Inplane Behaviour of Masonry.
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Transcript of Inplane Behaviour of Masonry.
/ / 13840 ISSN 10185593
Commission of the European Communities
industrial processes
INPLANE BEHAVIOUR OF MASONRY: A LITERATURE REVIEW
Report EUR 13840 EN
Blowup from microfiche original
ς
Commission of the European Communities
industrial processes
INPLANE BEHAVIOUR OF MASONRY: A LITERATURE REVIEW
A. ANTHOINE Commission of the European Communities
Joint Research Centre Institute for Safety Technology
Ispra Establishment 121020 Ispra (VA)
FINAL REPORT
DirectorateGeneral Science, Research and Development
Joint Research Centre
1992
PARL EUROP. Biblioth.
NC.EUR 13840 El[j
CI.
, ,¡!*d by the . ·;„ΐί ■■·.· W T h " EUROPEAN COMMUNITIES
DirectorateGeneral ; munlcat»ons, «formation Industries and Innovation
L2920 LUXEMBOURG
LEGAL NOTICE
M V'< Commission of the European Communities nor any person acting on behalf tru.· C υ; ission is responsible for the use which might be made of the following
information
Catalogue number: CDNA13840ENC
© ECSC — EEC — EAEC, Brussels  Luxembourg, 1992
I I I
INDEX Page
I INTRODUCTION 1
IITEST TECHNIQUES 2
Π.1 TESTS ON ELEMENTARY COMPONENTS Il.1.1Unit II.1.2 Mortar and grout II.1.3Steel
11.3.1 Tests under homogeneous state of stress
11.3.2 Tests under heterogeneous state of stress
ΙII ELASTICITY
IVFAILURE CRITERION
IV.1 UNIAXIAL FAILURE
V.l UNIAXIAL BEHAVIOUR
V.2 BIAXIAL BEHAVIOUR
V3 RATE EFFECT
VI FINITE ELEMENT MODELS
VI.1 MICROMODELLING
VI.3 MICROMACROMODELLING
VII CONCLUSION AND PERSPECTIVES
ACKNOWLEDGEMENTS
REFERENCES
2 2 2 3
II.2 TESTS ON SMALL ASSEMBLAGES (MICROELEMENTS) 3 II.2.1  Bed and head joints 3 II.2.2 Steelmortar and steelgrout interface "4 II.2.3 Masonry prisms 4
II.3 TESTS ON LARGE ASSEMBLAGES (MACROELEMENTS) 5 5
6
7
10
10 IV.2 BIAXIAL FAILURE 1 5 IV.2.1 Experimental approaches 17 IV.2.2 Empirical approaches 19 IV.2.3 Phenomenological approaches 22 IV.2.4 Theoretical approaches 25
V STRESSSTRAIN RELATIONSHIP 2 7
27
29
30
32
32
VI.2 MACROMODELLING VI.2.1  Concrete macromodels VI.2.2 Original macromodels 49
37 38
53
54
57
59
I INTRODUCTION
Masonry is one of the oldest building materials but also one of the least understood: for a long time,
its design has been based on empirical rules some of which may still be found in codes of practice. It has been
and still is extensively used throughout the world by remaining the most economically competitive material in
the case of low· and midrise buildings. Masonry infills are also widely used to enclose and partition space in
steel or reinforced concrete framed structures. Unfortunately, whether structural or not, masonry has a bad
reputation in seismically active areas, for being the main cause of casualties during earthquakes. However, it is
now admitted that this is mainly attributable to improper design, bad execution and/or bad maintenance and
that masonry can be as safe as concrete or steel if conceived, executed and maintained in a suitable manner.
Considerable research effort has thus been made in the past twenty years in order to achieve a better
understanding of the behaviour of masonry. This was meant to derive rational design provisions for new
buildings in plain, confined and reinforced masonry, as well as for old ones (strengthening and repair).
Experimental, analytical and numerical research has been carried out at increasingly complex level, from the
basic constituents and their interactions, through major structural elements such as shear walls and infilled
frames, to complete fullscale buildings. Some results of this research have been included as design guidelines
in the current versions of Eurocode 6 (common unified rules for masonry structures) and Eurocode 8
(structures in seismic regions  design); the chapter 1.36 of the latter is dealing with special provisons to be
adopted under earthquake actions when masonry constructions are built.
This report is meant to provide a review of the last twenty years literature on masonry research, with
special emphasis on seismic loads. Attention is focussed upon the inplane behaviour of masonry from a
macroscopic point of view; in other words, masonry is mainly considered as a bidimensional material defined
by average characteristics. The local effects (concentrated loads) as well as the outofplane behaviour
(transverse or eccentric loads and instability) are not considered here.
The report is divided into five chapters: first, the current test techniques which are used to identify or
validate masonry inplane models are briefly presented (chapter II). The three subsequent chapters deal with
three aspects of the inplane behaviour of masonry, namely elasticity (chapter III), failure criterion ^chapter
IV) and stressstrain relationship (chapter V). Special emphasis is placed on the anisotropic effects due to the
mortar joints, this being the main feature that distinguishes masonry from concrete. Finally, in chapter VI,
differentanalytical models are presented together with their numerical implementation when available.
Distinction is made between models which try to reproduce the actual composite texture of masonry (micro
models) and those which attempt to define an equivalent continuum (macromodels).
The inplane behaviour of masonry considered as a material, is of first interest since it governs the
behaviour of plane structures such as shear walls or infills when submitted to inplane vertical and/or lateral
loads. Nevertheless, this latter aspect (behaviour of masonry plane structures) which has been described in a
preliminary note by Stefanou [SS], will be comprehensively reviewed in a subsequent report.
II TEST TECHNIQUES
The mechanical properties of a composite material such as masonry may be accessed at different
levels by means of specific tests. Three main categories of test may be thus distinguished: tests on elementary
components (unit, mortar, grout and steel), tests on small assemblages of those elementary components
(unit+mortar, unit+grout+mortar, etc.), tests on large assemblages (panel). At each level, measurements may
be restricted to a specific characteristic (compressive or tensile strength in a given direction, Young's modulus,
etc.) or may provide overall information on the mechanical behaviour (complete stressstrain relationship
under monotonie or cyclic loading). In any case, the great variability of the materials requires tests to be
repeated several times. A brief overview of the most common test configurations is presented hereafter.
II.1 TESTS ON ELEMENTARY COMPONENTS
ILl.lUnit
The units are essentially tested in uniaxial compression perpendicular to the bed faces (fig. la), the main objective being the assessment of their vertical compressive strength fb as indicated in Eurocode 6 [23].
1 1
t
ι
lil a) b)
Figure 1: Testing of the units under uniaxial compression perpendicular (a) and parallel (b) to the bed faces.
The same test can be performed in direction parallel to the bed faces to evaluate the lateral compressive
strength which may be very different from the vertical one in the case of hollow units (fig.lb) [22],
Uniaxial tensile tests in direction parallel to the bed faces are also possible by means of brush platens
glued to the sides of the unit. Finally, biaxial tests (vertical compression  lateral tension) have also been
carried out [33] [37].
II. 1.2 Mortar and grout
The compressive strength of both materials is generally measured on small prisms or cubes submitted
to uniaxial compression (EC6Appendix 2). The tensile strength may be assessed through a direct tensile test
by means of brush platens or through a brazilian test or even through a flexural test as indicated in EC6
Appendix 2 (fig.2). As for concrete, biaxial and triaxial compression tests are also possible [33] [37].
f r r 111
b) ■ ^
Figure 2: Testing of mortar and grout in tension: uniaxial tension (a), brazilian test (b) and flexural test (c).
II. 1.3Steel
The mechanical characteristics of steel (elastic modulus, yield stress and plastic modulus) are usually
determined by uniaxial tensile tests.
II.2 TESTS ON SMALL ASSEMBLAGES (MICROELEMENTS)
II.2.1 Bed and head joints
LLU
rm
¿¿¿* ΑΛΛ
ΠΤΤΠ rm Figure 3: Couplets (a) and triplets (b) under various loading conditions.
4
The behaviour of the mortar joint when considered as an interface is often studied on elementary
assemblages such as couplets or triplets [1][30]. Different test configurations may be used in order to create
varying shear stress  normal stress combinations on the joint (fig.3). It is worth noting that, in all those tests,
the state of stress induced in the mortar joint is strongly heterogeneous which makes it difficult to interpret the
results. More homogeneous stress distributions may be obtained on larger test specimen (fig.4). Monotonie and
cyclic load histories may also be considered.
Hart Jekil locali i»·
y S N i
Figure 4: Test setup and induced stress distribution in the mortar joint (from[3]).
II.2.2 Steelmortar and steelgrout interface
Pullout tests may be carried out in different configurations in order to evaluate the strength
characteristics of the interface (fig.5).
y / y y / /__/ / ; \
ν y / / / YZZ.
Figure 5: Pullout tests in the bed joint (a) or in the grout (b).
II.2.3 Masonry prisms
Stacked bond or running bond prisms are mainly used to evaluate the vertical compressive strength of
masonry (fig.6). Owing to their small size (one or two units width, five units height) prisms can not be
considered as large assemblages and therefore are not fully representative. However, the problems associated
with the interpretation of the results have been extensively studied [22] [36] so that uniaxial compression of
prisms is generally considered as a reference test (EC6Appendix 2). Besides the ultimate load, the full stress
strain relationship may also be recorded [48].
*) vz. Ζ J
J y j
/
Figure 6: Stacked bond (a) and running bond (b) masonry prisms.
II.3 TESTS ON LARGE ASSEMBLAGES (MACROELEMENTS)
An assemblage may be considered as a macroelement if its planar dimensions are approximatly one
order of magnitude larger than the largest microdimension (unit size). This includes fullscale as well as
reducedscale or even miniaturized panels.
II.3.1 Tests under homogeneous state of stress
These tests are unique in that they allow the direct measurement of the average characteristics of
masonry under biaxial stresses. Compressive and/or tensile normal loads are applied uniformly to the edges of
the panel so that the stress distribution is on average homogeneous hence statically determinate throughout the
panel [20][29][49]. The principal stress directions are parallel to the edges and the eventual anisotropic
behaviour due to the joints may be evidenced by varying the layup angle θ (fig.7a). The uniaxial compression
test performed on a storey height wall (EC6Appendix 2) may be considered as a special case of the general
biaxial test.
n i l ι ιΛ ι 11
<ς
u n i u n i i y ^ e
ΐ τ r τ Î Î î î Î *v / / V ; / / / s /
θ= ,Ο
V
Figure 7: General biaxial test (a) and uniaxial vertical compression test (b) on a masonry panel.
II.3.2 Tests under heterogeneous state of stress
Those tests are generally meant to simulate the actual loading conditions encountered in certain shear
walls or infills in buildings [7][9] [21][26][29][54] [55]. By contrast with the former tests, the average state of
stress induced in the panel is heterogeneous and statically indeterminate (i.e. dependent on the average
behaviour law of masonry). Different loading conditions may be considered under monotonie or cyclic loading
histories (fig.8). Those test are better suited to providing information on the response of structural assemblages
than for revealing basic material properties.
i l i i i M
t/ >)/i' > A
riVftW* S'
ττττ ι ι r / ι ι
Figure 8: Tests of masonry panels under heterogeneous state of stress.
Ill ELASTICITY
The elastic properties of masonry have received little attention from research workers. Most emphasis
has been placed on its ultimate strength capacity since it is of fundamental importance for design codes based
on limit state analysis. However, in order to perform a realistic analysis of masonry structures (i.e. a finite
element analysis), knowledge of both strength and stiffness properties of masonry is required.
Most early studies have been limited to uniaxial compression tests perpendicular to bed joints since it
is the main type of loading to which masonry is subjected. The results of such experiments are uniaxial stress
strain curves (fig.9) from which one can derive different quantities such as initial, tangent or secant Young's
modulus, compressive strength, strain at failure, etc.
<Γ
Figure 9 : Uniaxial stressstrain curve of masonry under vertical compression.
Many authors have thus proposed empirical relationships giving the Young's modulus of masonry as a function
of its compressive strength or even of the compressive strength of the units. Though being incoherent from a
mechanical point of view, similar relationships are used in most codes of practice. Among the numerous
proposals, let's record that of Eurocode 6 :
E, = 1000 fk (1)
where E, is the short term secant modulus and ft the characteristic compressive strength of masonry.
As far as Poisson's ratio is concerned, the values proposed in the literature range from 0.10 to 0.40. In
Eurocode 6, it is defined through the shear modulus G by :
G = 0.4E, (2)
which is equivalent to a Poisson's ratio ν of 0.25.
As quoted before, those empirical values of E and ν are based on uniaxial compressive tests
8
perpendicular to the bed joints. They are therefore adapted to vertically loaded masonry structures. They may
not be relevant for biaxially stressed structures (shear walls, infilled frames, walls supported on beams) unless
masonry is approximately isotropic in its own plane as it is generally assumed. Little attention has been paid to
the validation of such an assumption which seems against evidence if referring to the composite nature of
masonry : mortar and units. These may have quite different elastic properties and are arranged in such a way
that the horizontal and vertical direction are obviously not equivalent. This may be even emphasized when the
units themselves are elastically anisotropic due to the presence of horizontal or vertical holes.
The attempts to characterize the degree of anisotropy of masonry are essentially experimental. For
example, biaxial tests have beeen carried out on grouted concrete masonry panels [29] and brick masonry
panels [18]. Distinct orthotropic properties were observed. However, because of the high variability in the
elastic behaviour, typical of brickwork, it was found that isotropic behaviour could be assumed without
significant error. Nevertheless, this has not been generalized to masonry manufactured from hollow bricks or
to ungrouted block masonry for which a higher degree of anisotropy is expected.
The prediction of the elastic properties of masonry from known geometrical and elastic characteristics
of units and joints has been considered idealistic for a long time. The main arguments against such an approach
are:
 the incomplete knowledge of the characteristics of the components (unit and mortar),
 the great variability of the properties of masonry owing to the variability of the constituents and of
their arrangement (joint width, distribution of flaws, etc.).
Nevertheless a theoretical approach of the mechanical behaviour of masonry is a valuable
complement of experimental studies. Even if it may fail in predicting the characteristics of masonry, it is of
great help when interpreting experimental results and predicting the influence of some significant parameter.
Existing theoretical approaches of the elastic properties of masonry are direct applications of the
homogenization theory developed for composite materials. As a matter of fact, masonry may be considered as
a particular periodic composite material where mortar and units play the role of matrix and inclusions
respectively. Up to now, only approximate evaluations of the elastic constants of masonry have been
performed:
 Liang et al. [34] homogenized masonry in two stages. First, only horizontal joints have been taken
into account, masonry being considered as a regular stacked system of two alternating isotropic layers which
were mortar and brick. Second, vertical joints have been incorporated by considering a similar system
composed of vertical alternating layers which were mortar and the material resulting from the previous
homogeneization stage.
 Maier et al. [35] used a similar procedure with either one or three intermediate stages, the difference
in the results being found insignificant
Both procedures are approximate in two different ways :
 masonry is considered under the plane stress assumption,
 the true geometrical arrangement of bricks and mortar joints is not fully taken into account.
Those two approximations may be removed by performing a true homogenization on a three
dimensional cell with exact limit conditions on the boundary. Such a procedure would provide the exact
homogenized elastic matrix for any kind of masonry (hollow brick, ungrouted, grouted, etc.).
10
IV FAILURE CRITERION
The knowledge of the failure criterion of masonry is fundamental if an ultimate limit analysis is to be
performed, as is recommended in most codes of practice.
IV. 1 UNIAXIAL FAILURE
Due to the ease of testing piers or panels of masonry under uniaxial loading, a large number of experimental investigations have been carried out in order to study the influence on the masonry compressive strength of parameters such as brick and mortar properties (strength and stiffness) and geometrical characteristics of units and joints. Numerous empirical formulae have thus been proposed, which give the compressive strength of masonry as a function of some of those parameters (see for example Tassios [56] or Stefanou [55]). According to Eurocode 6, the characteristic compressive strength of masonry f̂ , when not
determined experimentally, may be assumed to be given by :
fk=KfbafJ (3)
where fb and fm are the mean compressive strength of unit and mortar respectively and Κ, α, β are coefficients
which are not yet fixed. The values suggested in the preface of Eurocode 6 are :
α = 0.75 β = 0.25 Κ = 0.4ψ
where ψ is a factor given by
Ψ = 'ISY*3
if the compression strength of the unit fb is lower than 15 N/mm2 and mortar strength is not stronger than MIO,
otherwise ψ = 1.
One of the first attempts to establish a rational relationship between the compressive strength of masonry and the strength of its constituents (brick and mortar) is due to Hilsdorf [31]. He exhibited a particular local state of stress satisfying both the equilibrium equations (in an approximate manner) and the respective failure criteria of mortar and bricks. To do so, he assumed that, in axially loaded masonry, the horizontal joints of mortar were in triaxial compression whereas the units were in a state of vertical compression and bilateral tension (fig. 10). According to him, this state of stress is induced by the differing deformation characteristics of mortar and unit, the latter being suffer than the former. Moreover, it explains two recognized features of
11
masonry failure under vertical uniaxial compression:
failure is initiated by vertical cracking or splitting of the units,
external compressive stresses at failure exceed the uniaxial compressive strength of the mortar.
Pi
Figure 10: Local state of stress in masonry prisms under uniaxial vertical compression.
As a matter of fact, such an assumption on the local state of stress may also be considered as a consequence of the lowerbound theorem of limit analysis: let's only assume that the local state of stress is given by two constant tensors
0 0
0 0 o b 0 0 a b
and 0 0
0 0 0 (4)
in units and mortar respectively, without choosing a priori the signs of the lateral stresses Φ and σ™. Then the
equilibrium conditions are
^ = o zm = p (5)
if ρ is the applied vertical pressure, and
h bo b + h V = 0 (6)
if hb is the height of the bricks and hm the thickness of the mortar joints (hm < hb). Hence , for any value of o*, the following state of stress
σ° 0 0
0 a
b
0
0 0
p and
ab/oc 0 0 0 ab/a 0 0 0 p
(7)
12
in units and mortar respectively, where α = hm/hb < 1, is statically admissible. It must be noted however that
this is not completly exact since the boundary condition on the free lateral edges are not fulfilled.
Assume now that the failure criteria of bricks and mortar are both of the MohrCoulomb type :
(8)
for bricks, and
_£i _£i <1 (9)
for mortar, where σι and σ^ are respectively the major and minor principal stress and σχ·» < σς* are the uniaxial tension and compression strengths of bricks (similarly στ™ < arj"1 for mortar). Usually, bricks are stronger than mortar so that ac* is assumed to be higher than Oc"1.
According to the lowerbound theorem of limit analysis, the highest value of ρ for which the state of stress (7) satisfies (8) and (9), is a lower estimate of the collapse pressure. Hence, σ*> has to be chosen in order to maximize p. Conditions (8) and (9) lead to different inequalities according to the position of σ*> with respect to zero and αρ.
Figure 11: Domain of the (p,o*) plane where both failure criteria of mortar and brick are satisfied.
If those inequalities are represented in a (p,o*) plane (fig.l 1), it becomes obvious that the highest value of ρ is reached in the region where 0 < ο* < ρ at the intersection of the two lines :
«b «b
and
Ρ =1
sothat
13
GJ+CLGŢ b OJ+KGQ Pmax=Tb ^ T
= < TC
στ σ^
Oc °c
CT τ + KCTÇ
of where Κ = α — *  (10)
CTC
Thus, the positive sign of o* is not an assumption but a necessary condition to the maximization of p.
Though rudimentary, the model of Hilsdorf is rational and coherent with the observed mode of failure.
Furthermore, it leads to a simple expression of the compressive strength of masonry which depicts the known
influence of various parameters. From expression (10), it follows that masonry strength increases with
increasing compressive strength of brick and mortar, increasing tensile strength of brick and decreasing ratio of
joint thickness to height of the brick. However, comparisons with experimental results are not fully satisfying.
According to Hilsdorf, this may be due to an erroneous assumption of the failure criteria for brick and mortar
(the MohrCoulomb criterion may not be adequate).
In order to derive better information on those criteria, Khoo and Hendry [33] carried out strength tests
on brick and mortar under appropriate states of stress. They found that the experimental failure envelope of
brick was concave in shape and considerably different from the theoretical straight line of MohrCoulomb.
Following the reasoning of Hilsdorf, they superimposed the two experimental failure criteria in the (ρ,σ·b)
plane (fig. 12) and thus determined a better estimate of the compressive strength of masonry.
/r&r)
Figure 12: Superposition of the experimental failure criteria of brick fb and mortar f m in the (p,o*) plane.
Another source of error in the model of Hilsdorf, is that it is based on an internal state of stress which
does not fulfil all the boundary conditions. Furthermore, the vertical joints have been completly neglected. A
fully statically admissible state of stress for masonry in uniaxial vertical compression has been recently
proposed by Biolzi [8]. The head joints are partially taken into account since they are supposed to have no
14
strength at all. As Hilsdorf did, Biolzi applied the lowerbound theorem of limit analysis to determine a lower
estimate of the collapse pressure. The maximization was performed numerically. Unfortunatly, no comparison
with experimental data was performed because the tensile strength of brick and mortar were missing. However,
the results still depict the principal known features of the rupture behaviour of masonry.
All the aforementioned approaches are based on limit analysis which supposes a perfectly plastic
behaviour of both constituents. A quite different way to derive the compressive strength of masonry is, on the
contrary, to assume a perfectly brittle behaviour of both mortar and units. The same simplified statically
admissible state of stress proposed by Hilsdorf (expression (7)) may be used again, but the lateral stress in the
units o* is then derived from an elastic calculus.
Tassios [56] assumed that both mortar and units were isotropic elastic, En,, vm and Et,, Vb being their
respective Young's modulus and Poisson's ratio, so that their respective lateral strain ε„, and 6b were given by:
em
eb
=
=
aE„
— + E b
E,
VbP Eb
Ρ vmob
vbob
Eb
At the unitmortar interface, those strains were equal so that the lateral strain in the units o* could be
determined:
h a(vmEhvKEm) , v σ =7 \ , \—ρ (H)
The positive sign of d° was thus a consequence of the differing deformation characteristics of mortar and units
(En, < Eb, vm >vb). The local state of stress being completely known, failure was assumed to occur first in the
units as it was observed experimentally. Tassios chose a MohrCoulomb failure condition for the units
(expression (7)) and thus obtained:
Pm.x = a(vmEbvbEm)
,(lvm)Eb+a(lvb)EroJ
(12)
This expression shows that masonry strength increases with increasing compressive or tensile strength of the
unit and decreasing ratio of joint thickness to height of the unit. However, the compressive strength of mortar
has no influence except if it is implicitly related to the elastic characteristics: for example, in [56], ac1» was
assumed proportional to En,.
15
Atkinson and Noland [S] performed the same elastic calculus but came to the conclusion that failure
should occur first by crushing of the mortar rather than splitting of the units, which was against experimental
evidence. They obtained the reversed result by taking into account the nonlinear behaviour of mortar:
expression (11) was thus written in an incremental way, with stressdependant elastic characteristics for
mortar:
b _ a ( v m (  p ; am
) E b  v b E m (  p ; am
) ) Δ σ ° = 7 i r ' Δρ (13)
( l  v m (  p ; C Tm
) ) E b + a ( l  V b ) E m (  p ; om
)
Expression (13) led to lower lateral stress values in mortar and therefore to failure by splitting of the units but,
as in the previous approach, the compressive strength of masonry was independent of the strength of mortar.
Besides being cumbersome (the Young's modulus and Poisson's ratio of mortar had to be known for a wide
range of stresses), this theory was found to systematically underestimate the prism strength [37]. This was
probably attributable to the fact that the model did not take into account any redistribution of stress after initial
cracking (elasticbrittle behaviour), in contrast with the approach of Hilsdorf based on limit analysis (perfect
plastic behaviour).
IV.2 BIAXIAL FAILURE
The knowledge of the vertical uniaxial compressive strength of masonry is far from sufficient as soon
as biaxially loaded structures are considered : in shear walls, in infilled walls in framed construction or in walls
supported on beams, the internal state of stress is characterized by coexisting zones in biaxial compression,
tensioncompression or, more rarely, in biaxial tension. In this regard, Eurocode 6 only defines the
characteristic shear strength fvk in the bed joint direction. When not determined experimentally, fvk may be
taken as
fvk =fvk0 +0.4 a d £ 0.05 fd (14)
but not greater then the limiting value given in table 1. In (14), fvko ι the shear strength at zero a¿, is given in
table l. Od is the least design vertical (i.e. normal to the bed joint) compressive stress and fb is the vertical
compressive strength of the units. The first part of expression (14) is a Coulomb's condition which is mainly
governed by the mortar strength (see the values of fvko in table 1) : it corresponds to the shear resistance of the
bed joints. The limiting values ( 0.05 fb and those given in Table 1) depend exclusively on the unit strength.
They are therefore related to a possible failure of the units. Hence, two failure modes are implicitly considered.
However it is worth noting that the stress normal to the head joint does not appear.
16
Masonry Unit
Perforated Hollow & Cellular Concrete Units
Calcium Silicate Units
Clay Units of fb less than 15/œm
a
Clay units of fb greater than 1 SN/mm»
Mortar
M20,M15,M10
M5, M2
M20,M15,M10
M5, M2.
M20,M15,M10
M5, M2
M20,M15,M10
M5, M2
fvk0 N/H'
0.2
0.1
0.2
0.1
0.3
0.1
0.3
0.1
Limiting fvk N/mm'
0.8 but not greater than the strength of the unit along its length»
0.8
1.0
1.5
• When the strength of the unit tested along i t s length can be expected to be greater than 0.15 times the vertical strength perpend icu lar to the normal d i r e c t i o n of l a y i n g , by consideration of the pattern of holes in the unit, this l imit can be assumed to be satisfied.
Table 1 : Values of fvk0 and limiting values of fvk (from Eurocode 6).
To permit a realistic design (ultimate limit state) and analysis (finite element method) of biaxially loaded masonry structures, the whole inplane failure criterion of masonry is required. In contrast to its elastic behaviour, the strength characteristics of masonry are highly sensitive to the orientation of the principal stresses with respect to the joint direction. This anisotropy is mainly due to the relative weakness of the mortar when compared with the units, and it may be emphasized if the units exhibit anisotropic strength properties because of perforations. Thus, to define masonry failure, a threedimensional surface in terms of the two principal stresses σι £ o% and their respective orientation θ and Θ+90 to the bed joint, is required (fig. 13).
Figure 13: The principal stresses σι £ aj and their orientation to the bed joints.
17
IV.2.1 Experimental approaches.
There have been few attempts to obtain experimentally a complete inplane failure criterion for
masonry because of the difficulty in developing a representative biaxial test as well as the large number of
tests involved. The most reliable experiments are those performed on panels under globally homogeneous
stress states since they are directly translatable in terms of masonry strength.
Hegemier and al. [29] carried out a comprehensive series of biaxial tests on full scale grouted
masonry panels (both reinforced and unreinforced) and found the influence of bed joint angle to be minimal
with the behaviour being essentially isotropic. The resulting failure surface was very similar to that of concrete
(fig. 14). However, the authors pointed out that this isotropy could be destroyed by a nonjudicious selection of
block and grout strengths.
y·' så·? t 7$±Γ
•¿r*—u —
Figure 14: Biaxial failure criterion of concrete (a) and of grouted masonry (b) (from [29]).
Samarashinghe and Hendry [49] obtained a (σι, θ2, θ) failure surface in the tensioncompression
range, from tests on 1/6 scale brickwork panels (fig. 15). This surface was found to be concave in shape
Figure 15: Tensioncompression failure criterion projected on the (σι σχ) plane (from [49]).
18
especially for small values of Θ, which is an indication of brittle failure. Furthermore, the strength was found
to decrease strongly with increasing bed joint angle Θ.
Page [45] performed the same tests on half scale brickwork panels and found a roughly similar failure
surface (fig. 16). In all cases, failure was brittle and occured in planes normal to the panel by cracking either
the joints alone or in a combined mechanism involving both brick and joints.
Figure 16: Tensioncompression failure criterion projected on the (σι σ2) plane (from [45]).
The same author [44] carried out biaxial compressive tests on similar panels, in order to derive the
failure surface in the biaxial compression range (figl7). For most principal stress ratios, the bed joint
orientation did not play a significant role; generally, splitting failure occured in a plane parallel to the free
surface of the panel. However, when one principal stress dominated, failure occured in a plane normal to the
w ν VYA^* Ζ
0 6 0 « IO 12 14
Figure 17: Biaxial compression failure criterion in the (σι C2 θ) space (a)
and projected on the (σι σύ plane (b) from [44].
19
free surface, by cracking and sliding in the joints or in a combined mechanism involving both bricks and
joints.
Due to complexities of testing, Page could not perform similar tests in the biaxial tension range. He
therefore simulated them using an iterative finite element computer program where bricks were assumed to
remain elastic and mortar joints were modelled as line elements with limited strength capacities [43]. Besides
being markedly influenced by the joint angle Θ, the derived failure surface (fig. 18) was found to be critically
dependant upon the relationship between the shear and tensile bond strength of mortar joints. Despite its
seeming plausibility, this failure surface is not coherent since it predicts different values of the isotropic biaxial
strength (σι = oi) for different values of the angle θ (fig. 18b). This is in fact impossible; in a plane isotropic
state of stress, all directions are equivalent since they all may be considered as principal directions. It is worth
noting that, in the biaxial compression range (fig. 17), the experimental results of the same author do predict a
unique value for the plane isotropic strength.
Figure 18: Biaxial tension failure criteriain the (σι σι θ) space (a)
and projected on the (σι ci) plane (b) (from [43]).
IV.2.2 Empirical approaches.
By an empirical approach, is meant that the available data are merely fitted by a surface, without any
mechanical interpretation of the failure.
Based on the abovementioned experimental results, different empirical idealizations of the inplane
failure surface of masonry have been proposed in analytical forms particularly suitable for numerical
applications in linear and non linear finite element analysis. Usually, the required parameters of the proposed
models may be determined by a few experimental tests.
By applying a general method proposed by Capurso and Sacchi [13], Nova and Sacchi [40]
20
generalized the MohrCoulomb failure criterion with tension cutoff to orthotropic materials and, as a special
case, to transverse isotropic materials. Though masonry may not be considered as tranverse isotropic, the plane
stress version of this last criterion appeared to be in good agreement with the experimentals results of Page in
the biaxial compression range (fig. 19). Six parameters were required. Unfortunatly, the uniqueness of the plane
isotropic strength ( i.e. for σι = <sj) was only satisfied in tension.
Figure 19: Plane stress version of the transverse isotropic criterion
generalizing the MohrCoulomb's condition with tension cutoff.
An even simpler anisotropic biaxial failure criterion has been proposed by Bernardini et al. [7] : for a
given value of Θ, the failure surface was assumed to be an hexagon (fig.20) and the coordinates of its vertices
were assumed to vary linearly with Θ. Such a simplified failure surface required only four parameters but the
uniqueness of the plane isotropic strength ( i.e. for σι = a¿) was neither satisfied in compression nor in tension.
Figure 20: Simplified anisotropic failure criterion depending on four parameters (from [7]).
21
Dhanasekar et al. [20] found it more convenient to derive the biaxial failure criterion in terms of the stress system related to the direction of the joints : normal stress to bed joint σ„. normal stress to head joint σρ,
shear stress to bed and head joints τ (fig.21).
rn 
9
\ Figure 21: The stress system related to the direction of the joints.
The failure surface in the (ση σρ τ) space has been approximated by three intersecting elliptic cones (fig.22), so
that eighteen parameiers (six for each cone) were required. However, in order to reduce the number of
identifying tests, an approximate method of establishing a conservative failure surface from six tests has also
been proposed.
Figure 22: Failure surface in the (σ„ σρτ) space, idealized by three elliptic cones (from [20]).
When transformed back to the (σι G2 θ) space, the idealized surface exhibits definite corners which are
inconsistent with the smooth experimental surface but reasonable agreement has been achieved (fig.23). It is worth noting that if expressed in the (σησρτ) space, the failure surface cannot but admit a unique value for the
isotropic biaxial strength in tension as well as in compression.
22
e=h5 θ=22.5° θ*Οβ
Figure 23: Comparison of experimental and idealized failure surfaces in
the (σι σ2) plane, for different values of θ = 45,67.5 and 90 (from [20]).
1V.2.3 Phenomenological approaches.
By a phenomenological approach, is meant that the form of the failure criterion is a priori based on a
mechanical interpretation of the failure.
Most contributions towards establishing a rational biaxial failure theory of masonry are
phenomenological; experimental results are interpreted on the basis of a criterion which is assumed to be
suitable for the observed mode of failure. Differences between formulae are mainly due to different failure
hypotheses but, even under the same failure hypothesis, the values of the constants are strongly influenced by
the way the tests are performed and interpreted.
Most early studies have been focussed on the tensioncompression range since this state of stress is
characteristic of the central part of shear walls where failure initially occurs in many instances. At first, failure
was assumed to occur for a critical value of the principal tensile stress. To assess the validity of this
hypothesis, tests have been carried out on walls subjected to biaxial loading resulting in an heterogeneous
internal state of stress [9] [57]. For an appropriate value of the tensile strength, the experimental results
appeared to be in accordance with the given hypothesis. But, as noted by Benedetti and Casella [6], the values
of tensile strength found for a given material through the interpretation of experimental results differ from one
another and must be seen as values characteristic of the test and not of the material. This does not invalidate
the assumed failure hypothesis but is due to the fact that the principal stresses in the center of the panel are
calculated on the basis of simplified assumptions closely related to the loading conditions which vary from test
to test. However, the maximum tensile stress criterion is isotropic and therefore not suitable for masonry
whose strength is highly sensitive to the orientation of the principal stresses with respect to the joint direction.
Furthermore, it only accounts for failure by splitting though failure by joint separation was also observed under
the same loading conditions.
Still in the tensioncompression range, Yokel and Fattal [60] tried to take into account those two
possible failure modes by combining two failure criteria, one for each mode. For splitting, failure by a critical
23
linear combination of principal stresses appeared to be a better hypothesis than failure by a critical tensile
stress (this is still an isotropic criterion). The Coulomb's theory of friction was used to express bed joint
separation (head joints were not considered) i.e., failure was assumed to occur for a critical combination of
normal and shear stresses acting on the bed joint plane. This is an anisotropic criterion since it depends on the
bed joint orientation with respect to the principal directions of stress. It must be pointed out that both criteria
were stated in terms of nominal stresses i.e., average stresses acting on a representative volume of masonry. It
is therefore postulated that, under given nominal stresses, failure is not sensitive to the local stresses, as long as
a change in local stresses does not precipitate a change in the failure mode. In other words, both failure criteria
are characteristic of masonry as a whole and not of a constituent. In particular, the criterion for joint separation
should not be confused with the failure criterion of mortar : the latter would be stated in terms of the local
stresses which are undetermined unless further hypotheses are assumed (elastic behaviour for instance). Even
so, local stresses turn out to be quite different from nominal ones. A good example is the local state of stress
derived by Hilsdorf for masonry under nominal uniaxial compression (triaxial compression in mortar, biaxial
traction and uniaxial compression in units). Distinction between local and nominal stresses, or global and local
failure criteria , was clearly stated by Yokel and Fattal. However, some confusion about it will often be found
in subsequent studies. Yokel and Fattal confronted their criterion with test data derived from the failure of
brick walls under combinations of compressive diagonal and edge loads. Nominal stresses at failure were
derived on the basis of an elastic calculus. Whenever failure was caused by splitting or joint separation, load
capacity was reasonably predicted by the corresponding criterion for appropriate values of the constants
involved. However, the mode of failure itself was not predicted since the observation of it conditioned the
choice of the relevant criterion. In other words, failure did not coincide with the lower envelope of the two
criteria as it should do. Moreover, the anisotropic nature of masonry has been partially taken into account
through the criterion for bed joint separation but directional variations of splitting strength have been
completely neglected.
Following basically the reasoning of Yokel and Fattal but rectifying some of its deficiencies, Hamid
and Drysdale [28] derived a more general failure criterion for grouted and ungrouted concrete block masonry
under any state of biaxial stresses. This criterion was defined as the lower envelope of three conditions
expressed in terms of nominal stresses and related to three different modes of failure : splitting, bed joint
separation and head joint separation (fig.24). The splitting criterion was provided by a modified version of the
phenomenological orthotropic failure condition proposed by Hoffmann [32] for brittle composite materials, for
which five parameters were required. The failure criteria for bed and head joint separation have not been
completly postulated as they should have been. In each case, a kind of average has been made between the
Coulomb criterion used by Yokel and Fattal for the mortar joint, and the local criterion of the other constituent
involved in the failure mode : grout for bed joints in grouted masonry, block for head joints (fig.24). Such an
average was implicitly based on the assumption that the local normal stresses in both constituents of the joints
were equal to the nominal ones, which is known to be wrong in general. However, this allowed the
identification of some of the parameters of the proposed criterion from preliminary tests performed on the
24
constituents. To evaluate their failure criterion, Hamid and Drysdale used results from concrete block masonry
\\&k joiJ separaKo'
See) jotar ¿cadra Be>*
Figure 24: The three possible failure modes of masonry under biaxial stresses.
prisms tested under compressive or tensile loads for different orientations of the bed joint from the applied
axial load. The agreement was not as good as could be expected in view of the large number of parameters
introduced in the criterion (eight or ten depending whether ungrouted or grouted masonry was considered).
Ganz and Thurlimann [26] [27] proposed a failure criterion for perforated brick masonry as the lower
envelope of five conditions expressed in terms of nominal stresses. As in the former approaches, those
conditions were meant to represent specific failure modes involving either brick or mortar. However, they
were not postulated but derived from the failure criteria of the constituent materials by means of simplified
assumptions on the local state of stresses :
 the structure of the perforated brick has been divided into inplane elements (web elements) stressed
biaxially and transverse plate elements (rib elements) stressed essentially uniaxially (fig.25). Assuming a
square failure criterion without tensile strength for the brick material, an anisotropic failure criterion has been
obtained by combining those of the web and rib elements.
ÖEES 77/λ Sireacii tf «M/O//y
(Wtb Eltmtnl)
p a n Stressed Uniauollf (Rib Eumèni s i
0 ¿ U Mortar (HeaJ joint)
Figure 25: Local state of stress in bricks : view of a section parallel to the bed joints (from [26][27]).
 a Coulomb failure criterion with zerotension cutoff has been assumed for the shear resistance of
the bed joint (the head joints have been neglected), the local stresses in mortar being implicitly identified with
the nominal ones.
25
Theoretically, the lower envelope of the five conditions thus obtained (three for brick, two for mortar) should
have provided the failure criterion of masonry in terms of the strength characteristics of the constituents. In
fact, a simple check showed that the resulting criterion would not have been realistic; the masonry compressive
strength would have been equal to that of the brick against experimental evidence. In order to rectify this, the
authors injected masonry characteristics instead of those of brick. In other words, the four parameters of the
proposed failure criterion were fitted on experimental results obtained on masonry. The approach of Ganz and
Thurlimann is therefore more phenomenological than theoretical. Partial agreement was found with
experimental results. However, the shape of the failure surface in the (σ„ σρ τ) space roughly corresponded
with the idealized surface derived by Page in a purely empirical way (fig.26).
Figure 26: Comparison in the (ση σρ τ) space, of the empirical failure surface proposed
in [20] (a) and the phenomenological one proposed in [26] (b).
%
IV.2.4 Theoretical approaches
By a theoretical approach, is meant that the failure criterion of masonry is derived from the geometric
and mechanical characteristics of the constituents (unit, mortar, unit/mortar interface).
For the tensile strength of masonry normal and parallel to bed joints, Drysdale and Hamid [21] and
Tassios [56] proposed simple expressions based on the strength and geometric characteristics of the
constituents. Those expressions have been obtained by superposition of the individual strength of the
components involved in different failure modes observed in brazilian tests. For tension normal to bed joints,
the splitting crack passed along a bed joint/unit interface (fig.27a) so that the tensile strength was identified
with the tensile bond strength of the mortar/unit interface. For tension parallel to the bed joint, two failure
modes were envisaged: generally the splitting crack passed along the head joints on alternate courses (fig.27b)
so that the tensile strength was identified with an average of the tensile bond strength of the mortar/unit
interface, the tensile strength of the mortar and the tensile strength of the unit, weighted by the corresponding
areas; however, for stronger units and/or low compressive stress normal to the bed joint and/or small
overlapping of units in running bond, the splitting crack was expected to have a stepped appearance resulting
26
from tensile debonding at the head joint/unit interface and shear debonding along the bed joint (fig.27c) so that
another expression was proposed. The tensile strength was therefore given by the minimum of the two
formulae. Such a calculation may be considered as an application of the upperbound of limit analysis, the
tested mechanisms being the observed failure modes.
°> ±> c) ■
«
: > r . ,
" »
:
1 3
Figure 27: Failure modes for tension normal (a) and parallel (b,c) to bed joint
Recently, a failure envelope for masonry in the biaxial compression range, has been derived
numerically by Papa as a particular aspect of a more general study on masonry behaviour [47]. Simplified
homogenization procedures and continuum damage models have been implemented in a conventional finite
element code. Masonry was conceived as a twodimensional planestress periodic continuum composed of
linearelasticbrittle bricks and elastic mortar susceptible to damage. More details are given in chapter VI.3. As
far as failure is concerned, the resistance domain derived by Papa (fig.28) was qualitatively similar to the
experimental one obtained by Page (fig. 17). To our knowledge, this was the first attempt to derive the
complete biaxial failure criterion of masonry from the characteristics of the constituents. However, it required
the identification of many parameters (four for brick and seven for masonry) among which many were not
directly related to failure (elastic coefficients for instance).
Figure 28: Biaxial compression failure criterion projected on the (σι ai) plane (from [47]).
27
V STRESSSTRAIN RELATIONSHIP
V.l UNIAXIAL BEHAVIOUR
The stressstrain relationship of masonry under uniaxial compression is of the form given in figure 29.
Figure 29: Typical uniaxial stressstrain curve of masonry.
Besides the elastic and strength properties which have been reviewed in the previous paragraphs, the other significant parameters are the peak strain ε„„ the slope of the falling branch and the ultimate strain ε^ According to Eurocode 6, this stressstrain curve may be taken as parabolic rectangular as given in figure 30.
<f*
Ο.0Ο2. 0.0035" *
Figure 30: Idealized stressstrain relationship for the design of unreinforced masonry.
This is the same simplified design diagram as the one recommended for concrete in Eurocode 2. In particular, the values of ε,„ and ε„ (0.002 and 0.0035 respectively) are characteristic of concrete and do not depend either on the type of masonry considered (clay, stone or concrete units, grouted or ungrouted, etc), or on the direction of loading with respect to the bed joint. As a matter of fact, there has been little research into the specific stressstrain characteristics of masonry except for the strength value. Furthermore, there is no agreement about the values of the peak and ultimate strains in uniaxial compression, even for the same kind of masonry and the same direction of loading (normal to the bed joint usually).
28
According to Atkinson and Kingsley [4], clay and concrete masonry exhibit essentially the same
strain characteristics (ε™ = 0.0026, ε„ = 0.0038).
Turnsek and Cacovic [55] found higher values for brick masonry (£„, = 0.005, ε„ = 0.014).
Priestley and Elder [48] found that the behaviour of concrete masonry prisms under compression
could be adequately predicted by a modified version of the KentPark curve proposed for concrete. However,
they recommended lower values for the peak and ultimate strains (0.0015 and 0.0025 respectively).
Furthermore, they showed that the ductility of the stressstrain relationship (i.e. the ratio eje^ could be
effectively improved by the presence of thin stainless steel confining plates within the mortar bed, ultimate
strains greater than 0.012 being then observed (fîg.31).
Figure 31: Effect of confining plates on the stressstrain curve of concrete masonry prisms (from [27]).
Samaringhe et al. [50] established stressstrain curves for brick masonry in uniaxial compression for
varying bed joint orientation to the applied load(Fig.32).
Figure 32: Stressstrain curves in compression for brick masonry with varying bedjoint angles (from [50]),
29
The falling branches were missing since the tests were performed under increasing loading up to failure. The
peak strain values were considerably different according to the bed joint orientation, from 0.0003 for 67.5° to
0.004 for 0».
Recently, Naraine and Sinha [38] [39] studied the behaviour of brick masonry under cyclic
compressive loading both perpendicular and parallel to the bed joint As had been done for concrete, they
established an envelope stressstrain curve coinciding approximately with the monotonie curve, a common
point curve and a stability point curve (fig.33). Little difference was found between the two loading directions.
The values of the peak and ultimate strains were rather high (0.006 and 0.009 respectively).
A
Α«αΙ «Ιιαιη
Figure 33: Brick masonry stressstrain curve under cyclic compressive loading (from [38]).
V.2 BIAXIAL BEHAVIOUR
The available experimental data regarding the biaxial behaviour of masonry are focussed on the
strength characteristics. Little attention has been paid to the corresponding deformational characteristics.
Dhanasekar et al. [19] have explored the compressioncompression and the tensioncompression
ranges for brick masonry, up to peak load only (fig.34).
s> <ζ +
« I WO HO 1200 IOO0 1_ .L > J
 É l » · * wo no ixo m
& * * * wo KO 1300 m ' »
30
b 'η λ °f *
irfi εΛ.ιΰ ¿pio1· t. io"
Figure 34: Stressstrain curves for brick masonry in biaxial compression (a)
and tensioncompression (b) (from [19]).
The results were plotted with reference to the bed joint direction (σηε„, σρερ, τγ), not to the principal
loading directions as is usually the case for concrete. In biaxial compression, the curves exhibited a marked
nonlinearity before failure, whereas in tensioncompression, they remained essentially elastic up to failure.
V.3 RATE EFFECT
There has been little investigation into the influence of the rate of loading on the response of masonry.
Most available results concern uniaxial compressive tests performed on prisms at different loading rates
[36] [48]. Generally, as for concrete, a higher loading rate was found to result in a higher strength. As an
example, Priestley and Elder [48] noted that increasing the strainrate from 0.000005 to 0.005 s1 resulted in
about 20% increase in strength for unreinforced masonry prisms (fig .35). '
' 0 3.1 O.J 0.3 3.* 3.S ;.6 3.7 34 £ .¡Λ.
Figure 35: Comparison between stressstrain curves for low and high strain rates (from[48]).
31
In [29], Hegemier et al. reported the results of cyclic compressive tests performed at different
frequencies, on a concrete masonry panel, in the linear range (low stress level). The hysteresis loops were
narrow and remained invariant within a range extending from essentially quasistatic (0.005 Hz) to typical
expected mode frequencies for full scale structures (2.0 Hz). Therefore, except in the neighbourhood of a given
frequency, viscous damping was found inappropriate for implying strainrate dependence. This lead the
authors to question both the meaning and the value of the damping factor assumed in the seismic analysis of
masonry buildings: if structural damping originates from the material, then it should not be viscous and its
value should not exceed 2%. However, the high viscous damping currently used (8 to 10%) may be the result
of connection behaviour or some other aspect of the structure.
32
VI FINITE ELEMENT MODELS
In view of numerical analysis through a Finite element method, many different models have been
proposed for simulating the behaviour of masonry under biaxial stresses. Two main approaches may be
considered: micromodelling and macromodelling. The first aims primarly at representing the actual texture
of masonry, units and mortar joints being considered separately as homogeneous subregions each characterized
by distinct properties. The second attempts to define an equivalent continuum, the characteristics of which
permit the description of the global behaviour of masonry. The two approaches are complementary: on the one
hand, micromodelling is a theoretical approach which attempts to synthetize the behaviour of masonry from
the knowledge of the properties of each constituent and constituent interface; the necessary data have to be
extracted from smallscale laboratory tests (cf. II. 1 and II.2); for requiring refined finite element meshes, this
approach focuses on the detailed analysis of small structural elements (piers, shear walls) with particular
interest on strongly heterogeneous states of stress (connection elements, elements subjected to concentrated
loads). On the other hand, macromodelling is primarly phenomenological; the unknown parameters have to be
determined through tests performed on assemblages of sufficient size under homogeneous states of stress (cf.
II.3.1). For requiring coarser finite element meshes, such an approach is particularly suitable for the global
analysis of full structures but also for structural elements of sufficient size (shear walls) provided that local
effects are negligible.
VI. 1 MICROMODELLING
Particular attention has been paid to the mortar joints since they constitute a major source of nonlinearity and failure. Strictly speaking, in a planestress micromodel, both mortar joints and units should be represented by bidimensional elements for being considered at the same microlevel. Practically, because of the numerical problems related to their low thickness, joints have been generally modelled by onedimensional linkage elements whereas units were discretized using conventional plane stress elements (two or four elements for each brick). The scale of modelling was therefore slightly higher since the mortar joints were considered infinitely thin by comparison to the units. However, this was still micromodelling in the sense that bricks and mortar were considered separately.
In the model proposed by Page [41] [42], bricks were assumed isotropic elastic. Their elastic constants have been derived from uniaxial compressive tests: Young's modulus has been taken as the average of the moduli for load parallel and normal to the faces corresponding to the bed joint; an average experimental value of 0.167 has been used for Poisson's ratio. The linkage element representing mortar joints had limited strength properties and could deform both in the normal and shear direction according to two nonlinear stressstrain relationships independently of each other. Those relations have been determined indirectly from uniaxial compression tests performed on masonry panels with varying bed joint orientation: the elastic strain of the bricks was calculated on the basis of their average elastic properties and subtracted from the global measured strain (fig.36).
33
Figure 36: Stressstrain curves in compression (a) and shear (b) for masonry, bricks and mortar (from [42]).
The failure criterion of the joint element has been obtained directly from the same masonry tests conducted up
to failure: three linear bestfit curves have beeen used to describe the failure surface in the normal stress 
shear stress plane, one in the tension range and two in the compression range (fig.37).
1
X
..Á
*L
•
Figure 37: Assumed joint failure envelope in the normal stress  shear stress plane (from [42]).
Beyond failure, residual properties were assigned only to those joint elements which had failed under a
compressive normal stress: the normal stiffness remained unchanged while the shear stiffness was reduced
according the magnitude of the compressive stress present at failure.
Incorporated in an incremental finite element program, such a micromodel has been used to
reproduce an inplane bending test on a deep masonry beam under vertical load. Stress distributions have been
reproduced to a reasonable degree of accuracy, even for higher loads when substantial stress redistribution had
occurred. However, as the criterion for brick failure was not included, the ultimate load could not be predicted
since the final collapse involved both bricks and joints. Moreover, cyclic loadings could not be simulated
because unloading of the joint element had not been envisaged.
Chiostrini and Vignoli [15] proposed an even simpler micromodel. Bricks were again assumed
isotropic elastic whereas mortar joints were introduced using gapelements characterized by a stiffness in the
34
closed position and a friction coefficient for interface sliding. The physical constants were not derived from
elementary tests on the constituents but chosen in order to reproduce global experimental results. For suitable
values of the parameters, the overall behaviour of a masonry panel submitted first to a vertical pressure and
then to an increasing horizontal load has been perfectly reproduced (not predicted), together with the collapse
mechanism (fig.38). This was possible because bricks were not involved in the failure. As with the model of
Page, neither collapse loads involving bricks, nor cyclic responses could be predicted.
Figure 38: Analytical (+) and experimental (o) loaddisplacement curves
and predicted collapse mechanism (from [IS]).
Those two possibilities were included in the micromodel proposed by Arya and Hegemier [2] for
reinforced grouted concrete masonry. Since the basic constituents of the units were steel and concrete (block
and grout), it was found appropriate to use a material model initially developed for reinforced concrete. Thus,
grouted concrete blocks were assumed elasticbrittle in tension and elastic strainsoftening in compression.
The Von Mises yield criterion was used for failure under biaxial compressive stress whereas
the maximum tensile stress criterion was adopted for cracking due to tension. The material was assumed
isotropic elastic before the yield curve was reached. In the biaxial compression range, loading beyond the yield
curve was assumed to follow a nomothetic yield curve, shrinking to the origin with the increasing value of the
equivalent strain ε = (ε^ + t2i  £\t2)Xfl , until the collapse curve for a given value tp of ε (fig.39).
Figure 39: Initial (solid line), subsequent (dotted line) and ultimate (dashed line)
yield/failure surface for concrete masonry (from [2]).
35
A collapsed element was then considered as a void (no stiffness, no strength). Once it had reached the initial
yield curve, an element was also declared collapsed as soon as it was subjected to tensile stresses.
In tension cracking, the cracks were assumed smeared across the element and normal to the major
principal stress existing just prior to cracking. The elastic tensor of a cracked element was assumed
orthotropic with respect to the cracking direction: the Poisson's ratio and the stiffness normal to the cracks
were set to zero and the shear stress modulus was reduced by a factor depending on the opening of the cracks.
Simultaneously, the compressive strength parallel to the cracks was assumed to decrease with the opening of
cracks. When the cracks closed, the element was considered to regain its entire elastic stiffness except for the
shear modulus which was only partially recovered. Each element could have two sets of cracks, the second
being formed while the first was either open or closed. When both sets were open the elastic tensor was set to
zero.
The reinforcing steel was assumed elastic perfectly plastic in both tension and compression. In each
direction of reinforcement, the steel was replaced by an equivalent uniform layer with stiffness only in its own
direction. Perfect compatibility of displacements was assumed between steel and units. The effect of bond was
only taken into account in the strength of cracked units: when reinforced, the strength of cracked units was not
supposed to drop to zero instantaneously but to diminish exponentially with the plastic strain of the yielding
steel.
In a reinforced unit, the behaviour of each constituent (crushing or cracking of the unit, yielding of the
steel) was governed by the state of stress existing in the constituent, not by the total one in the element. Due to
the perfect compatibility of displacements, the total stress tensor increment was related to the strain tensor
increment by the sum of the constitutive tensors of the two constituents expressed in the same coordinate
system. Theoretically, the resulting constitutive tensor might have been singular (cracked element without
reinforcement or with yielded reinforcement) or even negative (strainsoftening). To circumvent numerical
instabilities, a null tensor was used for elements in the post yielding region, stresses being then governed by
the assumed shrinking yield curve, and a small artificial stiffness was assigned to those diagonal terms which
were theoretically zero.
A linkage element was used to represent mortar joints as well as groutblock interfaces. Perfect
adherence was assumed as long as the strength capacities of the interface given by a MohrCoulomb's
condition were not exceeded. The cohesion and the coefficient of friction were assumed to be decreasing
functions of the relative tangential displacements at the interface. When the failure criterion was violated
under tensile normal stress (debonding), complete separation of the interface was assumed. When the failure
criterion was violated under compressive normal stress (sliding or recontact), perfect normal contact was
assumed whereas tangential displacements were eventually adjusted for the failure condition to be again
respected.
This micromodel has been implemented into an incremental finite element code. The progressive
change in stiffness and strength characteristics of the structure due to cracking and/or crushing of units,
debonding, sliding arid/or recontact at interfaces and yielding of reinforcement required recomputation and
updating of the constitutive tensor at each load increment. The new constitutive tensor was determined by
36
iterating the equilibrium equations until obtaining a state of stress and strain compatible with the current
stiffness and strength characteristics of each constituent. The iterative process was terminated if either the
incremental displacements or the nodal forces converged in the sense of the Euclidian norm.
The behaviour of reinforced and unreinforced masonry walls under constant vertical pressure and
monotically increasing shear deformation has been simulated. The correlation between analytical and
experimental results was good: the predicted ultimate strength of the specimens was within 10% of the
experimental values. The brittle behaviour of masonry was correctly predicted (fig.40).
<9
3 «0 
8 (in.)
£> JV5
8 (in ) £frfrW Figure 40: Experimental and analytical loaddisplacement curves and critical deformed shape
for reinforced (a) and unreinforced (b) specimens (from [2]).
The reinforced specimen has also been subjected to a dynamic cyclic shear deformation of increasing
amplitude, still under constant vertical pressure. In the numerical analysis, only few cycles have been
simulated since the objective was to predict the failure envelope. Good correlation was achieved though the
load had been applied statically in the analysis (fig.41).
37
6 0 r
_ 401
ANALYSIS EXPERIMENTS
.03 .06 .09 δ ( ¡ η )
.12 .15
I ' Ι  ■.
I >*·
ι *■ ι
^ J y
«TT T x f >
f 't \ 1 >■ I
>. "" ' 1 ■ I
7 >* 7 x~ ] X 1 X.
/ >\ x 1
ν

f—f *^y s j V ■7 / 7 '·
H1
>; X X
y
' ' i
/. L
/ V ƒ > >* ;̂
Γ 7 
/ / ι
_*■ f / / ■
/ _?■ /_T"
> ■
'■· 1
■ — 1
. /
tørt· 3xiS
Figure 41: Experimental and analytical failure envelope for a reinforced specimen
under cyclic loading and selected successive deformed shapes (from [2]).
As already mentioned, micromodelling of masonry requires a refined knowledge of each constituent.
Sophisticated constitutive laws have thus been elaborated to describe complex mechanical events such as
cracking and crushing of the units and debonding, sliding or recontact of the joints. Unfortunately, such refined
idealizations may be partly disappointing for different reasons :
 The mathematical properties leading to convergence may not be assured for the solution process (softening,
vanishing stiffness).
 The introduced constitutive parameters are often numerous and difficult to identify because the smallscale
tests are not always representative and the tests results suffer from scattering.
 The computer capability is easily exhausted by the high number of degrees of freedom required to discretize
even small structures: generally it has only been possible to represent twenty to thirty bricks.
Consequently, modelling research has moved towards macromodels even for dealing with fairly
simple structures.
VI.2 MACROMODELLING
The existing macromodels may be divided in two main groups: those which are essentially similar to
the ones used for (reinforced) concrete, with slight modifications for characteristics peculiar to brickwork
(concrete macromodels), and those which have been specifically developed for masonry (original macro
models). Concrete macromodels are in fact quite suitable for fully grouted reinforced concrete masonry,
38
which is expected to behave similarly to reinforced concrete. However, they may fail in representing the
behaviour of unreinforced brick masonry since they never take into account the marked anisotropic effects due
to the mortar joints, whereas original macromodels have been elaborated specially to account for this
particular feature.
VI.2.1 Concrete macromodels
Modelling of concrete under biaxial stresses has been the object of intensive research for a long time.
A comprehensive review of current constitutive macromodels has been made by Chen [14]. This author
distinguished two main types of model: the elastic  hardening plastic  fracture models and the nonlinear
elastic  fracture models. In the former, the nonlinear response of concrete under biaxial compresive stresses is
described by the plasticity theory, whereas the nonlinear elasticity theory is used in the latter. In both cases,
the softening behaviour beyond failure is generally described by the smeared crack approach where concrete is
still assumed as a continuum after cracking; the other relevant theories (plastic softening, damage, discrete
cracking) are less frequently used.
More or less elaborated concrete models have been used to simulate the nonlinear behaviour of
masonry. In most cases, the original model was used without any fundamental modification, the various
parameters being merely adjusted to fit with the characteristics of masonry (elastic constants, strength values,
etc).
Ganju [25] used an early concrete model of the first type: the behaviour of unreinforced masonry was
assumed elastic  perfectly plastic  brittle in compression and elastic  brittle in tension. The yield surface was
given by the DruckerPrager condition. In the compression range, failure was assumed to occur for a given
maximum strain value. Unfortunately some information is missing in the referenced paper (failure criterion in
tension, postfailure behaviour, flow rule, etc).
Monotonie shear tests on vertically compressed panels have been simulated. The agreement between
theoretical and observed values of the ultimate load was reasonable in view of the coarse mesh adopted.
Shing et al. [54] used the same type of concrete model, but much more sophisticated, in the case of
reinforced masonry: the behaviour of an element was assumed elastic  hardening plastic  brittle in biaxial
compression and elastic  brittle otherwise. Crushing or cracking was assumed depending on whether the
ultimate compressive or tensile strain was reached (a stress criterion was also adopted for cracking). In the
post crackingfailure range, the smeared crack approach was used and the tensionstiffening effect was
simulated by allowing a gradual drop of tensile stress. No softening regime was assumed in the post crushing
failure range.
Reinforced masonry panels subjected to constant vertical pressure and cyclic lateral displacement of
increasing amplitude had been previously tested. However, only monotonie loading was applied in the
analysis: the resulting monotonie loaddisplacement curve was very close to the envelope of the
39
experimentally obtained hysteresis curve (fig.42), especially for panels exhibiting flexural behaviour (toe
crushing). For panels exhibiting shear behaviour (diagonal cracking), the diagonal crack opening could not be
satisfactorily modelled by the smeared crack approach: the shear cracking load appeared to be underestimated
while the ultimate shear strength was overestimated; furthermore, the analytical results still indicated a flexural
behaviour. According to the authors, this was due to the residual strength attributed to the cracked element; the
diagonal crack opening would have been probably better simulated by a discrete crack modelling.
ANALYSIS EXPERIMENT
TOE CRUSHING
0.00 0.30 1.00 U O 2.00 LATERAL DISPLACEMENT [ I N ]
Figure 42: Experimental and analytical envelopes of a wall exhibiting a flexural response (from [54]).
Calvi and Gobetti [11] used a concrete model of the second type based on the theory of
hypoelasticity: the nonlinear behaviour of unreinforced masonry was described by an incrementally linear
elastic stressstrain relationship with an isotropic tangential stiffness tensor dependant on the current state of
stress. This stress dependency was expressed through variable tangential bulk and shear moduli, so that stress
and strain increments could be easily decomposed into hydrostatic and deviatoric components related by
decoupled relations.
As for concrete, the tangential shear modulus G was assumed to be dependant on the second invariant
of the deviatoric stress tensor J2, and the favourable effect of the mean compressive stress on the shear stiffness
was added by introducing the first invariant of the stress tensor li:
G=G0^y24h (15)
where Go, γι and f¿ were positive constants to be experimentally determined. G could not become negative:
when it reached a given positive minimum, the material was assumed incompressible with no shear stiffness. In (20), the hydrostaticdeviatoric coupling introduced by l\ did not appear in the incremental relations since
an explicit NewtonRaphson method was used.
For the tangential bulk modulus K, two constant values were assumed, a positive one, K\, before
cracking and a negative one, K2, in the post cracking range; this was meant to represent the compaction of
uncracked masonry and the dilatation of cracked masonry under compression. In the transition zone, Κ did not
pass through zero but through infinity (isovolumetric behaviour).
40
The cracking criterion was defined as the lower envelope of two different conditions expressed in
terms of the principal stresses (σι > σί): on the one hand, when tensile, the major stress σι could not exceed
the tensile strength of masonry fCM and, on the other hand, a linear combination of σι and σ 2 could not exceed
the tensile strength of the bricks fd,. This latter condition had been derived from a roughly approximate elastic.
calculus: both masonry and bricks being assumed isotropic elastic, the tensile lateral strain in bricks (i.e.
parallel to σι) has been estimated as the difference between the lateral strain of masonry (CM = 02/EM ·
ÖIVM/EM) and that of bricks without mortar (% = o2/Eb  oiVb/Eb). The corresponding tensile lateral stress (σ
= Eb(£M  Eb)) being limited by the tensile strength of masonry, the following condition was obtained:
σ = ^ ^  σ 2 Ε
» ν Μ  Ε Μ ν „ σ ι = CiCT2 _ C 2 G i ( 1 6 )
where the coeficients Cj and C2 were positive. It must be pointed out that the tensile lateral stress in bricks
might have been derived in a more rigorous way from equilibrium and compatibility conditions, mortar and
bricks being assumed isotropic elastic (generalization to the biaxial case of the elastic calculus leading to
equation (11) in IV. 1); in that case, another linear combination of the principal stresses would have been
obtained with coefficients expressed in terms of the relative thickness a=hm
/hb and the Young's moduli and
Poisson's ratios of bricks and mortar:
Eb(l + a ) ( a E m ( l  v b v m ) + Eb ( l vS 1 ) ) a ( v m E b  v b E m )
( a E m + E b )2 ( c c E m v b + E b v m )
2 °2 a E r a ( l + v b ) + E b ( l  v m ) "
σ = , _ v
_ χ2 , H r ^   „"!■' .„ /".. ,a1=qCT2C2o1 (17)
where C'i and C'2 were also positive.
Nine input parameters were therefore required for the definition of the model: three for the shear
modulus (Go, γι. fi), two for the bulk modulus (Kj, K2) and four for the failure criterion (fCM, fcb. Q , C2).
They were identified by two experimental tests on masonry panels: a vertical compression test and a diagonal
compression test (brazilian test). Both tests were interpreted on the basis of simplified assumptions regarding
the state of stress and strain. The brazilian test was suitable to determine Go, γι, "ti and fcM, whereas Κι , K2
and C2 were derived from the vertical compression test provided that fcband C\ were already known.
In order to check the validity of the model, the two identification tests have been simulated
numerically. Good agreement between the experimental and analytical loaddisplacement curves was achieved
in both cases (fig.43 and 44).
A monotonie racking test realized previously has also been simulated. Go, γι and fz have been
obtained directly from the experimental loaddisplacement curve. Kj and fCM were available from direct
measurements. K2, fcb, C\ and C2 have been taken from other tests on solid brick masonry. Experimental and
analytical loaddisplacement curves were again in good agreement (fig.45).
41
i
Figure 43: Vertical load  vertical displacement curves for the vertical compression test
and deformed shape of the specimen (from [11]).
tfp
I ι
' * V
■v
\ 
\
\
r
1 Ul 
\X', \±
\
μΙ ' WY Ψ\
\
.
\ \
.
& Figure 44: Diagonal load  diagonal displacement curves for the brazilian test
and deformed shape of the specimen (from [11]).
pr "Œ
i ! /;/ 1/ TT
v \ y'
A>iyNi/_ τ
i/ ¡/i iL · /
Figure 45: Lateral load  lateral displacement curve for a monotonie racking test
and deformed shape of the specimen (from [11]).
The aforementioned model has been subsequently improved by Calvi et al. [12] to assess the
monotonie behaviour of strengthened masonry walls. The behaviour of the reinforcing bars was simulated by
truss elements having a bilinear constitutive law. Both unbonded and bonded reinforcement could be
simulated: in the first case, each bar was simulated by one truss connected to the masonry mesh only at both
42
extremities, whereas in the second case, the bar was divided into several trusses connected at the intermediate
nodes of the masonry mesh.
This model allowed the study of the effects of strengthening and enabled the most suitable
reinforcement scheme to be find. Racking tests on a coupled plain masonry wall subjected to different vertical
compression performed previously were used to identify the parameters of the masonry model. Then the same
kind of test could be simulated with different reinforcement schemes (vertical and/or horizontal, bonded or
unbonded).
The same model has been finally extended to the cyclic behaviour range by Calvi and Cantu [10].
Hydrostatic and deviatone behaviours, i.e. the bulk and shear moduli, were still considered separately, but
particular attention was paid to the shear modulus as the most significant parameter governing the response of
shear walls under cyclic lateral loading:
During loading (i.e. increase of J2), G was given by:
ο = γ3[θο^ώγ2ν^) (18)
where γ$ was a positive coefficient set to 1.0 for the first loading phase and actualized at the beginning of each
subsequent loading phase as follows:
Y3 = 1
CW (19)
where C was a constant characteristic of the masonry and W was the total previously dissipated energy i.e. the
total area of the previous hysteresis loops. This relationship had been derived from experimental observation.
When passing from loading to unloading (maximum of J2), the value of J2 is memorized (J2m*x) and a
hypothetical value of y¡ (γηβχΐ) lo be used in the next loading phase was guessed assuming a linear unloading
with the same shear modulus as at the beginning of the loading phase i.e. YJGQ (fig.46).
Figure 46: Assumed loadingunloading curve in the shear stress shear strain space (from [10]).
43
During unloading (decrease of J2), the shear modulus was given by:
G = /
Ï 3  " + Ynext
>Λ 1 —
V ¡max )) Go (20)
thus varying continuously from yjGo for J2=J2max. to Yne«Go for J2=0.
The transition between unloading and loadingftook place when J2 was a minimum. If the minimum of
J2 was zero (end of a halfcycle), the new value y3 of y¡ was then calculated on the basis of the actual
dissipated energy; the area of the actual latest loop being lower than the guessed one, the real value Y3 was
slightly higher than the guessed one 7next· Then, if the loading was applied in the same direction, the shear
modulus was given by the formula (18) with the new value Y3 of 73 (fig.47a). Conversely, if loading was
applied in the opposite direction, the residual deviatone strain value was memorized and G was fixed to a very
low value as long as the current deviatone strain remained between zero and the memorized value. This was
meant to take into account the closing of the cracks (pinching effect). As soon as the deviatone strain was zero
(i.e. the cracks closed), the shear modulus was again given by the formula (18) with the value Y3 calculated at
the end of the unloading phase (fig.47b).
Figure 47: Unloadingloading transition in the three possible cases: loading from zero J2 in the same (a)
or opposite (b) direction and loading from non zero J2 (c) (from [10]).
If the minimum of J2 was not zero (reloading before the end of a halfcycle), the new value 7*3 of y¡ was then
calculated on the basis of the incomplete actual loop (fig.37c): the choice of the measured area was arbitrary
but consistent with the model since it assumed the continuity between an infinitely small unloadingreloading
and a complete one.
The evolution of the bulk modulus was basically the same as for the monotonie model. The main idea
was to have an elastic hydrostatic behaviour (K=Kj) before cracking and to conserve a constant volume during
the unloading phases after cracking (K=°°).
The model was thus able to simulate the cyclic decay of both strength and stiffness (fig.48). However
it proved to be very sensitive to imposed load histories.
44
?\ί
300 .
200 ' t, 100 '
/ F
Figure 48: Analytical loaddisplacement response of a masonry panel
under cyclic diagonal loading (from[10]).
In the case of reinforced concrete masonry, Seible et al. [51] [52] used a monotonie model of the
second type: the nonlinear behaviour of reinforced masonry was described by a straindependant tensor
relating total stresses to total strains.
Masonry was considered to be an orthotropic material with reference to the principal stress axes
which in turn were assumed to coincide with the principal strain axes. The secant constitutive tensor had the
same form as the tangential one previously derived by Darwin and Pecknold [17] but the Poisson's effect was
completely neglected so that the stressstrain relationship expressed in the principal stress axes was:
(21) σι σ2 τ12 .
=
Et 0 0 0 E2 0 0 0 (E!+E2)/4
' ε ι ' ε2
.Ϊ12 .
where Ει and E2 were dependant on the principal strains ει and ε2. The Collins and Vecchio's modified
compression field theory [58] was used: except in the tensioncompression range, the principal directions were
considered independently so that each modulus (Ej or E2) was obtained as a function of the corresponding
strain (ει or ε2) from the uniaxial stressstrain law. The principal tensile stressstrain law (ει > 0) before
cracking was given by:
e ¡ =E
m e ¡ if ε^ε0 (22)
where ε„ was the cracking strain. After cracking fo > ε„), different tensionstiffening models were
considered:
σ· = E CT
1 m l + V200£i
(23a)
45
or
or
°i=£&*a*i)
^ E m E c x e x p 6<* )
(23b)
(23c)
where the coefficient α was increased with increasing reinforcement In [SI], the principal compressive stress
strain law (E¡ < 0) was given by:
f ' a i = E i ( e i ; e j ) e i =  ^ 2Ü_
<ι\
κ ε
ο Κ*ο) J
O i = 0
ß = supio.8 + 0 '.34ajll εο J
if 2 ε 0 <, E¿ £ 0
if £ i £ 2 e 0 £ 0 (24)
where f m and εο were the peak compression stress and corresponding strain. Such an expression was valid both
in the compressioncompression range (E¡ < 0 , Ej < 0) and in the tensioncompression range (E¡ < 0 , Ej > 0). In
the former case, the behaviour in both directions were independent of one another (β = 1.) whereas in the latter case, the compressive modulus E¡ in direction i was assumed to decrease with increasing tensile strain Ej in
direction j . In [52], two alternatives for the decreasing branch of the compressive stressstrain law were
considered:
ai=sup^0.1f^;0.64 îexp 'i—f 3íeie/)il : if E Í S E / = 1 . 6 E 0 ^ 0 (25a)
or, to take into account the confinement effect,
σ , =  ^ ík ß
D + (1D)exp ( E ( E i  E o ) ^
ε0 if Ej <, E0 <, 0 (25b)
where E and D were characteristic of the confinement.
The secant constitutive tensor of the reinforcement was given by:
Lxy
PxEsx 0 0 0 pyEjy 0
0 0 0 Yxy
(26)
where px, py were the steel ratios in the x, y directions and Esx, Esy the corresponding secant moduli. Each
46
secant modulus was dependent on the corresponding strain since a bilinear stressstrain curve was assumed for
steel:
if e y £ ε ^ ε γ σ = Ε8ε
a = fy+Ep(eey) if ε £ ε γ (27)
a = fy+Ε ρ (ε+ε γ ) if e£ey
where fy and ey were the yield stress and strain of steel, and E, and Ep were the elastic and elastoplastic
moduli.
Both materials, masonry and steel, being assumed perfectly bonded and therefore subjected to the
strain field, the secant constitutive tensor of reinforced masonry was obtained as the sum of the secant tensors
of the two constituents, provided they were expressed in the same axes of reference. The constitutive tensor of
masonry was therefore formulated in the xy plane by means of a proper coordinate transformation:
E l C2+E 2s 2 0 ( E , E 2 ) y
0 Eis2+E2c2 (E!E 2 )—
(E,E2)f (E,E2)f SllSi2 (28)
where s= sinØ, c=cos6 and θ is the angle between the principal stress axes and the reinforcement directions.
From the numerical point of view, tangent or initial stiffness methods were used to solve the non
linear equilibrium equations under prescribed load and/or displacements. Convergence was based on the ratio of the norm of residual forces to total forces.
This monotonic model has been used to reproduce or predict the loaddisplacement envelopes of different reinforced masonry structures (single storey wall, coupled shear wall, flanged wall) subjected to quasistatic or dynamic lateral cyclic loading. In [51], the first version of the compressive stressstrain law were used (fig.49).
OOO O.SO 1.00 mrcfct otspiACCucNT ( I N 

■ ■ ■ ■ ' ■ . . . .
m, ' ■ ■ ' ■ ' 1 1 . » É
1.00  0 50 0 00 OJO 1.00 lAfCRAl OUPlACCUCNf l»l
Figure 49: Experimental loaddisplacement curve and analytical envelope
in case of flexural (a) and shear (b) response of a shear wall (from [51]).
47
Excellent agreement was obtained in the case of ductile behaviour (flexural response of single walls, flange in
compression) but the lateral strength of specimens exhibiting nonductile behaviour were overestimated (shear
response of single walls, flange in tension). However, in both cases, the sequential crack pattern and yield
developments were well predicted.
In [52], the modified version of the compressive stressstrain law led to a better reproduction of the nonductile
behaviour especially when confinement in the bottom elements was considered (fig.50).
60'
3. 40· 2
Û 20
¡η ω 3 o
40
Analysis 1 : C  cracking Y — yielding TC — toe crushing
Flang· In comprealon
Flange in teniion
  Experiments — Analysis 1 (unconfined)
Analysis 2 (confined)
— r 2.0  1 0  1 0 1.0 0.0 1.0 ZO 3.0
TOP DISPLACEMENT (in)
Figure SO: Experimental and analytical loaddisplacement envelopes for a flanged wall (from [52]).
This model has been subsequently extended to fully cyclic loadings by Seible at al. [53].
Reinforcement was assumed to follow an elastoplastic law with strain hardening (fig.51).
Figure 51: Assumed cyclic behaviour for reinforcement.
As in the monotonie case, the cyclic constitutive behaviour of masonry was based on a uniaxial stressstrain
relationship, each principal direction being still considered independently except in the tensioncompression
range. The uniaxial cyclic behaviour of masonry was based on experimental work on concrete [59].
In direction i, the monotonie a¡e¡ curve used previously was assumed to provide an envelope and, at
any instant, the stressstrain relationship was completely defined by two envelope points A¡ and Q (fig.52).
48
The abscissa of A¡, e¡a, was also the maximum compressive strain reached during previous loading. The
residual strain e¡b (point B¡) under complete unloading in compression from point A¡ , could be deduced from
6ja according to a given unloadingreloading compressive modulus. The abscissa e¡c of point Q was defined as
the sum e¡b + e*¡c where e*¿c was the maximum tensile strain reached during previous loading. In tension,
unloading and reloading were assumed to follow the secant modulus from B¡ to C¡, thus accounting for opening
and closing of cracks.
Figure 52: Assumed cyclic uniaxial behaviour for concrete masonry (from [53]).
In the tensioncompression range, the coupling effect between the compressive direction j and the tension one i was still provided by the parameter β, but with reference to the relative tensile strain e¡ 1^:
β = sup 0.8 + 0 εο J
Since the principal axes were continuously rotating, the envelope points A¡ and C¡ in both principal
directions needed to be adjusted accordingly at each load step.
The proposed cyclic model has been used to simulate the cyclic response of single storey walls under
lateral loading. As in the monotonie case, a better agreement was obtained for flexural specimens (fig.53).
According to the authors, the larger discrepancies observed for shear specimens were due to the incapacity of
the model to account for partial closing of cracks, a phenomenon typical of the wide isolated cracks appearing
during shear response.
49
1204  I O Ο.β Ο.β  Ο Λ 0 .2 0.0 0.2 ΟΛ Ο.β Ο.β
LATERAL DISPLACEMENT (In)  1 . 0
 ι — ι — ι — ι — Γ Ο β Ο.β  Ο Λ Ò2 0.0 0.2 ΟΛ Ο.β Ο.β
LATERAL DISPLACEMENT (In)
Figure S3: Experimental and analytical loaddisplacement curves in the cases of
flexural (a) and shear response of a singlestorey wall (from [S3]).
VI.2.2 Original macromodels
The few macromodels which have been specifically developed for masonry are based on the
anisotropic failure criteria derived previously (cf. IV.2). Therefore, a common feature distinguishes them from
concrete macromodels: the anisotropy induced by the mortar joints is always and a priori taken into account
Given a failure criterion the simplest monotonie model that may be proposed is the linear elastic
brittle one: as long as the failure surface is not violated, masonry is supposed linear isotropic elastic; once the
failure surface has been reached, the residual stiffness and strength of the corresponding element is taken as
zero. Such a constitutive model may be implemented in an incremental finite element program where, at each
load increment, iterations are carried out to find a state of stress satisfying the equilibrium equations and
compatible with the strength and stiffness properties.
This model has been proposed and tested by Samaringhe et al. [SO], with the idealized failure surface
derived by Samaringhe and Hendry [49] in the tensioncompression range (fig.54). Because no complete
failure criterion was available at that time, no strength limitation was introduced in the biaxial compression
and biaxial tension ranges.
Shear walls subjected to vertical precompression and monotonie racking load applied at varying
height, have been used as a basis for comparison between theory and experiments. Since all the shear walls
failed suddenly in a brittle manner, it was found reasonable to identify the ultimate load with the first cracking
load. Failure resulted from the rapid propagation of the initial cracks through the tensioncompression region
so that close agreement was observed between experimental and theoretical results although biaxial
compression failure was not included in the model. Both location of the cracks and subsequent crack
propagation were well predicted in the tensioncompression region (fig.55).
50
m radians*
«L
Figure 54: Idealized failure surface for brickwork in the tensioncompression range (from [49]).
1 1
""1
i !
2 ? 2 3 _ _ _
Figure 55: Comparison between predicted (shade area) and experimental (dashed line) crack patterns
for varying height of loading (from [50]).
Page et al. [46] proposed a more elaborated monotonie model based on the complete idealized failure
surface derived by Dhanasekar et al. [20] in the stress system related to the joint orientation (ση σρ τ) (cf.
IV.2.2). Within the failure surface, isotropic elastic behaviour was assumed except in the biaxial compression
range where pronounced nonlinearity had been experimentally observed (cf. V.2). An elastoplastic model
based on an isotropic yield criterion was found to overestimate the strains normal to the joints (ε„ and ερ), and
underestimate the shear strain γ. To represent reasonably the nonisotropic plastic behaviour of masonry, it was
assumed that the plastic strain components in each direction (εΡη, εΡρ and 7P) were present at all stress levels
and related only to the corresponding stresses (ση, σρ and τ) by a power law. The variability of the data did not
warrant more complex relations. When the failure surface was reached, the stiffness tensor expressed in the
joint coordinate system and the stress components were actualized according the mode of failure (Table 2):
tension failure was assumed whenever there was a tensile stress on a joint. Under biaxial compression, shear
failure was assumed for low values of the normal compressive stress, otherwise crushing failure was assumed.
The reduction coefficients α and β appearing in the residual constitutive tensors were small coefficients which
provided an artificial means of controlling numerical instabilities during the development of failure zones
(theoretically, a zero value should have been assumed for both α and β).
51
Ho.
ι.
2.
3.
4.
Itode of Failure
Tension failure norma) to bed joint
Tension failure parallel to bed joint
Shear failure E ι̂ τ·
Biaxial tension or biaxial compression (crushing failure)
Modified Matrix
'<*.E 0 0
" E 0 0
1 υ 0
'«.E 0 0
0
E 0
0
«.E 0
V
1 0
0 et. E 0
[0]
0 0
at G
0 0 *.G
0 0
* < ^
0 Ü
A. G
Stress Components set to zero
Norma) stress (on) and shear stress (i)
Parallel stress (°μ) and shear stress (f)
None
Al) stress components
Table 2 : Modification of stiffness tensors for different modes of failure (from [46]).
This model has been incorporated in an elastoplastic incremental Finite element program. For the first
(small) increment, isotropic elastic behaviour was assumed throughout the structure. Where one of the
principal stresses was tensile, the local behaviour was assumed elasticbrittle otherwise elastoplasticbrittle
behaviour was assumed. Two regions of different behaviour were thus defined. For each successive load
increment, two successive sets of iterations were carried out, one to allow for the material plasticity and
another to allow cracking to progress. The two sources of nonlinearity, plasticity and failure, were treated
separately: during plastic iterations, the failure pattern was held constant; conversely, during the failure
iterations, the tangential stiffness of the uncracked plastified elements was held constant. Therefore, if
important stress redistribution occurred during failure iterations, the final state of stress in the elastoplastic
region might be inconsistent with the assumed stressstrain relationship. A better, but more cumbersome,
procedure would have consisted in repeating alternatively the two sets of iterations.
A monotonie racking test on a half scale infilled frame was used to test the adequacy of the model.
Two additional finite elements were used: a joint element to model the infillframe interface and a beam
element for the frame. The load deflection characteristics of the infilled frame were satisfactorily reproduced
by the model, even in the postcracking range (fig.56). Good agreement was also obtained on the ultimate load
and on the location and extend of the failure zone.
The finite element model was also used to determine the material characteristics which were more
critical in predicting the behaviour of infilled frames: the elastic modulus, the compressive strength and the
shear and tensile bond strength where found to have significant influence. Conversely, the small influence of
the parameters of the elastoplastic law indicated that an elasticbrittle behaviour could have been assumed in
the biaxial compression range without major error.
52
F I M
I M
'** M
*â
«
ta •
r /
/ y
/ . /
. · / V V
• f'
/ / >
/
/
f M I I
Figure 56: Experimental and theoretical loaddeflection curves of the infilled frame (from [46]).
Contro and Sacchi [16] proposed an anisotropic elastic plastic fracture model with zero strength in
tension and workhardening in biaxial compression. Attention was more focussed on the inelastic range than
on the postfailure one. The model was based on a simplified version of the anisotropic failure criterion
derived by Nova and Sacchi [40] as an orthotropic generalization of the MohrCoulomb condition (cf. IV.2.2).
For a given angle θ between one of the principal stress directions and the bed joint orientation, the failure
domain was bounded by six straight lines (fig.57) under the notension material hypothesis.
Figure 57: For a given angle θ : initial elastic (dashed line), current yield (solid line)
and ultimate (dotted line) surfaces (from [16]).
Though being possibly similar in shape to the failure domain, the initial elastic domain was assumed
rectangular in order to reduce the parameters to be determined. The current elastic domain was then defined as
the intersection of the failure domain and a rectangular domain evolving from the initial elastic one as a
consequence of plastic strains (fig.57). Only two sides were supposed to translate and interact with each other.
Their evolution was therefore governed by a second order matrix containing the hardening parameters
53
dependent only on Θ. The normality flow rule might not be necessarily adopted but was however suggested for
the sake of simplicity and for its capability to allow for material dilatancy since the yield criterion was of the
MohrCoulomb type.
The model being formulated within the framework of elastoplasticity, its numerical implementation
would not present any special difficulty. However, the authors pointed out several controversial questions
among which was the postfailure behaviour. Therefore the model has not been implemented so that no
comparison could be performed with experimental results.
VI.3 MICROMACRO MODELLING
The damage model proposed by Papa [47] and Maier et al. [35] may be qualified as a micromacro
model for combining the two levels of analysis. On the one hand, the material characteristics were defined at
the microlevel, masonry being conceived as a twodimensional planestress nonhomogeneous periodic
continuum composed of two phases: bricks were assumed isotropic linear elastic brittle, the failure threshold
being denned by the maximum tensile strains in tension and in compression; mortar was considered as an
isotropic elastic material susceptible to isotropic damage; perfect adhesive bond was assumed at the interface.
On the other hand, equilibrium equations and compatibility conditions were expressed at the macrolevel, i.e.
in terms of the average stresses and strains. The passage from the microlevel to the macrolevel was based on
a homogenization procedure: at each instant, the homogenized medium was orthotropic with an elastic
stiffness tensor defined by four independent parameters which could be derived from the "instantaneous"
moduli of mortar and bricks. Conversely, the passage from the macrolevel to the microlevel was based on a
"dcshomogenization" procedure (localization): at each instant, the local stresses in bricks and mortar were
derived from the overall ones in order to follow the stressstrain law of each constituent. The macrolevel
anisotropy due to the microlevel arrangement of the mortar joints was thus taken into account.
According to the chosen damage law, eleven or twelve parameters were required (four for the bricks
and seven or eight for the mortar). They could be identified by uniaxial tension and compression tests
performed on each constituent.
The model has been implemented with several variants of the damage law, in a conventional finite
element program using a predictorcorrector finiteincrement time integration scheme. In order to accelerate
the iterative process, simplified homogenization and localization procedures were used.
In the model, failure occurred when either a limit strain was reached in the bricks or a critical damage
value was reached in the mortar. A failure criterion of masonry has thus been derived (cf. IV.2.4).
The model has been also used to simulate cyclic compressive tests performed with increasing
amplitude on miniaturized masonry panels.
54
VU CONCLUSION AND PERSPECTIVES
Although there has been a great number of experimental studies, our knowledge of the inplane
behaviour of masonry as a material is still incomplete. This is mainly due to its variety and variability:
experimental results are strongly dependent on the type of masonry considered (stone, brick or concrete
masonry, solid, hollow or grouted units, etc) and suffer from scattering. They are also influenced by the test
procedures which are not standardized yet As a consequence, contradictory conclusions on several important
issues are not rare.
In such a context, analytical and numerical research has not progressed as much as for other familiar
building materials such as steel and concrete. Despite its original texture (periodic composite material),
masonry is generally treated as a "sort of like concrete" especially when reinforced, the main shortcoming of
the "concrete" models being that they do not take into account the anisotropic nature of masonry.
A way to achieve an enhanced description of the mechanical behaviour of masonry is to take
advantage of the homogenization techniques which have been extensively developed for composite materials.
Up to now, little work has been done in this sense: some encouraging results have been obtained recently in
different fields (elasticity [34], limit analysis [8], damage [35] [47]) but they are still approximate and/or
incomplete.
A more systematic and rigorous application of the homogenization theory to masony is clearly
needed. This can be performed within the framework of the finite element code CASTEM 2000 and could lead
to the implementation of new computationnal procedures. The following developments may be carried out:
1 Elasticity: derivation of the homogenized elastic constants on a 2D (plane stress) and a 3D cell with
exact boundary conditions; validation of the planestress assumption by comparing the 2D and 3D results.
2 Limit analysis: estimation of the homogenized failure criterion by the lowerbound and the upper
bound methods. The results can be immediately used to assess the ultimate strength of plane masonry
structures under static inplane load (bearing walls, shear walls, infills).
3 Crude nonlinear modelling: implementation of a linear elastic  brittle model based on the findings
of the two previous items.
At each level, the influence of the geometrical and mechanical parameters (bond pattern, unit shape,
mortar characteristics, unit material, etc.), as well as the possible imperfections (flaws, partial filling of joints,
etc.) could be studied in a systematic manner. This would provide valuable insights into the main parameters
affecting the overall behaviour of masonry before developing more elaborated models (e.g. elastoplasticity in
compression, damage in tension).
Such an analytical and numerical study must be supplemented by an experimental program involving
the three following aspects:
 characterization of the constituents (unit and mortar) and of their interface, in order to provide
realistic values of the parameters required in the homogenization process. This is a fundamental step of the
55
proposed program and may be achieved through the smallscale tests described in paragraph II1; however,
some additional work is needed to define representative tests for the unit/mortar interface.
 measurements of the overall characteristics of masonry (elastic coefficients, failure criterion) in
order to appraise the results of the homogenization process. For this purpose, the tests on large assemblages
under homogeneous state of stress are the most suitable. At the same time, useful information on the nonlinear
behaviour of masonry could be obtained.
 testing of fullscale plane masonry structures (walls or infills) under inplane loadings in order to
assess the ability of a given model to predict/reproduce the observed behaviour. The test setup should not
necessarily reproduce the actual working conditions of the structure but the loading conditions have to be
precisely known.
All tests should be performed with the same materials (type of unit, quality of mortar, bond pattern,
etc) in order to obtain a coherent set of data which will constitute a complete basis for future developments.
57
ACKNOWLEDGEMENTS
The author would like to thank Mr P.M. Jones and Mr P. Pegon for their helpful comments and suggestions.
59
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Books, Conferences, Journals:
Del Piero G., "Il consolidamento delle costruzioni" CISM, Udine (1983)
Del Piero G., "Le costruzioni in muratura" CISM, Udine (1984)
Sacchi Landriani G., Riccioni R. "Comportamento statico e sismico delle strutture murarie" CLUP.Milano (1982)
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International Brick Masonry Conferences
North American Masonry Conferences
International Journal of Masonry Construction
Masonry International
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