Engineering Structures 91 (2015) 96–111

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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct

Numerical analysis of the load-bearing capacity of brick masonry walls

strengthened with textile reinforced mortar and subjected to eccentric
compressive loading
Ernest Bernat-Maso a,⇑, Lluis Gil a, Pere Roca b
a
Department of Strength of Materials and Engineering Structures, Technical University of Catalonia, Colom 11, 08222 Terrassa, Spain
b
Department of Construction Engineering, Technical University of Catalonia, Jordi Girona 1-3, 08034 Barcelona, Spain

a r t i c l e i n f o a b s t r a c t

Article history: There is little research about the numerical modelling of brick masonry walls strengthened with Textile
Received 1 October 2014 Reinforced Mortar (TRM) against second order bending effects. Thus, a numerical model has been imple-
Revised 24 February 2015 mented to cover this knowledge gap. It has been validated by comparison with experimental cases and
Accepted 25 February 2015
applied to enhance the knowledge about the influence of different strengthening variables. The model is
Available online 13 March 2015
accurate at predicting the load-bearing capacity of walls and points out that strengthening both sides
might cause a remarkable increase of the load-bearing capacity and that the performance of the TRM
Keywords:
is greater for the most slender cases.
Masonry
Walls
Ó 2015 Elsevier Ltd. All rights reserved.
Finite element analysis
Textile reinforced mortar
Second order bending

1. Introduction better compatibility with the particular masonry to be strength-

ened. For example, lime-based matrixes are normally suitable for
The structural response of load-bearing brick masonry walls is historic masonry and cement-based ones are more proper for con-
characterised by the almost negligible tensile strength of this temporary masonry. Because of this physical compatibility, several
material. This property defines the most common structural failure authors [3–5] present the TRM as an appropriate strengthening
modes which affect these elements. Among them, the buckling– solution for masonry structures. The advantages of TRM in front
bending mixed failure is significant because of the risky and sud- of other possible strengthening solutions based on organic
den development of this collapse, which might be favoured by matrixes (e.g. Fibre Reinforced Polymer, FRP), have been reported
eccentric compressive loads [1]. Using composite materials to pro- in by Papanicolaou et al. [6,7]. The main differences between these
vide tensile strength and avoid this failure mode has been proved two strengthening systems arise from the organic/inorganic nature
to be effective in a previous research [2]. of the corresponding matrix.
Fibre Reinforced Polymer (FRP) and Textile Reinforced Mortar Load-bearing brick masonry walls were extensively used in
(TRM) are common strengthening solutions for concrete and residential buildings up to the first half of the 20th century world-
masonry structures. TRM (also called Fabric Reinforced wide. Although, these are the most common structural masonry
Cementitious Matrix, FRCM) is a composite material consisting of elements among the existing constructions, most of the recent
a textile grid embedded into an inorganic matrix. The textile grid experimental researches about using TRM to strengthen masonry
is commonly made of high strength materials like glass, basalt or structures have focused on arches and curved surfaces [5,8]. This
carbon fibres, whereas the inorganic matrix is usually a cement fact might be related with the remarkable shape adaptability of
or lime mortar. The physical properties of these inorganic matrixes the TRM system, the possibility of removing it in an easier way
allow a better physical compatibility with masonry substrates than than other strengthening options, like the FRP, and its mechanical
other adhesives (e.g. epoxy resins). In fact, different inorganic mor- compatibility with masonry. In particular, the relatively low modu-
tars are available so it is possible to select the one which provides lus of elasticity of the TRM (if compared with other strengthening
systems, like the FRP) makes it possible to strengthen ancient
structures without introducing extremely rigid elements, which
⇑ Corresponding author. Tel.: +34 937398728; fax: +34 937398994. might cause significant modifications of the structural response
E-mail address: ernest.bernat@upc.edu (E. Bernat-Maso). and introduce stress concentrators which might damage the

http://dx.doi.org/10.1016/j.engstruct.2015.02.032
0141-0296/Ó 2015 Elsevier Ltd. All rights reserved.
 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111 97

original structure. Thus, the TRM seems a suitable solution for the scattering of the experimental results on masonry walls, which
strengthening historical constructions where arches are common is around 30% (see [1]). Thus, limiting the relative error at a slightly
structural elements. lower value than the typical scattering seems a reasonable
The bibliographic research revealed that there is little available approach.
experimental information about the application of TRM on
masonry walls. The corresponding tests were focused on analysing 2. Scope and methods
the structural response against in-plane loads [3,4] or out-of-plane
cyclic loads [9]. However, in non-seismic areas the horizontal The work presented herein is the continuation of two previous
in-plane loading conditions might be not critical. Instead, the com- researches: one about testing and modelling unreinforced brick
pressive loads transmitted by roofs or slabs can be crucial if they masonry walls [1] and another based on the experimental and
are eccentrically applied on the wall’s thickness. These loads might analytical investigation about the structural response of TRM-
induce second order bending effects causing a flexural-buckling strengthened brick masonry walls [2]. It has to be highlighted that
mixed failure [1]. In this line, only a few experimental researches the scope of the current work is limited to brick masonry walls
have studied the second order bending effects on TRM strength- subjected to eccentric compressive loads which are strengthened
ened brick masonry walls [10]. with TRM to avoid the mixed buckling–bending failure. The
Despite the significance of this structural problem, it has to be numerical model described later on in Section 3 is aimed to
highlighted that performing experimental tests to analyse the complement the previous calculation models for unreinforced
load-bearing capacity of full scale masonry walls strengthened masonry walls [1] with the inclusion of the TRM strengthening
with TRM and subjected to eccentric compressive loads requires system. Hence, the main objective of this model is to predict the
a long-term and expensive research activity, and has been explored load-bearing capacity of these strengthened walls.
only by a few researches [2,10]. On the other hand, numerical mod- The applied methodology consisted of implementing a
elling techniques provide an interesting and inexpensive tool for numerical model for predicting the load-bearing capacity of
designing and assessing structures. However, no numerical model TRM-strengthened walls subjected to eccentric compressive loads,
for the analysis of brick masonry walls strengthened with TRM and validating this numerical tool by comparison with experimental
subjected to eccentric compressive loads, which might cause results, comparing the accuracy of this model with the accuracy
second order bending effects, is known by the authors. of an specific analytical method and finally, extending the experi-
The first researches dealing with the numerical simulation of mental results including the analysis of different theoretical cases
the structural response of the TRM as a sole material have been to enhance the understanding about the influence of different
recently published [11,12]. Other researchers have studied how strengthening variables of the TRM.
to model the TRM-masonry interaction using masonry prisms as
reference samples [13]. However, there are not known publications 2.1. Experimental cases for validation
about modelling full scale TRM strengthened brick masonry struc-
tures. Conversely, the numerical analysis of reinforced concrete The implemented numerical model has been applied to calcu-
elements strengthened with TRM has received significant atten- late the load-bearing capacity of 9 TRM-strengthened walls which
tion. In this line, some authors have dealt with numerical mod- were previously tested up to failure. This procedure has allowed
elling TRM-strengthened reinforced concrete beams subjected to assessing the accuracy of the proposed tool. The comparison walls
flexural loading conditions [14,15], other have been focused on were part of a larger experimental campaign and their mechanical
the shear strengthening [16] or on the numerical analysis of par- details are fully described in [2]. The main geometric characteris-
ticular construction elements made of reinforced concrete like tics of these structures and the corresponding typology of
slabs [17] or beam-columns joints [18]. To some extent, the mod- strengthening are summarised in Table 1. Table 2, which includes
elling of TRM reinforced concrete can be considered as a reference data previously presented in communications [1,2], provides all
case and may provide some insight at modelling TRM reinforced required information to follow the current work. The mechanical
masonry structures. properties of the bricks, bricklayering mortar, and strengthening
The simulations of FRP-strengthened masonry walls are refer- mortars is included in Table 2. The compressive strength of the
ence cases for the analysed structural problem. These have been used masonry was 10.8 MPa, its flexural strength was 0.36 MPa
addressed from different points of view: considering the in-plane and the Young’s modulus was 780 MPa, which is 72.22 times the
response [19], considering the out-of-plane behaviour [20], focus- compressive strength (see [1]). Table 3 summarises the properties
ing on arches [21] or vaults [22], or the simulation of particular of the fibre grids provided by the manufacturers.
applications of the FRP on masonry walls as a continuous grid [23]. The 9 TRM-strengthened walls were tested in a pinned-pinned
Thus, no specific research about the numerical simulation of configuration with the aims of clearly identifying the boundary
TRM-strengthened brick masonry walls subjected to eccentric
compressive loads has been found. Taking into account that assess-
ing the load-bearing capacity of these structures is an essential Table 1
part of the rehabilitation or strengthening interventions, which Geometric characteristics and strengthening system of the experimentally tested
walls.
are aimed to enhance the life-cycle of the buildings, the current
work presents the implementation of a numerical model aimed Wall Effective d TRM TRM Connectors # tTRM
to cover this knowledge gap. The proposed simulations have been height (mm) mortar fibre Fibre (mm)
(mm) grid grids
carried out using a commonly available commercial Finite Element
Analysis (FEA) package to spread the possibility of being used by W#21 1832 0.0 M G 0 1 13.0
W#22 1827 0.0 M G 0 1 8.0
practitioners. In addition, it has been intended to keep the model
W#23 1822 5.5 R G 0 1 9.5
as simple as possible while assuring the required accuracy. This W#24 1840 0.5 R G 0 1 9.0
simplicity is oriented to allow a wider utilisation among research- W#25 1828 0.5 R G 0 2 7.5
ers and practitioners and, specifically, to permit its implementation W#26 1823 6.0 M G 0 2 8.0
in other FEA packages. This accuracy level is estimated to corre- W#27 1822 9.0 X C 0 1 8.0
W#28 1828 0.5 X C 6 1 9.0
spond with a relative error of the predicted load-bearing capacity W#29 1827 8.5 X C 9 1 11.0
below 25%. This threshold value might be set by comparison with
98 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111

Table 2
Experimental and theoretical properties of the bricks and mortars used in the research. Coefficient of variation of the experimentally determined values in brackets. The
experimental information is detailed in [1] and [2].

Material Flexural strength, fx (MPa) Compressive strength, fcm (MPa) Bonding strength, fb (MPa) Modulus of elasticity, E (MPa)
a
Bricks 2.8 (0.28) 27.9 (0.19) – 10,000
Durland M7,5 mortar 1.3 (0.89) 3.7 (0.63) 0.36 (0.18) 3000 a
Portland-based mortar 8.1 (0.18) 42.2 (0.27) 2a 11,000 a

Lime-based mortar 6.6 (0.03) 14.5 (0.08) 0.8 a 8000 a

Pozzolana-based mortar 9.4 (0.10) 34.5 (0.08) 0.8 a 15,000 a

a
Data provided by the producer of the material.

Table 3
This analytical approach is based on the force and flexural equi-
Properties of the fibre grids used in the studied TRM solutions. Manufacturer provided librium equations and the strain compatibility expressions that
values. Table included in [2]. allow obtaining the axial (N) vs. bending moment (M) failure cri-
Fibre Ultimate elongation of Tensile strength of the Grid dimensions
teria assuming two possible failure modes: the tensile failure of
    the TRM and the compressive failure of the masonry.
grid the fibre (%) efibre TRM (kN/m) T TRM
ult ult

Glass 3 45 25 mm  25 mm
Carbon 2.1 160 10 mm  10 mm 2.3. Theoretical cases

After comparing the proposed numerical model with the

experimental cases and assessing its accuracy in comparison with
conditions and easing the comparison with the numerical model.
an analytical method, the implemented tool was used to predict
In addition, the strengthening system was always placed in the
the load-bearing capacity of a series of 14 theoretical walls. This
side of the wall subjected to tensile stresses because it was
second series of simulations was set to analyse different variables
assumed that the TRM would be more efficient when installed on
of the TRM strengthening system: the influence of placing two or
this side.
more fibre grids embedded into the strengthening layer, the effect
The name of the experimentally tested walls is summarised in
of strengthening both sides instead of strengthening only one side
the first column of Table 1. The thickness of the walls, without con-
and the influence of the slenderness of the wall (by considering
sidering the strengthening system, was 132 mm and their width
two heights, which correspond to the experimental M and H series
was 900 mm. The second column shows the experimentally
in [1]) on the effectiveness of the strengthening system. In addi-
obtained effective height, Hef, of each wall. Hef is the vertical dis-
tion, these theoretical cases were used to analyse the influence of
tance between the axes of rotation of the two hinges. d is the mea-
some simplifications assumed when modelling the TRM strength-
surement of the horizontal distance between the axes of rotation of
ened walls, as presented in Section 3.2.
the hinges. d provides an indirect way to measure the geometric
The geometric characteristics and the configuration of the
imperfections of the tested walls. The last five columns in
strengthening system used in each theoretical wall are sum-
Table 1 are dedicated to characterise the TRM and they include
marised in Table 4. It has to be highlighted that the thickness of
the type of mortar (M for the Portland based mortar, R for the
the TRM, tTRM, is kept constant (10 mm) when embedding one or
lime-based mortar and X for the pozzolanic mortar as presented
two fibre grids according with the experimental evidences pre-
in [2]), the type of fibre grid (G for glass and C for carbon), the pres-
sented in [2]. However, this thickness is doubled when placing four
ence of connectors, the number of fibre grids installed into the
fibre grids. In turn, one fibre grid is considered for each TRM layer
mortar layer and the real thickness of the TRM layer.
in the cases of the walls which are strengthened at both sides. The
Two failure modes were experimentally observed: the tensile
name of the walls presented in Table 4 should be understood as
failure of the TRM associated with a mechanism formation process
follows: the first letter is associated with the height of the wall
(W#21 and W#23) and the shear/compressive failure near the
(medium, M or high, H), the second symbol is a 0 if it is an unrein-
endings of the wall. Moreover, the real geometric imperfections
forced wall and it is an S if the wall is strengthened with TRM, the
were measured and introduced into the numerical model through
third symbol (optional) corresponds to the number of sides
the geometry definition. The measurement process consisted on
determining the distance between a vertically aligned profile and
the wall when it was placed in the testing position. A laser sensor Table 4
was used for this purpose and measurements were carried out Geometric characteristics and strengthening system of the theoretical walls.
every two masonry rows at the two largest edges of one of the
Wall Effective height # Strengthened # Fibre tTRM
two biggest faces of the walls. These measurements allowed deter- (mm) faces grids (mm)
mining the shape of the wall (imperfections respect the plane) and
H0 2900 – – –
the parameter d (imperfections respect the verticality of the wall). H0S 2900 – – –
HS11 2900 1 1 10.0
HS11S 2900 1 1 10.0
2.2. Analytical approach for comparison HS12S 2900 1 2 10.0
HS14S 2900 1 4 20.0
HS21S 2900 2 1 10.0
The results of the implemented numerical model at calculating
M0 1800 – – –
the load-bearing capacity of the experimental cases are also com- M0S 1800 – – –
pared with the results obtained by applying the analytical MS11 1800 1 1 10.0
approach recently presented in [2]. This analytical model is based MS11S 1800 1 1 10.0
on the assumption that the stresses are linearly distributed along MS12S 1800 1 2 10.0
MS14S 1800 1 4 20.0
the wall’s thickness, which is a non-conventional hypothesis.
MS21S 1800 2 1 10.0
However, it is supported by the good accuracy which was obtained.
 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111 99

strengthened with TRM, the fourth (optional) is the number of 3.1. Geometry
fibre grids installed in each strengthened side and the last symbol
is an S if the inclined contact (see Section 3.1) at the endings of the The geometric model is defined as a series of piled rectangular
wall is considered. This position is kept empty if this contact is not parts corresponding to the different masonry rows existing in the
considered. analysed wall. In addition, rectangular bodies are placed next to
Finally, it has to be noted that no connectors were considered the ones representing the masonry wall to describe the TRM layer
for the theoretical cases because the experimental evidence (see Figs. 1 and 2). Contacts between the different parts represent-
pointed out that these elements were not efficiently contributing ing the TRM layer are defined to allow modelling the tensile failure
at increasing the load-bearing capacity of TRM strengthened brick of the TRM observed in two experimental cases (see [2]). The inter-
masonry walls subjected to eccentric compressive loads. face between the brick masonry and the TRM is also described by
means of contact elements.
Two extra bodies, which are two triangles, one at each end of
3. Numerical model the wall (see Fig. 1), are defined to easily represent the hinges of
the pinned-pinned configuration considered for all experimental
The implemented model assumes the hypothesis of plane strain and theoretical cases. These triangles allow placing the load appli-
applied to a bi-dimensional (2D) description of a typical transverse cation point accurately and setting the rotation axes.
section of the wall. Such a model is a 2D simplification of the three- In addition, fictitious contacts are placed in an inclined line
dimensional (3D) real case. Using a 2D simplification reduces the crossing and dividing the masonry bodies near the extremes of
computational cost of the analysis and simplifies the definition of the walls (see Fig. 2). The purpose of these inclined contacts is
the materials and contacts. The plane strain hypothesis requires modelling the compressive/out-of-plane shear collapse mode,
a uniform definition of the problem along the direction which is which was experimentally observed in most of the TRM strength-
perpendicular to the simulated section. For the particular case of ened walls (see [2]). This failure mode was characterised by the
compressed masonry walls, this means that the load is uniformly opening of one diagonal crack, which always started from one
applied along the wall’s width. In addition, the strains which are extreme of the strengthened side of the wall and grew to the com-
oriented perpendicular to the analysis plane are supposed to be pressed side of the structure. The chosen orientation for these
negligible in both masonry and TRM. This hypothesis is closer than dividing lines was set to be constant and this corresponded to
the plane stress option for most of the real walls, which are wider the average inclination of the cracking lines which were observed
than the tested ones and are usually laterally restrained. In fact, the in the experimental collapses. Thus, only the most representative
plane stress approach might certainly be more representative of situation is taken into account for the geometric input of the
the tested comparison walls than the plane strain one, but the numerical model. For comparison purposes, wall W#21, which col-
aim of the proposed model is to cover the most representative lapsed due to the tensile failure of the TRM, was modelled with and
situation. Because of this, the plain strain simplification has been without these inclined fictitious contacts to assure that these do
adopted. This means that the model presented herein might be not influence on the results when the failure mode is not charac-
not suitable for calculating TRM-strengthened masonry piers. terised by the compressive/shear collapse.
In addition, simplified micromodelling has been used because The experimental cases have been modelled considering their
this approach does not require so much information of the material real geometry in a detailed way. The thickness of the TRM has been
properties as detailed micromodelling but performs better than considered constant along the wall’s height and equal to real TRM
macromodels. The detailed micromodelling option would repre- thickness at mid-height. In addition, the experimentally measured
sent bricks, mortars, fibre grids and their interfaces separately. alignment of the hinges (d in Table 1) and the real shape of the wall
The macromodelling technique would require a complex homoge- are used to create the model. The real shape of the wall, taking into
nisation process to obtain a unique equivalent material. In fact, the account its out-of-plane initial imperfections was determined by
proposed method is something intermediate. It models each brick measuring it at each two brick rows. Thus, the definition of the
together with the surrounding mortar as a homogenised material, geometry of the model has been carried out so as to capture the
called masonry, and uses the corresponding properties. Thus, no irregularities that might affect the development of second order
homogenisation process is required because the experimentally bending effects.
obtained properties of the equivalent material (masonry) are An ideal geometry definition is considered for modelling the 14
directly used. Similarly, the TRM is though as a homogeneous theoretical cases (132 mm thickness, 1 m width and two possible
material and the equivalent properties of this composite are used. heights, M or H). This decision allows comparing the considered
These properties gather the effect of the mortar, the fibre grid and strengthening variables without the interfering influence of the
the interfaces between them and with the masonry. The only par- geometric imperfections. Thus, these walls are perfectly vertical,
ticularity of the proposed approach is about the definition of the the hinges are horizontally aligned and the eccentricity of the load
tensile strength of the TRM, which is explained with detail in is always the same (20 mm). Two TRM thickness are considered for
Section 3.2.2. Contacts are defined between the objects that repre- simulating these walls, equal to 10 mm if one or two fibre grids are
sent each masonry row or each fictitious TRM row (see Sections 3.1 embedded into the TRM layer and 20 mm for the cases with 4 fibre
and 3.2). This approach saves over half of the contacts and the grids installed. The geometric variables are summarised in Table 4.
material definitions – in comparison with detailed micromodel –
and thus, eases the convergence of the numerical simulation. 3.2. Materials and contacts
In addition, the numerical model has been implemented to cal-
culate the structure by considering large displacements to accu- Strictly, the mechanical response of the brick masonry is ortho-
rately represent the buckling phenomena originated by the tropic because of the pattern of the joints (see the work by Milani
second order bending effects. et al. [24]). In addition, masonry shows different failure modes
This numerical tool uses ANSYSÒ v.12.1 and the materials, con- determined by the units, the mortar or the interfaces between
tacts and geometric definitions have been set as general as possible them, which are the components of this composite material. The
to assure that the model can be reproduced in general purpose failures that are linked to the interfaces refer to the joint opening
Finite Element Analysis (FEA) packages. or the join sliding mechanisms. In addition, there are several mod-
100 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111

Fig. 1. Typical geometric model for experimentally tested TRM-strengthened walls.

Fig. 2. Mesh and contacts distribution.

els which accurately represent the response of the masonry. grid (at larger loads, after cracking). Typical stress–strain relation-
Although these approaches could have been used in the imple- ships of the TRM in tension are presented in the code ACI 549.4R-
mented numerical approach (e.g. using a smeared crack approach 13 [27] (Fig. 5.3.2b of this standard) or by Bertolesi et al. [26]
from the research by Milani et al. [25]), they have been discarded among others. Fig. 3 is a sketch of the characteristic experimental
because of their complex application and their requirement of response of the TRM under tensile forces. This realistic complex
advanced simulation tools, which would be against the first aim constitutive model is not used in the simulation. Instead, the
of the implemented model: keeping it as simple as possible. adopted model is characterised by its simplicity. This assumption
Similarly, TRM, which is also a composite material, shows a is required for an efficient calculation of the load bearing capacity
complex structural response [11,12,26]. The compressive beha- of the TRM strengthened walls.
viour of the TRM is governed by the mortar matrix, whereas the Thus, masonry and TRM are modelled as homogeneous materi-
tensile response is influenced by (a) the mortar matrix (at low als, whose compressive response is defined with an isotropic linear
loads, before cracking and mobilising the fibre mesh), (b) the fibre elastic with perfect plasticity constitutive model. The typical com-
grid and (c) the adherence between the mortar matrix and the fibre pressive stress–strain diagram for masonry and TRM is shown in
 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111 101

experimental evidences in [28]. Nevertheless, a simple sensibility

analysis was done to study the influence of this parameter and to
assure that it was not crucial for the calculation of the load-bearing
capacity. The results showed that the influence of the Poisson’s
coefficient on the resistance of the TRM-strengthened walls is lim-
ited. This fact was noticed by using two different and extreme val-
ues of Poisson’s coefficient (0.10 and 0.45) to model the theoretical
case H0. The compared results pointed out that an increase of 350%
of the Poisson’s ratio was associated with a far littler increase of
23% of the ultimate load. Thus, slightly lower values of the load-
bearing capacity (approximately 5% lower, if assuming a linear
influence of the Poisson’s ratio on this result) might have been
obtained if considering a typical Poisson’s coefficient of 0.25
instead of a more realistic and adjusted to the studied masonry
one (t = 0.35). Hence, the value of the load-bearing capacity
depends on the Poisson’s coefficient but this dependence is less
Fig. 3. Sketch of the typical tensile stress–strain relation of the TRM.
significant than the influence of other parameters such as load
eccentricity, slenderness of the wall or Young’s modulus of the
masonry.
Fig. 4, left. The compressive strength and the Young’s modulus are Taking into account the scattering of the experimentally deter-
the only required parameters to characterise these responses. On mined properties of the masonry (specially the Young’s modulus
the other hand, the tensional behaviour of these materials is con- and the compressive strength), it was decided to perform a simpli-
trolled by contact elements, which have neither thickness nor elas- fied sensibility analysis. This consisted in modelling three times
tic properties. The sketch presented at the right side of Fig. 4 show every experimentally tested TRM-strengthened wall. The first time,
the used tensile constitutive law for the masonry and TRM. This is using the average and most representative values for the compres-
bilinear and it is defined by the tensile strength and the sive strength and the Young’s modulus. A pair of low values for
corresponding fracture energy, as detailed later on in this section. these parameters (E = 400 MPa, fc = 7 MPa) was considered in the
The current steel definition included in the FEA software is used second repetition, whereas the third time a couple of high values
to model the material of the parts corresponding with the hinges for these variables (E = 1100 MPa, fc = 15 MPa) was used. It has to
(end triangles in Fig. 1). be noticed that the low and high values are within the experimen-
Regarding the particular properties of the masonry, the follow- tal range obtained for these variables [2].
ing values are considered: the used density of the brickwork is On the other hand, the compressive behaviour of the TRM is
1732 kg/m3, the Young’s modulus of the masonry (E) is 780 MPa, associated with the properties of the strengthening mortar
the compressive strength of the masonry (fc) is 10.8 MPa (E/ (Young’s modulus and compressive strength). The present
fc = 72.22) and the corresponding Poisson’s coefficient is set to approach considers that the fibre grids do not influence on the
0.35. The three first values were experimentally obtained and the compressive response of the TRM. Knowing that the influence of
Poisson’s coefficient is based on the values found in the bibliogra- the Poisson’s coefficient of the masonry on the load-bearing capac-
phy. For instance, Bosiljkov et al. [28] proposed values of Poisson’s ity is limited, in the case of the TRM this parameter was set to 0.35,
ratio up to 0.4 when using mortar with low content of cement. like for masonry. The compressive strength (fcm) and Young’s
Moreover, Bosiljkov et al. [28] proposed a relationship between modulus (Em) used for modelling each strengthening mortar are
the modulus of elasticity and the Poisson’s ratio. It was noticed that summarised in Table 2. The values of the first two variables were
the highest Poisson’s values corresponded with the lowest values experimentally obtained and Em was provided by the mortar’s
of Young’s modulus. For this reason, a relatively high value of the manufacturer.
Poisson’s coefficient (t = 0.35) is used in the current research if The tensile responses of masonry and TRM are modelled using
compared with the one proposed in Eurocode 6, which is 0.25. In contacts. At the first calculation step, these contacts are ‘‘bonded’’.
this line, the use of t = 0.35 is completely supported by the It means that the corresponding nodes of the adjacent parts are set

Fig. 4. Generic compression (left) and tension (right) stress–strain response for the masonry and TRM used in the model. The compressive (fc) and tensile (fxt) strength in the
sketch is the one corresponding to each material as it is the Young’s modulus (E).
102 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111

to move equally and together. The stress increase in the contact, modes by defining fictitious contacts is the chosen approach to
which is associated with the step by step loading process described avoid using complex material definitions. All contacts are deeply
in Section 3.4, might cause the opening (debonding) of the contacts described in the next subsections and summarised in Table 5.
or their relative sliding accordingly with the bilinear constitutive
law shown in Fig. 4. These tensile and/or shear responses of the
3.2.1. Real contacts
contact elements are characterised by a cohesive zone model
There are real discontinuity surfaces between masonry rows
(CZM) that was originally proposed by Alfano et al. [29]. The
and between TRM and masonry. These interfaces should be mod-
CZM sets an elastic behaviour up to the brittle failure of the contact
elled with contacts. The experimental observation proves that
(see Fig. 4, right). When the tensile or tangential strength is
the opening between masonry rows is likely to occur together with
reached, the contact opens or slides while the corresponding stress
the tensile failure of the TRM. In contrast, no debonding process
decreases linearly with a slope that is defined by the corresponding
was observed between masonry and TRM during the tests, which
fracture energy.
consisted on applying eccentric compressive loads (see [2]). In
Modelling the contacts with the CZM allows using the relatively
addition, the sliding phenomenon was not experimentally noticed
complex contact technology but requiring just of a few parame-
for these contacts and therefore, enabling the sliding capabilities of
ters: the tensile or shear strength and the corresponding fracture
these contact elements is not considered to be necessary. Thus, the
energy. Thus, it is believed that using the CZM is the simplest
contacts between masonry and TRM are defined as ‘‘bonded’’. This
and most general way to represent the tensile response of the com-
definition forces the contact nodes of TRM and masonry to be
posite materials of the considered problem. In addition, using con-
always coincident, so moving equally. The same behaviour is
tacts is essential for the studied problem, which considers the
defined for the contacts between the first and the last masonry
second order bending effects.
rows with the corresponding triangular parts which represent
The tensile strength (fxt) and the critical fracture energy of the
the hinges.
first fracture mode (GIf) are required by the CZM to model the
The contacts between parts which represent masonry rows are
debonding/opening phenomenon. The maximum shear stress
initially defined as ‘‘bonded’’ but the possibility of debonding is
(smax) and the critical fracture energy of the second fracture mode
considered through the definition of a CZM. This tensile response
(GIIf ) are required to model the sliding failure case. Failure modes
is bilinear and follows the constitutive law previously presented
involving debonding and sliding are modelled by a mixed failure
(see Fig. 4, right). The sliding failure is not considered for these
criterion characterised by the power law energy criterion pre-
contacts because it was not experimentally observed. The values
sented in the following equation (Eq. (1)), where GI and GII are,
which have been used to define the debonding process (fxt and
respectively, the normal and tangential energy of fracture for the
GIf) are summarised in Table 5. The flexural tensile strength, fxt,
case of combined failure.
which was experimentally determined through bond-wrench tests,
! ! is used instead of the direct tensile strength. It is because the flexu-
GI GII ral tensile strength is considered to be more representative of the
þ ¼1 ð1Þ response of the masonry under the second order bending phenom-
GIf GIIf
ena. In fact, the contacts between masonry rows can be bended or
The definition of every contact between two parts requires set- compressed but not subjected to a tensile axial effort under the
ting a pair of objects: a contact object and a contact target. Then, considered loading configuration. Thus, using the flexural strength
the characterisation of the state of the contact (penetrating, in con- of the masonry to calculate the bending response of walls is gener-
tact, opening or sliding) is characterised by the relative position of ally accepted. The proposed formulation to design out-of plane
one of this objects respect to the other. loaded walls included in Eurocode 6 [30] is a reference example
Depending on the location of the contacts (see Fig. 2), these of this common assumption. Thus, taking into account that the
might be classified into real contacts and fictitious contacts. The studied walls are eccentrically compressed and their bending
first ones correspond with the joints of the masonry and the inter- response due to second order effects is distinctive, the flexural ten-
face between the TRM and the masonry. In contrast, the second sile strength is chosen as the most suitable variable to characterise
ones are set in order to allow the development of the experimen- the behaviour of the contacts between masonry rows. Finally, the
tally observed failure modes. Taking into account the real failure obtained results, in terms of load-bearing capacity, point out that

Table 5
Properties of the contacts used in the FEA.

Contact Walls Contact failure fxt GIf (N/ smax GIIf (N/
(MPa) m) (MPa) m)
Real. masonry joints All Debonding 0.36 13 – –
Real. masonry-TRM All None – – – –
Real. masonry-support All None – – – –
Fictitious. masonry–masonry All except H0, M0, HS11 and MS11 Debonding and 2.8a 100 0.56 20
(inclined) sliding
Fictitious. TRM–TRM W#21W#22 HS11 HS11S HS21S MS11 MS11S Debonding 8.1 295 – –
MS21S
W#23W#24 Debonding 6.6 240 – –
W#25 Debonding 12.0 440 – –
W#26 Debonding 11.3 412 – –
W#27 Debonding 20.0 733 – –
W#28 Debonding 17.8 652 – –
W#29 Debonding 14.5 533 – –
HS12S HS14S MS12S MS14S Debonding 9.0 330 – –
a
This value corresponds to the direct tensile strength, not to the flexural tensile strength.
 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111 103

the possible overestimation of the tensile strength of the contacts strength and the corresponding fracture energy (first and second
and the corresponding effect on the fracture energy, might not be a modes respectively). The tensile strength between two masonry
big deal because the numerical predictions tend to underestimate parts of the same row, which are separated by the inclined dis-
the capacity of the walls as presented in Section 4. continuity, is taken to be equal to the direct tensile strength of
The fracture energy, GIf, is estimated using equation (Eq. (2)). the ceramic pieces. This assumption is based on the idea that the
This expression has been adjusted using the experimental results bricks are not bended but subjected to direct tensile stresses on
presented in [31] and was previously used in [1]. the plane of the considered fictitious contact, which just represents
the connection between two parts of the same brick at each row. In
ðN=mÞGIf ¼ 36:65  f xt ðMPaÞ ð2Þ addition, this inclined contact is almost perpendicular to the most
bended plane orientation (horizontal) for the loading configuration
of the studied walls. Thus, the direct tensile strength is understood
3.2.2. Fictitious contacts to be more representative than the flexural one. The corresponding
Another two types of contacts, which do not correspond with fracture energy is also calculated using (Eq. (2)).
real joints, are considered in the proposed model. These contacts The maximum shear strength (smax) is analytically calculated
set fictitious joints with the aim of representing the real failure from the experimental data as showed below (Eq. (3)). The
modes without adding complexity to the material constitutive corresponding fracture energy is calculated using (Eq. (2)) where
equations. Table 5 summarises the values of the properties used the flexural strength is replaced by the shear strength. Using (Eq.
to define these fictitious contacts. (2)) for calculating the second mode fracture energy, GIIf , is a simpli-
The first typology of fictitious contact is used between pairs of fied approach, which has been assumed because a generally
TRM parts. These contacts are defined to allow the opening but accepted formulation to calculate this required parameter for
not the sliding between TRM parts. Thus, they make the TRM ten- inclined contacts on masonry structures has not been found.
sile failure (experimentally observed in walls W#21 and W#23)
T
possible. This tensile response follows a bilinear constitutive law smax ¼ ð3Þ
bm
presented in Fig. 4 (right), and it is characterised by the tensile
strength and the corresponding fracture energy. The first parame- The load components, which are normal (N) and tangential (T)
ter, fxt, is calculated as the maximum between: (a) the flexural ten- to the failure plane, are calculated from the maximum applied
sile strength of the strengthening mortar and (b) the direct tensile force (F) and the observed failure modes (inclination of the crack
strength of the fibre grid uniformly distributed on the area of the in walls W#22 and W#24 to W#29) as shown in Fig. 5. Then, the
TRM section. In the case (a) the tensile force associated with the tangential component (T) is supposed to be uniformly distributed
flexural tensile stresses on the mortar of the TRM is greater than on the area (b  m) of the observed discontinuity plane. This allows
the direct tensile resistance of the fibre grid embedded, which obtaining the maximum shear strength (Eq. (3)), where b is the
actually works in direct tension. Thus, in the case (a) the fibre grid width of the wall and m the length of the failure plane. The values
cannot bear the tensile force released by the mortar when it cracks used in the simulations are the average values for smax and the
by bending, so the mortar completely controls the failure of the corresponding GIIf .
TRM. Similarly to the case of the contacts between masonry rows, It has to be highlighted that the used CZM model considers both
the flexural tensile strength of the strengthening mortar is consid- fracture processes (opening and sliding of the contact) together, so
ered instead of the direct tensile strength. This assumption is based the shear response is sensible to the normal response and vice
on the same hypotheses than for the masonry: the mortar of the versa.
TRM might be compressed or bended but never subjected to a
direct tensile effort under the considered loading configuration, 3.2.3. Failure criterion
which tends to develop distinctive second order effects. Thus, the Each type of contact uses the corresponding failure criterion,
flexural tensile response of the mortar of the TRM is more repre- which is defined under the specifications of the CZM model.
sentative than a direct tensile strength for the studied cases. In Thus, the masonry joints fail in flexural tension when the
addition, these fictitious contacts might be understood as the natu- corresponding strength is reached although the model allows lar-
ral extension of the real contacts between masonry rows into the ger deformations but at a lower stress levels. Similarly, the TRM
TRM area, so using the same type of parameters is coherent.
Finally, the tendency of the model to underestimate the load-bear-
ing capacity of the studied walls (see Section 4) subtracts impor-
tance to a possible overestimation of the tensile strength (and
the corresponding fracture energy) of these fictitious contacts. On
the other hand, the case (b) assumes that the force associated with
tensile strength of the fibre grid (the mesh embedded into the TRM
mostly works in direct tension) is greater than the resistance
developed by the mortar in bending configuration. Thus, in the
case (b) the fibre grid can bear larger load in direct tension than
the mortar matrix up to its bending cracking time. It has to be
noticed that (b) is the most likely case for practical strengthening
applications. For these contacts the corresponding fracture energy
was calculated using the expression presented in (Eq. (2)) too.
The second type of fictitious contact is used in the two inclined
lines that divide the five masonry rows near each end of a single
wall. These contacts make it possible to simulate the compres-
sion/shear failure observed in most of the experimental tests on
TRM strengthened walls. These enable the debonding and the slid-
ing phenomena. Both tensile and shear responses are characterised
by bilinear constitutive laws defined by the tensile or tangential Fig. 5. Normal and tangential force components on the failure plane.
104 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111

fails in tension when the corresponding tensile strength is reached. The load is indirectly applied as a vertical descending displace-
The inclined contacts on masonry fail with a combination of the ment of the top vertex of the triangular part which represents the
tensile and shear stresses that causes the opening or sliding of upper hinge. This movement is applied at a constant rate (step by
the contact. Like for the previous cases, larger deformations are step calculation) and the force reaction at the application point is
allowed but with lower stress levels after reaching the maximum taken as the measurement of the applied load for comparing with
combined strength. All these failure criteria are local and refer to the experimental values. This displacement-controlled simulated
every node in the contact. In fact, the contact between each pair loading process makes it possible to calculate some steps after
of nodes technically fails when all the energy of fracture has been reaching the maximum load, clarifying the interpretation of the
dissipated, so there is no restrain for the independent displacement failure pattern.
of the initially coincident nodes. The global failure criterion, which The calculation is performed step by step. Each new displace-
is used to set the load-bearing capacity of every studied structure, ment increase is applied on the deformed configuration of the wall
is reached when the resultant deformation due to the opening or due to the previous loading steps. The number of calculation steps
sliding of the joints is not compatible with the internal equilibrium has to be little enough to complete the simulation with a reason-
of the walls. able cost (time elapsed). In addition, the first displacement
increase has to be larger than the gravity deformation to avoid
3.3. Mesh developing false tensile stress. However, the number of increments
has to be sufficiently refined as to accurately capture the wall’s
Fig. 2 shows a characteristic mesh used for the modelling. For response (approximately a minimum of 50 steps up to the col-
all cases masonry and TRM are meshed with 8-nodes uniform lapse). For all these reasons, the loading ratio was carefully
structured quadrilateral elements with quadratic integration of adjusted for each wall.
the displacements. The average size of the elements is 3 mm for
TRM and 5 mm for masonry, except the masonry of the theoretical 4. Results
walls with a height of 2900 mm (H series) which uses elements
with an average size of 20 mm to reduce the computational cost. In order to analyse the numerical results and to compare them
The parts representing the steel hinges (triangles) are meshed with with the experimental evidences, a set of dimensionless variables
an unstructured uniform mesh composed by 6-nodes triangular are defined. The first one is the dimensionless applied load / (Eq.
elements and 8-nodes quadrilateral elements with quadratic (4)), calculated as the ratio of the applied load, N, over Nu (Eq. (5)).
integration of the displacements and an average size of 5 mm.
Two types of elements are required to define the contacts. The / ¼ N=Nu ð4Þ
first one is assigned to the contact body and overlays the 2D solid
element, adopting the same shape and defining a boundary, Nu ¼ b  t  f c ð5Þ
whereas the second one is assigned to the target body. This last
typology of element is defined as a set of segment elements placed where Nu is the compressive strength of the cross section assuming
at the edge of the corresponding solid element. The contact occurs a uniform stress distribution on the width, b, and thickness, t, of the
when the contact element surface penetrates into one segment of wall with a stress value corresponding to the compressive strength
the target element. Both element types have three nodes and the of the masonry, fc.
assignation of contact or target elements to one side or other of Other calculated dimensionless variables are the mid-height
the contact is automatically done. lateral deformation, h, over the wall’s thickness, t; the vertical dis-
With the aim of adjusting the mesh size presented before, a placement of the top hinge, v, over the effective height, Hef, and the
sensibility analysis was done. This was performed under the ratio between the eccentricity, e over the wall’s thickness, t.
requirement of carrying out the calculation in a reasonable time. The results considered for a possible validation of the numerical
This means spending less than 45 min in an Intel(R) Core(TM)2 model are presented in Figs. 6 and 7. In Fig. 6, the vertical lines cor-
Duo CPU E8400 @3.00 GHz with 3.00 GB of RAM memory working respond to the range of possible values for the calculated load-
with Windows7 32 bits. bearing capacity depending on the considered compressive
A couple of cases were analysed to select the suitable mesh size. strength and Young’s modulus of the masonry (see Section 3.2).
Changing the mesh size of the masonry from 5 mm to 3.5 mm and Analysing the length of these vertical lines it is noticed that the
the TRM mesh size from 3 mm to 2 mm causes a change of the cal- influence of these variables is more significant for the walls which
culated maximum load-bearing capacity from 344.1 kN to were tested with the lowest eccentricities of the load. In addition,
344.0 kN for wall W#29, and changing the size of the masonry ele-
ments from 20 mm to 10 mm caused a difference less than 1% of
the load-bearing capacity of one of the most slender walls (HS11
theoretical wall). Hence, the influence of the size of the elements
is little for the selected sizes.

3.4. Boundary conditions and loading process

The boundary conditions of the model are defined to represent

the real boundary conditions of the experimental cases. These are
maintained for the theoretical cases and consisted of fixing the dis-
placement of the lower hinge by restraining the horizontal and ver-
tical displacements of the lowest vertex of the triangular part
which represents the lower hinge. Additionally, the horizontal dis-
placement of the top vertex of the triangular part representing the
upper hinge is fixed. These constrains allow the structure to rotate
around the restrained nodes while impeding any possible lateral
displacement at the wall’s ends. Fig. 6. Comparison of the numerical and experimental results for the TRMW walls.
 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111 105

Fig. 7. Experimental and numerical dimensionless force vs. lateral displacement at mid height curves.

the horizontal lines in Fig. 6 represent the calculated load-bearing circular points represent the experimentally determined load-
capacity when using the most representative values for the com- bearing capacity of every tested TRM-strengthened wall (labelled
pressive strength of the masonry and its Young’s modulus. The next to the circular point). A relative error of 19.5% is obtained
106 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111

when comparing the numerically predicted load-bearing capacities Analytical results [2] are compared with the experimental and
with the experimental evidences. The model tends to under- numerical ones to complement the evidences. These are all sum-
estimate the load-bearing capacity of the walls, especially in those marised in Table 6. It has to be noticed that the proposed numerical
cases whose eccentricity of the load is higher. Finally, the numeri- model tends to underestimate the load-bearing capacity of the
cal results plotted in Fig. 6 seem to point out that the eccentricity tested walls except for the cases strengthened with carbon fibre
of the compressive load influences on the load-bearing capacity meshes. In contrast, the analytical approach overestimates the
(the walls with major eccentricity are predicted to support minor load-bearing capacity. In addition, the results of the analytical
loads). However, this tendency is not clear for the experimental approach suggest that the failure pattern is always associated with
results. the compressive collapse of the masonry whereas the numerical
Fig. 7 shows the experimental and numerical lateral responses simulation takes into account different failure possibilities. Thus,
of the tested TRM-strengthened walls. The relation between the the simulation represents the observed failure modes in a more
dimensionless variables representing the lateral displacement at accurate way. The accuracy of both methods (relative error) at pre-
mid height and the applied force is represented. Two main results dicting the load-bearing capacity is in the same order of magni-
are obtained from these graphs: (1) the comparison of the stiffness tude, although the numerical model is applicable in a wider
of the lateral responses shows that the experimental behaviour is range of cases and affords the simulation of a larger variety of fail-
stiffer than the numerically predicted one; and (2) the characteris- ure modes. Finally, the dispersion of the numerically calculated
tic shape of the curves which are linear for the numerical model load-bearing capacity of comparable walls is greater than the par-
and a parabolic for the experimental data. ticular scattering of the experimental tests in this case (comparing
walls W#21 and W#22 between them and W#23 with W#24).
The numerical results for theoretical walls are represented in
Table 6 Figs. 8 and 9. The influence of considering the inclined contacts
Analytical and numerical results for the tested TRM strengthened walls. in those cases which are expected to fail due to mechanism forma-
tion (H0, HS11, M0 and MS11) is presented in Fig. 8. It is noticed
Analytical approach Numerical approach
that considering these fictitious contacts does not affect the struc-
Wall Mexp/ Manalytic/ Error N exp N FEA Error
max max tural response of these walls which are not expected to fail due to
(Nut/8) (Nut/8)a (%) (kN) (kN) (%)
compressive/shear forces near the endings. This result is obtained
W#21 0.648 0.765 18.1 299.7 264.6 11.7 by comparing the lateral and vertical response of the theoretical
W#22 0.828 0.841 1.6 328.6 208.2 36.7
case H0 with H0S, HS11 with HS11S, M0 with M0S and MS11 with
W#23 0.618 0.725 17.3 270.9 215.4 20.5
W#24 0.667 0.746 11.8 285.6 247.1 13.5 MS11S. In addition, Fig. 8 reveals that, according with the numeri-
W#25 0.787 0.906 15.1 414.0 263.0 36.5 cal results, the effectiveness of applying a TRM strengthening sys-
W#26 1.027 0.926 9.8 390.3 282.4 27.7 tem is focused on the lateral response of the walls and their load-
W#27 0.708 0.826 16.7 345.7 333.1 3.6 bearing capacity. In contrast, the vertical stiffness seems to be
W#28 0.575 0.806 40.2 313.5 381.1 21.6
W#29 0.865 0.844 2.4 330.2 344.1 4.2
unaltered by the application of a TRM system. Finally, it is observed
(see Fig. 8) that the developed lateral deformation at the collapse
Absolute average value of the 14.8 19.5
relative error (%)
instant is greater for the H series walls than for the M series walls.
Fig. 9 shows the influence of the number of fibre grids embed-
a
Corresponding to the maximum experimental axial force and according to the ded into the TRM layer on the wall’s response and the effect of con-
analytical formulation.
sidering the application of the strengthening system at one face or

Fig. 8. Influence of the inclined contacts on the structural response of the walls.
 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111 107

Fig. 9. Influence of the strengthening configuration on the structural response.

both faces. Theoretical walls of the H and M series are considered case). Thus, the major change is observed when strengthening both
and the influence of the slenderness is also studied. Observing faces of the walls, like for the vertical response. Moreover, the dif-
these graphs of the numerical results it is noticed that the vertical ferences between using 1, 2 or 4 fibre grids are more significant for
stiffness of the walls is increased when both faces are considered to the M series than for the H series. Finally, it is observed that the lat-
be strengthened. In contrast, this vertical stiffness is maintained for eral deformation of the wall MS21S at the maximum load is greater
the other studied variations. In addition, the model predicts a lat- than the corresponding value for the wall HS21S.
eral stiffness increase associated with the application of the TRM The analysis of the numerical results included in Table 7, which
at one side. However, there is almost no difference between plac- consider the influence of different strengthening configurations,
ing one or two fibre grids embedded into the mortar layer of the shows that the performance of the TRM strengthening system at
TRM. A slightly greater lateral stiffness is developed when placing increasing the load-bearing capacity is enhanced by making the
4 fibre grids (doubling the thickness of the mortar layer in this TRM layer thicker (compare the relative increase between HS11S
and HS12S with the increase showed in the case HS14S; and simi-
larly for M series) or applying the strengthening system at both
Table 7
faces of the wall. This last variation corresponds to the most effec-
Load-bearing capacity of the theoretical walls depending on the slenderness of the
wall and the strengthening pattern. tive one. Finally, the numerical predictions point out that TRM is
more effective at increasing the load bearing capacity of the most
Wall Inclined contact Load-bearing capacity Influence of
slender cases.
effect a (%) increase b (%) strengthening patternc
(%) Figs. 10 and 11 show the typical contour plots resulting from
the numerical simulations. The pressure distribution along
H0S 0.0
HS11S 0.1 133 fictitious contacts between TRM elements is presented in Fig. 10
HS12S 142 4 (left) for the case HS11S, which fails due to mechanism formation
HS14S 200 29 according with the results of the proposed model. In this image
HS21S 745 263 (Fig. 10 left), it might be observed that the tensile strength of the
M0S 0.0 TRM is simultaneously reached at the two contacts which are
MS11S 0.0 92 placed near mid-height. This fact corresponds with the experimen-
MS12S 100 4
tal experience, which suggests that the tensile failure of the TRM
MS14S 130 20
MS21S 439 181 usually happens at mid height.
a
Similarly, the tangential stress distribution along the fictitious
Comparison of cases with the same strengthening configuration with and
inclined contacts can be observed in Fig. 10 (right). This image cor-
without inclined contacts. H0S is compared with H0, HS11S is compared with HS11,
M0S is compared with M0 and MS11S is compared with MS11. responds to the theoretical wall MS12S, which fails due to masonry
b
Comparison with respect to the non-strengthened case. All analysed cases compressive/shear collapse near the ends of the wall, according
consider the inclined contact in the masonry. HS11S, HS12S, HS14S and HS21S are with the numerical results. In this figure, the circled parts of the
compared with H0S and MS11S, MS12S, MS14S and MS21S are compared with M0S. contact lines have reached the shear strength. This situation affects
c
Comparison with respect to the cases strengthened with one fibre grid
embedded into one mortar layer applied on one side of the wall. HS12S, HS14S and
more than a half of the contact line causing the failure of the wall.
HS21S are compared with HS11S and MS12S, MS14S and MS21S are compared with Finally, the masonry compressive failure is predicted for some
MS11S. theoretical walls, e.g. wall MS21S, shown in Fig. 11. The vertical
108 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111

Fig. 10. Distribution of the normal pressure in the fictitious contacts between TRM parts for wall HS11S (left) and distribution of tangential stresses in the fictitious inclined
contacts for wall MS12S.

stress distribution on the masonry area is plotted. This shows that The load-bearing capacity of the walls subjected to larger load
the compressive strength of the masonry has been reached at mid eccentricity tends to be underestimated by the proposed numerical
height of the wall. It has to be remarked that this failure mode was model. A conservative definition of the mechanical variables which
not experimentally observed. directly describe the bending response of the walls – mostly the
flexural strength of the masonry, fxt or the TRM defining variables
might explain this underestimation tendency.
5. Discussion In addition and according with the results from Fig. 6, the load-
bearing capacity calculated by the numerical model decreases with
Firstly, it is important to remark that the load-bearing capacity the increase of the eccentricity of the applied load. This result
predicted by the model correctly adjusts the experimental results. meets the typical response of the eccentrically loaded walls (see
The average error is within the typical experimental scattering of [1]), so the numerical model correctly reproduces the theoretical
destructive tests on masonry walls, around 30%, and it is better results. However, the experimental results do not follow this com-
than the error obtained with a similar model for non-strengthened mon tendency because these are influenced by the use of different
walls (see [1]). This fact agrees with the greater uniformity of the TRM systems. Thus, additional experimental evidences seem
structural response showed by the TRM-strengthened walls in necessary to clarify this experimental unexpected result.
comparison with the behaviour of unreinforced masonry walls. The greater lateral stiffness of the experimental response of the
The influence of the Young’s modulus and the compressive tested walls in comparison with the numerically obtained curves
strength on the capacity of the walls depends on the walls’ failure (Fig. 7) might be due to the possible underestimation of the value
mode. As can be seen in Fig. 6, the influence of both parameters is of the Young’s modulus of the masonry. This hypothesis would also
very significant in those cases with small load eccentricities, which explain the tendency of the model to underestimate the load-bear-
failed by shear/compression. In contrast, the response of the walls ing capacity of the walls. In addition, the difference between the
that fail due to bending/buckling phenomena, which are the ones shape of the numerical and experimental curves in Fig. 7 suggest
with the larger load eccentricities and/or slenderness, is less influ- that the implemented constitutive model for the masonry in com-
enced by the Young’s modulus and the masonry compressive pression might be too simple to fully predict the response of the
strength. In these cases, such parameters become less critical for wall. However, a more accurate description of this material (e.g.
the prediction of the ultimate response. However, an increase of parabolic) might be not necessary if the only aim of the model is
these parameters leads always to some increase of the resulting to predict the load-bearing capacity of TRM-strengthened walls.
load-bearing capacity. Thus, it would be interesting to improve the experimental
 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111 109

inorganic matrixes and the carbon fibre grids is not reliable. This
idea is supported by previous evidences (see [32]).
The results in Fig. 8 shows that the implemented fictitious
inclined contact is an effective and simple way to model the com-
pressive/shear failure of the masonry near the ends of the TRM-
strengthened brick masonry walls. When this type of contact is
appropriately defined, the compressive/shear failure is correctly
activated in the cases where it is actually observed (see Fig. 10
right), while it does not activate in the cases that are expected to
fail due to mechanism formation (Fig. 8).
Additionally, the numerical results of Fig. 8 shows that TRM
strengthening limits and even prevents the development of the
mechanism failure mode. The strengthening system reduces sig-
nificantly the lateral deformation of the walls and therefore limits
the second order bending effects, leading to an increase of the load-
bearing capacity. In contrast, TRM strengthening does not affect
the vertical wall deformability because this behaviour is mostly
controlled by the properties of the masonry. This theoretical rea-
soning is completely supported by the results of the implemented
numerical tool.
Moreover, it might be expected to observe an increase of the ver-
tical stiffness of the walls when both sides of the wall are strength-
ened, because in this case both masonry and TRM influence the
compressive response of the structure. This hypothesis is supported
by the numerical results plotted in Figs. 8 and 9, which show the
stiffer vertical response of the walls HS21S and MS21S.
As should be expected, the more slender walls (H series) show
greater lateral deformations than less slender cases (M series).
This is clearly observed in Figs. 8 and 9, showing the ability of
the implemented model at considering the effect of the main vari-
ables defining the studied problem.
In addition, the little difference between placing one, two or
four fibre grids in a TRM-strengthening layer, which has been
showed by the results from Table 7 and Fig. 9, might be related
with the fact that all these theoretical cases failed by a compres-
sive/shear collapse of the masonry near the wall’s ends. Taking into
account that this failure mode is controlled by the masonry, the
Fig. 11. Distribution of the vertical normal stress in the masonry for wall MS21S.
effect of modifying the characteristics of the TRM is limited and
only slightly affects the lateral deformation of these structures.
Thus, the numerical results help to understand that there is a
procedures in order to obtain a more realistic measurement of the remarkable difference between unreinforced masonry walls, which
Young’s modulus of the masonry to be used in the model. The next failed by mechanism formation, and TRM-strengthened masonry
step might be implementing more complex constitutive equations walls with the fibre grid applied on the side subjected to tensile
for representing the masonry response. efforts, which mostly failed by compressive/shear failure of the
Regarding the comparison of the numerical results with the masonry. However, the influence of placing extra fibre grids on
ones obtained using the referenced analytical method (Table 6), the strengthened side is not significant because the failure mode
it has to be noticed that this analytical approach uses experimental is controlled by the masonry.
data (the load-bearing capacity) as an input parameter to predict The simulations point out that strengthening both sides of the
the corresponding maximum bending moment which is compared walls increases their lateral stiffness. This response is related with
with the experimental one. In contrast, the numerical model only the increase of the thickness of the wall’s section and the reduction
uses the material properties and the geometric definition as input of the eccentricity of the load with respect to the original section.
data. Moreover, the analytical approach always considers the Finally, the failure mode changes when strengthening both sides,
masonry compressive failure for the analysed cases, whereas the which may turn from compressive/shear failure of the masonry
numerical model is able to predict the structural response con- near the endings of the wall to the masonry compressive collapse
sidering different failure modes and can predict the load-bearing at mid-height, according with the numerical results and the curves
capacity for a wider range of boundary conditions and loading pat- in Fig. 9. This change in the failure mode may explain the fact that
terns than the analytical tool. The better accuracy of the analytical the lateral deformation associated with the maximum load for the
approach for the analysed cases can be justified by these facts. wall of the M series (MS21S) is greater than the corresponding
The significant scattering of the numerical results (even greater deformation of the comparable wall of the H series (HS21S)
than the experimental ones according with the data in Table 6) is although this second case is characterised by a greater slenderness.
pointing out that the implemented model is extremely sensitive From Table 7, it can noticed that the most slender walls (H ser-
to the value of the load eccentricity and to the variables which ies), which are expected to be more critically affected by second
define the tensile response of the TRM. In addition, it is worth order effects, are the ones whose performance is increased more
noticing that the numerical model only overestimates the experi- significantly when applying a TRM-strengthening layer at the ten-
mental cases of the walls strengthened with carbon fibre grids. sile side, indicating that the proposed strengthening option is espe-
This result might point out that the adherence between the cially effective to avoid the bending–buckling failure. In turn,
110 E. Bernat-Maso et al. / Engineering Structures 91 (2015) 96–111

strengthening both sides of the wall might be considered as the an accurate analytical tool for calculating the energy of fracture, to
optimum intervention against bending–buckling failure as it limits include a more complex constitutive equation for the masonry, to
the lateral deformation and increases the load-bearing capacity. In consider a plane stress hypothesis or to analyse the influence of
addition to such advantage, applying TRM on both sides may be difference variables, including the angle of the fictitious contacts
necessary in practice due to the difficulty of foreseeing the value on masonry as further works to upgrade the presented model.
and sign of the load eccentricity or real walls, which may even vary To sum up, the proposed numerical model is useful at predict-
depending on the load conditions. ing the load-bearing capacity of the wide range of TRM-strength-
In Fig. 10 left, it is observed that the discretisation of the TRM ened brick masonry cases, which have been considered, and
influences the position of the joints that open when the mecha- provides comprehensive information about the mechanical
nism formation failure mode is developed. Regarding the compres- response of this type of structures.
sive/shear failure of the masonry near the endings of the wall, it
has been observed that this failure mode happens, in the numerical
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