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A micro-mechanical model for homogenisation of masonry

Article  in  International Journal of Solids and Structures · June 2002

DOI: 10.1016/S0020-7683(02)00230-5

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A Micro-Mechanical Model for the Homogenisation

of Masonry

A. ZUCCHINI 1
ENEA, INN.FIS.MACO, C.R.E. “E.Clementel”, v.Don Fiammell,2, I – 40129 Bologna, Italy.

P.B. LOURENÇO *
University of Minho, Department of Civil Engineering, Azurém, P – 4800-058 Guimarães, Portugal

1
E-mail: zucchini@indy.bologna.enea.it,Ph: +39 0516098256, Fax: +39 0516098062
*
E-mail: pbl@eng.uminho.pt, Ph: + 351 253 510200, Fax: + 351 253 510217
ABSTRACT

Masonry is a composite material made of units (brick, blocks, etc.) and mortar. For

periodic arrangements of the units, the homogenisation techniques represent a powerful tool

for structural analysis. The main problem pending is the errors introduced in the

homogenisation process when large difference in stiffness are expected for the two

components. This issue is obvious in the case of non-linear analysis, where the tangent

stiffness of one component or the tangent stiffness of the two components tends to zero with

increasing inelastic behaviour.

The paper itself does not concentrate on the issue of non-linear homogenisation. But as

the accuracy of the model is assessed for an increasing ratio between the stiffness of the two

components, the benefits of adopting the proposed method for non-linear analysis are

demonstrated. Therefore, the proposed model represents a major step in the application of

homogenisation techniques for masonry structures.

The micro-mechanical model presented has been derived from the actual deformations

of the basic cell and includes additional internal deformation modes, with regard to the

standard two-step homogenisation procedure. These mechanisms, which result from the

staggered alignment of the units in the composite, are of capital importance for the global

response. For the proposed model, it is shown that, up to a stiffness ratio of one thousand, the

maximum error in the calculation of the homogenised Young’s moduli is lower than five

percent. It is also shown that the anisotropic failure surface obtained from the homogenised

model seems to represent well experimental results available in the literature.

KEYWORDS

Composites, numerical techniques, homogenisation techniques, masonry

1
1. INTRODUCTION

Masonry is a composite material made of units and mortar, normally

arranged periodically. Utilising the material parameters obtained from

experiments and the actual geometry of both components, viz. units (e.g.

bricks, blocks or stones) and joints, it is possible to numerically reproduce

the behaviour of masonry structures, see e.g. Lofti and Shing (1994), and

Lourenço and Rots (1997). Nevertheless, the representation of each unit and

each joint becomes impractical in case of real masonry structures

comprising a large number of units.

The alternative is to describe the composite behaviour of masonry in

terms of macro or average stresses and strains so that the material can be

assumed homogeneous. This problem can be approached, basically, from

two directions. A possible direction is to gather extensive experimental data

that can be used confidently in the analyses. It is stressed that the results are

limited to the conditions under which the data are obtained. New materials

and/or application of a well known material in different loading conditions

might require a different set of costly experimental programs. Another

direction, adopted in this paper, is to seek a more fundamental approach

which resorts to homogenisation techniques. This approach, which aims at

describing the behaviour of the composite from the geometry and behaviour

of the representative volume element (or basic cell, see Fig. 1), grants us a

predictive capability.

The techniques of homogenisation, Bakhvalov and Panasenko (1989),

are currently becoming increasingly popular among the masonry

2
community. A method that would permit to establish constitutive relations

in terms of averaged stresses and strains from the geometry and constitutive

relations of the individual components would represent a major step forward

in masonry modelling. Given the difficult geometry of the masonry basic

cell, a close-form solution of the homogenisation problem seems to be

impossible, which leads , basically, to three different lines of action.

The first, very powerful approach is to handle the brickwork structure

of masonry by considering the salient features of the discontinuum within

the framework of a generalised / Cosserat continuum theory. This elegant

and efficient solution, Besdo (1985) and Mühlhaus (1993), possesses some

inherent mathematical complexity and has not been adopted by many

researchers, even though being capable of handling the unit-mortar interface

and true discontinuum behaviour. The step towards the practical application

of such an approach is still to be done.

A second approach, Anthoine (1995,1997) and Urbanski et al. (1995),

is to apply rigorously the homogenisation theory for periodic media to the

basic cell, i.e. to carry out a single step homogenisation, with adequate

boundary conditions and exact geometry. It is stressed that the unit-mortar

interface has not yet been accounted for by researchers. The complexity of

the masonry basic cell implies a numerical solution of the problem, which

has been obtained using the finite element method. The theory was thus

used by the cited authors to determine macro-parameters of masonry and

not, actually, to carry out analysis at the structural level. In fact, the rigorous

application of the homogenisation theory for the non-linear behaviour of

the complex masonry basic cell implies solving the problem for all possible
3
macroscopic loading histories, since the superposition principle does not

apply anymore. Thus, the complete determination of the homogenized

constitutive law would require an infinite number of computations.

The third approach can be considered as an “engineering approach” 1,

aiming at substituting the complex geometry of the basic cell by a simplified

geometry so that a close-form solution of the homogenisation problem is

possible. Keeping in mind the objective of performing analysis at the

structural level, Pande et al. (1989), Maier et al. (1991) and Pietruszczak and

Niu (1992) introduced homogenisation techniques in an approximate

manner. The homogenisation has generally been performed in two steps,

head (or vertical) and bed (or horizontal) joints being introduced

successively. In this case masonry can be assumed to be a layered material,

which simplifies the problem significantly. Lourenço (1996) further

developed the procedure, presenting a novel matrix formulation that allows

a much clearer implementation of linear elastic homogenisation algorithms

and also a relatively simple extension to nonlinear behaviour. Again, it is

stressed that the unit-mortar interface has not been accounted for by the

cited researchers.

The use of two separate homogenisation steps does not explicitly

account for the regular offset of vertical mortar joints belonging to two

consecutive layered unit courses. Moreover, the final result depends on the

order in which the two homogenisation processes are carried out.

1
“Engineering” is used here not in the sense that it is empiric or practical but in the sense

that must be engineered from reasoning.

4
Nevertheless, this simplified homogenisation approach has been used by

several authors and performs very satisfactorily in the case of linear elastic

analysis, Anthoine (1995) and Lourenço (1997). For the case of non-linear

analysis, where the ratio between the stiffness of unit and mortar becomes

larger, the simplified homogenisation approach leads to non-acceptable

errors and should not be used. Lourenço (1997) has shown that large errors

can occur in the standard two step homogenisation technique if there are

large differences of stiffness (>10) between unit and mortar. Anthoine

(1995,1997) has shown that the standard two step homogenisation technique

does not take into account the arrangement of the units in the sense that

different bond patterns (running bond and stack bond for example) may lead

to exactly the same result.

A different engineering approach has been proposed by Bati et al.

(1999), in which a close-form solution of the periodic arrangement of units

and mortar has been obtained, by substituting the parallelepiped-shaped

units by elliptic cylinders. This mathematically elegant solution does not

represent well the geometry and it is unclear if it represents an advantage

with regard to the standard two-step homogenisation technique.

The present paper presents a new micro-mechanical model, for

masonry in stretcher bond 2, to overcome the limitations of the standard two-

step homogenisation by a more detailed simulation of the interactions

2
“Stretcher bond” represents the typical arrangement of masonry units in a wall, with an offset half unit

for the vertical mortar joints belonging to two consecutive masonry courses. The “stacked bond”

arrangement, in which the vertical joints run continously through all the courses, is not allowed for

structural purposes by most masonry codes.

5
between the different internal components of the basic cell. The model can

still be considered as an engineering approach, in which an ingenious

observation of the behaviour of masonry leads to the simulation of

additional internal deformation mechanisms of the joints, that become more

and more important for increasing unit/mortar stiffness ratios. At this stage,

the unit-mortar interface is not considered in the model.

It is noted that micro-mechanical approaches that consider additional

internal deformation mechanisms have been derived independently by van

der Pluijm (1999) for the analysis of masonry subjected to flexural bending

and by Lopez et al. (1999) for the non-linear analysis of masonry walls

subjected to in-plane loading.

In this paper, the full three-dimensional behaviour will be considered

and attention will be given to a comparison between the results from a

detailed finite element analysis (FEA) and the proposed micro-mechanical

homogenisation model, in order to demonstrate the efficiency of the

proposed solution. Finally, the adequacy of the model to reproduce the

anisotropic failure surface of masonry will be discussed by means of a

comparison with available experimental results.

2. Descriptive analysis of masonry

As a consequence of the differences in stiffness between units and

mortar, a complex interaction between the two masonry components occurs

when masonry is deformed. The differences in stiffness cause a unequal

6
distribution of deformations over units and mortar, compared with the

average deformation of masonry composite. As a result the individual

(internal) stresses of units and mortar deviate from the average (external)

stresses of the composite.

For the purpose of understanding the internal deformational behaviour

of masonry components (units and mortar), when average deformations

occur on the boundaries of the basic cell, detailed finite element calculations

have been carried out for different loading conditions. For a clear discussion

of the internal distribution of stresses, a right-oriented x-y-z coordinate

system was defined, where the x-axis is the parallel to the bed joints, the y-

axis is parallel to the head joints and the z-axis is normal to the masonry

plane, see Fig. 2. This figure also shows the components considered in the

present paper. The cross joint is defined as the mortar piece of the bed joint

that is connected to the head joint.

The mesh used in the analyses is depicted in Fig. 3 and consists of

24 × 4 × 12 twenty-noded quadratic 3-D elements with reduced integration.

The unit dimensions are 210 × 100 × 52 mm3 and the mortar thickness is 10

mm. The assumption that the units are stiffer than the joints is usually made

by the masonry research community. In the present analysis, in order to

better understand the deformational behaviour of the mortar, the units are

considered infinitely stiff (for this purpose, the adopted ratio between unit

and mortar stiffness was 1000).

Fig. 4 illustrates the deformation corresponding to the analysis of the

basic cell under compression along the axes x, y and z, and under shear in

the planes xy, xz and yz. Loading is applied with adequate tying of the nodes
7
in the boundaries, making use of the symmetry and antisymmetry conditions

appropriate to each load case. Therefore, the resulting loading might not be

associated with uniform stress conditions or uniform strain conditions.

Linear elastic behaviour is assumed in all cases.

Fig. 4a demonstrates that, for compression along the x-axis, the unit

and the bed joint are mostly subjected to normal stresses, the bed joint is

strongly distorted in shear and the cross joint is subjected to a mixed shear /

normal stress action. While the cross joint effect can be neglected if the

cross joint is small compared to the basic cell, the shear of the bed joint

must be included in the micro-mechanical model of masonry for stiff units.

Fig. 4b demonstrates that, for compression along the y-axis, the unit

and the bed joint are mostly subjected to normal stresses, and the head and

cross joints are subjected to a mixed shear / normal stress action. These

relatively local effects are difficult to include in the micro-mechanical

model, have small influence on the overall behaviour of the basic cell and

will not be considered. This is confirmed by the results of Lourenço (1997)

where it was shown that the standard two-step homogenisation technique,

which neglects such effects, leads to almost exact results (errors smaller

than 2% for ratios unit / mortar stiffness up to 1000).

Fig. 4c demonstrates that, for compression along the z-axis, all

components of the basic cell are subjected to a truly homogeneous state of

normal stress. This again is confirmed by the results of Lourenço (1997)

where it was shown that the standard two-step homogenisation technique

leads to almost exact results (errors smaller than 0.2% for ratios unit /

mortar stiffness up to 1000).

8
 Fig. 4d demonstrates that, for xy shear, the unit and the head joint are

mostly subjected to shear stresses, the bed joint is strongly distorted in the

normal direction (tension) and the cross joint is subjected to a mixed shear /

normal stress. Due to antisymmetric conditions, the neighbouring basic cells

will feature normal compression in the bed joint. While the cross joint effect

can be neglected if the cross joint is small compared to the basic cell, the

normal stress of the bed joint must be included in the micro-mechanical

model.

The deformation of the basic cell under xz shear is shown in Fig.4e.

The cell components are mostly subjected to shear stresses, with unit and

head joint deformed in the horizontal plane, while the bed joint is distorted

also in the vertical plane. Therefore the shear stress σ yz cannot be neglected

in a micro-mechanical model.

Finally, the deformation of the basic cell under yz shear is shown in

Fig.4f. All cell components are mainly distorted by shear in the vertical

plane, while minor local stress components do not produce significant

overall effects.

3. Formulation of the micro-mechanical model

3.1 General

Lourenço (1997) has shown that large errors can occur in the standard

two-step homogenisation technique if there are large differences of stiffness

(>10) between unit and mortar. The micro-mechanical model presented in

9
this paper overcomes this limitation by a more detailed simulation of the

interactions between the different internal components of the basic cell.

The main idea of this approach, derived from observations of

deformations calculated with the finite element analyses shown in the

previous section, is that the standard two-step homogenisation technique

neglects some deformation mechanisms of the bed joint, that become more

and more important for increasing unit/mortar stiffness ratios, such as:

• vertical normal stress in the bed joint, when the basic cell is loaded with

in-plane shear;

• in-plane shear of the bed joint, when the basic cell is loaded with an

horizontal in-plane normal stress;

• out-of-plane shear σ yz of the bed joint, when the basic cell is loaded

with out-of-plane shear stress σ xz .

These mechanisms are due to the staggered alignment of the units in a

masonry wall and are neglected by the standard two-step homogenisation

techniques, which are based on the assumption of continuous perpendicular

head joints.

Due to the superposition principle, which applies in linear problems,

the elastic response of the basic cell to a generic load can be determined by

studying six basic loading conditions: three cases of normal stress and three

cases of simple shear. In the present formulation, for each loading case and

each basic cell component, suitably chosen components of the stress and

strain tensors are assumed to be of relevance for the stress-strain state of the

basic cell, while all the others are neglected, see Fig. 6 and Fig. 8 for

10
examples. Equilibrium is, of course, ensured for all loading cases. The

number of unknowns of the problem is larger than in the usual

homogenisation procedure in order to take into account the above "second

order" effects. The unknown internal stresses and strains can be found from

equilibrium equations at internal interfaces between basic cell components,

with a few ingenious assumptions on the cross joint behaviour and on the

kinematics of the basic cell deformation, see Fig. 5 for the adopted

geometric symbols. The equivalent properties of an homogenised material

are then easily derived from the internal stresses and strains, by forcing the

macro-deformation of the model and of the homogenised material to be the

same, meaning that both systems must contain the same strain energy.

3.2 Young’s moduli and Poisson’s coefficients

The Young's moduli and the Poisson's coefficient of an equivalent

orthotropic material can be derived from the elastic strains of the basic cell

loaded with a uniform normal stress on the two faces perpendicular to a

given axis (x, y or z). All other stresses vanish on the boundary. Fig. 9 shows

the case of uniform loading in the horizontal in-plane direction (x-axis). In

this case all shear stresses and strains inside the basic cell are neglected,

except the in-plane shear stress and strain (σxy and εxy) in the bed joint and in

the unit. Non-zero stresses and strains are assumed to be constant in each

basic cell component, except the normal stress σxx in the unit, which must

be a linear function of x to account for the effect of the shear σxy in the bed

joint, and the shear stress σxy in the unit, linear in y.

11
 With these hypotheses, the following relations hold for the stresses at

internal or boundary interfaces:

l −t
(1) Interface brick − head joint σ xx2 = σ xxb − σ 1xy
2h
( 2) Interface brick − bed joints σ byy = σ 1yy
l −t
(3) Right boundary hσ xx2 + 2tσ xx
3
+ h(σ xxb + σ 1xy ) = 2(h + t)σ xx
0

2h
( 4) Upper boundary lσ byy + tσ 2yy = (l + t)σ 0yy
(5) Front boundary 2 thσ zz2 + 2(l − t)tσ 1zz + 2lhσ zzb + 4t 2 σ zz3 = [ 2th + 2(l + t)t + 2lh]σ zz0

and for the strains:

(6) Upper boundary 2tε1yy + hε byy = hε 2yy + 2tε yy

3

( 7) Right boundary tε xx2 + lε xxb = 2tε xx

3
+ (l − t)ε1xx
(8) Front boundary ε zzb = ε1zz
( 9) Front boundary ε zzb = ε zz2

where l is half of the unit length, h is half of the unit height and t is half of

the bed joint width. Unit, bed joint, head joint and cross joint variables are

indicated throughout this paper respectively by the superscripts b, 1, 2 and

3, respectively. σ xxb and ε xxb are the mean value of the normal stress

σ xx and normal strain ε xx in the unit. σ xx0 , σ yy0 , σ zz0 are the uniform

normal (macro) stresses on the faces of the homogenised basic cell,

respectively in the x-, y- and z-direction. The equilibrium of the unit (Fig. 7)

yields:

(10) hσ bxx1 + (l − t )σ1xy = hσ bxx2

where we assume that the shear acts only on the bed-unit interface (l-t).

12
If σ bxx is linear in x, its mean value in the mid-unit (equal to the mean value

in the unit) is:

σ bxx1 + σ bxx2
(11) σ xx
b
=
2

From Eqs.(10,11) we get :

l−t
σ xx
b1
= σ xx
b
− σ1xy
(12) 2h
l −t
σ bxx2 = σ xx
b
+ σ1xy
2h

which have been used in Eqs. (1,3). The couple required for the momentum

equilibrium of one fourth of the unit in the basic cell (Fig. 6) derives from

the neighbouring cell along y-axis. The symmetric unit quarter of the cell

above (Fig. 7) reacts at the centre line of the unit with a couple due to a self-

equilibrating vertical stress distribution, which is neglected in the model.

In Eqs.(1-9) the unknown stresses and strains in the cross joint can be

eliminated by means of the following relations:

E2 2 E3 1 E1 1
(13) ε 3yy = ε yy σ zz3 = σ zz ε xx
3
= ε xx
E3 E1 E3

(14) σ xx3 = σ 1xx

Eqs.(13) assume that the cross joint behaves as a spring connected in

series with the bed joint in the x-direction, connected in series with the head

joint in the y-direction and connected in parallel with the bed joint in the z-

direction. Eq.(14) represents the equilibrium at the cross-bed joint interface.

It can be noted that the stress-strain state in the cross joint does not play a

13
major role in the problem, because of its usually small volume ratio, so finer

approximations are not considered.

Introducing Eqs.(13-14) in Eqs. (1-9) results in the elimination of all

unknowns related to the cross joint. Further coupling with the nine elastic

stress-strain relations in the unit, head joint and bed joint, namely,

ε xxk =
1 k
Ek
[ ]
σ xx − ν k (σ yyk + σ zzk )

(15) ε yyk =
1 k
Ek
[
σ yy − ν k (σ xxk + σ zzk ) , ] k = b,1,2,

ε zzk =
1 k
Ek
[ ]
σ zz − ν k (σ xxk + σ yyk )

yields a linear system of 18 equations. The unknowns are the six normal

stresses and strains of the three components (unit, head joint and bed joint)

and the shear stress and shear strain in the bed joint, amounting to a total of

20 unknowns.

Two additional equations are therefore needed to solve the problem.

The equations can be derived introducing the shear deformation of the bed

joint: the elastic mismatch between the normal x strains in the unit and in

the head joint is responsible for shear in the bed joint because of the

staggered alignment of the units in a masonry wall. This mechanism is clear

in Fig. 9 (where only the horizontal displacements have been magnified for

sake of clearness) and leads to the approximated relation 3:

If the assumed linear behaviour of ε xx in x is taken into account, it would lead

3 b

l−t
to ε xx = ε xx + σ xy
b2 b 1
, but usually the second term in the right-hand side can be
2hE b
neglected.

14
 1 ∆x 2 − ∆xb ε xx2 × t − ε xx
b2
× t ε xx2 − ε xxb
(16) ε 1xy = × = ≅
2 2t 4t 4

This relation holds in the hypothesis that the bed joint does not slip on the

unit. With the additional elastic relation:

(17) σ 1xy = 2G1 ε 1xy

a system of 20 equations and 20 variables is finally obtained. This linear

system of equations can be solved numerically 4 to give the internal stresses

and strains for uniaxial load in the i-direction, given by

(18) σ ii0 = 1 , σ 0jj = 0 ( j ≠ i ) , i, j = x, y, z

where i represents the three orthogonal directions associated with the axis x,

y or z. The shear stress in the unit can be found by means of the internal

equilibrium equation :

∂σ xxb ∂σ xy ∂σ xzb
b

(19) + + = 0,
∂x ∂y ∂z

which leads to :

 y
(20) σ xyb = σ 1xy 1 − 
 h

4
It is noted that an explicit symbolic solution does exist and has been obtained.

Nevertheless, the complexity of the solution precludes its use for practical purposes. The

system of twenty equations can be easily reduced to a system of nine equations, which

can be solved with any efficient linear solver.

15
 The homogenised Young's moduli and Poisson's coefficients of the

basic cell are finally:

σ ii0 ε 
(21) Ei = , ν ij =  jj  , i, j = x, y, z
ε ii  ε ii  i

where

l − t + 2tE1 / E3
ε xx = ε 1xx
l +t

ε yy2 ( h + 2tE2 / E3 ) + hε yyb

(22) ε yy =
2( t + h )

ε zz = ε zzb

and the subscript i in the Young’s modulus E and the Poisson’s ratio

calculation ( ) i indicates that the values are calculated for uniaxial loading

in the i-direction (i = x, y, z).

3.3 In-plane shear modulus G xy

The homogenised shear modulus Gxy can be calculated by loading the

basic cell with simple in-plane shear by means of suitable load and

displacement fields. All external loads are zero on the basic cell boundary,

except a uniform shear stress σ xy0 applied on the upper and lower face, and

the equilibrium reactions σ xy on the left and right face. In this case the

model neglects all stresses (and corresponding strains), except the in-plane

shear in each basic cell component and the normal vertical component

σ 1yy in the bed joint. Non-zero stress and strain components are assumed to

be constant in each basic cell component, except σ xy in the unit, which

16
must be a linear function of x to account for the effect of the normal stress

σ 1yy in the bed joint. The deformation of the basic cell is approximated as

shown in Fig. 10, with the bed joint in traction. Note that in the

neighbouring basic cells (along x-axis) the bed joint is in compression, due

to the antisymmetric loading conditions.

The internal stresses can be related by the equilibrium at adequately

isolated parts of the composite:

Upper boundary t σ xy2 + lσ xyb = (t + l) σ xy

0

l 1
(23) Interface brick − head joints σ xy2 = σ xyb + σ yy
2h
Interface brick − bed joints σ xyb = σ 1xy

where σ xyb is the mean value of σ bxy in the unit.

The normal strain ε1yy can be derived from the geometric considerations in

Fig. 10, where all the geometric quantities can be defined. Neglecting

second order terms, it is straightforward to obtain:

t
∆y + ∆y
2t ′ − 2t l ∆y ∆y
(24) ε 1yy ≅ ≅ , ε xy2 − ε xyb = +
2t 2t 2t 2l

which lead to :

(25) ε 1yy = ε xy2 − ε xyb

and, introducing the linear elastic relation between stress and strain, finally:

(26) σ 1yy = E1 (ε xyb − ε xy2 )

Eqs.(23,26), combined with the shear stress-strain relations

(27) σ xyk = 2Gk ε xyk (k = b,1,2)

17
yield the shear stresses in the basic cell components:

lE1 + 4hGb
σ xy2 =σ xy
0
= kσ xy
0

l  Gb
2

lE1 + 4hGb + E1  − 1
l + t  G2 
t+l t 2
(28) σ 1xy =σ xy
b
=σ xy
0
− σ xy
l l
2h
σ 1yy = (σ xy2 − σ xyb )
l

The shear strains of the basic cell components and of the homogenised

material, according to the deformation shown in Fig. 10, are related by the

strain-displacement relations :

1  ∆x 1 ∆y  1  ∆x 2 ∆y 
ε 1xy ≅  −  , ε xy2 ≅  + 
2  2t l  2 h t 
(29)  ∆x 1 
 + ∆x 2 
1  ∆x 2 ∆y 
≅ 
2
ε xyb ≅  −  , ε xy
2 h l  (
2 h+t )
which lead to

1  b h t2 
ε xy = (lε xy + tε xy )
2
+ tε xy + (ε xy − ε xy )
1 2 b

h + t  l + t 
(30)
l +t

The shear strains ε xyi in the above equation can be calculated from the shear

stresses given in Eq.(28) by means of the elastic relations of Eq.(27),

resulting, finally, in the homogenised shear modulus Gxy

σ xy l (t + l )(t + h )
0

G xy = =
tl (t + h ) (t + l − kt )(lh − t 2 ) t (t + l )(t + l − kt )
( 31)
2ε xy
k + +
G2 Gb G1

where k is defined in Eq.(28).

3.4 Out-of-plane shear modulus G xz

18
 To calculate the homogenised shear modulus Gxz, simple out-of-plane

shear conditions in the xz-plane are imposed to the basic cell. Right and left

faces are loaded with a uniform shear σ 0xz , while all other boundary stresses

are zero, except the equilibrium reactions σ xz on front and rear face. Only

out-of-plane shear stresses σ xz in each basic cell component and σ1yz in the

bed joint (and corresponding strains) are taken into account in the model,

while all others are neglected. Non-zero stress and strain components are

assumed to be constant, except σ bxz which varies linearly in x to account for

the effect of σ1yz in the bed joint.

The deformation of the basic cell in this case is approximated as shown

in Fig. 11, where one side has been fixed for the purpose of graphical

clarity. The shear strain ε1yz , with geometric considerations, can be found to

be:

t
∆z 2 − ∆zb
1 ∆z l
(32) ε 1yz = ≅
2 2t 4t

The following relations also hold:

(l − t ) 1
Interface brick − head joint σ xz2 = σ xzb − σ yz
2h
 (l − t ) 1 
(33) Right boundary h  σ xzb + σ yz  + 2t σ xz3 + hσ xz2 = 2(t + h)σ xz0
 2h 
Interface cross − bed joints σ xz = σ 1xz
3

Interface brick - bed joint Δz b = Δz1

By means of the shear stress-strain relations

( 34) σ xzk = 2Gk ε xzk (k = b,1,2)

19
and of the kinematic relations

1 ∆z1 1 ∆z 2 1 ∆z b
( 35) ε 1xz ≅ , ε xz2 ≅ , ε xzb ≅
2 l 2 t 2 l

Eqs.(32,33) yield :

t+h
ε 1xz = ε xzb =σ xz0
2(tG1 + hGb )
4hGb + (l − t )G1
(36) ε xz2 = ε xzb
4hG2 + (l − t )G1
1 2
ε 1yz = (ε xz − ε xzb )
2

and the homogenised shear modulus can be finally found as:

(37) G xz =
σ xz0 σ0
= xz 2
t +1
=
(t + l )(tG1 + hGb )
(t + h ) t 4hGb + (l − t )G1 + l 
2ε xz 2 tε xz + lε xzb  
 4hG2 + (l − t )G1 

3.5 Out-of-plane shear modulus G yz

The basic cell in this case is assumed to be in simple out-of-plane shear

(in the plane yz) by means of appropriate boundary conditions. The external

load is a uniform shear stress σ 0yz applied on upper and lower face of the

basic cell, while equilibrium reactions σ yz act on front and rear face, where

the boundary condition uy=0 is imposed. Only the shear stresses σ yz (and

corresponding strains) are taken into account in the model. It can be argued,

from the deformation shown in Fig. 12 (where one side has been fixed for

the purpose of graphical clarity), that

Upper boundary tσ yz2 + lσ byz = (t + l )σ 0yz

(38) Interface brick − bed joints σ byz = σ 1yz
Interface brick - head joints ε yzb = ε yz2
20
Combining these equations with the stress-strain relations

(39) σ yzk = 2Gk ε yzk (k = b,1,2)

yields:

Gb b Gb 0 l +t
(40) ε 1yz = ε yz = σ yz
G1 G1 2(lGb + tG2 )

The homogenised strain is

tε 1yz + hε yzb
(41) ε yz =
t+h

and the homogenised shear modulus G yz is finally given as:

σ yz0 t + h lGb + tG2

(42) G yz = = G1
2ε yz t + l tGb + hG1

4. ELASTIC RESULTS

The model described in the previous section has been applied to a real

masonry basic cell and compared with the results of an accurate finite

element analysis (FEA). This was considered a better evaluation of the

analytical model that comparing analytical results with experimental results.

In fact, the analytical model needs material data for the components and this

type of data, at least for the mortar, always result from debatable

assumptions or debatable interpretation of experimental results at the

composite level (the curing conditions of mortar inside the composite are

impossible to replicate, leading to meaningless results if the mortar

specimens have been cured outside the composite). In the finite element

21
analysis and the analytical model, the properties of the components can be

taken absolutely equal.

The same elastic properties have been adopted for the bed joint, head

joint and cross joint (E1 = E2 = E3 = Em, ν1 = ν2 = ν3 = νm). Different

stiffness ratios between mortar and unit are considered. This allows to

assess the performance of the model for inelastic behaviour. In fact, non-

linear behaviour is associated with (tangent) stiffness degradation and

homogenisation of non-linear processes will result in large stiffness

differences between the components. In the limit, the ratio between the

stiffness of the different components is zero (or infinity), once a given

components has no stiffness left. The unit dimensions are 210 × 100 × 52

mm3 and the mortar thickness is 10 mm. The material properties of the unit

are kept constant, whereas the properties of the mortar are varied. For the

unit, the Young's modulus Eb is 20 GPa and the Poisson's ratio νb is 0.15.

For the mortar, the Young's modulus is varied to yield a ratio Eb / Em

ranging from 1 to 1000. The Poisson's ratio νm is kept constant to 0.15.

The adopted range of Eb / Em is very large (up to 1000), if only linear

elastic behaviour of mortar is considered. However, those high values are

indeed encountered if inelastic behaviour is included. In such case, Eb and

Em should be understood as linearised tangent Young’s moduli, representing

a measure of the degradation of the (tangent / secant) stiffness matrices

utilised in the numerical procedures adopted to solve the non-linear

problem. Note that the ratio Eb / Em tends to infinity when softening of the

mortar is complete and only the unit remains structurally active.

22
 The elastic properties of the homogenised material, calculated by means

of the proposed micro-mechanical model, are compared in Fig. 13 with the

values obtained by FE analysis. The agreement is very good in the entire

range 1≤ Eb / Em ≤1000. Fig. 14 gives the relative error of the elastic

parameters predicted by the proposed model and show that it is always ≤

6%. The thinner curves in Fig. 14 ("simplified model") give the results of a

simplified model (Ex only), derived from the model presented in the paper,

but where the deformation mechanisms of the bed joint, mentioned in

Sec.3.1, have not been taken into account. The simplified model therefore

neglects the main effects due to the misalignment of the units in the

masonry wall and coincides with the full model when the units are aligned

in the wall. The simplified model is, therefore, closer to the standard two-

step homogenisation referred to in Chapter 1. Fig. 14 also includes the

results of the standard two-step homogenisation of Lourenço (1997). A non-

acceptable error up to 45% is found in such case, for the homogenisation of

the elastic Young’s modulus along the x direction. Directions y and z are not

shown in the picture for the sake of clarity of the picture. Less pronounced

differences are found in these directions as the unit geometry if oriented in

the x direction and the running bond reduces largely the influence of the

head joint for homogenisation in the y direction, see Lourenço (1997).

For large ratios Eb / Em the simplified model predicts value of Ex, vxz and

Gxz much smaller than the actual values obtained by FE analysis. The large

and increasing errors of the simplified model on these variables (up to 50%)

indicate that for very degraded mortar the neglected deformation

mechanisms of the bed joint contribute significantly to the overall basic cell
23
behaviour. In the proposed micro-mechanical model the in-plane shear

resistance of the bed joint ( σ 1xy ) is responsible for the increased stiffness in

the x-direction (up to 50%), which could not be accounted for only with the

normal stresses in the unit and in the mortar. This increase of the stiffness

in x yields also higher Poisson's coefficient in y and z. The vertical normal

stress in the bed joint ( σ 1yy ) contributes to the in-plane shear stiffness, while

the out-of-plane shear ( σ 1yz ) can double (for very large ratios Eb / Em) the

shear resistance of the basic cell to a shear load σ xz0 calculated with the

simplified model.

5. A HOMOGENISED FAILURE CRITERION

Failure of quasi-brittle materials such as concrete and masonry is a

difficult issue. Even for apparently simple loading conditions such as

uniaxial compression, failure mechanisms denoted as Mode I, Mode II or

local crushing are the object of a long-going debate among researchers, see

van Mier (1998) for a discussion. For masonry under uniaxial compression,

a lot of researchers claim that mortar is subjected to triaxial compression

and the unit is in a mixed uniaxial compression - biaxial tension, see e.g.

Hendry (1998). The assumption that failure of masonry is governed solely

by the tensile failure of the unit, induced by the expansion effect of mortar,

is certainly highly debatable because the influence of the micro-structure

24
(voids, inclusions, etc.) is also a key issue. A discussion on these aspects is

out of the scope of the present paper and will not be carried out.

The sole objective of this section is to demonstrate that the shape of the

anisotropic failure surface based on the micro-mechanical homogenised

model is reasonable and seems to be able to reproduce experimental results

available in the literature. A direct connection to the triggered failure modes

is not the issue here. Currently, a research project being carried out at

University of Minho is addressing these issues.

The homogenised micro-mechanical model allows to calculate not only

the homogenised material properties of the basic cell, but also stresses and

strains in each basic cell component. Making use of the superposition

principle, holding up to failure if an elastic-brittle behaviour is assumed for

mortar and unit, the stress distribution for an arbitrary loading case can be

derived by linear combination of the solutions of the six basic problems

presented in Chapter 3.

Then, the failure load for the homogenised cell results from reaching the

failure criteria of any of the two components. For the purpose of this

section, the simplest failure criteria can be considered for the unit and

mortar. Assuming that both materials are isotropic, the Rankine yield

surface has been assumed to describe the tensile behaviour, while the classic

von Mises criteria has been adopted to describe the compression behaviour,

see Fig.14. These are defined by

Rankine: σ kp = σ tk

25
 (43)

Von Mises : σ Mises

k
= σ ck ; k=1,2,b

where σ kp is the maximum principal tensile stress, σ Mises

k
is the equivalent

von Mises stress, and σ tk , σ ck are the tensile and compressive strengths of

the component k. It is stressed that von Mises is hardly acceptable as a

failure criterion for frictional materials subjected to general three-

dimensional stress states, which is not the case here. On the contrary, it can

approximate failure in the compression-compression regime or the tension-

compression regime for plane stress problems, as adopted here. It has been

used for this purpose by a number of authors.

Fig. 16 shows the resulting failure surfaces in the plane stress space

( σ 1 , σ 2 ) for a test case, where the principal loading stress directions

coincide with the material axes, i.e. only in-plane normal stresses σ 1 , σ 2

and no shearing are applied on the cell faces. The material and geometric

parameters for unit and mortar, which are defined in the picture, aim at

reproducing the results from Page (1981,1983). In the micro-mechanical

model, the principal directions in the bed joint do not coincide with the

material axes even in the absence of shear loading, due to presence of shear

in the bed joint. The intersection of all failure surfaces (the thicker line in

Fig. 16a which is reproduced in Fig. 16b) is the failure surface of the

homogenized basic cell. In the unit, due to the variation of σ xx

b
with x , the

compression failure criteria is applied to the point which leads to a

maximum of the von Mises stress, which can be easily calculated.

26
 The stresses in Fig. 16 have been normalised by the mortar tensile

strength ( σ tm ) for the purpose of comparison with experimental results. It

can be noted that the plot of the yield stress in the unit of Fig. 16a is not a

perfect ellipse (check top and bottom parts): actually it is the intersection

(worst value) of two different von Mises ellipses, corresponding to the

maximum and minimum values of the stress σ xx

b
, which has been assumed to

vary linearly with x in the unit. For a given stress path, the failure loads and

the type of failure mechanism depend strictly on geometry, on elastic

material properties and above all on the relative material strengths of the

different cell components. Note that the direction of the maximum principal

stress in each component does not correspond always to the same material

direction, but does change with the load ratio σ 1 / σ 2 . Additionally, the

tensile stress of the unit in the compression-compression range is σ zz as the

lateral expansion in z of the mortar (prevented in x and y by the biaxial

compression) is the cause of a tensile stress state of the unit in the direction

z.

According to the proposed model, Fig. 16b shows that, for the selected

material and geometric properties, failure by tension of the head joint is

expected in the tension-compression range, while tension in the bed joint is

the cause of the failure in the compression-tension range. In the

compression-compression range, three mechanisms are responsible of the

failure of the cell for decreasing σ 1 / σ 2 ratios: tensile failure in the bed joint

(for very high ratios), compressive failure in the head joint and compressive

failure in the bed joint. Again, it is believed that these conclusions are
27
debatable and more research is needed on the issue of compressive failure of

masonry.

Nevertheless, a comparison between the results obtained with the micro-

mechanical model and the experimental results of Page (1981,1983) are

given in Fig. 17. The agreement in the actual values is misleading as the

parameters of the micromechanical model were fitted to obtain the actual

uniaxial strengths exhibited in the experiments. Nevertheless, very good

agreement is found in the shape of the yield surface, indicating that the

proposed model can be used as a possible macro-model to represent the

composite failure of masonry. Such an approach might reduce the effort to

develop and implement specific complex macro-models for the composite

behaviour of masonry such as in Lourenço et al. (1998).

It is stressed that the present work is, at this stage, mostly fundamental

and represents a contribution to researchers working in the homogenisation

field. Homogenisation methods represent powerful tools available for

analysts, but are not yet fully developed. The aim of this section is only to

demonstrate that an anisotropic failure criterion similar to the criteria

observed experimentally can be obtained. Given the difficulties in

adequately measuring mortar and interface properties, i.e. the absence of

adequate experimental values to assess the model, and the actual simplicity

of the model, the analytical results presented seem of value to the authors.

Finally, it must be stressed that failure by tension of the head joints will

not imply necessarily the failure of the composite system in the macroscale,

as adopted in this paper. For the simplified approach used here, this seems

the most reasonable assumption (i.e. if the weakest link fails, the system
28
fails). The issue of actual non-linear behaviour of the components with

progressive stiffness degradation must be assessed elsewhere. The definition

of failure is a tricky issue for a composite material such masonry. The well-

know experimental results of Page (1981,1983) indeed result from a

definition of failure in compression as early splitting of the bed joints in

tension, in the case of compression parallel to the bed joints, see Dhanasekar

et al. (1985).

6. CONCLUSIONS

This paper presents a novel micro-mechanical homogenisation model for

masonry, which includes additional deformation modes of the basic cell.

From a comparison with the results obtained in a detailed finite element

simulation of the basic cell, it is demonstrated that relatively small errors

occur in the homogenisation process, by including these mechanisms. The

proposed one-step homogenisation represents a major development with

respect to the standard two-step homogenisation process, head and bed

joints being introduced successively, in which very large errors occur for

large differences between the unit and mortar stiffness, Lourenço et al.

(1998).

Finally, it is shown that the anisotropic failure surface obtained from the

proposed micro-mechanical model, assuming elastic-brittle behaviour of

unit and mortar, seems to, qualitatively, reproduce well the experimental

results available for the composite behaviour of masonry. The quantitative

29
assessment of the model cannot be addressed at this stage, due to the

reduced experimental data available. It is expected that interface behaviour

and progressive stiffness degradation must be included in the simplified

homogenisation techniques to assess the their quantitative performance.

Acknowledgements

A. Zucchini was partially supported by GIANO project funded by EU

and MURST.

P.B. Lourenço was partially supported by project PRAXIS-C-ECM-

13247-1998 funded by the Portuguese Science and Technology Foundation

(FCT).

7. REFERENCES

Anthoine, A., 1995. Derivation of the in-plane elastic characteristics

of masonry through homogenization theory, International Journal of Solids

and Structures 32(2), 137-163.

Anthoine, A., 1997. Homogenisation of periodic masonry: Plane

stress, generalised plane strain or 3D modelling?, Communications in

Numerical Methods in Engineering 13, 319-326.

Bakhvalov, N., Panasenko, G., 1989. Homogenization: Averaging

processes in periodic media. Kluwer Academic Publishers, Dordrecht, The

Netherlands.

30
 Bati, S.B., Ranocchiai, G., Rovero, L., 1999. A micromechanical

model for linear homogenization of unit masonry, Materials and Structures

32, 22-30.

Besdo, D., 1985. Inelastic behaviour of plane frictionless block

systems described as Cosserat media, Archives in Mechanics 37(6), 603-

619.

Dhanasekar, M., Page, A.W. and Kleeman, P.W., 1985. The failure of

brick masonry under biaxial stresses, Proc. Intsn. Civ. Engrs., Part 2, 79,

295-313.

Hendry, A.W., 1998. Structural Masonry. Macmillan Press, United

Kingdom.

Lofti, H.R., Shing, P.B., 1994. Interface model applied to fracture of

masonry structures, Journal of Structural Engineering ASCE 120(1), 63-80.

Lopez, J., Oller, S., Oñate, E., Lubliner, J., 1999. A homogeneous

constitutive model for masonry, International Journal for Numerical

Methods in Engineering 46, 1651-1671.

Lourenço, P.B., 1996. A matrix formulation for the elastoplastic

homogenisation of layered materials, Mechanics of Cohesive-Frictional

Materials 1, 273-294.

Lourenço, P.B., 1997. On the use of homogenisation techniques for

the analysis of masonry structures, Masonry International 11(1), 26-32.

Lourenço, P.B., Rots, J.G., 1997. A multi-surface interface model for

the analysis of masonry structures, Journal of Engineering Mechanics ASCE

123(7), 660-668.

31
 Lourenço, P.B., Rots, J.G., Blaauwendraad, J., 1998. Continuum

model for masonry: Parameter estimation and validation, Journal of

Structural Engineering ASCE 124(6), 642-652.

Maier, G., Papa, E., Nappi, A., 1991. On damage and failure of unit

masonry. In: Experimental and numerical methods in earthquake

engineering. Balkema, Brussels and Luxenbourg, pp. 223-245.

van Mier, J.G.M., 1998. Failure of concrete under uniaxial

compression: An overview. In: Fracture mechanics of concrete structures,

Vol. 2. Aedificatio, Freiburg, pp. 1169-1182.

Mühlhaus, H.-B., 1993. Continuum models for layered soil and blocky

rock. In: Comprehensive rock engineering, Vol. 2. Pergamon Press.

Page, A.W., 1981. The biaxial compressive strength of brick masonry,

Proceedings from the Institution of Civil Engineers - Part 2 - 71, 893-906.

Page, A.W., 1983. The strength of brick masonry under biaxial

compression-tension, International Journal of Masonry Construction 3(1),

26-31.

Pande, G.N., Liang, J.X., Middleton, J, 1989. Equivalent elastic

moduli for unit masonry, Computers and Geotechnics 8, 243-265.

Pietruszczak, S., Niu, X., 1992. A mathematical description of

macroscopic behavior of unit masonry, International Journal of Solids and

Structures 29(5), 531-546.

van der Pluijm, R., 1999. Out of plane bending of masonry: Behaviour

and strength. Ph.D. Dissertation, Eindhoven University of Technology, The

Netherlands.

32
 Urbanski, A., Szarlinski, J., Kordecki, Z., 1995. Finite element

modeling of the behavior of the masonry walls and columns by

homogenization approach. In: Computer methods in structural masonry - 3,

Books & Journals International, Swansea, pp. 32-41.

33
Fig. 1 – Basic cell for masonry and objective of homogenisation

34
 y
z x

Unit

Bed joint

Head joint

Cross joint

y
z x Basic cell (R.V.E.)

Fig. 2 – Definition of (a) masonry axes and (b) masonry components considered in the analysis: unit, head joint, bed joint and cross joint

35
 y

x
z

Fig. 3 – Finite element mesh for the basic cell adopted in the analyses

36
 y

z x

(a) X direction

37
 y

x
z

(b) Y Direction

38
 y

x
z

(c) Z Direction

39
 y

x
z

(d) XY Direction

40
y
x

(e) XZ Direction

41
 y
x

(f) YZ Direction

Fig. 4 – Deformed configuration resulting from the finite element analysis on the basic cell: (a) compression x, (b) compression y, (c)

compression z, (d) shear xy, (e) shear xz and (f) shear yz.

42
 t l

Head Joint
Unit h
Cross Joint
y 2 b

3 1
Bed Joint 2t
x 1 3
z
b 2
Cross Joint
Unit h

Head Joint

l t
Fig. 5 – Adopted geometry symbols.

43
 Cross joint (3)

σ xx0 σ xx0 Unit (b)

Head joint (2)

Bed joint (1)

(a) (b)

σ xx2 σ xx2 σ xxb1 σ xxb 2

σ 1xy σ xyb
σ xx3 σ xx3 σ 1xx σ 1xx σ xx3

σ xyb σ 1xy

σ xxb 2 σ xxb1 σ xx2 σ xx2

(c)
Fig. 6 – Normal stress loading parallel to the x axis: (a) equivalent homogenised cell; (b) assumed deformation behaviour; (c) assumed involved stress components

44
 σ
σ bxx2

σ xx
b

σ bxx1 x
l/2 l 2l
σ xyb σ xyb

σ xxb1 σ xxb1

σ xyb σ xyb

σ byy

σ xxb1 σ xxb 2

σ xyb

Fig. 7 – Normal stress loading parallel to the x axis: unit equilibrium (couple moment equal to self-equilibrating vertical stress distribution)
45
 σ yy0
Cross joint (3)

Unit (b)
Head joint (2)

Bed joint (1)

σ 0
yy

(a) (b)

σ yy2 σ byy

σ yy2 σ byy

σ 3yy σ 1yy σ 3yy

σ 3yy σ 1yy σ 3yy

σ byy σ yy2

σ byy σ yy2
(c)

Fig. 8 – Normal stress loading parallel to the y axis: (a) equivalent homogenised cell; (b) assumed deformation behaviour; (c) assumed involved stress components

46
 ∆x2

2t

∆xb

x
t
Fig. 9 – Model assumptions for compression along the x axis

47
h

∆y

2t 2t' ∆=(t+∆x1)tgα≈ t tgα

α ∆y
y t ∆x1 ∆x2
h

l t

Fig. 10 – Model assumptions for xy shear

48
t l

α ∆zb

∆z ∆z2

Fig. 11 – Model assumptions for xz shear

49
 h

2t

h y

z
∆zb ∆z1 ∆zb

Fig. 12 – Model assumptions for yz shear

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Titolo:
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Titolo:
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(c)
Fig. 13 – Comparison between the micro-mechanical model and FEA results for different stiffness ratios: (a) Young’s moduli, (b) Poisson’s ratio

and (c) Shear moduli

53
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(b)

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 Titolo:
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(c)
Fig. 14 – Comparison between the micro-mechanical model and FEA results for different stiffness ratios: (a) Young’s moduli, (b) Poisson’s ratio

and (c) Shear moduli

56
 σ2

σt

−σc
σt σ1

−σc

Fig. 15 – Composite von Mises-Rankine failure criteria in the principal stress space.

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 Titolo:
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(b)
Fig. 16 – Calculated micro-mechanical failure criterion for masonry under biaxial in-plane loading (principal axes coincident with material axes):

(a) complete failure modes of the unit and mortar and (b) composite masonry failure.

59
 Titolo:
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Fig. 17 – Comparison between micro-mechanical failure criterion and experimental results of Page (1981,1983).

60

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