Analysis and modelling of the in-plane shear

Construction
and Building
MATERIALS
Construction and Building Materials 20 (2006) 308–321
www.elsevier.com/locate/conbuildmat
Analysis and modelling of the in-plane shear behaviour
of hollow brick masonry panels
A. Gabor *, E. Ferrier, E. Jacquelin, P. Hamelin
Laboratoire Mécanique Matériaux, Université Claude Bernard Lyon1, 82, Boulevard Niels Bohr, Campus de la DOUA,
69622 Villeurbanne Cedex, France
Received 5 July 2002; received in revised form 31 October 2004; accepted 31 January 2005
Available online 21 March 2005
Abstract
The paper presents a numerical and an experimental analysis of the in-plane shear behaviour of hollow brick masonry panels.
The non-linear behaviour of masonry is modeled considering elastic-perfectly plastic behaviour of the mortar joint. Experimental
methodology consists in the diagonal compression of considered masonry walls. Numerical and experimental results are compared
and discussed. The efficiency of modelling to forecast a representative failure mode of the masonry is investigated.
2005 Elsevier Ltd. All rights reserved.
Keywords: Hollow brick masonry; Finite element modelling; Diagonal compression test
1. Introduction
Masonry buildings constitute a significant part of the
building patrimony of the world. Structural walls of
these buildings were principally designed to resist gravity loads. Horizontal loads, induced by earthquakes,
generate severe in-plane and out-of-plane forces in these
walls. The behaviour and damage pathology of these
structures under seismic loads is very large [3], and it
manifests by slight fissures or ruination. The structural
element which has an important role in improving seismic force resistance is the shear wall.
The structural shear walls of a masonry building subjected to horizontal loading commonly present two
types of failure. The first one is out-of-plane failure,
where cracks appear along the horizontal mortar joints.
The second one is in-plane failure, generally characterized by a diagonal tensile crack. If the out-of-plane failure is avoided, then the structural resistance is mainly
influenced by the in-plane behaviour of the shear wall.
*
Corresponding author.
E-mail address: [email protected] (A. Gabor).
0950-0618/$ - see front matter 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.conbuildmat.2005.01.032
Because of the small height/width ratio (62), relatively
large shear stresses could develop, being favourable to
a non-ductile behaviour [11]. Besides, the brittle behaviour of masonry units and mortar reduce the energy dissipation capabilities of the masonry elements. Therefore,
the knowledge of the parameters which govern the shear
behaviour is important when the seismic behaviour
assessment of masonry structures is investigated.
Masonry can be considered as a composite material
built of brick units and mortar joints. There is a large
number of parameters that take part in the mechanical
behaviour of masonry: mechanical properties of brick
and mortar, geometry of bricks, joint arrangement,
etc. The evaluation of the influence of these parameters
on the overall behaviour of a masonry panel is not simple, so masonry is often assumed to be isotropic elastic
(e.g., ancient design codes, or more recently, Eurocode
6 [2]).
The evolution of finite element techniques permits a
more refined analysis. The numerical implementation
has been principally devoted to develop reliable interface models via adapted constitutive laws or incorporating fracture mechanics and plasticity theory concepts.
A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
309
Nomenclature
Eb
Emort
Eprism
hb
hmort
dvb
dvmort
dvprism
elastic modulus of bricks
elastic modulus of mortar
elastic modulus of masonry prism
height of the brick
apparent thickness of the mortar joint
vertical displacement of the brick
vertical displacement of the mortar joint
vertical displacement of the masonry prism
An interesting approach developed by Lourenço [9,8]
considers the mortar joint as the weakest element of
the brickwork, where all type of plastic deformation
take place. The mortar joint is modelled by an interface
element, using multisurface plasticity in order to describe compression, shear and tensile behaviour. A similar approach is developed in [4] considering the
constitutive equation of the interface in terms of two
internal variables representing the frictional sliding
and the mortar joint damage. This model has a brittle
response under tensile stresses and presents frictional
dissipation possibilities together with stiffness degrading
under compressive stresses. These characteristics are
used to describe hysteretic in-plane behaviour of masonry walls.
The numerical procedures mentioned above are
highly efficient and take into account most of the phenomena that occur in the masonry behaviour. Nevertheless, their numerical implementation is quite laborious.
In order to avoid this inconvenience, one can adopt several points of view. For instance, the application of
homogenization could eliminate this disadvantage: we
can mention approaches based on modelling the cracking of masonry [6,10] or using constitutive equations
built on anisotropic elasto-plasticity [7].
The present paper proposes a simple numerical modelling and experimental approach to investigate the
shear behaviour for the hollow brick masonry. The
numerical procedure is based on the finite element modelling which considers the local mechanical parameters
of bricks and mortar. For this purpose, we experimentally determine these parameters in compression and in
shear for a given masonry. The observed behaviour is
implemented, via adapted constitutive laws, in a commercial finite element modelling code (ANSYS). Afterward, we study the accuracy of the model to foresee
the behaviour of masonry panels. Once this stage
accomplished, we validate the modelling by experimental testing. The procedure consists in the diagonal compression of masonry panels, which produces stress states
susceptible to appear under seismic loads. The interest
of the modelling is also to verify if the chosen experimental testing procedure will reproduce the attended
ehb
evb
evmort
ehprism
evprism
/
smax
sres
horizontal strain in the brick
vertical strain in the brick
vertical strain in the mortar joint
horizontal strain in the masonry prism
vertical strain in the masonry prism
friction angle
maximum shear stress
residual shear stress
failure mode: cracking along the compressed diagonal.
Modelling and experimental results are compared, for
the applied load interval, considering the strain and
stress evolution in the masonry and the global response
in displacement.
2. Local mechanical parameters of compression and shear
behaviour
We can suppose that for the early stages the behaviour of a masonry panel under in-plane loading is linear
elastic. For this purpose, we are brought to experimentally assess and to implement in the finite element modelling the elastic mechanical properties (e.g., elastic
moduli, PoissonÕs ratio) of bricks and mortar. We preferred to measure these elastic parameters on masonry
prisms rather than on individual specimens of bricks
or mortar for several reasons. First, given the relatively
small vertical size of bricks, the confining effect induced
by the loading platens of the experimental apparatus
artificially increases the strength of the tested specimens,
so stress–strain diagrams could be false. Second, the
mortar characteristics in joints could be different from
those measured on cylindrical prisms. This difference is
caused by the differentiated drying and water absorption
at the mortar block interface [11,1].
The test specimens were made following the instructions given in RILEM technical recommendation [12]
concerning the determination of masonry strength in
compression (Fig. 1). Masonry prism can be considered
as extracted from a real masonry wall. The prisms were
constructed in a single wythe of 210 · 100 · 50 mm hollow clay bricks with a ready-to-use cement mortar. The
bricks were laid in running bond with an apparent mortar joint of 10 mm. Both bricks and mortar are currently
available in the market. Joint thickness and brick dimensions are the same as for the masonry panels employed
later for the experimental test.
Vertical and horizontal strain in masonry units were
measured directly by strain gages, while in the whole
prism, they were measured by LVDT extensometers
(Fig. 1). The compression stress is calculated as the ratio
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A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
Fig. 1. Geometrical configuration of masonry prism for compression
test and distribution of measurement devices.
crossed also the joined bricks (Fig. 2). Given the fact
that strain gauges were placed at a zone predisposed
to fissuring we can draw the compression stress (r)horizontal strain ðehb Þ diagram of the brick (Fig. 3). This
reveals that until cracking the brick behaviour is
quasi-elastic. At cracking, horizontal deformation
develops instantaneously, without relevant increasing of
the compression stress. Elastic (secant) modulus of the
bricks in compression is evaluated on the base of the
compression stress (r)-vertical strain ðevb Þ curve (see
Fig. 3) for 30% of the ultimate stress. At the analysis
of this diagram, we can notice that both masonry and
brick has linear elastic behaviour; elastic modulus of
bricks is greater than that of masonry. This agrees with
the results given in Hendry et al. [5].
Vertical strain in mortar ðevmort Þ can be obtained considering the strain measured for the brick ðevb Þ and for
the whole prism ðevprism Þ. This is done using the hypothesis that the total displacement of the prism ðdvprism Þ is
given by the sum of the displacements of bricks ðdvb Þ
and mortar ðdvmort Þ. In the computation the apparent
thickness of mortar joint, (hmort) and of brick (hb) are taken into account. Considering the same compression
stress in all components, the elastic modulus of mortar
(Emort) is determined:
evprism ¼
Fig. 2. Fissuring pattern of masonry prism in compression.
)
dvb þ dvmort
hb
a¼
hb þ hmort
hmort
evmort ¼ aðevprism evb Þ þ evprism ;
rprism ¼ rb ¼ rmort
of the applied force and the gross surfaces of bricks.
Considering the cracking mode, we noted the apparition
of vertical cracks along the head joints. The cracks
)
Emort ¼
Eprism Eb
:
aðEb Eprism Þ þ Eb
Compression test on masonry prism
15
h
Compression stress (MPa)
εb
εvprism
εvmort
v
εb
10
h
εprism
εvprism
εvb
εvmort
εhprism
5
εhb
0
0
1
2
3
Strain (mm/mm)
Fig. 3. Stress–strain diagram for compression test on masonry prism.
4
–3
x 10
ð1Þ
A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
Table 1
Elastic characteristics of bricks and mortar
Elastic modulus (MPa)
PoissonÕs coefficient
Brick
Mortar
13,000
0.20
4000
0.20
Determined values for brick and mortar obtained
from the average values of three test results are presented in Table 1.
Moreover, compression tests were also made on five
70 · 140 mm mortar cylinders, in order to compare its
behaviour obtained for the two types of tests. The elasticity modulus measured on the cylinders has been considered as the secant modulus at 30% of the ultimate
stress (Fig. 4). The average value of the five test series
gives a modulus Emort equal to 8300 MPa, which is
two times greater than that determined on masonry
prisms. Besides, the behaviour is also different: for the
same loading level, in the first case it is linear and here
is parabolic, similar to a concrete specimen stress–strain
curve.
Henceforth, in the finite element modelling, we will
consider the elastic parameters determined on the masonry prisms because they are measured in real loading
conditions and characterize the real behaviour of the
masonry assemblage.
In order to determine shear behaviour parameters, a
test method using small specimens in double shear has
been employed. This test method is adapted from a RILEM recommendation [13]. The experimental device is
conceived in such a manner that it can simultaneously
311
apply a static horizontal confinement load and a steadily
increasing vertical shear load to the specimen (see
Fig. 5). Loading forces are controlled by force cells; axial load is maintained constant during tests using a special device. Relative displacement between two adjacent
bricks is measured by LVDT devices. Given the fact that
parameters which are assessed will be used in twodimensional modelling, they are referred to the gross
geometrical dimensions. Therefore, shear and confinement stresses are calculated considering apparent contact surfaces between bricks. Totally, 18 small
specimens were tested, for confining stresses regularly
varying from 0 to 1.5 MPa.
In order to describe this behaviour we present on
Fig. 6 two test results for r = 0.6 MPa confinement
stress. Mechanical behaviour of prisms is characterized
by a very rigid behaviour for the elastic domain, with little displacement (order of microns). When the maximum
shear stress is reached, a softening behaviour begins
continued with a sliding motion between the adjacent
bricks. As we can see, the shape of the curves is the
same, but the two maximum shear stresses are relatively
dispersed. The size and the regularity of the distribution
of mortar cores strongly influence the level of the maximum shear stress. We mention that mortar cores have
been formed by the filling of the brick cavities with fresh
mortar, but this took place not in a compact and uniform manner. Externally, the fissuring manifests at the
brick/mortar interface (Fig. 7(a)), but at the inspection
we remark that the maximum shear stress corresponds
to the moment when mortar cores or brick wallettes
are fractured. The rupture of brick wallettes can be
Fig. 4. Stress–strain diagram for compression test on mortar cylindrical specimen.
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A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
Fig. 5. Experimental setup for shear test on masonry prisms.
Shear stress vs. displacement diagram
2.5
τ1
τ2
shear stress τ (MPa)
2
σ
1.5
1
lateral confinement stress σ=0.6 MPa
0.5
elastic domain limit
0
0
0.5
1
1.5
displacement (mm)
Fig. 6. Measured shear stress–displacement diagram for r = 0.6 MPa
confinement stress.
described as one of short beams where shear behaviour
principally dominates, whereas the rupture of mortar
cores is characterized by pure shear loading of a cubic
prism (Fig. 7(b)). Besides, the rupture of internal wallettes generally occurs on the total height of the brick.
The cores contribute to a high resistance in shear of
the masonry work. The shear resistance of brick masonry could be related to the gross surface area of the
bricks.
Considering the evolution of the maximal shear stress
(smax) as function of the confinement stress (r), we could
remark that it has a very slight increasing (Fig. 8). The
confinement stress has little effect on the maximum shear
stress. This is explained by the fact that interface behaviour is governed by the blocking mechanism between the
brick wallettes and the mortar cores. The lateral confinement stress serves only to hold in place the bricks, assuring therefore the correct work of the cores, until sliding
occurs. The maximum shear stress varies between 1.63
and 2.2 MPa with an average value of 1.85 MPa.
The sliding motion at the brick/mortar interface can
be considered as a frictional one, described by the evolution of the residual shear stress (sres) in function of the
confinement stress (r) (Fig. 9). The friction coefficient
can be deduced upon this curve and it corresponds to
an angle equal to / . 40.
We can remark that dispersion of experimental data
is important. This is due, on the one hand, to the intrinsic inhomogeneity of the masonry assemblage, and on
the other hand, to the uniformity and size of the formed
mortar cores. Nevertheless, the dispersion of the obtained results is in an acceptable interval, comparable
to that found in the literature.
Given these parameters, we will investigate a modelling approach using Mohr–Coulomb friction law with
cohesion in order to describe the mortar joint and the
mortar/brick interface behaviour. As we can see, the
shear behaviour is characterized by a peak shear stress
and a sliding behaviour of the interface. The parameters
related to this behaviour in a Mohr–Coulomb formulation will be the maximum shear stress or the cohesion c,
the residual friction angle / and a dilatancy angle w
(Fig. 10). The dilatancy angle measures the uplift of
one brick over the other upon shearing; it decreases to
zero with increasing shear slip and confining pressure.
Given the fact that masonry joints have extremely low
dilatancy [9] the model is formulated with zero dilatancy
angle. The assumption of zero dilatancy angle affects the
results only marginally.
3. Analysis of the behaviour of hollow brick masonry
3.1. Geometrical characteristics of studied masonry panels
The masonry panels were constructed in order to correspond to the diagonal compression test criteria [14].
The geometrical dimensions of masonry panels are
A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
313
Fig. 7. (a) Failure mode at the brick/mortar interface. (b) Dislocation mechanism of mortar cores and fracture of brickÕs internal wallettes.
Evolution of maximal shear stress τ
max
2.8
Maximal shear stress τ
max
2.1
1.4
0.7
0
0
0.4
0.8
1.2
1.6
Lateral confinement stress σ (MPa)
Fig. 8. Peak values of shear stresses in function of lateral confinement stress.
chosen in a way that they contain a sufficient number of
bricks and mortar joints. Therefore, a panel could represent a real masonry wall. Specimens have to be approximatively square and having a length at least four times
the length of a brick. In these conditions, the dimensions
of masonry panels are 870 · 840 · 100 mm (Fig. 11). In
order to put real boundary and loading conditions, we
consider that wall angles are embedded in ‘‘virtual’’ solid loading shoes (Fig. 11).
3.2. Numerical implementation
The pursued approach considers fully elastic bricks
with appropriate elasto-plastic model for the mortar
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A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
Evolution of the residual shear stress τres
2.4
(MPa)
2
Residual shear stress τ
res
1.6
1.2
0.8
0.4
0
0
0.4
0.8
1.2
1.6
Lateral confinement stress σ (MPa)
Fig. 9. Residual stress vs. lateral confinement stress diagram. The slope gives the friction angle.
Fig. 11. Boundary and loading conditions of the masonry wall.
Fig. 10. Typical shear stress–displacement diagram [9].
joint. The behaviour of the mortar is responsible for the
overall non-linear behaviour of the masonry assembly.
The masonry is regarded as a material realized by a regular inclusion of blocks into a matrix of mortar. The
mortar is considered as a net which perfectly bonds to
bricks.
For the implementation of the Mohr–Coulomb
parameters, an elastic-perfectly plastic formulation
(Drucker–Prager) is adopted. The material parameters
c and / are introduced in such a way that the circular
cone yield surface of the Drucker–Prager model corresponds to the outer aspices of the hexagonal Mohr–
Coulomb yield surface. In the plane stress case, Drucker–PragerÕs constitutive law reduces to a single continuous yield surface, whose equation reads
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 2
ðr þ r2y rx ry Þ þ s2xy þ aðrx þ ry Þ k 6 0:
ð2Þ
3 x
Parameters k and a are related to the friction angle /
and the cohesion c of the considered material:
2 sin /
a ¼ pffiffiffi
;
3ð3 sin /Þ
6c cos /
k ¼ pffiffiffi
:
3ð3 sin /Þ
ð3Þ
These parameters determine the yield stresses in uniaxial
tension and compression rt and rc:
rt ¼
p1ffiffi
3
k
;
þa
rc ¼
p1ffiffi
3
k
:
a
ð4Þ
A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
315
lengths as in reality. The time step loading permits to
evaluate the evolution of strain and stress distribution
in the masonry panel.
We analyse the modelling results obtained for hollow
brick masonry panels submitted to predominant shear
test. Afterward, we compare these results with those obtained by the experimental testing program.
3.3. Modelling results
Fig. 12. Meshing of the masonry panel.
For modelling, ANSYS 5.4 commercial software is
used. Four node, plane elements (plane42) are employed
to discretize all components of the brickwork. Mesh size
is imposed by the relative dimensions of units and mortar: the size of the elements of the mortar is uniform and
equals to the thickness of the joint, whereas element size
of bricks becomes coarse in their interior (Fig. 12). The
numerical implementation is done using Euler backward
scheme and Newton–Raphson iteration. For each load
step, the maximal number of equilibrium iteration is
100; last loading force corresponds to the step when
unconvergence occurs. The modelling considers real
boundary and loading conditions; surface loads and degrees of freedom constraints are applied on the same
The objective of the modelling is to verify the capability of the experimental testing program to reproduce the
desired failure mode and to evaluate the stress and strain
state in the masonry, for a considered load interval and
for different boundary conditions. Thus, we built a model
considering that degrees of freedom constraint and loads
are applied on the 10th part (r = 1/10) of the length of
the wall (Fig. 11), as required by the RILEM technical
recommendation [14]. In these circumstances, the finite
element modelling returned that plastic strain will
develop only in the neighborhoods of the bearing zones
(Fig. 13), without any plastic deformation of the central
region of the wall along the compressed diagonal. The
shear stress has also its maximum values at the aforementioned areas (Fig. 14). Therefore, we changed the
boundary conditions from 1/10 to 1/6th of the length
of the wall, in order to see if we can produce the failure
along the diagonal.
For this latter case, the analysis of strain distribution
during loading shows that plastic strain occurs at 90% of
ultimate load in the center of the masonry panel and at
the edges of loading shoes (Fig. 15). Also, we remark
that for ultimate loads, the plastic strain in the center
Fig. 13. Distribution of plastic shear strain in the masonry panel for r = 1/10 boundary condition. The light zones indicates where shear plastic strain
is maximum.
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A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
Fig. 14. Distribution of plastic shear stress in the masonry panel for r = 1/10 boundary condition. The light zones indicates where shear stress is
maximum.
Fig. 15. Finite element modelling results (r = 1/6). Plastic shear strain for the applied force F = 19,000 daN.
of the panel develops more than at the edges of loading
shoes. Thus, failure of the wall is caused by the development of strain in this zone (Figs. 16 and 17). Given the
fact that plastic deformation occurs in the lately stages
of loading and that masonry has generally a brittle
behaviour, we can anticipate that failure will appear
suddenly.
In order to evaluate modelling efficiency for the global response, force–strain diagrams are drawn, for different values of the maximum shear stress (smax) see
Fig. 18. Lowest, average and maximum values are taken
into account. Analyzing the force–deformation diagrams we remark that the masonry panel has a quasielastic global behaviour until near the ultimate loading
force (Figs. 23 and 24). At 92% of this ultimate force
we remark a sudden change of the global stiffness which
predicts the degradation of mechanical properties and
the failure. We remark also that the increase of the cohesion extends the elastic limit of the panel but does not
affect the post elastic plateau.
A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
317
Fig. 16. Finite element modelling results (r = 1/6). Plastic shear strain for the applied force F = 20,000 daN.
Fig. 17. Finite element modelling results. Ultimate shear strain distribution (r = 1/6, F = 21,031 daN).
3.4. Testing of masonry panels
In order to set-up an appropriate test, which produces a predominant shear stress state in a masonry panel, it is important to know how the shear stress in a
panel is distributed and how it can be experimentally
generated. If we consider a square homogeneous elastic
masonry element, submitted only to shear stresses (Fig.
19), the created principal stresses will be inclined by 45
to the head and bed joint axes. One of the stresses is
compressive and the other is tensile; their values are
equal to the initial shear stress. It is assumed that the
failure will occur if the principal tensile stress reaches
the diagonal tensile strength of the masonry covering
both the sliding failure in bed joints and the cracking
of units.
Thus, from the experimental point of view, the simultaneous compression and tension along the diagonals of
the wall produce a pure shear stress state which engenders the failure by cracking along the compressed diagonal. However, this type of biaxial test is not easy to
set-up. Consequently, it is suitable to exclude the tensile
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A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
Strain evolution along compressed diagonal
300
τ
=2.2 MPa
max
τmax=1.63 MPa
200
Force (kN)
τmax=1.4 MPa
100
0
0
200
400
600
Strain (µm/m)
800
1000
Fig. 18. Finite element modelling results. Force-strain diagrams.
load along of one the diagonals and keep only the compression. Hence, the compression load applied along the
diagonal will produce stresses equivalent to a biaxial
compression and shear stress state along head and bed
joints (Fig. 20). The generated compression and shear
stresses have the same intensity and are equal to the half
of the compression stress along the diagonal.
The recommendations of RILEM propose such an
inclined compressive loading test on masonry elements
in order to estimate the diagonal tensile strength. The
employed experimental setup consists in a high stiffness
Fig. 19. (a) Homogeneous square panel submitted to pure shear stress.
(b) Stress state generated in a square panel submitted to diagonal
compression.
rigid frame which permits to apply the considered compression load via a 1000 kN hydraulic jack (Fig. 20).
Applied force and the displacement along the two diagonals are measured using load cell and respectively
LVDT extensometers. The load is continuously increased until rupture.
Two masonry panels were tested, considering the two
boundary conditions r = 1/10 and 1/6. For the former
Fig. 20. Experimental setup for diagonal compression test.
A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
319
case we obtained a localized failure, with a marked
sliding in joints and crushing at the one of the bearing
zones (Fig. 21). At the same time, for the latter panel
we noticed a generalized type of failure (Fig. 22). The
failure has been very sudden, without any predictive fissures on the surface of the wall. Nevertheless, the analysis of the specimen after rupture reveals that cracks
propagated along the compressed diagonal and that
they were present in the mortar joints and in the bricks
(see Fig. 22).
3.5. Evaluation of results
Fig. 21. Cracking and failure of the wall for r = 1/10 boundary
condition.
Fig. 22. Cracking and failure of the wall for r = 1/6 boundary condition.
The experimental force–strain diagram has the same
shape as that obtained by the modelling: the slope of
the curve for the elastic domain coincides well, but the
post-elastic level is less accentuated (Figs. 23 and 24).
The Table 2 compares maximal values of the modelling
and those of the test. The maximum difference between
the strain measured experimentally along the compressed diagonal and that obtained by the modelling is
6% for the elastic zone and 25% at the ultimate load.
Concerning the strain along the stretched diagonal
the discrepancy is in the range of 2–6%. Besides, the
ultimate forces are quasi identical and elastic limit
forces coincide with a good approximation (below
2%). The observed differences are relatively small
and are in the range of usual values found in the literature. The computed values and the strain distribution map obtained by numerical way show the
capability of the modelling to forecast the non-linear
behaviour of the masonry.
Fig. 23. Numerical and experimental results for hollow masonry. Strain evolution along compressed diagonal.
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A. Gabor et al. / Construction and Building Materials 20 (2006) 308–321
Fig. 24. Numerical and experimental results for hollow masonry. (b) Strain evolution along stretched diagonal.
Table 2
Numerical and experimental results
Elastic limit
Force
(daN)
Ultimate
Strain
(lm/m)
Force
(daN)
Strain
(lm/m)
Compressed diagonal
Experimental
18,825
Numeric
18,800
378
447
21,525
21,031
540
708
Stretched diagonal
Experimental
Numeric
98
101
21,525
21,031
190
238
18,825
18,800
4. Conclusions and outlook
In this paper, we presented a finite element modelling
approach for the study of the behaviour of hollow brick
masonry under predominant shear stress. The accomplishment of the modelling in the prevision of the nonlinear behaviour depends on the right choice of the
implemented mechanical parameters. The principal
parameters of the chosen formulation have been the
elasto-plastic properties of the mortar joint: cohesion
and residual friction. These parameters have been determined by simple static tests of representative masonry
assemblages. The obtained numerical results have been
experimentally validated in the case of diagonal compression test of masonry panels. We analyzed also the
influence of the boundary conditions on the behaviour
of the masonry panel. Comparing these results, we remark that finite element modelling approaches with a
good accuracy the behaviour of masonry panels: ultimate loads, ultimate strains, plastic strain evolution
and failure modes are represented with very good
approximation.
Nevertheless, we can remark some difficulties concerning the reproducibility of experimental results, especially for the determining of local mechanical
parameters. This is due to the intrinsic properties of
the materials and to the particularity of hollow brick
masonry. In order to overcome this situation, it is essential to create and complete an accurate database obtained by experimental means. In this context, further
theoretical and experimental research will be conducted
not only on the characterization of local behaviour but
also on the description of the in-plane global behaviour
under simulated seismic load. Therefore, we can succeed
to results that can provide accurate guidelines for retrofitting interventions with innovant and high performance materials.
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