1
EXPERIMENTAL AND NUMERICAL ANALYSIS OF BENDING-BUCKLING MIXED FAILURE
OF BRICKWORK WALLS
Ernest Bernata*, Lluis Gila, Pere Rocab and Cristián Sandovalc
a Department
of Strength of Materials and Engineering Structures
Technical University of Catalonia UPC. BarcelonaTech.
ETSEIAT, Colom 11. 08222 Terrassa, Spain
b Department
of Construction Engineering
Technical University of Catalonia UPC. BarcelonaTech.
ETSECCPB. Jordi Girona 1-3. 08034 Barcelona, Spain
c
Department of Structural and Geotechnical Engineering, and School of Architecture
Pontificia Universidad Católica de Chile.
Casilla 306, Correo 22. Santiago, Chile.
ABSTRACT
The eccentric in-plane loading of masonry walls involves complex bending performance that
includes second-order effects. In this work, a bidimensional (2D) simplified micro-model for
the analysis of this type of failure is developed. An experimental investigation based on 20
tests of full-scale unreinforced masonry walls is performed. The tests are characterised by
slenderness and load eccentricity. The analytical methods of Eurocode-6 and ACI-530 are
compared with experimental data from the present investigation, other experimental results
available in the literature and simulation results from the numerical model.
KEYWORDS
Buckling failure, unreinforced brickwork wall, experimental tests, simplified micro-modelling
*
Corresponding author:
Tel.: +34 937398728; fax: 937398994
e-mail address: ernest.bernat@upc.edu
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1. INTRODUCTION
Although the use of structural masonry has been dramatically reduced since the widespread
introduction of concrete and steel structures, there is still a large number of existing buildings
in use that are structurally composed of load-bearing brick masonry walls. Assessing the
strength of these buildings is a part of the maintenance work required to enhance their useful
life. Among the possible failure modes, buckling failure must be addressed to verify the safety
of the walls from sudden collapse.
The in-plane load-bearing capacity of unreinforced brickwork walls depends on both the
material properties and the geometry of the structure. The typical effects of eccentricity and
slenderness have been widely analysed and are reported in many works [1]. The behaviour of
eccentrically loaded unreinforced masonry walls is complex because of the non-linear
compressive behaviour of the masonry and its fragile response to tension. Although the tensile
strength of masonry walls is neglected in most codes (see [2] or [3]), some authors (see [4], [5]
or [6]) have reported that this property has a significant influence, even at small values, on the
load-bearing capacity of slender or eccentrically loaded walls.
To understand the structural behaviour of compressed brickwork walls, several experimental
research studies have been published reporting empirical analytical calculation methods for
the assessment of the wall load-bearing capacity. The application range of such methods is
limited to specific support and load conditions [4, 7-13]. Recently, the development of modern
computational methods (e.g., FEA) has fostered the proposal of a large number of numerical
models to simulate the behaviour of masonry. Most of the proposed computational methods
for the analysis of masonry structures have been formulated using micro-modelling [14-16] or
homogenised macro-modelling approaches [17-20].
In the experimental field, several studies have reported on the load-bearing capacity of
masonry walls. The works of da Porto [21] and Cavaleri [22] are especially significant, with
respect to the study of the out-of-plane behaviour of load-bearing walls. The similarity
between full-scale and small-scale tests has been widely studied ([23]), and full research
programmes have been completed considering only small-scale walls ([6, 23-25]). Compressive
full-scale tests on masonry walls or columns are not very common. Among the studies on fullscale walls, those of Watstein & Allen [26] and Kirtschig & Anstötz [27] are remarkable because
of the use of specific systems to model neatly hinged supports (i.e., rotationally free) at the
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wall ends. In these cases, the clarity and simplicity of the support condition allow the study of
the influence of the load eccentricity and the wall slenderness with no uncertainty in the
actual boundary conditions.
One of the parameters that affect the buckling failure and the load-bearing capacity of
masonry walls is the masonry modulus of linear deformation (Young’s modulus). Several
recent studies that focussed on full-scale specimens have provided meaningful results on the
relationship between the compressive strength of masonry and its modulus of linear
deformation ([28]). With respect to material properties determination, the work of
Maurenbrecher ([29-32]) is significant. Maurenbrecher ([29-32]) studied the effect of building
procedures on the properties of the resulting masonry, particularly the compressive strength
(fc) and the modulus of linear deformation (E). Maurenbrecher ([29-32]) also analysed the
influence of the load eccentricity on the masonry compressive strength (fc). Gazzola [33] and
Sinha [34] have investigated the flexural strength of masonry to understand out-of-plane
behaviour.
Most of the experimental studies to date have been focussed on the derivation of empirical
analytical formulations for the estimation of the load-bearing capacity of walls (see [27]). One
of the first analytical formulations, presented by Yokel ([13, 35]), has been the basis of several
later proposals.
Alternative formulations that cover a wider range of situations and are meant to be used with
the aid of computers are becoming more common in recent times ([10, 28]). Because of the
growing complexity of the analytical formulations, modern computer methods are becoming
popular alternatives. Many finite element approaches have been proposed for the analysis of
masonry wall structures, including simplified methods, such as one-dimensional (1D) beam
descriptions [36], and more realistic and sophisticated applications, such as the micro-models
presented by Lourenço [37], Salerno [38] and Martini’s three-dimensional (3D) models [39]. As
a compromise, there is a growing tendency to work in homogenised 2D models.
Homogenisation techniques for masonry have been studied by many researchers ([15, 40-44]).
2. SCOPE AND METHODS
The greater the slenderness and the load eccentricity are, the greater the effect of the tensile
strength and the boundary conditions on the load-bearing capacity [35]. Typical failure is
because of second-order out-of-plane bending caused by the load eccentricity and possible
geometric irregularities.
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Only a few studies on full-scale unreinforced masonry walls under eccentric compressive load
have been performed to investigate this type of failure. Moreover, the previous researchers
did not focus on the effects of the tensile strength on the buckling response of the walls and
the corresponding masonry tensile strength was not measured. In some cases, the support
conditions at the wall ends are not accurately reported. These limitations make it difficult to
use the data produced by the previous tests as a basis to calibrate and validate newly
developed numerical methods. Furthermore, although some extensive research has been
successfully undertaken in this area recently ([6, 45]), full-scale test results are required to
complement the reduced-scale test data. Therefore, the first objective of the present research
is to perform experimental tests on full-scale walls to complement the data available in the
literature and to analyse the effects of slenderness and eccentricity on the load-bearing
capacity of masonry walls. This study includes the characterisation of the material mechanical
properties of both the components and the masonry as a composite material.
Some authors ([6, 46, 47]) have demonstrated that the most widely utilised structural masonry
codes are too conservative when applied to the analysis of very slender or very eccentrically
compressed unreinforced masonry walls. Comparing the experimental values from the tests
with the result of the application of widely used standards is another specific objective of the
present work.
Numerical models have been proposed to overcome the limitations of the standards in
calculating the structural response of masonry walls. However, most of the proposed
numerical methods have been implemented in-house, using proprietary software that is not
freely available to outside practitioners. Conversely, the present research is based on a finite
element approach that is available in a commercial software environment. Validating and
proving the accuracy of this available numerical method is another objective of this research.
In addition to the experimental results obtained in the present study, the model is also
validated using experimental data available in the literature.
The available data are extended by using the numerical tool to perform a comprehensive
numerical simulation. The experimental and numerical results are compared to two standards
(EC-6 [2] and ACI 530 [3]) to assess the accuracy of the standards.
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3. EXPERIMENTAL TESTS
To continue the previous work on small-scale masonry models [6] and to enhance the analysed
cases contrasting the results with tests on full-scale walls, a complete experimental
programme at full scale was executed.
Twenty masonry walls were tested under eccentric in-plane compression to achieve out-ofplane failure from the combination of second-order bending (which would cause a rapidly
increasing out-of-plane deformation of the walls) and buckling (which would explain a sudden
failure due to the applied axial load associated with a geometric instability). In addition, the
components’ (bricks and mortar) mechanical properties and the masonry compound
properties were characterised. In particular, 131 tests were performed to determine the
compressive strength of the mortar, and 249 tests were performed to assess the flexural
strength of the mortar. Totals of 19 and 42 tests were performed to characterise the
compressive strength and the flexural strength, respectively, of the bricks. The water
absorption of the ceramic was measured by testing 29 samples. Also 29 samples were tested
to find the compressive strength and Young’s modulus of the masonry, and 32 more were
tested to measure the masonry tensile strength.
In the next subsections, the experimental programme is summarised. Information is included
on the preparation of the samples, the test setup and the results.
3.1.
Material properties and production of test walls
Two different types of Portland cement mortar were used to fabricate the walls. Both mortars
were acquired as commercial products, to be mixed with water according to the
manufacturer’s instructions. Standardised tests to determine the compressive and flexural
strengths were performed as described in [48]. The results are summarised in Table 1. Lowcompressive-strength mortar was used to simulate ancient brickwork.
One type of solid-clay brick, with dimensions of 280x132x45 mm3, was used. Tests to
determine the compressive strength (fcb = 27.93 MPa) and water absorption (from initial dry
state, 1.46 mg/(mm2·min), and after 1 minute wetted, 0.65 mg/(mm2·min)) were performed as
described in [49] and [50], respectively. A three-point bending test was used to determine the
flexural strength (ftb = 2.81 MPa). The tensile strength of the bricks was determined from the
flexural values according to the method proposed in [24].
The masonry was tested to obtain the compressive strength, fc, the modulus of linear
deformation in the direction perpendicular to the bed joints, E, and the bonding strength
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between bricks and mortar joints, fxt. Specimens consisting of 5 stacked bricks were used to
obtain values for fc and E, and samples of two stacked bricks were used for the other tests. The
values obtained for each wall series are summarised in Table 2. The results show that the value
of the modulus of linear deformation is low compared with the correlations between E and fc
provided by the codes (E/fc = 1000 in [2] and E/fc = 700 in [3]). The measured values are closer
to those found by other researchers (see [31, 51-53] and Table 3), although the measured
values are still significantly lower. Using low-strength mortars and actual brick-laying
conditions (less controlled than in typical laboratory conditions) may explain the unexpectedly
low modulus of deformation.
Five different geometries of full-scale unreinforced masonry walls (H, F, M, T, S) were tested as
summarised in Table 4. Twenty walls were tested. All walls were built by professional
bricklayers in the vertical position to emulate the actual fabrication process. The bricks were
wetted before being placed in position to assure good adherence. The alignment of each row
was checked during the construction. The walls were fabricated on an auxiliary steel tool
utilised for the transport, elevation and positioning into the test setup. Walls W#1 to W#9
were fabricated and air-cured under indoor atmospheric conditions, whereas the rest of the
walls were fabricated and air-cured under outdoor atmospheric conditions.
3.2.
Wall test setup and procedure
The test setup is described in Figure 1. To produce accurate and clear boundary conditions, the
walls were prepared with a pair of steel hinges as part of the test setup. The lower hinge was
laid on the floor (H and F series) or placed over a structural steel beam (M, S and T series) to
increase the height of the wall’s base, whereas the upper hinge was connected to a hydraulic
jack with a compression load capacity of 1000kN. The upper hinge could move only in the
vertical direction to allow for free descending displacement on the top of the wall during the
test.
A steel beam was installed between the jack and the upper hinge to distribute the load along
the wall width. The connection between the wall and the test machine was made using two
steel plates at each end. These plates held the wall in contact with the hinges and allowed
different load eccentricities to be set. Any eccentricity between 0 mm and 80 mm, at 5 mm
intervals, was possible. The contact between the wall and the test system consisted of a 20mm M2 mortar layer to distribute the load and avoid points of high stress. The details of the
connection system are shown in Figure 2. Eight potentiometers were installed to control the
rotation of both hinges (two sensors for each hinge) and the descending movement of the
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upper hinge (four sensors on the steel distribution beam). Depending on the test, two or four
laser displacement sensors were used to measure the out-of-plane deformation. The midheight deformation was always measured with the deformation at ¼ and ¾ height in the case
of walls W#10 to W#20.
The tests were force-controlled. All data were recorded at 50 Hz. An ordinary video camera
was used to record all tests, and a high-velocity camera was used to record the W#10 to W#20
tests to capture the details of the failure mode.
The testing was executed as follows: a) The wall was placed in the test position using an
overhead travelling crane and a pallet truck, affixing the wall to the loading system in the
theoretically desired position; b) The actual geometry was measured with a laser sensor,
including local and global geometric deviations. Two out-of-plane measurements, one at each
border of the wall, were obtained every 3 masonry rows; c) The sensors were put in place; d)
The surrounding area was protected from the projection of rubble; and e) The test proceeded
until the wall collapsed. The load increased continuously at a uniform rate until the sudden
collapse point.
3.3.
Experimental results
Table 5 summarises the test conditions and results for the wall series. The geometry type (as
outlined in Table 4) and slenderness (Eq. 1) are indicated for each wall. The mid-height
eccentricity ratio (em/t), calculated as the ratio of the initial mid-height eccentricity (em) to the
thickness (t) of the wall, is also listed. A dimensionless measure of the hinge vertical alignment
(d/t), describing the overall initial rotation of the wall, is also calculated and shown in Table 5.
Figure 3a) shows the geometric configuration of a typical wall of the current campaign. With
respect to the experimental results, the load-bearing capacity of the wall, Pmax, and the ratio
(Eq. 2) of the maximum load to the maximum theoretical uniformly distributed load (Eq. 3), are
included. The mid-height out-of-plane deformation at the maximum load, hmax, and the
descending displacement of the upper hinge at the maximum load, vmax, both measured from
the effective height (Hef), are shown in the last columns of Table 5. Hef is the real distance
between the axes of the hinges.
=Hef/t
(Eq. 1)
ф=Pmax/Pu
(Eq. 2)
Pu =b·t·fc
(Eq. 3)
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The walls were tested so that the eccentricities at the lower and upper ends were equal.
Although a constant eccentricity along the wall height was expected under these conditions,
additional accidental eccentricities appeared because of construction imperfections. The
actual wall shape was measured after the placement of each wall in the test position. The
actual mid-height eccentricity, em, was calculated based on the measurements. Construction
imperfections also caused the hinges to be imperfectly aligned, requiring the measurement of
the initial horizontal distance between the upper and bottom axes (d). In Table 5, Table 6,
Table 7 and Table 8, if d is positive the upper hinge was displaced in the out-of-plane direction
(the right side in the configuration of Figure 1) with respect to the lower hinge. The
contribution of the initial alignment (d) to the eccentricity at mid-height was systematically
estimated as d/2 for the present experimental campaign.
The walls were compressed until failure. Three different collapse mechanisms were observed.
The most common failure mechanism observed was the formation of a single horizontal crack
at approximately mid-height with a sudden out-of-plane displacement and collapse of the
entire wall. Tensile failure caused the formation of a natural joint and was the mechanism
leading to the corresponding wall collapse. This mechanism was observed for all walls (Figure
4) except W#4, W#14 and W#17. Walls W#14 (M series) and W#17 (T series) failed in a mixed
mode, i.e., a combination of joint opening at the tension side with masonry crushing at the
compression side (Figure 5). Wall W#4 failed by out-of-plane shear near the upper edge of the
wall. The results of Wall W#5 were not included in the analyses because they show an
anomalous response that is inconsistent with the rest of the H-series walls.
The experimental results are summarised in Figure 6. As expected, the results show that the
load-bearing capacity decreases significantly with increases in the load eccentricity and the
slenderness. For an actual eccentricity of approximately 16% of the wall thickness, changing
the slenderness from 9.1 to 14.1 or 21.7 causes a loss of the load-bearing capacity of 35.5% or
96.3%, respectively. Similarly, the load-bearing capacity of the walls in the H series (with a
theoretical slenderness of approximately 21) decreases by approximately 75% if the
eccentricity is increased from 0.3% to 16.1% of the wall thickness. The load-bearing capacity
decreases by approximately 80% if the eccentricity is reduced from 5% to 25.6% of the wall
thickness for M-series walls.
The results in Figure 7 exhibit a linear relationship between the vertical displacement of the
walls and their maximum load-bearing capacity. As the results in Figure 8 show, the
descending displacement of the wall top section at the instant of failure depends on the initial
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eccentricity at mid height (which, if the alignment of the hinge is not taken into account, is
equal to e/t-d/2t). Practically, the experiments show that, for a fixed slenderness, the vertical
displacement required to cause the collapse of the wall decreases with an increase in
eccentricity. Slenderness also affects this relationship. The results differ between walls from
series H, M and S, as shown in Figure 8. For example, an increase of the normalised initial
eccentricity (without the hinge-misalignment effects) from 4.2% to 20.8% causes a reduction in
the vertical displacement needed to reach failure from 2.5% of the effective height to 0.5% in
H-series walls (i.e., a reduction of 80% ). Similar behaviour is observed for the M-series walls.
As shown in Figure 8, an increase in the normalised initial eccentricity from 3.1% to 24.8%
causes a 63% reduction in the vertical descending movement needed to cause the failure of an
M-series wall. Similar to the results in Figure 7, these results are consistent with the fact that
most of the walls, as mentioned, failed because of geometric instability.
The experimental results show that the initial shape (eccentricity) affects both the loadbearing capacity and the vertical deformation at the instant of failure. This effect increases as
the slenderness of the wall increases.
The results are also scattered. The scatter could be attributed to the lack of homogeneity in
the component material properties, the manual procedure used to build the walls and/or the
variation in the curing processes. Tests performed on the material components (summarised in
Table 1) resulted in coefficients of variation slightly higher than those found in literature. The
vertical position of the walls during the construction and the manual brick laying may also
explain some of the scatter compared with other test studies (e.g., [6]). In addition,
imperfections in the wall geometry were measured in the present study and may partially
explain the scatter in the results. These imperfections are acceptable according to the current
manufacturing standards. The air curing conditions may also explain part of the scatter. Some
of the walls were built and cured in summer outdoor conditions, whereas others were
produced in winter conditions, both indoors and outdoors.
The scatter in the results is comparable to the scatter typically observed in similar studies [26,
27]. Although the scatter is significant, these tests more closely represent actual masonry
structures than strict laboratory-controlled tests. The data obtained in this study are
representative of real structures. Nonetheless, scatter should be taken into account when
analysing and comparing the results in section 5.
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4. NUMERICAL TOOL AND STANDARD CODES
4.1.
Features of the finite element analysis method
Although there are widely accepted models to represent the behaviour of masonry (see [1420, 54-58]), a different approach, intended to be easily implemented in commercial software
(ANSYS© 12.1), is preferred because of its wider availability to practitioners. The model used in
this study is a 2D plane-strain simplified micro-model that combines solid elements to
represent the units with interface joint elements to represent the bed joints. As it is normally
done when using simplified micro-models, the unit volume is increased to represent the
surrounding mortar joints. Hence, the model is composed of extended units that model both
bricks and mortar and zero-thickness interface elements that simulate the discontinuities. The
interface elements do not affect the response to compression but can open and rotate under
tension. In compression, the units are assumed to behave elastically up to the compression
limit; subsequently, perfect plasticity is applied. Under tension, the response is controlled by
the interface joint elements, which have a perfectly brittle response. Hence, the non-linear
inelastic behaviour of the material in tension is concentrated in the joint elements.
Because the interface elements do not deform in compression, the masonry deformation
modulus is determined by the deformability of the extended units. The Young’s modulus used
for the extended units is equal to the value determined experimentally on masonry specimens.
The contact elements are characterised by a cohesive zone model (CZM) that was originally
proposed by [59] to describe tensile behaviour. The CZM defines elastic behaviour (using the
same E as in the masonry homogenised elements) up to the fragile failure of the joint. When
the tensile strength (brick-mortar interface) is reached, the contact opens while the normal
tensile stress decreases linearly. The parameters required to model the joints in the CZM are
the ultimate tensile stress (fxt) of the brick-mortar interface and the fracture energy of the first
fracture mode (GfI) of the brick-mortar interface. The model includes large deformations to
allow second-order bending effects and to allow the expected buckling failure. The load is
indirectly applied as an imposed vertical descending displacement of the upper end of each
wall. The analyses are performed by increasing the displacement step-by-step and determining
the post-critical behaviour.
All simulated cases are defined with a hinge test setup. In Figure 3, the test configuration and
model geometry used for each case is presented. Images a), b) and c) of this figure show the
real test configuration, with the actual shape of the hinges, for the walls of the current
experimental campaign (a), the experimental campaign presented in [27] (b) and the cases in
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[26] (c). The second column of images in Figure 3 (d, e, f), shows the geometric definition of
the model used for the same experimental campaigns. Finally, illustrations g) h) and i) of the
Figure 3 (cases g), h) and i)) represent the model used for each case including the boundary
conditions, the masonry rows definition and the loading process.
As can be seen in Figure 3 d), e) and f), the hinged condition of the wall has been simulated in a
simplified way by means of stiff triangular objects placed at each end of the walls. These
triangular objects have been defined with a much larger stiffness than that of masonry. This
simplified way of modelling the hinges has been successfully used in similar applications (see
[6]). These objects provide a clear position for the load so that the eccentricity can be
modelled accurately in all cases.. A minimum 1-mm eccentricity is always used to provide an
initial imperfection to activate the geometric non-linear response. All the models (see Figure 3
g), h) and i)) have been defined with the actual geometry, measured as discussed in section 3.2
for the present experimental campaign and available in [26,27] for the comparison cases
tested by other authors.. Table 6-8 summarise all geometric data used in numeric models. The
description of the geometry (see Figure 3 d), e) and f)) takes into consideration the eccentricity
at each end of the wall (eb and et which are the eccentricity at the bottom and top of the wall
respectively), the hinge alignment (d) and the actual out-of-plane geometric imperfections
which determine the mid-height eccentricity, em. The effective height (Hef), wall thickness (t)
and average masonry row height (hrow) are also presented in Table 6-8.
A mapped mesh of 8-node quadrangular elements is used in combination with the interface
elements. The parameters required by the numerical model are the modulus of elasticity (E),
the Poisson’s coefficient (υ) and the compressive strength (fc) of the masonry. The model
requires also the bonding strength (fxt) of the brick-mortar interface and the fracture energy of
the brick-mortar interface of the first fracture mode (GfI) as input data. In the lack of direct
experimental information, the fracture energy has been estimated based on a linear fitting,
based on experimental values from [60], relating the fracture energy of the brick-mortar
interface, GfI with the corresponding bonding strength fxt. The resulting correlation allows the
estimation of the fracture energy as GfI= 36.65· fxt, where GfI is given in N/m and fxt in
MPa.Table 9 summarises the values of all variables used for each simulation case. Table 6 and
Table 7 show the comparison of the numerical predictions with all the experimental results for
the cases taken from the literature [26, 27]. Figure 9 and Figure 10 show a graphical
comparison between the experimental load-bearing capacity of the walls presented in [26, 27]
and the numerical results for different slenderness and eccentricities. Table 8 presents the
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geometric data for the simulation of the walls tested in the current research. It must be noted
that previous researches do not provide comprehensive information about the lateral or
vertical displacement of the walls at the collapse load. In contrast, the work herein presented
includes a comparison of the numerical predictions of the lateral deformation of the walls at
mid-height (Figure 8) and the vertical descending movement of the top of the specimen with
the corresponding experimental values (Figure 7).
4.2.
Application of the standard codes EC-6 and ACI-530
The load-bearing capacity of the walls is calculated using two standard codes, EC-6 and ACI530 ([2, 3]), which have been applied to calculate the load-bearing capacity of walls in the
literature ([26, 27]). The comparison of the standards’ analytical results with the experimental
results is used to determine the accuracy of these two widely used codes.
The codes are compared using the experimental values of the modulus of linear deformation,
E, and the masonry compressive strength, fc (Table 9). Safety factors are not considered
because the purpose of the calculation is the direct comparison with experimental data.
5. COMPARISON OF RESULTS AND DISCUSSION
In this section, the experimental results obtained by the present study and by other
researchers are compared with the predictions of the numerical model and with codes EC-6 [2]
and ACI-530 [3].
Experimental results from other researchers [26, 27] are analysed, and the numerical model is
validated by comparison to these tests. These studies involve 28 different combinations of
slenderness and eccentricity (see Table 6 and Table 7). All of them are compression tests on
full-scale unreinforced masonry walls hinged at both ends. The same numerical approach is
compared with the experimental results of the tests described in section 3. The goal of this
comparison is to evaluate the accuracy of the simplified micro-model in predicting the loadbearing capacity of walls.
The methods proposed by EC-6 [2] and ACI-530 [3] for the determination of the wall capacity
are also applied and compared with the experimental and numerical results.
Experimental data from two sources, namely Watstein & Allen [26] and Kirstchig & Anstötz
[27] were chosen for the comparison because of the clear boundary conditions of the tests
(the walls were hinged at both ends), the detailed information regarding the material
properties and the fact that all walls were at full scale.
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Table 9 summarises the parameters used in each simulation. For Poisson’s coefficient, which is
of little relevance for the type of 2D modelling utilised, a common value taken from the
literature [61, 62] was used.
The brick-mortar bonding strength (fxt) for the walls tested in [27] was estimated because it
was not provided by the authors. In this case, the value of fxt was chosen to satisfy the ratio
fc/fxt equal to the experimentally obtained value in the present study (fc/fxt = 40). The rest of
the parameters in Table 9 were obtained from the experimental tests.
The series of walls labelled as “HB” in [26] was chosen for comparison. The numerical model
provided a very satisfactory prediction of the corresponding ultimate wall load-bearing
capacities. The average relative error was 10.5%. Some calculated results were higher than the
experimental ones and others were lower, so the model appears to be balanced (see Figure 9).
It predicts without either a conservative or an unsafe bias. However, the results with the least
error were in the cases with the lowest eccentricities. For e=0, the error was 6.4%; for e=t/3,
the error was 19.5 %. The slenderness did not seem to affect the accuracy of the model, as
good results were obtained regardless of the value of the slenderness. On the whole, the
model slightly overestimates the load-bearing capacity of the walls. By contrast, the EC-6 code
produces conservative results, underestimating the capacity of the walls for all cases of [26]
(with an average relative error of 62.1%), while the ACI-530 code shows greater accuracy, with
an error of 45.3%, and a tendency to underestimate the load-bearing capacity of the most
eccentrically loaded walls. Conversely, the ACI-530 code overestimates the load-bearing
capacity of the concentrically loaded walls.
In Figure 10, the results from tests on walls with calcium silicate units reported in [27], are
compared with the results obtained with the numerical model. Good agreement between
experimental and numerical results is shown, with an average relative error of 28.4%. This
error is within the range of the scatter in the experimental results in [27]. The agreement is
particularly good in the cases with lowest eccentricities and/or slenderness (4.2% for e=0 and
𝜆=5.6) in contrast with the trend obtained for the results reported in [26]. The numerical
model seems to overestimate the load-bearing capacity of the walls tested by Kirtsching and
Anstötz [27]. The EC-6 standard code conservatively underestimates the strength of almost all
walls compared with the data in [27] (Figure 10). In all the comparisons, EC-6 overestimates
the load-bearing capacity in only two cases, those with the highest eccentricity. The ACI-530
standard code overestimates most of the ultimate loads. It shows more accurate results for
the cases with low load eccentricity and overestimates the load-bearing capacity for the cases
14
with the greatest load eccentricity. Compared with these data, the average relative error of
the EC-6 standard code is 32.3% and that of the ACI-350 standard code is 28.9%.
The comparison of the numerical model with the results from [26, 27] shows that the largest
influence of the Young’s modulus (E) over the axial load-bearing strength of the walls occurs
when the load is eccentrically applied and second-order bending appears. As the Young’s
modulus increases, the deformation of the wall from second-order effects decreases because
of the increased stiffness of the wall. A larger axial load is thus needed to develop secondorder bending failure, and buckling becomes more likely. Similarly, larger flexural brick-mortar
bonding strength (fxt) contributes by providing more strength against second-order bending
failure. The load-bearing capacity of the walls can also be improved by increasing the masonry
compressive strength (fc). However, the effect of the compressive strength is more apparent
when the predominant failure mode is material crushing, as in the case of walls with low load
eccentricity (e) and a low slenderness ratio (𝜆=Hef/t).
Once validated, the numerical model is applied to predict the response of the H-, M- and Sseries experimental walls presented in section 3.3 (except for wall W#5, whose inconsistent
response has been previously discussed). Table 9 summarises the parameters used in the
simulations. Comparisons between calculated load-bearing capacity and experimental results
are shown in Figure 11, Figure 12 and Figure 13. The best agreement is obtained for the most
slender walls (H series, Figure 11), with an average relative error of 38.4%. Although these
results are not as good as those obtained for the experimental results of previous studies ([26,
27]), the accuracy is acceptable because it is within the range of the common scatter obtained
in the experimental tests. As mentioned previously, experimental results of tests on masonry
walls that rely on the flexural strength of the material normally show highly scattered results,
e.g., [33]. In fact, the scatter in the experimental results in this study is larger than the average
relative error of the numerical model. For H-series walls, the results show no general over- or
underestimation. The best results are obtained for the most slender walls. This is the same
trend that is observed in the modelling tests from [26]. The comparison of a linear fit of the
experimental results to a linear fit of the numerical model results (see Figure 11) show that the
model correctly predicts the behaviour of the more slender walls (the H-series walls).
For the M- and S-series walls, the numerical results are consistently conservative. The error
decreases as the eccentricity increases. This result indicates that the numerical model is more
applicable for slender and eccentrically loaded walls. The average relative errors are 43.2% and
61% for M- and S-series walls, respectively. The errors in the most eccentric tests are 5.1% and
15
52.8% for M- and S-series walls, respectively. The ultimate load calculated with the numerical
model follows the trend of the experimental load measurement (see the linear fit in Figure 12
and the results in Figure 13) and is accurate within the limits of the experimental scatter. The
behaviour of the model could be improved by implementing a constitutive equation to account
for the over-strengthening of masonry in compression, e.g., the increase of the compressive
strength for eccentric loading noticed by other authors [52]. The inclusion of the overstrengthening in compression would cause higher load-bearing capacities when the crushing
failure process is significant, as in the case of less slender walls subjected to concentric or
moderately eccentric loads.
In general, acceptable agreement is obtained between the experimental results and the
numerical predictions. With respect to the application of standards [2, 3], in the case of the
experimental programmes considered, EC-6 always underestimates the load-bearing capacity
of the walls, with an average relative error of 96.0%. The best agreement is observed for the
less slender walls (S series), with an error of 88.6%. The results from ACI-530 produce an
average relative error of 48.3%, producing conservative results for almost all cases. ACI-530
yields the best results for walls of the H series with an average relative error of 34.6%.
6. CONCLUSIONS
Experimental results on full-scale brickwork walls, tested with in-plane eccentric compression
loads, are presented. The experiments comprise tests on masonry walls, changing the
parameters of slenderness and load eccentricity. All walls are hinged at both ends. This study
completes a previous work on small-scale models.
The experimental results show considerable scatter. Nonetheless, the relationships with
eccentricity and slenderness exhibit the same trends reported in previous research.
The collapses are caused by geometric instability, except for walls W#4, W#14 and W#17, as
discussed in section 3.3. This type of failure is observed in 17 of the 20 tests. As expected, this
collapse mode is more evident in the case of the most slender walls or the most eccentrically
loaded walls.
Using two mechanical hinges in the experimental design is advantageous for the purpose of
validating the numerical tools because the boundary conditions are more precisely known and
can be more accurately modelled.
16
A numerical model for the prediction of the ultimate load-bearing capacity of masonry walls is
chosen and validated by comparison with the experimental results. The experimental results
are also compared with the methods proposed by standards ACI-350 and EC-6 for the
calculation of the load-bearing capacity of masonry walls.
The numerical model yields better results than the two standard codes for all the wall series
tested. Moreover, it is the most balanced method, with predictions of the load-bearing
capacity both over- and underestimating the experimental results. The EC-6 predictions are
conservative for all cases except for those shown in Figure 11. Generally, ACI-530 is more
accurate for cases with low eccentricity and low slenderness. In other words, ACI-530 is more
applicable to cases when the failure is directly associated with masonry crushing. The
numerical model is more accurate in the cases with high eccentricity and more slenderness, as
observed by comparing Figure 11 with Figure 13. The model is less accurate in cases with low
eccentricity and/or less slenderness. This reduction in accuracy may be caused by the
simplicity of the compressive behaviour model (linearly elastic and perfectly plastic). Future
work to include an improved compressive stress-strain relationship in the model is
recommended. Further future works could also be oriented toward incorporating the out-ofplane shear in the analysis of unreinforced masonry walls.
To obtain acceptable results with a finite element analysis, a realistic Young’s modulus must be
used because it determines the deformation leading to the geometric instability associated
with buckling failure. The formulations included in the standard codes ([2, 3]) are highly
dependent on the value of Young’s modulus. Given the variability and limited information on
the value of Young’s modulus in masonry, it is necessary to do further work on the study of
this essential parameter and its influence of the behaviour of slender and eccentrically loaded
walls.
These results emphasise that a realistic description of the joint behaviour is essential to
accurately predict the load-bearing capacity of unreinforced masonry walls, especially when
the failure mode is primarily second-order bending.
7. ACKNOWLEDGEMENTS
The authors wish to acknowledge Dr. Marco Antonio Pérez, Mr. Juan José Cruz, Mr. Francesc
Puigvert and Mr. Christian Escrig for providing essential support in the experimental
programme and Dr. Jordi Marcé for his assistance in the development of the numerical model.
17
This study has been partially performed within the project BIA 2006-04127, funded by the
Spanish Ministry of Science and Innovation, and the project MURETEX, funded by CDTI Spanish Ministry of Economics and Competitiveness, whose assistance is gratefully
acknowledged.
8. REFERENCES
[1] Mann W. Basics of design of masonry walls subjected to vertical loads and buckling
according to the present draft of Eurocode EC6 and comparison with experimental data.
Proceedings of the 9th International Brick/Block Masonry Conference. Berlin: Deutsche
Gesellschaft fur Mauerwerksbau eV; 1991. p. 1281-91.
[2] Standardization ECf. Eurocode 6: Design of masonry structures. Part 1-1: General rules for
buildings. Rules for reinforced and unreinforced masonry. 1997:150.
[3] Committee MSJ. Building Code Requirements for Masonry Structures. ACI 530-05.
Farmington Hills: American Concrete Institute; 2005.
[4] Lu M, Schultz A, Stolarski H. Analysis of the Influence of Tensile Strength on the Stability of
Eccentrically Compressed Slender Unreinforced Masonry Walls Under Lateral Loads. J Struct
Eng. 2004;130:921.
[5] Brencich A, Corradi C, Gambarotta L, Mantegazza G, Sterpi E. Compressive strength of solid
clay brick masonry under eccentric loading. Proceedings British Masonry Society2002. p. 3746.
[6] Sandoval C, Roca P, Bernat E, Gil L. Testing and numerical modelling of buckling failure of
masonry walls. Constr Build Mater. 2011;25:4394-402.
[7] Dajun D. Studies on brick masonry under compression. Mater Struct. 1997;30:247-52.
[8] Defalco A, Lucchesi M. No tension beam-columns with bounded compressive strength and
deformability undergoing eccentric vertical loads. IJMS. 2007;49:54-74.
[9] Hansen L, Nielsen MP. Stability of masonry columns. orbitdtudk. Lyngby: Technical
University of Denmark, Department of Civil Engineering, Section for Structural Engineering;
2003. p. 147.
[10] Mura I. Stability of nonlinear masonry members under combined load. Comput Struct.
2008;86:1579-93.
[11] Romano F, Ganduscio S, Zingone G. Cracked nonlinear masonry stability under vertical and
lateral loads. J Struct Eng. 1993;119:69-87.
[12] Schultz A, Mueffelman J, Ojard N. Critical axial loads for transversely loaded masonry
walls. 12 th Int Brick/Block Masonry Conf Proc2000. p. 1633-46.
[13] Yokel FY. Strength of load bearing masonry walls. J Struct Div. 1971;97:1593-609.
[14] Orduña A. Seismic assessment of ancient masonry structures by rigid blocks limit analysis.
at www civil uminho pt/masonry: PhD Thesis 2003. University of Minho, Portugal. Available at
www. civil. uminho. pt/masonry; 2003. p. 171.
[15] Sacco E. A nonlinear homogenization procedure for periodic masonry. Eur J Mechs - A/Sol.
2009;28:209-22.
[16] Zucchini A, Lourenço PB. Mechanics of masonry in compression: Results from a
homogenisation approach. Comput Struct. 2007;85:193-204.
[17] Ainsworth M, Mihai LA. An adaptive multi-scale approach to the modelling of masonry
structures. Int J Numer Method Eng. 2009;78:1135-63.
[18] Chen S, Moon F, Yi T. A macroelement for the nonlinear analysis of in-plane unreinforced
masonry piers. Eng Struct. 2008;30:2242-52.
[19] Mistler M, Anthoine a, Butenweg C. In-plane and out-of-plane homogenisation of
masonry. Comput Struct. 2007;85:1321-30.
18
[20] Ramalho M, Taliercio A, Anzani A, Binda L, Papa E. A numerical model for the description
of the nonlinear behaviour of multi-leaf masonry walls. Adv Eng Softw. 2008;39:249-57.
[21] da Porto F, Mosele F, Modena C. Experimental testing of tall reinforced masonry walls
under out-of-plane actions. Constr Build Mater. 2010;24:2559-71.
[22] Cavaleri L, Failla A, La Mendola L, Papia M. Experimental and analytical response of
masonry elements under eccentric vertical loads. Eng Struct. 2005;27:1175-84.
[23] Roca P. Simplified methods for assessment of masonry shear-walls. 6º Congresso
Nacional de Sismologia e Engenharia Sísmica2004. p. 101-18.
[24] Charry J. Estudio experimental del comportamiento de paredes de obra de fábrica de
ladrillo ante la acción de cargas laerales. 2010.
[25] Murthy C, Hendry a. Model experiments in load bearing brickwork. Build Sci. 1966;1:28998.
[26] Watstein D, Allen MH. Structural performance of clay masonry assemblages built with
high-bond organic-modified mortars. Second International Brick Masonry Conference:
Structural Clay Products Institute; 1970. p. 99-112.
[27] Kirtschig K, Asntötz W. Buckling tests on masonry. Proceedings of the 9th International
Brick/Block Masonry Conference. Berlin: IBMAC; 1991. p. 202-9.
[28] Knutsson H. Vertical load bearing masonry - The danish approach. Mason Int. 1991;5:23-6.
[29] Maurenbrecher AHP. Effect of the test procedures on compressive strength of masonry
prisms. 2nd Canadian Masonry Symposium. Ottawa: National Research Council Canada; 1980.
p. 119-32.
[30] Maurenbrecher AHP. Compressive Strength of eccentrally loaded masonry prisms. In:
NRCC, editor. Third Canadian Masonry Symposium. Edmonton: National Research Council
Canada; 1983.
[31] Maurenbrecher AHP. Axial Compression Tests on Masonry Walls and Prisms. Third North
American Masonry Conference. Arlington, Texas: National Research Council Canada; 1985. p.
17.
[32] Maurenbrecher AHP. Compressive strength of hollow concrete blockwork. Proceedings of
the Fourth Canadian Masonry Symposium. New Brunswick: Institute for Research in
Construction of National Research Council Canada; 1986. p. 997-1009.
[33] Gazzola EA, Drysdale RG, Essawy AS. Bending of concrete masonry wallettes at different
angles to the bed joints. Proceedings of the Third North American Masonry COnference.
Arlington, Texas: Construction Research Center Civile Engineering Department University of
Texas; 1985. p. 15.
[34] Sinha B, Ng C, Pedreschi R. Failure criterion and behavior of brickwork in biaxial bending. J
Mat Civil Eng. 1997;9:70-5.
[35] Yokel FY. Stability and load capacity of members with no tensile strength. J Struct Div.
1971;97:1913-26.
[36] Dawe JL, Liu Y. Analytical modeling of masonry load-bearing walls. Can J Civil Eng.
2003;30:795-806.
[37] Lourenço PB. Analysis of masonry structures with interface elements. Theory and
applications. TU-DELFT report no 03-21-22-0-011994. p. 34.
[38] Salerno G, de Felice G. Continuum modeling of periodic brickwork. Int J Solids Struct.
2009;46:1251-67.
[39] Martini K. Finite element studies in the two-way out-of-plane failure of unreinforced
masonry. In: Institute EER, editor. Proceedings of the 6th National Conference on Earthquake
Engineering. Seattle1998. p. 1-12.
[40] Cecchi A, Sab K. A homogenized Reissner–Mindlin model for orthotropic periodic plates:
Application to brickwork panels. Int J Solids Struct. 2007;44:6055-79.
[41] Mercatoris BCN, Bouillard P, Massart TJ. Multi-scale detection of failure in planar masonry
thin shells using computational homogenisation. Eng Fract Mech. 2009;76:479-99.
19
[42] Milani G, Lourenço PB, Tralli A. Homogenised limit analysis of masonry walls, Part II:
Structural examples. Comput Struct. 2006;84:181-95.
[43] Milani G, Lourenço PB, Tralli A. Homogenised limit analysis of masonry walls, Part I: Failure
surfaces. Comput Struct. 2006;84:166-80.
[44] Zucchini A, Lourenço PB. Homogenization of masonry using a micro-mechanical model:
Compressive behaviour. In: Soares CAM, editor. III European Conference on Computational
Mechanics Solids, Structures and Coupled Problems in Engineering. Lisbon, Portugal:
ECCOMAS; 2006.
[45] Reyes E, Casati M, Gálvez J. Experimental scale model study of cracking in brick masonry
under tensile and shear stress. Mater Construcc. 2008;58:69-83.
[46] Jager W, Bergander H. Comparison of buckling safety of masonry walls according to EC6
and German standards. Proc Br Mason Soc. 1998:279-83.
[47] Lijdens A, Villegas L. Aspectos del diseño de muros portantes de fábrica simple según
diferentes normas: excentricidades, reducción de la capacidad portante, e hipótesis de cargas
a considerar. Rev Obras Pub. 1994;141:43-53.
[48] 83 CAC. UNE-EN 1015-11:2000/A1:2007. Métodos de ensayo de los morteros para
albañilería. Parte 11: Determinación de la resistencia a flexión y a compresión del mortero
endurecido. Madrid: AENOR; 2007.
[49] 41 CAC. UNE-EN 772-1:2002. Métodos de ensayo de piezas para fábrica de albañilería.
Parte 1: Determinación de la resistencia a compresión. Madrid: AENOR; 2002.
[50] 41 CAC. UNE-EN 772-11:2001/A1:2006. Métodos de ensayo de piezas para fábrica de
albañilería. Parte 11: Determinación de la absorción de agua por capilaridad de piezas para
fábrica de albañilería, en hormigón, piedra natural y artificial, y de la tas. Madrid: AENOR;
2006.
[51] Brencich A, Corradi C, Gambarotta L. Eccentrically loaded brickwork: Theoretical and
experimental results. Eng Struct. 2008;30:3629-43.
[52] Brencich A, Felice GD. Brickwork under eccentric compression: Experimental results and
macroscopic models. Constr Build Mater. 2009;23:1935-46.
[53] Brencich A, Gambarotta L. Mechanical response of solid clay brickwork under eccentric
loading. Part I: Unreinforced masonry. Mater Struct. 2005;38:257-66.
[54] Attard M, Nappi A, Tin-Loi F. Modeling Fracture in Masonry. J Struct Eng. 2007;133:1385.
[55] Brasile S, Casciaro R, Formica G. Multilevel approach for brick masonry walls – Part I: A
numerical strategy for the nonlinear analysis. Comput Method Appl M. 2007;196:4934-51.
[56] Brasile S, Casciaro R, Formica G. Multilevel approach for brick masonry walls – Part II: On
the use of equivalent continua. Comput Method Appl M. 2007;196:4801-10.
[57] Lourenço PB, Rots JG. Multisurface interface model for analysis of masonry structures. J
Eng Mech. 1997;123:660-8.
[58] Milani G, Lourenço PB, Tralli A. 3D homogenized limit analysis of masonry buildings under
horizontal loads. Eng Struct. 2007;29:3134-48.
[59] Alfano G, Crisfield MA. Finite Element Interface Models for the Delamination Anaylsis of
Laminated Composites: Mechanical and Computational Issues. Int J Numer Method Eng.
2001;50:1701-36.
[60] Pluijm R. Material properties of masonry and its components under tension and shear. In:
Neis VV, editor. 6th Canadian Masonry Symposium. Saskatoon, Saskatchewan, Canada1992. p.
675-86.
[61] Oliveira D. Mechanical characterization of stone and brick masonry. report 00-Dec/E-4,
Universidade do Minho, Departamento de Engenharia Civil, Guimarães, Portugal. Guimarães,
Portugal2000. p. 1-77.
[62] Rosas Rodríguez JG, Villegas L, Lorenzo Esperante D. The numerical models front to the
behaviour of masonry elements tested in laboratory. Informes Construcc. 2008;53.
20
Tables
Table 1. Characterisation test results of mortar, corresponding to the walls in which each
mortar was used. The coefficient of variation of the resulting value is in parentheses.
Table 2. Characterisation test results of masonry.
Table 3. E/fk ratio from literature and experimental tests
Table 4. Theoretical geometry of the tested full-scale unreinforced masonry walls.
Table 5. The results of the tests on unreinforced masonry walls under eccentric compressive
loads
Table 6. Values of the parameters for the numerical model
Table 7. Geometric values used in the numerical simulation of the experimental campaign
from [26]
Table 8. Geometric values used in the numerical simulation of the experimental campaign
from [27]
Table 9. Geometric values used in the numerical simulation of the current experimental
campaign
Compressive strength
Flexural strength
Mortar # Tests Value (MPa) # Tests Value (MPa) Walls
M1
81
3.18 (0.33)
47
1.24 (0.30)
1
M2
168
3.70 (0.63)
84
1.25 (0.89)
2-20
Table 1. Characterisation test results of mortar, corresponding to the walls in which each mortar was used. The
coefficient of variation of the resulting value is in parentheses.
Wall
#
1
Mortar
Modulus of linear
Compressive strength
Bonding strength
deformation, E (MPa)
fc (MPa)
fxt (MPa)
18.2
0.23
M1
2-5
6-9
12.9
780
M2
13.7
10-20
0.36
10.8
Table 2. Characterisation test results of masonry.
Source of information
[31] Clay brickwork series 2
[31] Clay brickwork series 1
[51] Concentric loading
[52] Concentric Brickwork type 1
[52] Concentric Brickwork type 2
[52] Concentric Brickwork type 3
Modulus of linear
Compressive
deformation, E (MPa)
strength, fc (MPa)
21700.00
8675.00
2172.50
1867.00
1700.00
2313.00
26.04
13.82
12.92
13.50
13.25
7.53
E/fc
833.33
627.50
168.19
138.30
128.30
307.04
21
[51] Concentric load
Mortar M1. Wall 1
Mortar M2. Walls 2-5
Mortar M2. Walls 6-9
Mortar M2. Walls 10-20
1440.00
780.00
780.00
780.00
780.00
11.70
18.20
12.90
13.70
10.80
123.08
42.86
60.47
56.93
72.22
Table 3. E/fk ratio from literature and experimental tests
Series Height, H (mm) Width, b (mm) Thickness, t (mm) Walls #
H
2700
900
132
1-3,5-9
M
1650
900
132
10-16
S
1000
900
132
18-20
F
2700
900
132
4
T
1800
900
270
17
Table 4. Theoretical geometry of the tested full-scale unreinforced masonry walls.
em/t (%) d/t (%) Pmax (kN) Ф(%) hmax/Hef·10-3 vmax/Hef·10-3
Wall
Geometry
𝜆
W#1
H
22.3
4.2
0.0
169.3
7.8
4.5
2.5
W#2
H
22.2
14.9
0.0
65.7
4.3
5.9
1.4
W#3
H
22.1
8.5
7.6
133.8
8.7
9.2
2.4
W#4
F
15.0
3.9
4.9
578.6
37.8
12.7
10.8
W#5
H
22.1
10.6
5.3
239.8
15.6
5.1
2.2
W#6
H
21.7
16.1
-9.5
30.0
1.8
4.9
0.5
W#7
H
21.8
5.4
-4.5
134.7
8.3
6.7
1.9
W#8
H
22.3
0.3
6.1
129.4
8.0
5.6
1.5
W#9
H
21.9
9.8
-6.4
109.8
6.7
4.5
1.3
W#10
M
14.1
8.1
8.0
423.9
33.0
8.2
6.0
W#11
M
14.3
18.1
7.6
371.2
28.9
9.2
5.5
W#12
M
13.9
5.0
3.8
471.1
36.7
8.9
6.0
W#13
M
14.1
25.6
1.5
83.8
6.5
9.4
2.2
W#14
M
14.1
16.0
0.0
518.5
40.4
9.0
5.4
W#15
M
14.0
24.3
2.3
236.7
18.4
10.1
3.2
W#16
M
14.1
12.5
-2.3
408.2
31.8
8.5
4.5
W#17
T
6.8
66.9
9.1
491.1
38.3
8.8
4.8
W#18
S
9.1
15.8
2.3
803.4
62.6
2.9
5.4
W#19
S
9.2
22.9
0.0
686.1
53.5
4.5
8.7
W#20
S
9.3
25.0
2.3
152.6
11.9
7.6
4.3
Table 5. The results of the tests on unreinforced masonry walls under eccentric compressive loads
22
Case
et (mm) eb (mm) em (mm) d (mm) Hef (m) t (mm) hrow aprox. (mm)
HB-10-0
1.15
HB-20-0
0
HB-30-0
0
2.12
2.98
HB-40-0
3.95
HB-10-1/6
1.15
HB-20-1/6
HB-30-1/6
15.5
0
0
-15.5
2.12
2.98
HB-40-1/6
3.95
HB-10-1/3
1.15
HB-20-1/3
HB-30-1/3
31.0
-31.0
HB-40-1/3
93
73
2.12
2.98
3.95
Table 6. Geometric values used in the numerical simulation of the experimental campaign from [26]
Case
et (mm) eb (mm) em (mm) d (mm) Hef (m) t (mm) hrow aprox. (mm)
5.6-0
11.1-0
18.8-0
0.635
0
0
1.25
0
2.12
27.7-0
3.12
5.6-1/8
0.635
11.1-1/8
18.8-1/8
14.4
14.4
27.7-1/8
18.8-1/4
2.12
0
5.6-1/4
11.1-1/4
1.25
14.4
28.8
28.8
3.12
0.635
2.12
3.12
5.6-1/3
0.635
18.8-1/3
38.3
38.3
105
1.25
28.8
27.7-1/4
11.1-1/3
115
1.25
38.3
2.12
27.7-1/3
3.12
Table 7. Geometric values used in the numerical simulation of the experimental campaign from [27]
Case
W#1
et (mm) eb (mm) em (mm) d (mm) Hef (m) t (mm) hrow aprox. (mm)
20
20
5.6
0.0
2.95
132
60
23
W#2
20
20
19.6
0.0
2.93
W#3
10
10
1.2
10.0
2.92
W#4
0
0
0
6.5
1.98
W#5
5
5
7.0
7.0
2.92
W#6
25
25
33.7
-12.5
2.86
W#7
10
10
13.1
-6.0
2.87
W#8
10
10
-7.6
8.0
2.94
W#9
5
5
21.5
-8.5
2.89
W#10
10
10
0.2
10.5
1.87
W#11
20
20
13.9
10.0
1.89
W#12
10
10
1.6
5.0
1.84
W#13
30
30
31.8
2.0
1.86
W#14
20
20
21.2
0.0
1.86
W#15
30
30
29.0
3.0
1.85
W#16
20
20
19.5
-3.0
1.86
W#17
80
80
88.0
12.0
1.91
W#18
20
20
20.8
3.0
1.20
W#19
30
30
30.3
0.0
1.22
W#20
30
30
33.0
3.0
1.22
282
132
Table 8. Geometric values used in the numerical simulation of the current experimental campaign
Experimental reference
E (MPa) fc (MPa)
[26]
25400
44.81
[27]
8540
12.2
W#1
W#2, W#3 and W#5
W#6-W#9
W#10-W#16 and W#18-W#20
υ
0.3
18.2
780
12.9
13.7
10.8
Table 9. Values of the parameters for the numerical model
0.35
fxt (MPa) GfI (N/m)
2.98
110
0.3
7
0.23
8
0.36
13
24
Figures
Figure 1. Test setup a) for walls from series S, b) for walls from series M and c) for walls from
series H
Figure 2. Detail of the connection between the wall and the test system at the lower end (left)
and the upper end (right)
Figure 3. Test configuration for the presented campaign (a), the campaign from [27] (b) and
the campaign from [26] (c); geometric definition for the presented campaign (d), the campaign
from [27] (e) and the campaign from [26] (f); model used for the simulation of the walls from
for the presented campaign (g), the campaign from [27] (h) and the campaign from [26] (i).
Figure 4. Failure mode of wall W#10. Opening of a joint and mechanism formation
Figure 5. Failure mode of wall W#14. Crushing of the compressed side mixed with the tensile
opening of two joints to form a mechanism
Figure 6. Experimental relationship between eccentricity and load-bearing capacity for three
different theoretical values of slenderness corresponding to wall geometries H, M and S.
Figure 7. Experimental relationship between downward displacement at maximum load and
load-bearing capacity for three different theoretical values of slenderness corresponding to
wall geometries H, M and S.
Figure 8. Experimental relationship between actual eccentricity (not taking into account hinge
alignment) and the descending vertical displacement at maximum load for three different
theoretical values of slenderness corresponding to wall geometries H, M and S.
Figure 9. Comparison of the experimental results of [26] with the numerical simulation and
application of the standard codes
Figure 10. Comparison of the experimental results of [27] with the numerical simulation and
application of the standard codes
Figure 11. Comparison of the experimental results of H-series walls with the numerical
simulation and application of the standard codes
Figure 12. Comparison of the experimental results of M-series walls with the numerical
simulation and application of the standard codes
Figure 13. Comparison of the experimental results of S-series walls with the numerical
simulation and application of the standard codes
25
Figure 1. Test setup a) for walls from series S, b) for walls from series M and c) for walls from series H
Figure 2. Detail of the connection between the wall and the test system at the lower end (left) and the upper end
(right)
26
Figure 3. Test configuration for the presented campaign (a), the campaign from [27] (b) and the campaign from [26]
(c); geometric definition for the presented campaign (d), the campaign from [27] (e) and the campaign from [26] (f);
model used for the simulation of the walls from for the presented campaign (g), the campaign from [27] (h) and the
campaign from [26] (i).
27
Figure 4. Failure mode of wall W#10. Opening of a joint and mechanism formation
Figure 5. Failure mode of wall W#14. Crushing of the compressed side mixed with the tensile opening of two joints
to form a mechanism
28
Figure 6. Experimental relationship between eccentricity and load-bearing capacity for three different theoretical
values of slenderness corresponding to wall geometries H, M and S.
Figure 7. Experimental relationship between downward displacement at maximum load and load-bearing capacity
for three different theoretical values of slenderness corresponding to wall geometries H, M and S.
29
Figure 8. Experimental relationship between actual eccentricity (not taking into account hinge alignment) and the
descending vertical displacement at maximum load for three different theoretical values of slenderness
corresponding to wall geometries H, M and S.
30
Figure 9. Comparison of the experimental results of [26] with the numerical simulation and application of the
standard codes
Figure 10. Comparison of the experimental results of [27] with the numerical simulation and application of the
standard codes
31
Figure 11. Comparison of the experimental results of H-series walls with the numerical simulation and application of
the standard codes
Figure 12. Comparison of the experimental results of M-series walls with the numerical simulation and application
of the standard codes
32
Figure 13. Comparison of the experimental results of S-series walls with the numerical simulation and application of
the standard codes