortar joints) among blocks is assumed. In otherwords, masonry has been modelled as a skeleton in which the interactions between the rigid blocks are representedby forces and moments which depend on their relative displacements and rotations.

Two models have been proposed. The former is a discrete element model and the latter is a continuous model. Thediscrete model is based on the implementation, in viscoelastic field, of a numerical model already formulated in thecase of regular periodic running bond masonry in linear elastic field (Cecchi and Sab, 2004). The blocks which formthe masonry wall are modelled as rigid bodies connected by viscoelastic interfaces (mortar joints).

The continuous model is based on the homogenization procedure. Explicit equations for constitutive functions areobtained under assumptions of rigid blocks and mortar joints modelled as interfaces (Cecchi and Tralli, 2012); henceboth models start from the same assumptions. But as is well known, the homogenization model describes masonry asan equivalent continuum, whereas the discrete model units and mortar joints are distinctly described.

An extensive experimental campaign has been carried out to analyze the sensitivity of masonry global behavior totexture (running bond and header bond) and the size of the mortar joints (thin joints and thick joints). A comparisonbetween the continuous homogenized and discrete element model is crucial in order both to calibrate the reliability ofthe two models and their field of applicability, and to evaluate the role of the bond on the structural masonry behaviorand sensitivity to mortar thickness in the time masonry behavior. In addition, as is well known, brick size (heterogene-ity dimension) is significant if compared with the wavelength of the macroproblem. For this reason experimentationdevoted to this topic has been carried out by assuming two different values of panel dimensions.

2. DISCRETE MODEL

A standard running bond periodic masonry is considered. Block dimensions are a (height), b (width), and s (thickness).Let yi;j be the position of the generic block Bi;j (Fig. 1) in the 3D Euclidean space. It is clear that j can actuallytake arbitrary values, while i is such that i+ j is even. Assuming the rigid block hypothesis, the displacement of eachblock is a rigid body motion referred to the motion of its center and defined by the following expression (Cecchi andSab, 2004):

u(t) = ui;j(t) +i;j(t) (y yi;j); 8y 2 Bi;j (1)where ui;;j(t) and i;;j(t) are the translation and rotation vectors of block Bi;;j , respectively, and yi;;j is the positionvector of the center of the block.

Considering the regularity of the masonry structure, the Bi;;j block interacts with the Bi+k1;j+k2 block by meansof six k1;k2 joints as follows:

1. If k1, k2 = 1, then k1;k2 is a horizontal interface;

FIG. 1: Masonry structure.

Journal for Multiscale Computational Engineering

Discrete Element Model for in-Plane Loaded Viscoelastic Masonry 157

2. If k1 = 2 and k2 = 0, then k1;k2 is a vertical interface. For example, the interfaces of the B0;0 block are

1;1 =

0@ b=2 y1 0y2 = a=2s=2 y3 s=2

1A ; +1;1 =0@ 0 y1 b=2y2 = a=2

s=2 y3 s=2

1A1;+1 =

0@ b=2 y1 0y2 = a=2s=2 y3 s=2

1A ; +1;+1 =0@ 0 y1 b=2y2 = a=2

s=2 y3 s=2

1A2;0 =

0@ y1 = b=2a=2 y2 a=2s=2 y3 s=2

1A ; 2;0 =0@ y1 = b=2a=2 y2 a=2

s=2 y3 s=2

1A :(2)

For mortar, a linear viscoelastic nonaging material subject to a uniaxial stress history starting at time t = 0, accordingto Boltzmanns principle of superposition, is characterized by a strain at any time t > 0 given by the followingexpression (Park and Schapery, 1999):

"(t) =

Z t0

J(t ) _() d; (3)where J is defined as creep compliance. Stress histories, including jumps, can be taken into account in Eq. (3),providing that time derivatives are intended in the distribution. Then, if a material is subject to a prescribed uniaxialstrain history, the stressstrain relation may be written as

(t) =

Z t0

E(t ) _"() d; (4)

where E is the relaxation function.An overview of the mechanical models which may be adopted for describing the viscoelastic behavior of materials

can be found in the works of Choi et al. (2007) and Taliercio (1991). In the present work, the so-called generalizedMaxwell rheological model (Kaliske and Rothert, 1997) is adopted and the following expression represents the relax-ation function:

E(t) = E0 +nXi=0

[1 Ei exp(t=i)] = E1 +nXi=0

Ei exp(t=i); (5)

which can be alternatively written as

E(t) = E0

1

nXi=0

ei [1 exp(t=i)]!: (6)

where E0 and E1 are the instantaneous and equilibrium (or delayed) moduli, respectively; Ei represents the relax-ation moduli; and i are the relaxation times, and then ei = Ei=E0 represents the dimensionless relaxation moduli.The series in Eq. (5) is usually referred to as Prony (or Dirichlet) series.

Hence if the mortar joint is modelled as an interfacesuch a problem has been studied in linear elasticity byKlarbring (1991) by means of perturbative techniquesthe deformation between two blocks may be written as afunction of the [[u]] displacement jump. The constitutive prescription for the contact is a linear viscoelastic relationbetween the tractions on the block surfaces and the jump of the displacement field:

(t)n =

tZ0

K(t )[[u()]] d; on k1;k2 ; (7)

where (t) is the stress tensor, n is the normal to k1;k2 , [[u(t)]] is the jump of the displacement field at k1;k2 ; andK(t) is given by

Kij(t) =1

eamiklj(t) nknl; (8)

Volume 12, Number 2, 2014


where am(t) is the viscoelastic stiffness tensor of the mortar and e is the thickness of the real joint. It will be assumedin the sequel that the mortar is isotropic, and then Eq. (8) becomes

K(t) =1

e

Em(t)

2(1 + m)

I+

1

1 2m(n n)

; (9)

where Em(t) and m are the relaxation function and the Poisson ratios of the mortar (Klarbring, 1991; Avila-Pozoset al., 1999). Note that the K tensor has a diagonal form in this case. Only the mortar Youngs modulus is assumed tobe subject to viscosity, as it is shown in Eqs. (5) and (6).

Considering the in-plane case, the vector of degrees of freedom is u = (ui;j1 ; ui;j2 ;

i;j3 )

T, where u1 is the in-plane

horizontal displacement, u2 is the in-plane vertical displacement, and 3 is the rotation respect to the y3 axis. Theinteractions between the blocks through the interfaces are represented by elastic forces f i;j and couple ci;j that mustbe found. According to Cecchi and Sab (2004), the following notations are introduced:

k1;k21 = ui+k1;j+k21 ui;j1 + k2a

i+k1;j+k23 +

i;j3

2; (10)

k1;k22 = ui+k1;j+k22 ui;j2 k1

b

2

i+k1;j+k23 i;j32

; (11)

k1;k23 =

i+k1;j+k23 i;j3 : (12)

2.1 Horizontal Interfaces (k1 = 1; k2 = 1)Let e be the thickness of the real horizontal joint, Sh = b s=2 the area of the horizontal interface, and Ih3 = b3s=96 itsinertia with respect to the y3 axis (that is orthogonal to middle 2D plane of masonry). By denoting with [[u]] the jumpof the displacement field at the k1;k2 interface, the following expression of the horizontal interface elastic energymay be obtained:

W k1;k2 =1

2

Zk1;k2

[[u]]K[[u]] =G(t)

2e

Sh

k1;k21

2+K(t)

2e

Sh

k1;k22

2+ Ih3

k1;k23

2(13)

where K(t) and G(t) are, respectively, the bulk and shear relaxation moduli of vertical and horizontal interfaces andare given by the following expressions:

K(t) =Em(t)

(1 + m)(1 2m) ; G(t) =Em(t)

2(1 + m): (14)

In the present work, the viscoelastic behavior of the bed joints is assumed coincident to one of the head joints and isdefined by the relaxation function Em(t).

The forces and the moment that the Bi+k1;j+k2 block applies to the Bi;j block are

(f1)k1;k2i;j =

@W k1;k2

@ui;j1=

G(t)

eSh

k1;k21 (15)

(f2)k1;k2i;j =

@W k1;k2

@ui;j2=

K(t)

eSh

k1;k22 (16)

(c)k1;k2i;j =

@W k1;k2

@i;j3=

K(t)

eIh3

k1;k23

G(t)

2eShk2a

k1;k21 +

K(t)

4eShk1b

k1;k22 (17)



2.2 Vertical Interfaces (k1 = 2; k2 = 0)Let e be the thickness of the real vertical joint; Sv = a s the area of the vertical interface, Iv3 = a3s=12 its inertiawith respect to the y3 axis. The following expression of the vertical interface elastic energy may be obtained:

W k1;k2 =1

2

Zk1;k2

[[u]]K[[u]] =G(t)

2e

hSv(

k1;k22 )

2i+K(t)

2e

hSv(

k1;k21 )

2 + Iv3(k1;k23 )

2i

(18)

The forces and the moment, for the in-plane case, that the Bi+k1;j+k2 block applies to the Bi;j block are

(f1)k1;k2i;j =

@W k1;k2

@ui;j1=

K(t)

eSv

k1;k21 (19)

(f2)k1;k2i;j =

@W k1;k2

@ui;j2=

G(t)

eSv

k1;k22 (20)

(c)k1;k2i;j =

@W k1;k2

@i;j3=

K(t)

eIv3

k1;k23 +

K(t)

4eSvk1b

k1;k22 (21)

At present, one can easily check that the in-plane elastic actions Felastic may be written

Felastic = @W=@u = K u = Fext: (22)

Here W is the total elastic energy, Fext is the vector of the applied in-plane actions, and K is the in-plane stiffnessmatrix. Further details are reported in the Appendix.

3. HOMOGENIZED MODEL FOR RIGID BLOCKS CONNECTED BY VISCOELASTIC INTERFACES

The homogenization procedure is based on the same geometry of the discrete system shown in Fig. 1, where thetexture pattern may represent a running bond or a header bond brickwork; moreover, the block Bi;;j together with thesix surrounding blocks forms a representative volume element (RVE).

The displacement of the block Bi;;j is the rigid body motion defined by Eq. (1). The constitutive law for anyinterface between adjoining blocks k1;k2 is supposed to be a linear viscoelastic relationship between the tractionsover the block surfaces and the jump in displacement [[u(t)]] across k1;k2 [Eq. (7)].

Approximate expressions for the macroscopic creep coefficients can be obtained using suitable kinematic andstatic fields over any RVE of masonry. From here onwards, macroscopic variables are defined as the volume averagesover any RVE of the corresponding microscopic variables. Let E(t) be the macroscopic in-plane strain tensor in thehomogenized equivalent medium. The set KC [E(t)] of the macroscopic displacements and rotations [U(t);(t)]kinematically compatible with E(t) is introduced:

KC [E(t)] =[U(t);(t)] ; ui;j(t) = E(t) yi;j + v(t)i;j ; i;j = !i;j ; [v(t);!(t)] 2 L2 : (23)

Here (v(t);!(t)) denote translations and rotations, respectively, defining any in-plane rigid body motion of blockBi;j . Similarly, let (t) be the macroscopic in-plane stress tensor. The set SC ((t)) of the macroscopic forces andcouples (F(t);C(t)) statically admissible with any macroscopic stress (t) is defined as

SC [(t)] =n[F(t);C(t)] ; f i;jk1;k2 = n

i;jk1;k2

and Ci;jk1;k2 = ci;jk1;k2

o: (24)

Here f i;jk1;k2 are interface tractions and ci;jk1;k2

couples on block Bi;j .



According to the kinematically admissible solution and following Cecchi and Sab (2004), approximate expressionsfor the homogenized relaxation coefficients, denoted by AFijkl, can be obtained and read:

AF1111 =4K 0v

eh

a+ eh+

b+ ev

a+ ehK 00h

ev

a+ eh

4ev

b+ eveh

a+ eh

; (25)

AF2222 =K 0heh

a+ eh

; (26)

AF1212 = K00h

K 00veh

b+ ev+

b+ ev

a+ ehK 0h

ev

b+ ev

eh

a+ eh

K 0h

ev

b+ ev+

4a+ eh

b+ evK 00h

eh

b+ ev+

4(a+ eh)2

(b+ ev)2K 00v

ev

b+ ev

; (27)

AF1212 = K00h

K 00veh

b+ ev+

b+ ev

a+ ehK 0h

ev

b+ ev

eh

a+ eh

K 0h

ev

b+ ev+

4a+ eh

b+ evK 00h

eh

b+ ev+

4(a+ eh)2

(b+ ev)2K 00v

ev

b+ ev

; (28)where K(t) and G(t) are, respectively, the bulk and shear relaxation moduli of vertical and horizontal interfaces[Eq. (14)]. The same coefficients were determined by Cecchi and Tralli (2012), adequately taking into account thethickness of the mortar joints by substituting e=b and e=a with e=(b+ e) and e=(a+ e), respectively. However in thepresent work, Eqs. (25)(28) are adopted without any modification; in this way the same definition adopted for theinterfaces in the discrete model is used.

In a dual manner, according to the statically admissible solution, the approximated macroscopic creep coefficients,denoted by CFijkl(t), are given by

CF1111(t) =4e

a

e

bJ 0v(t)J

00h (t)

4e

aJ 00h (t) +

b

aJ 0v(t)

e

a

; (29)

CF2222(t) =e

aJ 0h(t); (30)

CF1212(t) =

e

a

h4e

bJ 0h(t)J

00v (t) + 4J

0h(t)J

00h (t)

e

a+ J 00h (t)J

00v (t)

e

a

i4J 0h(t)

e

a+ J 00h (t)

b

a

e

a

; (31)

CF1122(t) = 0: (32)Here, J 0h(t) = Jmh (t), J 0v(t) = Jmv (t), J 00h (t) = 2Jmh (t)(1 + mh ), and J 00v (t) = 2Jmv (t)(1 + mv ), Jmh (t) and Jmv (t)being the creep functions of the bed and head joints, respectively.

4. NUMERICAL EXAMPLES

A numerical experimentation has been carried out with twofold aim: (i) to evaluate the sensitivity of the discretemodel to several parameters and its reliability to take into account the in-time masonry behavior; and (ii) to evaluatethe sensitivity of the discrete and homogenized model to the size of heterogeneity (block dimensions) by reference tothe size of the panel.



4.1 Sensitivity to Masonry Pattern

Two different types of masonry panel textures are considered. Both types are composed by UNI bricks (25055120mm), with bed and head mortar joints having the same thickness e. The first panel-type texture [Fig. 2(a)] is charac-terized by a running bond pattern with six blocks in horizontal direction, whereas the second one is characterized bya header bond pattern [Fig. 2(b)] with 12 blocks in horizontal direction, in order to represent the same panel length L.The panel is also characterized by 15 courses in vertical direction.

Two experimentations have been proposed: (i) for different mortar joint thicknesses, 2, 10, and 15 mm representingthin, standard, and thick joints, respectively; and (ii) for running and header bond patterns [Figs. 2(a) and 2(b),respectively].

Then six different panels are proposed (Table 1) and the corresponding geometric parameters are resumed in thefollowing table. The mechanical characteristics of the mortar are Em(t = 0) = 7700 MPa and m = 0.2, according toexperimental data proposed by Brooks (1990). The linearviscous function of mortar is assumed as defined in Eqs. (5)or (6), and Table 2 shows several values of Em(t) adopted in the following examples.

4.1.1 Elementary Cases

Panel Subject to a Vertical Compressive Load.In the first example (ex. 1) each panel is simply supported at the baseand is subject to a vertical load distribution q1 applied at the top (Fig. 3). It must be noted that in the proposed discretemodel, loads are applied to the centers of the blocks of the top course (y2 = H a=2) and restraints are applied to thecenters of the blocks of the bottom course (y1 ! y2 = a=2).

Figure 4 shows the vertical displacement of the blocks of the two different pattern types having e = 10 mm, subjectto the distributed vertical load. It is clear that each course of blocks is characterized by an almost constant verticaldisplacement, and then the displacement of a generic block of the top course may be taken as reference displacement

(a) (b)FIG. 2: Masonry panel textures considered for the numerical examples. Running bond pattern (a), header bond pattern(b).

TABLE 1: Case studies considered varying the texture of the paneland mortar joint thickness

Case Panel type e [mm] L [mm] H [mm] s [mm]1 1 2 1510 853 1202 1 10 1550 965 1203 1 15 1575 1035 1204 2 2 1462 853 2505 2 10 1550 965 2506 2 15 1605 1035 250



TABLE 2: Elastic moduli of mortar adopted in the numerical examplest [days] 0 20 50 100 150 200 250 300

Em(t) [MPa] 7700 2309 2208 2062 1944 1847 1769 1705

(a) (b)FIG. 3: Panel subject to a vertical distributed load applied at the top: (a) continuous model and (b) discrete model.

(a) (b)FIG. 4: Map of the vertical displacement of the blocks for the panel with two texture patterns subject to a verticalcompressive load at t = 0: (a) case 2 and (b) case 5.

for each case. Figure 5 shows the vertical displacement at the top of the panel with respect to time and varyingthickness of the mortar joints for the running bond pattern. Numerical results are represented by continuous lineswith triangles. As expected, vertical displacements increase due to the relaxation of the elastic modulus of the mortar.Moreover, Fig. 5 shows that vertical displacements are larger if the thickness of mortar joints increases from 2 to15 mm.

It must be noted that numerical results turn out to be quite coincident for both texture pattern types; In fact, in thiscase, the behavior of the panel depends only on the horizontal mortar joints, which are the same for both patterns.Then, each discrete model behaves similarly to a multilayer model. Results obtained with the discrete model, varyingthe elastic modulus of the mortar over time, may be compared with the analytic solution of a homogeneous platein-plane stress compressed vertically:

u2(y1; H; t) =Q1H

AF2222(t)(Ls)=

q1(Ls)H

AF2222(t)(Ls)=

q1H

AF2222(t); (33)



FIG. 5: Vertical displacement at the top of the running bond panel subject to a vertical distributed load. Continuouslines with triangles for the results obtained with the discrete model; dashed lines for the analytic solution.

whereAF2222(t) is given by Eq. (26). Analytic solutions are added to Fig. 5 with dashed lines for each joint thickness. Ifthe joint thickness is thin (e = 2 mm), numerical results appear closer to the analytic solution; however, errors betweennumerical results and analytic solutions are the same for the three thickness cases and are close to 7%. Moreover, thediscrete model is more rigid than the homogeneous one.

In the equation above, Q1 represents the total vertical load; however, if the same pressure q1 = Q1=(Ls) isapplied at the top of each panel type, the expression shows clearly that vertical displacement of the homogeneousmodel depends only on AF2222(t), which is the same for both patterns considered. Then the corresponding analyticsolutions are coincident (Fig. 6, dashed lines), whereas numerical solutions for the two pattern types are very closebut not coincident (Fig. 6, lines with triangles), as it has been denoted in Fig. 5.

Panel Subject to a Horizontal Compressive Load.In the second example (ex. 2), the panel is simply supportedat the left edge (y1 = 0) and it is subject to a compressive distributed horizontal load q2 applied at the right edge (y1 =L). In the discrete model, loads are applied at the centers of the even blocks at the right edge of the panel [Fig. 7(b)],whereas restraints are applied at the centers of the even blocks at the left edge of the panel [Fig. 7(a)].

FIG. 6: Vertical displacement at the top of both running and header bond panels subject to a vertical distributed load.Continuous lines with triangles for the results obtained with the discrete model; dashed lines for the analytic solution.



(a) (b)

FIG. 7: Panel subject to a horizontal distributed load applied at the right edge; (a) continuous model and (b) discretemodel.

Figure 8 shows the horizontal displacement of the two different pattern types having e = 10 mm at t = 0, subjectto the distributed horizontal load. In this case the displacement of the even blocks along the right edges are almostconstant and the corresponding value may be assumed as reference result. Different from the previous case, thebehavior of the panels under the lateral load is strictly dependent on the vertical joints, which are two times largerwith the header bond texture pattern with respect to the running bond texture pattern. Then the second texture patterntype turns out to be more deformable with respect to the first one, as shown in Fig. 9 by continuous lines with trianglesfor the horizontal displacement of the right edge of the panel. Moreover, displacements over time follow the relaxationof the elastic modulus of the mortar.

Results obtained with the discrete model, varying the elastic modulus of the mortar over time, are compared withthe analytic solution of a homogeneous plate compressed horizontally:

u1(L; y2; t) =Q2L

AF1111(t)(H s)=

q2(H s)L

AF1111(t)(H s)=

q2L

AF1111(t); (34)

where AF1111(t) is given by Eq. (25). The expression above clearly shows that horizontal displacement depends onlyon AF1111(t), which is smaller for the panel with a header bond texture pattern, characterized by a = 120 mm, with

(a) (b)FIG. 8: Map of the horizontal displacement of the blocks for the panel with two texture patterns subject to a horizontaldistributed load at t = 0: (a) case 2 and (b) case 5.



FIG. 9: Horizontal displacement at right edge of a panel subject to a horizontal distributed compressive load. Con-tinuous lines with triangles for the results obtained with the discrete model; dashed lines for the analytic solution.

respect to the one with a running bond texture, having a = 250 mm. Then the numerical results are in quite goodagreement with the analytic solutions, with errors close to 12% for the running bond pattern and close to 7% for thehead bond pattern. In Fig. 9 analytic solutions have been added with dashed lines, and similar to the previous case,the discrete model is more rigid than the homogeneous one.

Panel Subject to a Horizontal Shear Load.In the third example (Fig. 10), a horizontal distributed shear load q3applied at the top course of the panel is considered (y2 = H a=2) for the discrete model. Horizontal displacementsare constrained at the bottom of each panel (y2 = a=2), and vertical displacements are constrained at the centers ofthe even blocks along the lateral edges (y1 = 0, y1 = L).

Figure 11 shows the horizontal displacements due to the shear load at t = 0 for the two different texture patternsalready considered in the previous examples. In this case the horizontal displacement of each course of blocks isalmost constant for both texture patterns; then the value assumed at the top course may be considered as a referencesolution for the discrete model. Figure 11 also shows that the horizontal displacements of the two cases are quitesimilar if the same distributed load q3 is applied. This aspect is also found if the corresponding analytic solution isconsidered:

(a) (b)FIG. 10: Panel subject to a horizontal distributed shear load applied at the top; (a) continuous model and (b) discretemodel.



(a) (b)FIG. 11: Map of the horizontal displacement of the blocks for the panel with two texture patterns subject to ahorizontal shear load at t = 0: (a) case 2 and (b) case 5.

u1(y1; H; t) =Q3H

AF1212(t)(Ls)=

q3(Ls)H

AF1212(t)(Ls)=

q3H

AF1212(t); (35)

The expression shows that the horizontal displacement depends on the shear coefficient AF1212(t) [Eq. (28)] of thehomogenized model, which is quite similar for both texture patterns considered. Numerical results are quite smallerthan analytic solutions (Fig. 12), with errors close to 7%. Similar to the first example, the effect of vertical mortarjoints is not important and both pattern types present similar behavior.

Panel Subject to a Vertical Shear Load.Finally, a vertical distributed shear load q4 applied at the right edgeof the panel is considered. Vertical displacements are forbidden at the centers of even blocks along the left edge,whereas horizontal displacements are restrained at the top and bottom block courses (Fig. 13). As in the previouscase, by applying the same distributed force to both texture patterns, vertical displacements turn out to be quite similar(Figs. 14 and 15) and the corresponding analytic solutions depend on the elastic parameter AF1212(t) [Eq. (28)]:

FIG. 12: Horizontal displacement at the top of a panel subject to a horizontal distributed shear load. Continuous lineswith triangles for the results obtained with the discrete model; dashed lines for the analytic solution.



(a) (b)FIG. 13: Panel subject to a vertical distributed shear load applied at the right edge: (a) continuous model and (b) dis-crete model.

(a) (b)FIG. 14: Map of the vertical displacement of the blocks for the panel with two texture patterns subject to a verticalshear load at t = 0: (a) case 2 and (b) case 5.

FIG. 15: Vertical displacement at the right edge of a panel subject to a vertical distributed shear load. Continuouslines with triangles for the results obtained with the discrete model; dashed lines for the analytic solution.



u2(L; y2; t) =Q4L

AF1212(t)(H s)=

q4(H s)L

AF1212(t)(H s)=

q4L

AF1212(t): (36)

4.1.2 A Meaningful CaseThe six cases previously described are analyzed when the panel is simply supported at the basethe central node ofthe base is hinged so as to prevent rigid motionand subject to three loads: a vertical distributed load at the top q1for y2 = H , a horizontal distributed load at two lateral edges q2 for y1 = 0 and y1 = L, and a concentrated force F3 atthe top for y1 = L/2 and y2 = H [Fig. 16(a)]. It is noteworthy that under these load conditions, the principal stressesare both due to compression; hence the no-tension behavior of masonry may be discounted.

In this case an analytical solution does not exist; hence a standard 2D finite element (FE) model is introduced forstudying the behavior of the homogenized panel. Then, moduli in Eqs. (25)(28) are adopted as viscoelastic propertiesof the plane FEs. Similar to the previous examples, the discrete model is characterized by load and restraints applied tothe centers of the blocks [Fig. 16(b)], whereas in the 2D FE model, loads and restraints are applied along the edges ofthe panel; Moreover, in the discrete model, the hinge that prevents rigid motions is applied to the two blocks close tothe midpoint of the base course, whereas the concentrated force F3 is applied to the two blocks close to the midpointof the top course [Fig. 16(b)]. Then, some differences between the two models are expected, especially for the runningbond pattern, which is characterized by a few blocks along the y1 direction.

The models are applied for t = 0 and t = 300 days, respectively. Two load combinations are considered for thisnumerical example (Cecchi and Tralli, 2012):

1. Combination 1 (CMB1): panel subject to distributed loads q1 and q2.2. Combination 2 (CMB2): panel subject to distributed loads q1, q2 and concentrated force F3 (Fig. 16).For combination 1, an analytical solution exists. In the following, the "22 strain distribution is evaluated along two

panel sections AA and BB (Fig. 16) placed respectively at y2 = H/3 and 2H/3, in order to evaluate the diffusionof the concentrated force F3 into the discrete and continuous models.

Figures 17(a) and 17(b) show the vertical displacement of the two different pattern types having e = 10 mm att = 0, subject to the loads in combination 2. Figure 17(b) shows clearly that vertical displacements for the panel witha head bond pattern are more concentrated close to the blocks where F3 is applied.

Figures 18(a)18(d) show the "22 strain distribution for the running bond pattern under both load combinations,at t = 0 and 300 days for thin and normal mortar joints, respectively. Continuous lines with triangles represent resultsobtained with the discrete model, whereas dashed lines represent results obtained with the 2D FE model. As expected,strains in CMB1 are constant along both sections, and the results obtained with the two models are almost coincident

(a) (b)FIG. 16: Panel subject to three loads; (a) continuous model and (b) discrete model (running bond pattern).



(a) (b)FIG. 17: Map of the vertical displacement of the blocks for the panel with two texture patterns subject to CMB2 loadsat t = 0: (a) case 2 and (b) case 5.

(a) (b)

(c) (d)

FIG. 18: Panel sections A A and B B: trend in "22 strain in combination 1 and 2 at t = 0 and t = 300 for thepanel with running bond pattern.



for each case considered. Moreover, the relaxation over time is evident and an "22 absolute value increase from5 106 to 2 105 [Figs. 18(a) and 18(b), respectively) for the thin mortar joint case and from 2 105 to 104[Figs. 18(c) and 18(d), respectively] for the standard mortar joint case. In CMB2, "22 strains vary significantly alongthe y1 direction and assume large values close to the section midpoint, due to the effect of the concentrated force.Furthermore, strains are larger along section B B with respect to section A A. It is clear that the 2D FE modelgives continuous values for "22 along y1, whereas the discrete model gives "22 values at the centers of the blocks;However, with the running bond pattern, characterized by six blocks along y1, "22 strains obtained with the discretemodel are not coincident to the ones obtained with the 2D FE model and differences are larger along section B B,which is close to the application point of the concentrated force. Differences between the two models do not varysignificantly if thin or normal mortar joints are considered; hence in this case the moduli in Eqs. (25)(28) may beadequate to represent the behavior of the homogenized model.

Similar to the previous case, Figs. 19(a)19(d) show the "22 strain distribution for the head bond pattern underboth load combinations, at t = 0 and 300 days for thin and normal mortar joints. Continuous lines with trianglesrepresent results obtained with the discrete model, whereas dashed lines represent results obtained with the 2D FE

(a) (b)

(c) (d)

FIG. 19: Panel sections A A and B B: trend in "22 strain in combination 1 and 2 at t = 0 and t = 300 for thepanel with head bond pattern.



model. The relaxation over time is evident and the panel with standard mortar joints is obviously more deformablewith respect to the panel with thin mortar joints. It must be noted in this case that the strains obtained with the discretemodel are almost coincident with those obtained with the 2D FE model for both load combinations and both horizontalsections. Therefore it is clear that the models have a similar behavior due to the large number of blocks along the y1direction with respect to the running bond pattern; hence a head bond pattern is characterized by a large capacity ofload diffusion. A similar aspect was underlined by Cecchi and Di Marco (2002) by comparing a stack bond patternwith a running bond pattern. Furthermore, if vertical stresses 22 are evaluated along AA and B B sections, thetrends in each load combination do not vary over time, as it has been found by Cecchi and Tralli (2012). For example,Figs. 20(a) and 20(b) show vertical stresses of the panels over time, characterized by standard mortar joints having arunning bond pattern and head bond pattern, respectively. In this case, differences between the discrete models andthe homogeneous models are smaller than those found for the strains.

4.2 Sensitivity to Size of Heterogeneity

In Section 4.1.2, the numerical homogeneous solutions for the panels characterized by a running bond pattern turnedout to be quite far from the results obtained with the corresponding discrete models. In that case the size of the brickswas not significantly small with respect to panel size. Then in this section, a panel having width equal to 2L, heightequal to 2H , and made by UNI bricks following a running bond pattern is considered. In this case the ratio betweenpanel width and brick width is larger than 10. The viscoelastic behavior of the new panel is compared with that ofthe smaller panel having the same texture pattern [Fig. 2(a)]. The viscoelastic properties of mortar joints are the samefor both panels. The load combinations and restraints defined in Section 4.1.2 are applied and, for simplicity, onlystandard mortar joints are considered. Figures 21(a) and 21(b) show clearly that differences between the discretemodels and the homogeneous models are smaller than those found with the small panel. In particular, differences overtime in the middle of section AA in CMB2 decrease from 11% with the small panel [Figs. 18(c) and 18(d) to 0.3%with the big one [Fig. 21(a)].

5. CONCLUSIONS

In this work, a comparison between a numerical discrete model and a continuum homogenized model was performedfor studying the behavior of masonry subject to simple in-plane loads over time. Both models started from the same

(a) (b)

FIG. 20: Panel sections A A and B B: trend in 22 stress in combinations 1 and 2 for the panel with runningbond pattern (a) and head bond pattern (b).



(a) (b)

FIG. 21: Panel sections A A and B B: trend in "22 strain in combination 1 and 2 at t = 0 and t = 300 for thepanel having dimensions 2L and 2H .

assumptions of rigid blocks and mortar joints modelled as interfaces (Cecchi and Tralli, 2012). The well-knownMaxwell rheological model (Kaliske and Rothert, 1997) was adopted for representing the relaxation of the mortarelastic modulus Em. Pure compressive loads, shear loads, and both horizontal and vertical loads were applied to thepanel, and a parametric analysis was carried out to investigate the effect of several parameters: (i) mortar-to-brickthickness ratio; and (ii) masonry texture (running versus header bond).

The discrete model turned out to be simple and effective for representing the behavior of masonry panels over time,and block displacements over time followed the relaxation behavior of the mortar. Numerical results have been foundto be in agreement with analytical solutions of the homogenized continuum model and differences did not vary forincreasing mortar joint stiffness due to the same hypothesis adopted for both models. Differences between numericaland analytical solutions turned out to be close to 7% in most of the simple cases considered. Moreover, as expected,the differences in the texture pattern were significant for the panel subject to a horizontal compressive load, whereasblock displacements obtained with the other load cases were found to be quite coincident for both texture patterns.

Considering a more complex load case characterized by distributed and concentrated forces, the texture patternassumed great importance related to the capacity of diffusion of the concentrated load over time. The discrete modelfor the panel with a head bond pattern gave results quite coincident to the continuous results given by the homogenized2D FE model, whereas the strains obtained with the discrete model for the running bond pattern were quite differentfrom those of the continuous model. Moreover, the last load case allowed us to evaluate the effect of the size ofheterogeneity with respect to panel dimensions. The homogenized solutions turned out to be close to the discrete onesif the size of the brick is small (b less than L/10) compared to panel dimensions.

ACKNOWLEDGMENT

The research project reported in this paper was conducted thanks to financial support from PRIN20102011, no.2010NRBMTP of the Italian Ministry for University and Research (MIUR).

REFERENCESAllen, M. P. and Tildesley, D. J., Computer Simulations of Liquids, Oxford, NY: Oxford Science Publications, 1994.Avila-Pozos, O., Klarbring, A., and Movchan, A. B., Asymptotic model of orthotropic highly inhomogeneous layered structure,

Mech. Mater., vol. 31, pp. 101115, 1999.



Berto, L., Saetta, A., Scotta, R., and Vitaliani, R., Shear behaviour of masonry panel: Parametric F. E. analyses, Int. J. SolidsStruct., vol. 41, no. 1617, pp. 43834405, 2004.

Binda, I., Anzani, A., and Gioda, G., An analysis of the time dependent behaviour of masonry walls, in Proc. of the 9th Int1.Brick/Bloc Masonry Conf., DGFm, Berlin, vol. 2, pp. 10581067, 1991.

Brooks, J. J., Composite modelling of masonry deformation, Mater. Struct., vol. 23, pp. 241251, 1990.Brooks, J. J. and Abdullah, C. S., Composite models for predicting elastic and long-term movements in brickwork walls, Proc. Br.

Mason. Soc., vol. 1, pp. 2644, 1986.Cecchi, A. and Di Marco, R., Homogenized strategy toward constitutive identification of masonry, J. Eng. Mech., vol. 128, no. 6,

pp. 688697, 2002.Cecchi, A. and Sab, K., A comparison between a 3D discrete model and two homogenised plate models for periodic elastic

brickwork, Int. J. Solids Struct., vol. 41, pp. 22592276, 2004.Cecchi, A. and Tralli, A., A homogenized viscoelastic model for masonry structures, Int. J. Solids Struct., vol. 49, no. 13, pp.

14851496, 2012.Choi, K. K., Lissel, S. L., and Reda Taha, M. M., Rheological modeling of masonry creep, Can. J. Civ. Eng., vol. 34, no. 11, pp.

15061517, 2007.Kaliske, M. and Rothert, H., Formulation and implementation of three-dimensional viscoelasticity at small and finite strains,

Comput. Mech., vol. 19, no. 3, pp. 228239, 1997.Klarbring, A., Derivation of model of adhesively bounded joints by the asymptotic expansion method, Int. J. Eng. Sci., vol. 29, pp.

493512, 1991.Lourenco, P. and Rots, J. G., Multi surface interface model for the analysis of masonry structures, J. Eng. Mech., vol. 123, pp.

660668, 1997.Luciano, R. and Sacco, E., Homogenization technique and damage model for old masonry material, Int. J. Solids Struct., vol. 34,

no. 24, pp. 31913208, 1997.Owen, D. R. J. and Hinton, E., Finite Elements in Plasticity: Theory and Practice, Swansea, UK: Pineridge Press Limited, 1980.Papa, E. and Taliercio, A., A visco-damage model for brittle materials under monotonic and sustained stresses, Int. J. Numer.

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method based on Prony series, Int. J. Solids Struct., vol. 36, no. 11, pp. 16531675, 1999.Taliercio, A., An overview of masonry creep, in C.A. Brebbia, Ed., Structural Studies, Repairs and Maintenance of Heritage

Architecture XI, Southampton, Great Britain: WIT Press, pp. 197208, 1991.

APPENDIX

A.1 Numerical Procedure

In order to determine internal forces and displacements of masonry panels subject to external forces and modelledby the discrete model, a molecular dynamics method (Allen and Tildesley, 1994; Owen and Hinton, 1980) has beendeveloped in the perspective of linear and nonlinear analysis with dynamic loading. In this case, the equation to besolved for each value assumed by the elastic modulus of the mortar over time is

u = (ui;j1 ; ui;j2 ;

i;j3 )

T ; (A1)

M (@2u=@t2) +K u = Fext; (A2)where Fext are the applied actions, M is the (diagonal) mass matrix, and K the stiffness matrix.

The same procedure was described by Cecchi and Sab (2004) in a more general manner. To solve the dynamicEq. (A2), the predictorcorrector algorithm GEAR of order 2 is used. Vectors u(t), v(t), and a(t) denote, respectively,



the displacement, the velocity, and the acceleration at time t. Using a Taylor expansion, the correspondent predictorvectors at time t + t are

up(t+ t) = u(t) + t v(t) + 12t2 a(t) + o(t3); (A3)

vp(t+ t) = v(t) + t a(t) + o(t2); (A4)ap(t+ t) = a(t) + o(t): (A5)

Using the balance equation, the real accelerations a may be found and the error in the predictor time step a may becalculated:

a =M1(Fext K up); (A6)a(t+ t) = a ap(t+ t): (A7)

Finally, the corrector time step is introduced:

u(t+ t) = up(t+ t) +1

4t2 a c0; (A8)

v(t+ t) = vp(t+ t) +1

2t a c1; (A9)

a(t+ t) = ap(t+ t) + a c2; (A10)where c0 = 0, c1 = 1, and c2 = 1. The time-step integration is stopped when the error between internal and externalforces is lower than a predefined tolerance:

err = jjKup Fextjj < toler: (A11)The t time step must be much smaller than a critical value TC , calculated as a function of mass and stiffness propertiesof the block, as it was denoted by Cecchi and Sab (2004):

t =TC100

; TC =

rm

kn; m = a b s; kn = Sh K(t)

e; (A12)

where is the density of the block.The numerical procedure starts from the geometrical description of a generic masonry wall. Each block is identi-

fied by the position of its center. Figure A1 shows the wall considered in numerical examples with the running bondtexture pattern. Odd courses present n = 6 blocks, whereas even courses present n+1 blocks. Moreover, even coursesare characterized by first and last blocks that are half-blocks due to the running (or header) bond texture pattern ofmasonry.

The following steps are proposed (Cecchi and Sab, 2004):

FIG. A1: Geometrical description of the wall with the running bond texture pattern.



Definition of geometrical and mechanical quantities; Construction of mass tensor for the generic i block; Imposition of the boundary condition on forcesapplied loadsand on displacementconstraint degrees of

freedom;

Step 0: the initial displacements, velocities, and accelerations are set to zero; Step i: Computation of the predicted displacements, velocities, and accelerations [Eqs. (A3)(A5)]; Evaluation of elastic forces and moments at the interfaces according to the procedure in Section 2. If the static equilibrium according to Eq. (A11) is satisfied, then stop; Evaluation of real acceleration at the i-step according to Eq. (A10); Evaluation of the corrected displacements, velocities, and accelerations [Eqs. (A8)(A10)]; Go to step i+ 1.In order to quicken the convergence, the evaluation of the kinetic energy has been performed at each step. If at the

i+1 time step the kinetic energy is smaller than the kinetic energy at the i step, then the velocity vector is set to zero.The flowchart described above is performed for each value of mortar elastic modulus Em(t) over time, adopting

values collected in Table 2. Hence, the mechanical quantities of the problem are modified, taking into account therelaxation of the modulus for the mortar, and the corresponding block displacements, velocities, and accelerations aredetermined. In particular, K(t) and G(t) follow the relaxation of the mortar and consequently, the time step t of eachanalysis is different for each Em(t) value.


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DISCRETE ELEMENT MODEL FOR IN-PLANE LOADED VISCOELASTIC MASONRY